Symmetry in Crystallography - iycr2014 · The action of a group G on a mathematical structure M...

Symmetry in Crystallography

Denis Gratias and Marianne Quiquandon

IRCP Chimie-ParisTech, Paris

24 Avril 2015

1 / 36Symmetry in Crystallography

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Outline

Which tool ?

Which space ?

What physical significance ?

What next ?


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Which tool ?

Which tool ?

Symmetry Invariance Group Theory


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Which tool ?

Definition

A group is a set G, together with an internal law that combines anytwo elements a and b to form another element, denoted a b orsimply a b that is in G.To qualify as a group, the set and the operation, (G, ), must satisfy four requirements :

Closure : For all a, b in G, the result of the operation, c = a b, is also in G.

Associativity :For all a, b and c in G, (a b) c = a (b c).

Identity element :There exists an element e in G, such that for every element a in G, the equatione a = a e = a holds. Such an element is unique (see below), and thus one speaks of the identityelement.

Inverse element :For each a in G, there exists an element b in G such that a b = b a = e, where e isthe identity element.


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Which tool ?

Group action theory

The action of a group G on a mathematical structure M whoseautomorphism group is Aut(M), is defined by the homomorphism :

G Aut(M) such that a,b G, (ab) = (a).(b)

Example :

e a b ca e c bb c e ac b a e

1 on the set M = {1} 0 : 1 1 1 1

1 on the set M = {1,1}

1 : 1 1 1 12 : 1 1 1 13 : 1 1 1 1


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Which tool ?

Standard representation

G Aut(Rn)

where Aut(Rn) are the isometries of Rn.

G (g) r = (g|t)r = gr + t

or in reciprocal space :

G (g) |q = (g|t)|q = |gqe2igq.t

(g|t) =q|gqe2igq.tq|


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Other representations : Phases trans-formations G H


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Group action and crystallography

Let g.m stands for (g)m, the transform of m M by g G :

the orbit G.m is the set of the transforms of m by G ;

the elements of G which leave m M unchanged make a subgroup Gm called the stabilizer or little groupof m

By restriction, the action G Aut(M) defines an action on M for any subgroup H of G ; then

Hm = H Gm .


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Which space ?A. Bienenstock and P.P. Ewald, Acta Cryst 15, 1253 (1962)

J.W. Jeffery Acta Cryst 16 1239 (1963)

D.S. Rokhsar, D.C. Wright and N. D. Mermin, Acta Cryst A44197 (1988)

D.A. Rabson, N.D. Mermin, D.S. Rokhsar and D.C. Wright, Rev.Mod. Phys. (July 1991)

D. Mermin in Quasicrystals : the state of the art, Edts D.P.DiVincenzo and P.J. Steinhardt, Series on Directions inCondensed Matter Physics 16 (1999), p 137195


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Symmetry is best defined in reciprocalspace

D. Mermin

Any geometric operation that transforms a solid into another one withidentical correlation functions to any finite order n can be qualified asa symmetry operation of the solid in the sense that the operationleads to a solid that is physically indistinguishable from the first one.For a crystal of lattice :

%(r) =

qi(qi )e2iqi .r

the correlation function to order N is defined by :

cN(r1, r2, . . . , rN) =%(r r1) . . . %(r rN)d3r


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Symmetry is best defined in reciprocalspace

Correlation functions are best expressed in Fourier space :%(r r1) . . . %(r rN)d3r =

q1,...,qn

(q1) . . . (qN)e2i(q1.r1+...+qn.rn)

e2i(q1+...+qN ).r d3r

The integral is non zero only for wave vectors q1, . . . ,qN such thatq1 + . . .+ qN = 0.


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Indiscernability principle

There is an indiscernability between two densities %(r) and %(r) if allthe correlation functions to any finite order are equal between the twostructures making them physically undistinguishable .This imposes the product of the Fourier coefficients on a closed pathto be equal :

(q1) . . . (qN) = (q1) . . . (qN), with q1 + . . .+ qN = 0


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Remarkable phases relations (1)Because the density %(r) takes real values, (q) = (q) and thus :

|(q)|2 = (q)(q) = (q)(q) = |(q)|2

Two undistinguishable densities have Fourier coefficients that differonly by a phase factor :

(q) = (q)e2i(q) (1)

Then :

(q)(q) = (q)e2i(q)(q)e2i(q) = (q)(q)

and thus :(q) = (q)


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Remarkable phases relations (2)Next, considering the closed path { q1,q2,(q1 + q2)}, we observethat :

(q1)(q2)((q1 + q2)) = (q1)e2i(q1)(q2)e2i(q2)

((q1 + q2))e2i((q1+q2))

= (q1)(q2)((q1 + q2))

and therefore :

(q1) + (q2) + ((q1 + q2)) = (q1) + (q2) (q1 + q2) = 0

so that :(q1 + q2) = (q1) + (q2) (2)

The phase factors between the Fourier coefficients of twoundistinguishable densities are linear forms in q.


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Remarkable phases relations (3)

Considering now g an operation of the point group of the crystal, theindiscernability principle imposes :

(gq) = e2ig(q)(q) (3)

Using similar arguments as previously with the closed paths{gq,gq} and {gq1, gq2,g(q1 + q2)} and noting thatg(q1 + q2) = gq1 + gq2 we obtain :

g(q) = g(q), g(q1 + q2) = g(q1) + g(q2) (4)


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Remarkable phases relations (4)

Also, explicit writing of the product gh of two operations h and g ofthe indiscernability group leads to the group compatibility relation :

gh(q) = g(hq) + h(q) (5)

and, finally, two distributions related by :

g(q) = g(q) + (gq) (q) (6)

are equivalent (gauge invariance).


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N-dim crystallography

P. M. de Wolff, Acta Cryst., A33, 493 (1977)

A. Janner and T. Janssen, Phys. Rev. B, 15, 643 (1977)

T. Janssen, Acta Cryst. A, 42, 261-271 (1986).

D. Levine and P. J. Steinhardt, Phys. Rev. Lett. 53 (26),2477-2480 (1984)

M. Duneau M and A. Katz, Phys. Rev. Lett. 2688 (1985)

P. A. Kalugin, A. Y. Kitayev and L. S. Levitov J. Physique Lett. 46L-601-607 (1985) ;

V. Elser, Acta Cryst A 42 36-43 (1986)

P. Bak, Phys. Rev. B 32, 5764 (1985)


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% = ( ) % = ( ) = 1 , = = 1


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Superspace groups

The agreement of the superspace group approach with theindiscernability criteria is trivial : with no loss of generality, the basiclinearity in q of the relations between phases shows that they areassociated with scalar products q.t in the superspace :

g(q) = gq.t and (q) = q.t (7)

Formula (7) are sufficient to fulfill all the phase relations imposed bythe indiscernability principle.


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Superspace groups are indiscernabi-lity groups

The indiscernability group of a quasicrystal or incommensurate phase the Fourier carrier of which is a Z-module of rank N is theN-dim superspace group :

any two locally isomorphic structures have same correlationfunctions to any finite order and are therefore physicallyundistinguishable from each other ;

the operations that transform into each other two locallyisomorphic structures (obtained by two distinct but parallel cuts)are indiscernability operations.

the superspace group is the pertinent symmetry tool to beconsidered in physics for the purpose of describing thethermodynamical aspect of phases transformations betweenquasicrystals and crystals.


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Examples

Standard and generalized Penrose ti-lings. . .


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Examples

. . . are not equivalent !


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Examples

Symmetry of translation


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Examples

Y. Meyer harmonious ensembles

28 / 36Symmetry in CrystallographyN

Further examples

Substitution tilings


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Further examples

The hidden mirror


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What next ?

The Pinwheel tiling of J. H. Conway


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Robert V. Moody, Derek Postnikoff and Nicolae Strungaru, Ann. Henri Poincare 7 (2006), 711730


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Which tool ?DefinitionGroup action

Which space ?What physical significance ?ExamplesFurther examplesWhat next ?

Symmetry in Crystallography - iycr2014 · The action of a group G on a mathematical structure M...

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