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CRYSTALLOGRAPHIC OPERATIONS

Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain

miércoles, 9 de octubre de 13

Bilbao Crystallographic Server http://www.cryst.ehu.es

C´esar Capillas, UPV/EHU 1 SYMMETRY OPERATIONS AND THEIR MATRIX-COLUMN PRESENTATION

miércoles, 9 de octubre de 13 Mappings and symmetry operations

Definition: A mapping of a set A into a set B is a relation such that for each element a ∈ A there is a unique element b ∈ B which is assigned to a. The element b is called the image of a.

! ! The relation of the point X to the points X 1 and X 2 is not a mapping because the image point is not uniquely defined (there are two image points).

The five regions of the set A (the triangle) are mapped onto the five separated regions of the set B. No point of A is mapped onto more than one image point. Region 2 is mapped on a line, the points of the line are the images of more than one point of A. Such a mapping is called a projection.

miércoles, 9 de octubre de 13 Mappings and symmetry operations

Definition: A mapping of a set A into a set B is a relation such that for each element a ∈ A there is a unique element b ∈ B which is assigned to a. The element b is called the image of a.

An leaves all distances and angles invariant. An ‘isometry of the first kind’, preserving the counter–clockwise sequence of the edges ‘short–middle–long’ of the triangle is displayed in the upper mapping. An ‘isometry of the second kind’, changing the counter–clockwise sequence of the edges of the triangle to a clockwise one is seen in the lower mapping.

A parallel shift of the triangle is called a . Translations are special . They play a distinguished role in crystallography.

miércoles, 9 de octubre de 13 Description of isometries

coordinate system: {O, a, b, c}

isometry: ~ X X (x,y,z) (x,y,z)~ ~ ~ ~ x = F1(x,y,z)

miércoles, 9 de octubre de 13 Matrix-column presentation of isometries

linear/matrix translation part column part

matrix-column Seitz symbol pair miércoles, 9 de octubre de 13 Short-hand notation for the description of isometries

(W,w) ~ isometry: X X

notation rules: -left-hand side: omitted -coefficients 0, +1, -1 -different rows in one line

examples: -1 1/2 -x+1/2, y, -z+1/2 1 0 { x+1/2, y, z+1/2 -1 1/2

miércoles, 9 de octubre de 13 EXERCISES

Problem 2.19

Construct the matrix-column pair (W,w) of the following coordinate triplets:

(1) x,y,z (2) -x,y+1/2,-z+1/2 (3) -x,-y,-z (4) x,-y+1/2, z+1/2

miércoles, 9 de octubre de 13 Combination of isometries (U,u) ~ X X

(V,v) (W,w) ~ X

miércoles, 9 de octubre de 13 Inverse isometries

(W,w) X ~ X ~ X (C,c)=(W,w)-1

I = 3x3 identity matrix (C,c)(W,w) = (I,o) o = zero translation column (C,c)(W,w) = (CW, Cw+c) CW=I Cw+c=o

C=W-1 c=-Cw=-W-1w

miércoles, 9 de octubre de 13 Matrix formalism: 4x4 matrices

augmented matrices:

point X −→ point X˜ :

miércoles, 9 de octubre de 13 4x4 matrices: general formulae point X −→ point X˜ :

combination and inverse of isometries:

miércoles, 9 de octubre de 13 EXERCISES

Problem 2.19 (cont.)

Construct the (4x4) matrix-presentation of the following coordinate triplets:

(1) x,y,z (2) -x,y+1/2,-z+1/2 (3) -x,-y,-z (4) x,-y+1/2, z+1/2

miércoles, 9 de octubre de 13 Crystallographic symmetry operations

Symmetry operations of an object The isometries which map the object onto itself are called symmetry operations of this object. The symmetry of the object is the set of all its symmetry operations.

Crystallographic symmetry operations If the object is a crystal pattern, representing a real crystal, its symmetry operations are called crystallographic symmetry operations.

The equilateral triangle allows six symmetry operations: rotations by 120 and 240 around its centre, reflections through the three thick lines intersecting the centre, and the identity operation.

miércoles, 9 de octubre de 13 Crystallographic symmetry operations

characteristics: fixed points of isometries (W,w)Xf=Xf geometric elements

Types of isometries preserve handedness

identity: the whole space fixed

translation t: no fixed point x˜ = x + t

◦ rotation: one line fixed φ = k × 360 /N rotation axis

screw rotation: no fixed point screw axis screw vector

miércoles, 9 de octubre de 13 Crystallographic symmetry operations

miércoles, 9 de octubre de 13 Crystallographic symmetry operations

Screw rotation n-fold rotation followed by a fractional p translation n t parallel to the rotation axis

Its application n times results in a translation parallel to the rotation axis

miércoles, 9 de octubre de 13 do not Types of isometries preserve handedness

centre of roto-inversion fixed roto-inversion: roto-inversion axis

inversion: centre of inversion fixed

reflection: plane fixed reflection/mirror plane

glide reflection: no fixed point glide plane glide vector

miércoles, 9 de octubre de 13 Symmetry operations in 3D Rotoinvertions

miércoles, 9 de octubre de 13 Symmetry operations in 3D Rotoinvertions

miércoles, 9 de octubre de 13 Symmetry operations in 3D Rotoinvertions

miércoles, 9 de octubre de 13 Crystallographic symmetry operations Glide plane

reflection followed by a fractional translation 1 2 t parallel to the plane

Its application 2 times results in a translation parallel to the plane

miércoles, 9 de octubre de 13 Matrix-column presentation of some symmetry operations Rotation or roto-inversion around the origin:

W11 W12 W13 0 0 0

W21 W22 W23 0 0 = 0 ( W31 W32 W33 0 ) 0 0 Translation:

1 w1 x x+w1 1 w2 y = y+w2 ( 1 w3) z z+w3 Inversion through the origin:

-1 0 x -x -1 0 y = -y ( -1 0 ) z -z miércoles, 9 de octubre de 13 Geometric meaning of (W , w) W information

(a) type of isometry

order: Wn=I rotation angle

miércoles, 9 de octubre de 13 EXERCISES Problem 2.19 (cont.)

Determine the type and order of isometries that are represented by the following matrix-column pairs: (1) x,y,z (2) -x,y+1/2,-z+1/2 (3) -x,-y,-z (4) x,-y+1/2, z+1/2

(a) type of isometry

miércoles, 9 de octubre de 13 Geometric meaning of (W , w) W information

Texto(b) axis or normal direction :

(b1) rotations:

=

(b2) roto-inversions:

reflections:

miércoles, 9 de octubre de 13 EXERCISES Problem 2.19 (cont)

Determine the rotation or rotoinversion axes (or normals in case of reflections) of the following symmetry operations

(2) -x,y+1/2,-z+1/2 (4) x,-y+1/2, z+1/2

rotations: Y(W) = Wk-1 + Wk-2 + ... + W + I

reflections: Y(-W) = - W + I

miércoles, 9 de octubre de 13 Geometric meaning of (W , w) W information

(c) sense of rotation: for rotations or rotoinversions with k>2 det(Z):

x non-parallel to

miércoles, 9 de octubre de 13 Fixed points of isometries solution: (W,w)Xf=Xf point, line, plane or space no solution Examples: What are the -1 0 0 0 x x fixed solution: 0 1 0 0 y = y 1/2 z points of this 0 0 -1 ) z ( isometry?

-1 0 0 0 x x NO solution: 0 1 0 1/2 y = y ( 0 0 -1 1/2) z z

miércoles, 9 de octubre de 13 Geometric meaning of (W,w) (W,w)n=(I,t) (W,w)n=(Wn,(Wn-1+...+ W+I)w) =(I,t)

(A) intrinsic translation part : glide or screw component t/k (A1) screw rotations:

=

(A2) glide reflections: w

miércoles, 9 de octubre de 13 Glide or Screw component (intrinsic translation part)

(W,w)k= (W,w). (W,w). ... .(W,w)=(I,t)

(W,w)k=(Wk,(Wk-1+...+ W+I)w) =(I,t)

screw rotations : t/k=1/k (Wk-1+...+ W+I)w

glide reflections: w

miércoles, 9 de octubre de 13 EXERCISES Problem 2.20 (cont.)

Determine the intrinsic translation parts (if relevant) of the following symmetry operations

(1) x,y,z (2) -x,y+1/2,-z+1/2 (3) -x,-y,-z (4) x,-y+1/2, z+1/2

screw rotations: t/k=1/k (Wk-1+...+ W+I)w

glide reflections: w

miércoles, 9 de octubre de 13 Fixed points of (W,w)

Location (fixed points x F ):

(B1) t/k = 0:

(B2) t/k ≠ 0:

miércoles, 9 de octubre de 13 EXERCISES Problem 2.19 (cont.)

Determine the fixed points of the following symmetry operations:

(1) x,y,z (2) -x,y+1/2,-z+1/2 (3) -x,-y,-z (4) x,-y+1/2, z+1/2

fixed points:

miércoles, 9 de octubre de 13 P21/c (No. 14)

EXAMPLE

Matrix-column presentation

Geometric interpretation miércoles, 9 de octubre de 13 www.cryst.ehu.es

miércoles, 9 de octubre de 13 Crystallographic databases

Group-subgroup relations Structural utilities

Representations of point and space groups

Solid-state applications

miércoles, 9 de octubre de 13 Crystallographic Databases

International Tables for Crystallography

miércoles, 9 de octubre de 13 EXERCISES Problem 2.19

Construct the matrix-column pairs (W,w) of the following coordinate triplets:

(1) x,y,z (2) -x,y+1/2,-z+1/2 (3) -x,-y,-z (4) x,-y+1/2, z+1/2 Characterize geometrically these matrix-column pairs taking into account that they refer to a monoclinic basis with unique axis b,

Use the program SYMMETRY OPERATIONS for the geometric interpretation of the matrix-column pairs of the symmetry operations.

miércoles, 9 de octubre de 13 Bilbao Crystallographic Server Problem: Geometric SYMMETRY Interpretation of (W,w) OPERATION

miércoles, 9 de octubre de 13 Problem 2.19 SOLUTION

(i)

1 0 -1 0 (W,w)(1)= 1 0 (W,w)(2)= 1 1/2 1 0 -1 1/2

-1 0 1 0 (W,w)(3)= -1 0 (W,w)(4)= -1 1/2 -1 0 1 1/2

(ii) ITA description: under Symmetry operations

miércoles, 9 de octubre de 13 CO-ORDINATE TRANSFORMATIONS IN CRYSTALLOGRAPHY

miércoles, 9 de octubre de 13 International Tables for Crystallography (2006). Vol. A, Chapter 5.1, pp. 78–85. 5.1. Transformations of the coordinate system (unit-cell transformations)

International Tables for Crystallography (2006). Vol. A, Chapter 5.1, pp. 78–85.BY H. ARNOLD 5.1. Transformations of the coordinate system (unit-cell transformations) 5.1.1. Introduction 5.1.3. General transformation BY H. ARNOLD There are two main uses of transformations in crystallography. Here the crystal structure is considered to be at rest, whereas the (i) Transformation of the coordinate system and the unit cell coordinate system and the unit cell are changed. Specifically, a 5.1.1. Introductionwhile keeping the crystal at rest. This aspect forms the5.1.3.main topicGeneralof pointtransformationX in a crystal is defined with respect to the basis vectors a, b, c the present part. Transformations of coordinate systems are useful and the origin O by the coordinates x, y, z, i.e. the position vector r There are two main uses of transformationswhen nonconventionalin crystallography.descriptionsHereof athecryscrystaltalstructurestructure areis consideredof point Xtoisbegivenat rest,by whereas the (i) Transformation of the coordinateconsidered,systemfor instanceand the inunitthecelstudyl coordinateof relationssystembetweenanddifferentthe unit cell are changed. Specifically, a while keeping the crystal at rest. This aspect forms the main topic of point X in a crystal is defined with respect to the basis vectorr sxa, b,ycb zc structures, of phase transitions and of group–subgroup relations. ˆ ‡ ‡ the present part. Transformations Unit-cellof coordinatetransformatsystemsionsare usefuloccur pandarticularlythe originfrequentlyO by the cwhenoordinates x, y, z, i.e. the position vector r x when nonconventional descriptionsdifferentof asettingscrystalorstructurecell choicesare of ofmonoclinic,point X is orthorhombicgiven by or considered, for instance in the study of relations between different a, b, c y : rhombohedral space groups are to be compared or when ‘reducedr xa yb zc ˆ †0 1 structures, of phase transitions andcells’of aregroup–subgroupderived. relations. ˆ ‡ ‡ z B C Unit-cell transformations occur p(ii)articularlyDescriptionfrequentlyof the symmetrywhen operations (motions) of an x @ A different settings or cell choices of monoclinic, orthorhombic or The same point X is given with respect to a new coordinate system, object (crystal structure). This involves the transformation of thea, b, c y : rhombohedral space groups are tocoordinatesbe comparedof aorpointwhenor ‘reducedthe components of a position vector whileˆ i.e.†0the 1new basis vectors a0, b0, c0 and the new origin O0 (Fig. cells’ are derived. keeping the coordinate system unchanged. Symmetry operations are 5.1.3.1),z by the position vector (ii) Description of the symmetry operations (motions) of an B C treated in Chapter 8.1 and Part 11. TheTheysaareme pointbrieflyXreviewedis given winith respect@ toAa new coordinater0 xsystem,0a0 y0b0 z0c0: object (crystal structure). This involvesChapterthe5.2.transformation of the ˆ ‡ ‡ coordinates of a point or the components of a position vector while i.e. the new basis5.1. vectorsTRANSFORMATIONSa0, bCo-ordinateIn0, cthis0 andsection,OFtheTHEnewtheCOORDINATEorelations rigintransformationO0 between(Fig.SYSTEMthe primed and unprimed 5.1.3.1), by the position vector quantities are treated.1 keeping the coordinate system unchanged. Symmetry operations are P QÀ : 5.1.2. Matrix notation ˆ treated in Chapter 8.1 and Part 11. They are briefly reviewed in r0 x0a0 yThe0b0 generalz0c0: transformation 3-dimensional(affine transformation) spaceof the Chapter 5.2. Throughout this volume, matrices are written in the followingˆ ‡coordinate‡ system consists of two apÃarts, a linearaÃ0 part and a shift In this section, the relations betweenof origin.theTheprimed3 3andmatrix(aunprimed, b,Pc),of b origintheà linearP O:bpartÃ0 point:and the X(3x, y,1 z) notation:  † 0 1 ˆ 0 1  † quantities are treated. column matrix p, containing the componecà nts ocÃf0 the shift vector p, 5.1.2. Matrix notation The general transformationdefine(affinethe transformationtransformation)uniquely.of the@ It Ais represented(P,p@) A by the symbol As (1 3) row matrices: coordinate system consists of two(P, p).arts, a linearTheseparttransformatiand aonshiftrules apply also to the quantities covariant Throughout this volume, matrices areÂwritten in the following ! ! ! ! ! ! (a, b, c) the basis vectorsof origin.of directThespace3 3 matrix P of the linear withpartrespectand(taheto,theb3 ,basisc1 ),vectors originaÃ, bà O’:, cà and pointcontravariant X(x ,ywith,z ) notation:  † (i) The linearespectr parttoimpliesa, b, c, whichaÂchangeare† writtenof oasrientationcolumn matrices.or lengthTheyoarer the (h, k, l) the Miller indicescolumnof amatrixplanep(or, containinga set of thebothcomponeof thentsbasisindicesof tvectorsheofshifta dairectionvector, b, c,ini.e.pdirec, t space, [uvw], which are transformed planes)Fig. 5.1.3.1.in directGeneraldefinespaceaffinethetransformation,ortransformationthe coordinatesconsistinguniquely.of a shiftItofisoriginrepresentedby by the symbol As (1 3) row matrices: of a frompointO toinO0 reciprocalby(Pa, shiftp). vectorspacepTransformationwith components p1 and p2 and amatrix-columnchange a0, b0, c0 a, bpair, c P (P,p)  of basis from a, b to a0, b0. This implies a change in the coordinates of † ˆ † u0 u (a, b, c) the basis vectors of direct space the point X from(i)x, yTheto x0, lineay0. r part implies a change of orientation or length or P11 P12 P13 v0 Q v : (h, k, l) the Miller iAsndices(3 of1) aorplane(4 (or1) columna set ofmatrices: (i) linear part: change of orientation or length:0 1 ˆ 0 1   both of the basis vectors a, b, c, i.e. a, b, c Pw021 P22 wP23 x x=y=z the coordinates of a point in direct space 0 1 planes) in direct space or the coordinatesAlso, the inverse matrices of P and p are needed. They are ˆ † @ A @ A ˆ † a0, b0, c0 a, b, c P P31 P32 P33 of a point inaÃreciprocal=bÃ=cà space the basis vectors of reciprocal space1 In contrast to all quantities mentioned above, the components of a † Q PÀ † ˆ † B C (u=v=w) the indices of a direction inˆdirect space P11 P12 positionP13 vectorPr11ora the@Pcoordinates21b P31cof, a pointA X in direct space x, y, z dependˆ also on‡the shift‡of the origin in direct space. The As (3 1) or (4 1) column matrices:p p p p theandcomponents of a shift vector from   1= 2= 3 a, b, c P21 P22 generalP23 (affine) transformationP12a P22bis gPiven32cby, ˆ † 1 0 1 x x=y=z the coordinates of a point in directoriginspaceO to the new originq OP0 À p: ˆ † ‡ ‡ ˆ † ˆ À P31 P32 P33 P13a P23b P33c : aÃ=bÃ=cà the basis vectorsq qo1=fqr2eciprocal=q3 spacethe components of an inverse origin B C x0 ‡ x ‡ † † ˆ † The matrix q consists of the components of the negative@shift vector A (u=v=w) the indices of a direction in directshiftq whichspacefromreferoriginto theOcoordinate0 to originsystemO, wa0,ithb0, cP0,11i.e.a ForP21a bpurePlinear31c, transformation,y0 Qthe yshift vectorq p is zero and the 1 ˆ ‡ ‡ 0 1 ˆ 0 1 ‡ q PÀ p (ii) origin shiftsymbol by a ishifts (P, ovector). p(p1,p2,p3): p p1=p2=p3 the components of a shift vector from q q1a0 q2b0 q3c0: P a P b P c, z0 z ˆ † ˆ À ˆ ‡ ‡ 12 22 32 B C B C origin O towthe neww1=worigin2=w3 O 0 the translation part of a symmetry ‡The determinant‡ of P@, detAP , s@houldA be positive. If det P is ˆ † Thus, the transformation (Q, q) is the inverse Ptransformationa P b of P cthe origin † hasQ11 x Q12y Q13z q1 † q q q q the components of an inverse originoperation W in direct space 13 negative,23 a right-handed33 : coordinateO’ system‡ is‡transformed‡ into a 1= 2= 3 (P, p). Applying (Q, q) to the basis vectorsO’ =a , bO, c +and ‡pthe origin‡ † Q21x Q22y Q23z q2 : ˆ † x x=y=z=1 the augmented 4 1 column matrix0 of0 0 left-handed one (orcoordinatesvice versa). ˆIf (p0det1,pP2,p‡ 3)0, inthe‡ new‡basis1vectors shift from origin O 0 to origin O, Ow0,iththe old basisForvectorsa purea, blinear, c withtransformation,origin O are obtained.the shift vector p is zero and the 1 ˆ †  † are linearly dependentthe olda ndcoordinatedo notQ 31formx† ˆsystemQa32ycompletQ33 z eqc3 oordinate q P p the coordinatesFor a two-dimensionalof a pointtransformatiin directonspaceof a0 and b0, some B ‡ ‡ ‡ C À symbol is (P, o). @ A ˆ À elements of miércoles,Q are9 de octubreset de 13as follows: Qsystem.33 1 and w w1=w2=w3 the translation part of a symmetry The determinant of P, det P , ˆshould be positive.Example If det P is As (3 3) or (4 4) squareQ13 Qmatrices:23 Q31 Q32 0. In this chapter, transformations in three-dimensional space are ˆ † ˆ ˆ ˆ ˆ † † operation W in direct 1space The quantitiesnegative,which transforma right-handedin the samecoordinateway as thesystembasis is transformedIf no shift of originintoisaapplied, i.e. p q o, the position vector P, Q PÀ linear parts of an affine transformation; treated. A change of the basis vectors in two dimensions, i.e. of the x x=y=z=1 the augmented ˆ4 1 column matrixvectors ofa, b, c areleft-handedcalled covariantone (orquantitiesvice versandaare). Ifwrittenbasisdet Pasvectorsrow0, thea randnewof pointb,basiscanX isvectorsbetransformedconsideredby as aˆ three-dimenˆ sional ˆ †  † if matrices.P is appliedTheyareare:to alinearly1 3dependentrow matrix,and do not form† ˆ a complete coordinate the coordinates of a point in directQ mustspacebe applied t o a 3† 1 column transformation with invariant c axis. This is achieved by setting the Miller indicessystem.of a plane (or a set of planes), (hkl), in direct x x0 matrix, and vice versa  † P33 1 and P13 P23 P31 P32 0. As (3 3) or (4 4) square matrices: space and In this chapter, transformations inˆ three-dimenˆsionalˆrs0paceˆa, bare, c PˆQ y a0, b0, c0 y0 : W the rotation part of a symmetry (ii) A shift of origin is definedˆ by† the0shiftz 1 ˆvector †0 z 1 P, Q ÂP 1  linear parts of an affine transformation; treated. A change of the basis vectors in two dimensions, i.e. of the 0 À operationthe coordinatesW in directof a pointspacein reciprocal space, h, k, l. @ A @ A ˆ if P is applied to a 1 3 row matrix, basis vectors a and b, can be considered as a three-dimenp p1asionalp2b p3c: P p † theBothaugmentedare transformedtransformationaffineby 4 4wtrans-ith invariant c axis. This is achievedIn this case,ˆbyr settingr‡0, i.e. the‡position vector is invariant, although Q must be Papplied to a 3 1 column  † the basis vectorsˆ and the components are transformed. For a pure ˆ o 1  †formation matrix,P w1ithha,nkod, lP 0,h0,, kP,0l P: P TheP basis0. vectors a, b, c are fixed at the origin O; the new basis matrix, and viceversa  33 0 0 ˆ0 13 23† 31 32 shift of origin, i.e. P Q I, the transformed position vector r0 ˆ † ˆ ˆ † ˆ ˆvectorsˆ are fixedbecomesat the new originˆ Oˆ0 which has the coordinates W the rotation part ofQa symmetryq theUsually,augmentedthe Milleraffineindices4are made4 trans-relative prime before and after Q (ii) A shift of†origin1 is definedp1b,yp2t,hep3 shiftin thevectorold coordinate system (Fig. 5.1.3.1). ˆ o 1 formationthe transformatimatrix,on. with Q PÀ operation W in direct space r x q a y q b z q c   The quantities which are covariantˆ wpith respectp1a top2theb basisp3c: 0 1 2 3 P p the augmented affineW 4w 4 trans-the augmented 4 4 matrix of a ˆ ‡For a pure‡ origin shift, the basisˆ vectors‡ † ‡do not‡ change† ‡ their‡ † lengths P W vectors a, b, c are contravariant with respect to the basis vectors r q1a q2b q3c  † symmetry operation  in† direct space (cf. ˆ ‡ ‡ ‡ ˆ o 1 formation matrix,ˆ woith o1 0, 0,a0Ã, bÃ, cà of reciprocalThe basisspace.vectors a, b, c are fixedor orientations.at the originInOthis; thecase,newthebasistransformation matrix P is the unit    ˆ Chapter† 8.1 and Part 11). x p1 a y p2 b z p3 c Q q the augmented affine 4 4 trans-The basis vectorsvectorsof arereciprocalfixed sapacet thearenewwrittenoriginmatrixas a Ocolumn0Iwhichand thehassymbolthe coordinatesof theˆ pureÀ s†hift‡ becomesÀ † ‡(I , pÀ). † Q  † 1 matrix and theirp transformation, p , p in theisoachievedld coordinateby the msystematrix Q: (Fig. 5.1.3.1). r p1a p2b p3c: ˆ o 1 formation matrix, with Q PÀ 1 2 3 ˆ À À À   ˆ aÃ0 aà 78 W w the augmented 4 4 matrix of a For a pure origin shift, the basis vectors do not changeHere thetheirtransformedlengthsvector r0 is no longer identical with r. W symmetry operation  in† direct space (cf. 0 bÃ0 1 Q0 bà 1 ˆ o 1 or orientations.ˆ In this case, the transformation mItatrixis convenientP is thetounitintroduce the augmented 4 4 matrix Q   cÃ0 cà Chapter 8.1Copyrightand Part 11). 2006 International UnionBmatrix Cof CrystallographyI andB theC symbol of the pure shift becomeswhich is(Icomposed, p). of the matrices Q and q in the followi †ng manner @ A @ A Q11 Q12 Q13 aà (cf. Chapter 8.1): 78 0 Q21 Q22 Q23 10 bà 1 ˆ Q11 Q12 Q13 q1 Q31 Q32 Q33 cà B CB C Q q Q21 Q22 Q23 q2 @ Q a Q b AQ@ c A Q 0 1 11 à 12 à 13 à ˆ o 1 ˆ Q31 Q32 Q33 q3 Copyright  2006 International Union of Crystallography ‡ ‡   Q21aà Q22bà Q23cà : B 0 0 0 1 C ˆ 0 ‡ ‡ 1 B C Q31aà Q32bà Q33cà @ A B ‡ ‡ C with o the 1 3 row matrix containing zeros. In this notation, the @ A  † The inverse transformation is obtained by the inverse matrix transformed coordinates x0, y0, z0 are obtained by 79 EXAMPLE

miércoles, 9 de octubre de 13 EXAMPLE

miércoles, 9 de octubre de 13 Transformation matrix-column pair (P,p)

1/2 1/2 0 1/2 (P,p)= -1/2 1/2 0 1/4 ( 0 0 1 0 )

1 -1 0 -1/4 -1 (P,p) = 1 1 0 -3/4 ( 0 0 1 0 )

miércoles, 9 de octubre de 13 Transformation of the coordinates of a point X(x,y,z): -1 -1 x (X’)=(P,p) (X) x’ P11 P12 P13 p1 =(P-1, -P-1p)(X) y’ = P21 P22 P23 p2 y z’ ( P31 P32 P33 p3 ) z

special cases -origin shift (P=I): -change of basis (p=o) :

Transformation of symmetry operations (W,w): (W’,w’)=(P,p)-1(W,w)(P,p)

Transformation by (P,p) of the unit cell parameters: metric tensor G: G´=Pt G P

miércoles, 9 de octubre de 13 Short-hand notation for the description of transformation matrices

Transformation5.1. TRANSFORMATIONS matrix:OF THE COORDINATE(aSYSTEM,b,c), origin O 1 P QÀ : ˆ

aà aÃ0 P11 P12 P13 p1 bà P bÃ0 : 0 1(P,pˆ 0)=1 P21 P22 P23 p2 cà cÃ0 @ A @ A These transformation rules apply also to(the quantitiesP31 covariantP32 P33 p3 ) with respect to the basis vectors aÃ, bÃ, cà and contravariant with respect to a, b, c, which are written as column matrices. They are the indices of a direction in direct space, [uvw], which are transformed Fig. 5.1.3.1. General affine transformation, consisting of a shift of origin by (a’,b’,c’), origin O’ from O to O0 by a shift vector p with components p1 and p2 and a change of basis from a, b to a0, b0. This implies a change in the coordinates of u0 u the point X from x, y to x0, y0. v0 Q v : 0 1 ˆ 0 1 -written by columnsw0 w Also, the inverse matrices of P and p are needed. They are @ A @ A 1 In contrast to all quantities mentioned above, the components of a Q PÀ -coefficients 0, +1, -1 notationˆ rules: position vector r or the coordinates of a point X in direct space and x, y, z depend also on the shift of the origin in direct space. The -differentgeneral (affine) transformation columnsis given by in one line 1 q PÀ p: ˆ À x x The matrix q consists of the components of the negative shift vector-origin shift0 q which refer to the coordinate system a , b , c , i.e. y0 Q y q 0 0 0 0 1 ˆ 0 1 ‡ q q1a0 q2b0 q3c0: z0 z ˆ ‡ ‡ B C B C @ A @Q11Ax Q12y Q13z q1 Thus, the transformation (Q, q) is the inverse transformation of ‡ ‡ ‡ (P, p). Applying (Q, q) to the basis vectors a , b , c and the origin Q21x Q22y Q23z q2 : 0 0 0 ˆ 0 ‡ ‡ ‡ 1 O0, the old basis vectors a, b, c with origin O are obtained.1 -1 -1/4 Q31x Q32y Q33z q3 For a two-dimensional transformation of a0 and b0, some B ‡ ‡ ‡ C example: @ A elements of Q are set as follows: Q33 11 an1d -3/4 a+b, -a+b, c;-1/4,-3/4,0 ˆ Example Q13 Q23 Q31 Q32 0. { Theˆ quantitiesˆ ˆwhich ˆtransform in the same way as the basis 1If no shift0 of origin is applied, i.e. p q o, the position vector vectors a, b, c are called covariant quantities and are written as row r of point X is transformed by ˆ ˆ matrices. miércoles,They are: 9 de octubre de 13 the Miller indices of a plane (or a set of planes), (hkl), in direct x x0 space and r0 a, b, c PQ y a0, b0, c0 y0 : ˆ † 0 1 ˆ †0 1 z z0 the coordinates of a point in reciprocal space, h, k, l. @ A @ A Both are transformed by In this case, r r0, i.e. the position vector is invariant, although the basis vectorsˆ and the components are transformed. For a pure h0, k0, l0 h, k, l P: shift of origin, i.e. P Q I, the transformed position vector r0 † ˆ † becomes ˆ ˆ Usually, the Miller indices are made relative prime before and after the transformation. r0 x q1 a y q2 b z q3 c The quantities which are covariant with respect to the basis ˆ ‡ † ‡ ‡ † ‡ ‡ † vectors a, b, c are contravariant with respect to the basis vectors r q1a q2b q3c ˆ ‡ ‡ ‡ aÃ, bÃ, cà of reciprocal space. x p1 a y p2 b z p3 c The basis vectors of reciprocal space are written as a column ˆ À † ‡ À † ‡ À † matrix and their transformation is achieved by the matrix Q: r p1a p2b p3c: ˆ À À À aÃ0 aà Here the transformed vector r0 is no longer identical with r. bÃ0 Q bà 0 1 ˆ 0 1 c c It is convenient to introduce the augmented 4 4 matrix Q B Ã0 C B à C which is composed of the matrices Q and q in the followi †ng manner @ A @ A Q11 Q12 Q13 aà (cf. Chapter 8.1): 0 Q21 Q22 Q23 10 bà 1 ˆ Q11 Q12 Q13 q1 Q31 Q32 Q33 cà B CB C Q q Q21 Q22 Q23 q2 @ Q a Q b AQ@ c A Q 0 1 11 à 12 à 13 à ˆ o 1 ˆ Q31 Q32 Q33 q3 ‡ ‡   Q21aà Q22bà Q23cà : B 0 0 0 1 C ˆ 0 ‡ ‡ 1 B C Q31aà Q32bà Q33cà @ A B ‡ ‡ C with o the 1 3 row matrix containing zeros. In this notation, the @ A  † The inverse transformation is obtained by the inverse matrix transformed coordinates x0, y0, z0 are obtained by 79 EXERCISES Problem 2.20

The following matrix-column pairs (W,w) are referred with respect to a basis (a,b,c):

(1) x,y,z (2) -x,y+1/2,-z+1/2 (3) -x,-y,-z (4) x,-y+1/2, z+1/2

Determine the corresponding matrix-column pairs (W’,w’) with respect to the basis (a’,b’,c’)= (a,b,c)P, with P=c,a,b.

Determine the coordinates X’ of a point X= 0,70 with respect to the new basis (a’,b’,c’). 0,31 0,95

miércoles, 9 de octubre de 13