Topology of Crystal Structures: Applications of Graph Theory

(the vector method)

Jean-Guillaume Eonjgeon@iq.ufrj.br

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

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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)