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Topology of Crystal Structures: Applications of Graph Theory
(the vector method)
Jean-Guillaume [email protected]
Institute of Chemistry, Federal University of Rio de Janeiro, Rio de Janeiro - Brazil
Manila, Workshop on Modern Trends in Mathematical Crystallography, May 20 2017
Outline
Part 1: basic concepts
• Some graph theory• Underlying nets and topology• Labelled quotient graphs• Symmetry• Crystallographic nets• Barycentric embeddings
Part 2: advanced topics
• Minimal nets• Rings and generators• Nets as projections• Non-crystallographic nets• Building units• Groupoids
Topology of Crystal Structures
Basic concepts
Crystallography and crystallochemistry
(NH4)3PMo12O40
Atomic positionsChemical bondsTopology
Electron density maps
Crystal structure and electron density in the apatite-type ionic conductor
La9.71(Si5.81Mg0.18)O26.37
Roushown Ali, Masatomo Yashima, Yoshitaka Matsushita, Hideki Yoshioka, Fujio Izumi
Journal of Solid State Chemistry 182 (2009) 2846–2851Electron density isosurface (0.5 × 10−6 e·pm−3) in Ca24Al28O64]4+·4e− (maximum entropy method analysis) superposed on the atomic structure of the unit cell
K. Hayashi, P.V. Sushko, Y. Hashimoto, A.L. Shluger, H. HosonoNature Communications 5 (2014) 3515
periodic nets and quotient graphs
100010
001
Quotient
Unit cell:
One Rhenium atomThree oxygen atomssix Re – O bonds
Chung S.J., Hahn Th., Klee W.E. Acta Cryst. A40, 42-50 (1984)
Combinatorial and Embedding Topology
Strong distortion Hopf link
Graph theory (Harary 1972)
A graph G(V, E, i) is defined by:– a set V of vertices,– a set E of edges,– an incidence function i from E to the set of
couples {x, y} of elements of V.
u
v
w
i (e1) = {u, v}i (e2) = {v, w}i (e3) = {w, u}
V = {u, v, w} E = {e1, e2, e3}
e1e2
e3
Graphs with an orientation (σ)
• For G = (V, E, i), let Gσ = (V, Eσ, iσ) such that:• σ : Eσ E is a two-to-one, onto mapping• Preimages of e = {u, v} are noted e+and e-
• iσ(e+) = (u, v) and iσ(e-) = (v, u)
• Notations: e+ = uv, e- = vu
u
v
w
i (e1) = (u, v)i (e2) = (v, w)i (e3) = (w, u)
V = {u, v, w} E+ = {e1, e2, e3}
e1e2
e3
e1+ = e1 = uv
e1- = vu
u
v
w
i (e1) = {u, v}i (e2) = {v, w}i (e3) = {w, u}
V = {u, v, w} E = {e1, e2, e3}
e1e2
e3
G
GσE- = {e1
-, e2-, e3
-}
Graphs and multigraphs
loopMultiple edges
Simple graph: graph without loops or multiple edges
Subgraphs
H G
V(H) V(G)E(H) E(G)iH is a restriction of iG
Induced subgraph
V(H) = S V(G)E(H) maximal in E(G)
Order, size and degrees
• Order of G: |G| = number of vertices of G• Size of G: ||G|| = number of edges of G• Degree of a vertex u: d(u) = number of edges
incident to u (loops are counted twice)• Regular graph of degree r: d(u) = r
Walks, paths and cycles
• Walk: alternate sequence of vertices and edges mutually adjacent
• Closed walk: the last vertex is equal to the first = the last edge is adjacent to the first one
• Path: a walk which traverses only once each vertex
• Cycle: a closed path
• Forest: a graph without cycle
a.b.j.i walk
a.b.c path
b.j.d.e.f.g.h.i closed walk
b.j.i cycle
Edges only are enough to define the walk!
Nomenclature
• Pn path of n edges• Cn cycle of n edges• Bn bouquet of n loops• Kn complete graph of n vertices• Kn
{m} complete multigraph of n vertices with all edges of multiplicity m
• Kn1, n2, ... nr complete r-partite graph with r sets of ni vertices (i=1,..r)
C5
P3
B4
K3{2}K4
K2, 2, 2
Connectivity
• Connected graph: any two vertices can be linked by a walk
• Point-connectivity: κ(G), minimum number of points that must be
withdrawn to get a disconnected graph
• Line-connectivity: λ(G), minimum number of lines that must be
withdrawn to get a disconnected graph
Determine κ(G) and λ(G).
G
λ(G) = 3
κ(G) = 2
Component = maximum connected subgraph
Tree = component of a forest
T1 T2
F = T1 U T2
Spanning graphs
G
T: spanning tree of G
Morphisms of (oriented) graphs
• A morphism between two graphs G(V, E, i) and G’(V’, E’ i’) is a pair of maps fV and fE between the vertex and edge sets that respect the incidence relationships:
for i (e) = (u, v) : i’ {f E (e)} = ( f V (u), f V (v))or: f(uv) = f(u)f(v)
• Isomorphism of graphs: 1-1 morphism
Aut(G): group of automorphisms
• Automorphism: isomorphism of a graph on itself.
• Aut(G): group of automorphisms by the usual law of composition, noted as a permutation of the edges (invariant by reversal of the signs!)
u
w
e1e2
e3
v
C3
Generators for Aut(C3)
(e1, e2, e3)
(e1, e3-) (e2, e2
-){
Special case of the loops
• Inversion of a loop: (e+, e-)• 4th order permutation of two loops: (e1
+, e2+, e1
-, e2-)
e1e2
Reversal of the signs: K2{3}
e1+
e1-
Forbidden permutation: (e1+, e2
+)( e1-, e3
-)
e3-
e2-
Permitted permutation: (e1+, e2
+)( e1-, e2
-)
Quotient graphs
• Let F be a subgroup of Aut(G) that acts freely on G,
• Denote [x] the orbit of x in V or E by F in G.• The quotient G/F is the “graph of the orbits”:
V(G/F) = {[v] for v in V}E(G/F) = {[e] for e in E}[e] = [u][v] for e = uv
• The graph homomorphism q : q(x) = [x] is called the natural projection of G on G/F
e3
e1
e2
u v
w
R: generated by (e1, e2, e3)( noted R = <(e1, e2, e3)> )
C3
[u]R
[e1]R
B1 = C3/R
qR
u v
wx
e1
e2
e3
e4
R = <(u, v)(x, w)>
Find the quotient graphs C4/R, C4/S and C4/T
C4[u]R
[x]R[e2]R
[e3]R
[e1]R
qR
S = <(u, v, w, x)>T = <(v, x)>
Compare degrees in C4 and its quotients
Free action automorphisms
f in Aut(G) acts freely on the graph G if there is no fixed element in V or E by f
(U,V,W) and (A,B,C,D) act freely(V,W), (B,D) and (A,D)(B,C) do not
Periodic nets
• A net is a simple, 3-connected graph, which is locally finite (WE Klee, 2004).
• (N, T) is a p-periodic net if:– N is a net,– T < Aut(N), is free abelian of rank p,– The number of (vertex and edge) orbits in N by
T is finite.
Then, T acts freely on N
Crystallographic nets
• Crystallographic net: simple 3-connected graph whose automorphism group is isomorphic to a crystallographic space group.
• n-Dimensional crystallographic space group: a group Γ with a free abelian subgroup T of rank n which is normal in Γ, has finite factor group Γ/T and whose centralizer coincides with T (T is maximal abelian).
Example: the square net
V = Z2
E = {pq | q-p = ± a, ± b}
a = (1,0), b = (0,1)
T = {tr: tr (p) = p+r, r in Z2}
C3
Squarenet
<(U,V,W)>
T
Natural projection : examples
ReO3: vertex and edge lattices
O1 O1O2
O2
O3
O3
Re e1e2
ReO3 : quotient graph
K1,3(2) K1,3
(2)
O1
e2 e1
O2O3
e3
e4 e5
e6Re
The β-W net
At
Bt Ct
A
B C K3
(2) K3(2)
Simple structure types
Structure Quotient graph NaCl K2
{6}
CsCl K2{8}
ZnS (sphalerite) K2{4}
SiO2 (quartz) K3{3}
CaF2 (fluorite) K1,2{4}
Drawbacks of the quotient graph
• Non-isomorphic nets can have isomorphic quotient graphs• Different 1-complexes with isomorphic nets can have non-
isomorphic quotient graphs (non-crystallographic nets for different choice of the translation group)
• Different 1-complexes can have both isomorphic nets and isomorphic quotient graph (non-ambiently isotopic embeddings)
Non-isomorphic nets with isomorphic quotient graphs
A
BC
D
A AB B
C
C
D
D
Vector method: the labelled quotient graph as a voltage graph
A
BC
D
AAB
B
C
C
D
D
A
D
BC
A(01)
B(10)
01
10
b
a
b
a
1001
Labelled Quotient Graphs of Periodic Nets
(N, T) : periodic net
A net N A translation group T
N/T : Labelled Quotient Graph
Two vertex-lattices Three edge-lattices Label vectors in T
The kagome net (kgm)
A unit cell in an embedding of kgmand a fundamental transversal withits boundaries.
The fundamental transversal and
the labelled quotient graph obtainedafter closing half-edges.
β-W net: Find the labelled quotient graph
at
bt ct
t in Z2, i=(1,0), j=(0,1)V = {at , bt , ct}E = {atbt , atct , btct , atbt+j , atct+j , ctbt+i}
a
b c10
01 01
“Voltage” or “edge-coloured” graphs
• Graphs with an orientation :for e = uv define e- = vu (and e+ = e)
• A voltage of a graph G by a group A is a mapping : E A with (e-) = [(e)]-1
• Derived graph G :V(G) = {(v, a) in V A}E(G) = {(e, a) in E A}(e, a) = (u, a)(v, ab) for e = uv and (e) = b
• A acts freely on G : f(x, a) = (x, fa) for x in V or E and f in A
Nets and topology: why 3-connectivity?
Derived graphs : examples
10
01
C3
A = <C3>
A = Z2
Cycle and cocycle spaces on Z
• 0-chains: ∑λiui for ui V
• 1-chains: ∑λiei for ei E• Boundary operator: ∂e = v – u for e = uv• Coboundary operator: δu = ∑ei for ei = uvi
• Cycle space = Ker(∂) (cycle-vector C : ∂C = 0)• Cocycle (cut) space = Im(δ) (cocycle-vector k = δu)• dim(cycle space) = #E - #V +1 (cyclomatic number)
λi in Z
Cycle and cocycle vectors
A
B
C
D
1 2
3
4
56
Cycle-vector: e1+e2-e3
Cocycle-vector: δA = -e1+e2+e4
∂e1 = A - D
Find a basis for: 1) the cycle-space and 2) the cocycle-spaceof the cartesian product K2 x K3 (trigonal prism)
Cycle and cut spaces on F2 = {0, 1}(non-oriented graphs)
• The cycle space is generated by the cycles of G• Cut vector of G = edge set separating G in
disconnected subgraphs• The cut space is generated by δu (u in V)
u
v
w
Find δ(u + v + w)
δu δu + δvδu + δv + δw
Minimal nets
• Cyclomatic number g = number of independent cycles of G = number of chords of a spanning tree T
• Choose A = Zg free abelian group of order g• Voltage assignment :
for each chord e of T, (e) is a different generator of Zg
for e in T, (e) = 0
• Minimal net M[G] = derived net G
• The translation subgroup of M[G] ≡ Cycle-space of G
Example 1: graphite (K2(3))
1-chain space: 3-dimensional
Two independent cycles: a = e1-e2, b = e2-e3
One independent cocycle vector: e1+ e2+e3
e1
e2
e3
ab
1
2A B
3
Sr[Si]2 = srs : K4
Edge space: 6-dimensional
Three independent cycles: a , b , cThree independent cut vectors : coboundary of three vertices
a
b
c
Hyperquartz : K3{2}
Archetype: 4-dimensionalTetrahedral coordinationCubic-orthogonal family:(e1-e4), (e2-e5), (e3-e6), ∑ei
Lengths: √2, √2, √2, √6Space group: 25/08/03/003
Edge space: 6-dimensional
Four independent cycles: e1-e4 , e2-e5 , e3-e6 , e1+e2+e3
Two independent cut vectors
1
2
3
4
5
6
A
B C
Symmetry
• : Space group of the crystal structure• T: Normal subgroup of translations• /T: Factor group• Aut(G): Group of automorphism of the
quotient graph G• /T is isomorphic to a subgroup of Aut(G)
Symmetry of minimal nets
• Translation subgroup isomorphic to the cycle space of the quotient graph G (dimension equal to the cyclomatic number)
• Factor group: isomorphic to Aut(G)• Unique embedding of maximum symmetry:
the archetype (barycentric embedding)
Voltages, permutations and crystal symmetry
a
b
C4 = (a, b, -a, -b)ab
-a
-b
C2 = (a, -a)(b, -b)
Reflections
a
bab
σ = (a, -a)
Reflections
a
bab
σ = (b, -b)
Reflections
a
bab
σ = (a, b)(-a, -b)
Reflections
a
bab
σ = (a, -b)(b, -a)
ReO3: /T Aut(G) Oh
• C4 Rotation: (e1, e3, e2, e4)• C3 Rotation: (e1, e3, e5) (e2, e4, e6)• Reflection: (e1, e2)
O1 O1O2
O2
O3
O3
Re e1e2 O1
e2 e1
O2O3
e3
e4 e5
e6Re
Geometric crystal class:minimal nets
• Generators and conjugacy classes in Aut(G) as edge permutations
• Linear operations in the cycle-space• Matrix invariants: order, trace, determinant
1
2
3
A B
Generators of Aut(K2{3}):
• (e1, e2, e3)• (ei , -ei)(A, B)• (e1, -e3)(e2, -e2)(A, B)
Linear operations in the cycle space (point symmetry):
(a, b) = (e1 – e2, e2 – e3) → (e2 – e3, e3 – e1) = (b, - a - b)
In matrix form: 0 -1
1 -1 γ{(e1, e2, e3)} =
0 1
1 0 γ{(e1, -e3)( e2, -e2)} =
-1 0
0 -1 γ{(ei, -ei)} =
p6mm
When the net is not a minimal net:
Geometric crystal class
The point group of the net is the subgroupof Aut(G) which respects the subspace ofthe cycle-space with zero net voltage.
Check the space group of 4.82
A
B
1 2
3
4
56
DC
Kernel: c = e4 + e6 + e3 – e2
Generators:(e4,e6,e3,-e2)(e1,-e5,-e1,e5)(A,B,C,D)(e4,-e6)(e2,e3)(e1,-e1)(A,C)
Cycle basis:a = e1 + e4 + e5 – e3
b = e1 + e2 - e5 + e6
Notice: a b c┴ ┴
Point symmetry
(a, b) → (-e5 + e6 + e1 + e2, -e5 – e4 – e1 + e3) = (b, -a)
(a, b) → (-e1 – e6 + e5 – e2, -e1 + e3 - e5 – e4) = (-b, -a)
p4mmGenerator 0 -1 0 -1 1 0 -1 0
Translation (0 0) (0 0)
Walk: A → B → C → D → A Final translation : c = 0 e4 e6 e3 -e2
Find the space group of quartz
1
2
3
4
5
6
a = 1000 = e1 - e4
b = 0100 = e2 - e5
c = 0010 = e3 - e6
d = 0001 = e4 + e5 + e6
Hyper quartz = N[K3(2)]
Quartz = N[K3(2)]/<1110>
A
B C
Isomorphism class of a net
ij
i jj
Find the labelled quotient graph for the double cell
Isomorphism class of a net
i
jj
1 3 2
4
Automorphisms of the quotient graph G that leave the quotient group of the cycle space by its kernel invariant can be used to extend the translation group if the generated group acts freely:
Kernel: e1-e2
Extension: (e1, e2)(e3, e4)
Isomorphism class of a net
a
bbe1 e4 Kernel: e1-e2
Extension: c = (e1, e2)(e3, e4)(A, B)
b
A B
{A, B}
{e1, e2} {e3, e4}
c2 = ac
e2
e3
Show that cristobalite has the diamond net.
100
101
001
010
010
e1
e8
e7
e6e5e4e3
e2
A
C
B
D
Kernel: e2 – e1 = e7 – e8
e3 – e4 = e6 – e5
(1,8)(2,7)(3,6)(4,5)(A,D)(B,C)
{A,D} {B,C}
{1,8}
{2,7}, 100
{4,5}, k
e8 – e4 + e1 – e5 = 001 → e1 – e5 + e8 – e4 = 001
2k = e4 – e8 + e5 – e1 = 0 0 -1{3,6}(-{4,5}) = 010: voltage for {3,6} = 010 + k
Representation and embedding
• Euclidian representation, p : V E En
for u V, p(u) is a point of En
if e = uv, p(e) is the line p(u)p(v) We note p(e) the vector p(v) - p(u) in Rn
• Embedding: p is injective on Vp(e) p(e´) = unless e ~ e´
K4
Representation with crossing in E2 Embedding in E2
Minimal net: integral embedding
• m = |E(N/T)|• Embedding (p) in the Euclidian space Em with
orthonormal basis {ei, i=1,...m}• For ei E(N/T) set p(ei) = ei
Fix an origin in the vertex set: p(u) = O• For v V(N), let w be a walk from u to v
with projection in N/T: q(w) = xiei
• Set p(v) = O + xiei
Euclidian representations
• Barycentric representation:– : V(N) → E (Euclidian space)– r(u)(u) = (v) r(u) = degree of u
• Periodic representation: –* : T → T* (translation group in E)–[t(u)] = t*[(u)] for any t T, u V(N)
• Determination from the labelled quotient graph:
- the vertex method
- the edge method
v u
Barycentric representation: the vertex method
A
1 0
0 -1
B Neighbours of A : B , B and B00 00 10 0-1
Barycentric equation: 3 (A ) = (B ) + (B ) + (B ) 00 00 10 0-1
3 (A ) = (B ) + [(B ) + 10] + [(B ) + 0-1]
(A ) = 0 (B ) =
00 00 00 00
00 a
b
e1
e2
e3
00
Vertex method and Laplacian matrix
L = (Lij)Lii = - d(Vi)
Lij = m if ViVj E(G) with multiplivity m, for i j
(Vi) = Sum of voltages over ingoing edges at Vi
(V): column vector of Vi
((V)): column vector of (Vi)L.(V) = (a(Vi)) det(L) = 0 !
Solved by eliminating the last row of L and imposing V1 = 0
10 01
A
B C
Draw a barycentric embedding of the 2-periodic net derived from the following labelled quotient graph.
Example
10 01
A
B C
-4 2 2 A -1 -1L = 2 -3 1 (V) = B ((V)) = 1 0 2 1 -3 C 0 1
Set A at the origin:
B 2 2 -1 -1 -3/8 -1/8C -3 1 1 0 -1/8 -3/8= =
-1 a b
A
BC
Example
Periodic embeddings: principles
Edge-space = Cycle-space Cut-space
G: n vertices, m edges
E: m x 1 column vector (edges)B: m x 1 column vector (cycle and cut bases)K: m x m matrix expressing the vectors of B as combinations of edges
B = K.E E = K-1.B
Barycentric embeddings
Embedding = Choice of the basis B in Rp :
• Cycle vectors : given by the net voltages over the cycles of G
• Cut vectors : zero vectors
Archetype = projection of the integral embedding on the cycle-space
Graphite layer : (63)
A
1 0
0 -1
B
e1
e2
e3
A B
1 -1 0K = 0 1 -1 1 1 1
1 0B = 0 1 0 0
2/3 1/3 1/3 1 0 2/3 1/3E = K-1.B = -1/3 1/3 1/3 0 1 = -1/3 1/3 -1/3 -2/3 1/3 0 0 -1/3 -2/3
a
b
e1
e2
e3 2/3 1/3E = -1/3 1/3 -1/3 -2/3
Graphite layer : (63)
a.b = -1
a.a = b.b = 2 (a, b) = 120◦
Atomic positions
p6mmGenerator 0 -1 -1 0 0 1 1 -1 0 -1 1 0
Translation (0 0) (0 0) (0 0)
A: x + TB: x + e1 + T
After inversion: A → -x + T = B = x + 2a/3 + b/3 + T -2x = 2a/3 + b/3 + T
With t = b , x = -a/3 -2b/3 = 2a/3 + b/3
A: 2a/3 + b/3 + TB: a/3 + 2b/3 + T
Topology of Crystal Structures
Advanced topics
Nets as quotients of the minimal net
100
010001
<111> = translation subgroup generated by 111 N/<111>: T = <{100, 010}>
N (T = Z3) N/<111>-1 -1 0
Embeddings as projections of the archetype
A
B
C
D
1 2
3
4
56
3-d archetype,2-d structures: 1-d kernel
3-cycle: 3.92, 93 4-cycle: 4.82
Nets as quotients (projections)
A
BC
D01
10
A
BC
D001
010
100
A
D
BC
1001
A
D
BC
010001
100
=̂
Sr[Si ]
( 4 32)2
I1
(4.8 )4
2
p mm(3.9 ,9 )
3 1
2 3
p m
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
Nets as quotient graphs : kernel of the projection
• (N, T) a periodic net,• G = N/T, its quotient graph,• Voltage assignment : N ~ G.• M[G] : minimal net, with translation group R.• w : closed walk of G with zero net voltage (w) = 0.• Lifting w in M[G] defines (w) in R• S = {(w) : (w) = 0}, subgroup of R• N ~ M[G]/S• S is generated by strong rings of N
S : Kernel ofthe projection
Rings and strong rings in graphs
• A ring is a cycle that is not the sum of two shorter cycles
• A strong ring is a cycle that is not the sum of (an arbitrary number of) shorter cycles
Example 1: the square net
A cycleA strong ring
Sum of three strong rings
Example 2: the cubic net
A strong ring: S1
A ring: R
A 6-cycle: C = S2 + S3
R = S1 + C
Rings are elements of the cycle-space: what is their origin, from the point of view of the labelled quotient graph?
Rings of the derived graph
• Net voltage on a walk :(w) = (e1) (e2 )...(en ) for w = e1e2...en
• Cycles of G project on closed walks of G with net voltage 0• No cycle in the covering tree!• Strong ring : cannot be written as a sum of smaller cycles
Trivial rings
• G quotient graph with voltage assignment in the free group Fn
• Rings in M[G] are commutators of generators of Fn
[a,b]= aba-b-
Nets as quotients: 1-periodic examples
S = <b-2a>
M[B2] = square net
N = M[B2]/S
Square net :
F2, relator [a,b]
Quotient net :
F2, relators [a,b], a2b-2 Check it!
a b
1 1
N
N/<a-b>
A 2-periodic case: tfa and tfa/<001>
10 01
001
100 010
Group presentation
Free group <a,b> <a,b>/C(r)
C(r) : consequence of the relation r, subgroup of <a,b>, contains
r, its conjugates xr = xrx-1, their inverses and combinations
Cycle-space : isomorphic to the kernel C(r)
Strong rings = shortest basis vectors of C(r)
Cyclomatic number and generators
a, b: free generators Commutation relationship: ab = ba
Abelian group of rank 2
Commutator: r = [a, b] = aba-1b-1
[a, b2] = ab2a-1b-2 = (aba-1b-1)b(aba-1b-1 )b-1 = [a, b]b[a, b] a consequence of [a, b]
Trivial rings : hcb
A single commutator [a, b]
A single strong ring, up to translation
A faithfull representation
Fundamental rings : hex
Relator: abc (abc = 1 or c = b-1a-1 )
Two fundamental rings and no trivial ring:
abc acb
[a, b] , [a, c], [b, c] 4-cycles
3-cycles
acb → bac ( = b(acb)b-1 ) [a, b] = (abc)(bac)-1
Strong rings in kgm
Relators: ac-1 , bd-1 , [a, b]
[a, b] 8-cycle
[a, b] = aba-1b-1 = adc-1b-1 6-cycle
Nets defined by a graph and relators,classification of nets
• G a graph with cyclomatic number n • Choose : voltage assignment in A = Zn • S subgroup of A defined by a set of relators• N = M[G]/S• Nets N are ordered by set-inclusion of S
subgroups in A.
Known nets derived from K3{2}
Quartz adb-c- Sum of the 3 2-cycles
NbO adb-c Sum of 2 3-cycles
Moganite adc- Sum of 2 2-cycles
afw bc- One 3-cycle
Kagome ad , bc- Edge disjoint 3-cycles
b-W adb- , bc- Two tangent 3-cycles
Lattice classification of nets derived from K3{2}
• Periodic nets (N,T) such that• Aut(N) is not isomorphic to any isometry group
in the Euclidian space.
(Hidden symmetries = inconsistent with translations)
Non-crystallographic nets
Bounded automorphisms: B(N)
• g Aut(N), such that:
{d(x, g(x)) for x V(N)} is bounded
• B(N) is normal in Aut(N).• For crystallographic nets: B = T (and conversely)
Example
A0
B0
t
= (A0, B0)
t (Xi) = Xi+1 for X = A, B-1t T
Cu carboxylates in MOF structures
: transposition of two Cu centers
System of imprimitivity for B(N)
• Partition of V(N) into finite subsets such that:• For any g B(N) and v (u),• g(v) and g(u) belong to the same cell (block) of
Ai
Bi
blocks:{Ai, Bi} {Ci}
Ci
Theorem
In any non-crystallographic net, the set F(N) of bounded automorphisms of finite order is a normal subgroup of the automorphism group: F(N) < Aut(N)
Orbits by F(N) provide a system of finite blocksof imprimitivity.
F(N) infinite,one fixed point-lattice
F(N) infinite,no fixed point-lattice
F(N) finite,no fixed point-lattice
F(N) and orbits
Equitable and equivoltage partitions
• A partition of the vertex set V(G) of a graph G with cells Ci is equitable if the number of neighbours in Cj of a vertex u in Ci is a constant bij, independent of u;
• A vertex partition of a labelled quotient graph is equivoltage if:– it is equitable and– there is a 1-1 mapping between the stars of any two vertices of the
same cell, which respects the voltages.
Fundamental theorem of NC nets
Any non-crystallographic net can be represented by a labelled quotient graph with an equivoltage partition. Any periodic barycentric representation of the NC net displays vertex collisions; vertices that are equivalent under bounded automorphisms of finite order project on the same Euclidian point.
: Equivoltage partition
N
N/
N/T
(N/)/T = (N/T)/
qT
q´T
q q
: system of imprimitivity derived from F(N)
A 1-periodic example
q
1
1
11
1
qT
q´T
q
Sequence of partitions
1q2
1 = {{A},{B},{C,D}} 2 = {{A},{B,E}}
1
1
1A
B
C
D
1
1
q1
A
E
B
q
A 2-periodic case[E] [D] [C]
[I] [A] [B]
00 10
10 01 01 01 00
00 10
10 01
{A,C,E}
{B,D,F}
Creating a NC net from K3(2)
A
CB
10
01
10
01
= {{A},{B,C}}
4/4/o19 (Sowa, 2012)
ths
Surgery Techniques
Vertex identificationEdge subdivision
Elementary steps: vertex or edge addition and deletion
Combined operations:
Vertex decoration
Vertex decoration
Decoration – contraction
Edge subdivision Vertex deletion
Vertex identification
Edge addition
ContractionDecoration
Graphs with crossings
=
Vertex identification K5
Decoration – Contraction in periodic nets
Building units and underlying nets
Labelled quotient graph ofbuilding units: the voltagerank may be 0, 1, 2 or 3.
Labelled quotient graph ofThe underlying net: the voltagerank may be 0, 1, 2 or 3.
Octahedra chains in rutile
sql – like interlinking of octahedra chainsrtl (rutile) labelled quotient graph
Centered sql
From kgm to hca
Change of origin for A’ and C lattices
From kgm to hca
Relationships between hcb and hca
edge graph
?
?
NiAs
Labelled quotientgraph of nia
Labelled quotientgraph of MoS2 layers
Labelled quotientgraph of MgI2 layers
Bi-layers
MoS2 MgI2 (kgd net)
From hcb to MoS2
From hcb to kgd
001 hcb stacking in nia
4 parallel hcb layers glued atequivalent vertices in MoS2 bi-layers
non-equivalent vertices in MgI2 bi-layers
Garnet A3B2(XO4)3
Unconnected BO6 and XO4 units,
Two 3-periodic interpenetratedcomponents for the A-O subnet.
Na3Sc2(VO4)3
Garnet (80 vertices – 192 edges)
Labelled quotient graph of one A-O component
T = A3O6 cages
Decorated srs net
Substitution of double A-O-A edges by single A-A edges with same net voltage lcv
Garnet: from srs to A-O subnets
T-decorated srs
Contraction (orange edges)
Change of origin
Contraction(orange edges)
Labelled quotient graph of lcv
Garnet: local structure around A3O6 cages
srs-like interlinking of A3O6 cages Top and side views of 4-helicoidal chainsTilt of A3O6 cages: arccos(√3/3) = 54.74o
Distortion of AO8 cubic coordination:B and X interlinking of the twoInterpenetrated lcv A-O subnets
Q = 21°
= p/2 - arccos(1/3) = 19.47o
Groupoids
T = <a, b> infinite translation group in the plane
a
b
Objects: S = [0,4] x [0,4], finite area
Translation in finite objects? (What is the symmetry of the chess board?)
G = {(x,t,y): x, y S, t T, x = ty}
(x,u,y)(y,v,z) = (x,uv,z){
Groupoids of action
Global symmetry Partial symmetry Local symmetry
m, t
m, g
gt t, g m
G=(T,S) T=<a,b>=Z2 , S={A,B,C,D}
g=(C,b,B), h=(B,a,A) : gh=(C,ba,A)
Space Groupoid
A groupoid is formed from those symmetry operations (isometries) of a crystal structure built by isomorphic substrutures:
- Local Symmetries: symmetry operations of the substructure,
- Partial Symmetries: symmetry operations mapping one substructure onto another,
- Global Symmetries: symmetry operation of the entire structure.
Structural analysis of pyroxenes
Mg1 Mg2+ 4 e 0. 0.90420(10) 0.25 Mg2 Mg2+ 4 e 0. 0.28560(10) 0.25 Si1 Si4+ 8 f 0.29223(6) 0.09198(7) 0.24570(10)O1 O2- 8 f 0.11300(10) 0.0828(2) 0.1380(3)O2 O2- 8 f 0.3636(2) 0.2577(2) 0.3179(3)O3 O2- 8 f 0.35430(10) 0.0065(2) 0.0258(3)
MgSiO3
HT clinoenstatiteSpace group: C2/c (Nº 15)
a = 9.5387b = 8.6601c = 5.2620b = 108.701
Structural module in pyroxene (HT clinoenstatite, MgSiO3)in the plane (b-a,c) (primitive cell)
Simplified structure (Mg2 deleted)
Stacking of modules in the direction a+b
a+b
Interlinking of chainsanalysis through the labelled quotient graph
AB(-100)
B(-110)
B
C(010)
C(-110)
C
A
B
C
HT clinoenstatite
kgd(kagome dual)
Quotient graph of the pyroxene module in pyroxenes (primitive cell)
001
-110
Layer group of the module
(b-a, c, a+b) (a-b, c, a+b)
(a, b, c) (b, a, c)
(x, y, z) (-x, y+1/2, z)
(mx0 1/2 0)
(b-a, c, a+b) (b-a, -c, -a-b)
(a, b, c) (-b, -a, -c)
(x, y, z) (x, -y, -z)
2x
Layer group p2/b11 (monoclinic/rectangular) Nº 16
Crystal class 2/m (C2h)
Layer group
001
-110
LT clinoenstatite: quotient graph and double modular structure in the plane (b, c)
Module HT clinoenstatite
LT clinoenstatite versus HT clinoenstatite
BA
D
FC
EAutomorphism without fixed point:
= (A, B)(C, F)(D, E)
Does not change voltages over cycles
represents a translation of the periodicnet mapping one module on the other
Distortion of the Crystal structure Along the respective direction
Interlinking of chains and alternance of modules in LT clinoenstatite
A
B
EC
DF
Action of the automorphism on the edges
a
bc
u
vw
= 010; 2b = -110 g = 010; 2 = 110
a
b
c
u
v
w
-11 -11
01-10
kgd
New translation: = ½ ½ 0
Module HT clinoenstatite
Protoenstatite: quotient graph and double modular structure in the plane (b, c)
BA
D
FC
E
Protoenstatite versus clinoenstatite
= (A, B)(C, F)(D, E)
Does not change voltages in the plane(a,b) but inverses the c axis.
The linear part corresponds to a reflection
Of the periodic net mappingone module on the other
Alternance of modules in protoenstatite
A
B
B
A
E
Proto
LT-clino
D
C
F
E
D
C
F
TranslationAssociated to
:
½ ½ 0
Topology = kgd
Protoenstatite: inversion of chain in successive modules by n(½ ½ 0)[0,0,1]
c
a
b
Ortoenstatite: quotient graph and quadruple modular structure in the plane (b, c)
A
E
F
D
L
K
B
G
H
C
J
I
Interlinking of chains and alternance of modules in ortoenstatite
= (A,D,B,C)(E,L,G,J)(F,K,H,I) : automorphism of the reduced quotient graph Q, conserves voltages along directions 100 e 010
a
b
c
u
v
w
a
b
v
u
w
c
a
a
a
b
b
b
v
v
v
u
u
u
w
ww
c
c
c = 010; 4b = -120 g = 010; 4 = 120
-11 -11
01-10
kgd
New translation: t = ¼ ½ 0
Q
Sequence of modules in ortoenstatite
a
b t
does not represent anautomorphism of the net:
inverses the c axis, and infinite linear chains, alternatively, two modules yes, two modules no.
represents a partial symmetry operation in a groupoid.
Space groupoid of ortoenstatite
• Translation t = t(¼ ½ 0)• Glide reflection = (¼ ½ 0)[0,0,1]• Modules : mk, k Z• mk = t(k-k/2)k/2.m0 = k.m0
• k/2: largest integer inferior or equal to k/2• Layer group of module mk: k.p2/b11.k-1 • Partial symmetry operations from mk to ml: l.p2/b11.k-1 • Only 2k act upon the complete structure: 2k = (t)k → glide reflection a → Pbca
Thank you for your attention
Main references
• Beukemann A., Klee W.E., Z. Krist. 201, 37-51 (1992)• Blatov V.A., Acta. Cryst. A56, 178-188 (2000)• Blatov, V.A., Shevchenko, A.P., Proserpio, D.M., Cryst. Growth Des. 14, 3576-
3586 (2014)• Carlucci L., Ciani G., Proserpio D.M., Coord. Chem. Rev. 246, 247-289 (2003)• Chung S.J., Hahn Th., Klee WE., Acta Cryst. A40, 42-50 (1984)• Delgado-Friedrichs O., Lecture Notes Comp. Sci. 2912, 178-189 (2004)• Delgado-Friedrichs O., O´Keeffe M., Acta Cryst. A59, 351-360 (2003)• Eon J.-G., J. Solid State Chem. 138, 55-65 (1998)• Eon J.-G., J. Solid State Chem. 147, 429-437 (1999)• Eon J.-G., Acta Cryst A61, 501-511 (2005)• Eon, J.-G., Acta Cryst A67, 68—86 (2011)• Eon, J.-G., Acta Cryst A72, 268-293 (2016)• Gross J.L, Tucker T.W., Topological Graph Theory, Dover (2001)• Harary F., Graph Theory, Addison-Wesley (1972)• Klee WE., Cryst. Res. Technol., 39(11), 959-960 (2004)