SPACE GROUPS
International Tables for Crystallography, Volume A: Space-group Symmetry
Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain
Bilbao Crystallographic Server http://www.cryst.ehu.es
C´esar Capillas, UPV/EHU 1 SPACE GROUPS
Crystal pattern: A model of the ideal crystal (crystal structure) in point space consisting of a strictly 3-dimensional periodic set of points Space group G: The set of all symmetry operations (isometries) of a crystal pattern
Translation subgroup H G: The infinite set of all translations that are symmetry operations of the crystal pattern
Point group of the The factor group of the space group G with respect to the translation space groups PG: subgroup T: PG ≅ G/H INTERNATIONAL TABLES FOR CRYSTALLOGRAPHY VOLUME A: SPACE-GROUP SYMMETRY Extensive tabulations and illustrations of the 17 plane groups and of the 230 space groups
•headline with the relevant group symbols; •diagrams of the symmetry elements and of the general position; •specification of the origin and the asymmetric unit; •list of symmetry operations; •generators; •general and special positions with multiplicities, site symmetries, coordinates and reflection conditions; •symmetries of special projections; GENERAL LAYOUT: LEFT-HAND PAGE General Layout: Right-hand page Primitive and centred lattice basis in 2D {a1, a2}: two translation vectors, linearly independent, form a lattice basis Primitive basis: If all lattice vectors are expressed as integer linear combinations of the basis vectors If some lattice vectors are expressed as Centred basis: linear combinations of the basis vectors with rational, non-integer coefficients
Conventional centred basis (2D): c
Number of lattice points per primitive and centred cells Crystal families, crystal systems, conventional coordinate system and Bravais lattices in 2D 3D-unit cell and lattice parameters lattice basis: {a, b, c} C B unit cell: the parallelepiped A defined by the basis vectors primitive P and centred unit cells: A,B,C,F, I, R
number of lattice points per unit cell Crystal families, crystal systems, lattice systems and Bravais lattices in 3D HEADLINE BLOCK Short Hermann- Crystal class Mauguin symbol Schoenflies (point group) Crystal symbol system
Number of Full Hermann- Patterson space group Mauguin symbol symmetry HERMANN-MAUGUIN SYMBOLISM FOR SPACE GROUPS Hermann-Mauguin symbols for space groups
The Hermann–Mauguin symbol for a space group consists of a sequence of letters and numbers, here called the constituents of the HM symbol.
(i) The first constituent is always a symbol for the conventional cell of the translation lattice of the space group
(ii) The second part of the full HM symbol of a space group consists of one position for each of up to three representative symmetry directions. To each position belong the generating symmetry operations of their representative symmetry direction. The position is thus occupied either by a rotation, screw rotation or rotoinversion and/ or by a reflection or glide reflection.
(iii) Simplest-operation rule: pure rotations > screw rotations; pure rotations > rotoinversions ‘>’ means ‘has priority’ reflection m > a; b; c > n 14 Bravais Lattices crystal family Symmetry directions
A direction is called a symmetry direction of a crystal structure if it is parallel to an axis of rotation, screw rotation or rotoinversion or if it is parallel to the normal of a reflection or glide-reflection plane. A symmetry direction is thus the direction of the geometric element of a symmetry operation, when the normal of a symmetry plane is used for the description of its orientation. Hermann-Mauguin symbols for space groups Example: Hermann-Mauguin symbols for space groups
secondary tertiary primary direction direction direction PRESENTATION OF SPACE-GROUP SYMMETRY OPERATIONS IN INTERNATIONAL TABLES FOR CRYSTALLOGRAPHY, VOL. A 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 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 Matrix formalism
linear/matrix translation part column part
matrix-column Seitz symbol pair Space group Cmm2 (No. 35) How are the symmetry operations Diagram of symmetry elements represented in ITA ? 0 b
a
Diagram of general position points General Position General position
~ (i) coordinate triplets of an image point X of the original point X= x under (W,w) of G y z ~ -presentation of infinite image points X under the action of (W,w) of G
(ii) short-hand notation of the matrix-column pairs (W,w) of the symmetry operations of G
-presentation of infinite symmetry operations of G (W,w) = (I,tn)(W,w0), 0≤wi0<1 Space Groups: infinite order
General position Coset decomposition G:TG
(I,0) (W2,w2) ... (Wm,wm) ... (Wi,wi)
(I,t1) (W2,w2+t1) ... (Wm,wm+t1) ... (Wi,wi+t1)
(I,t2) (W2,w2+t2) ... (Wm,wm+t2) ... (Wi,wi+t2) ...... (I,tj) (W2,w2+tj) ... (Wm,wm+tj) ... (Wi,wi+tj) ......
Factor group G/TG
isomorphic to the point group PG of G
Point group PG = {I, W2, W 3,…,Wi} Example: P12/m1 Coset decomposition G:TG
Point group PG = {1, 2, 1, m}
General position ¯ TG TG 2 TG 1 TG m (I,0) (2,0) ( 1¯ ,0) (m,0)
(I,t1) (2,t1) (1¯ , t1) (m, t1)
(I,t2) (2,t2) (1¯ , t2) (m,t2) ...... (I,tj) (2,tj) (1¯ , tj) (m, tj) ......
-1 n1 n1/2 at 1 n2/2 -1 n2
n3/2 inversion centres (1,t): -1 n3 EXAMPLE Coset decomposition P121/c1:T Point group ? General position
(I,0) (2,0 ½ ½) (1,0) (m,0 ½ ½) (I,t1) (2,0 ½ ½+t1) (1 ,t1) (m,0 ½ ½ +t1) (I,t2) (2,0 ½ ½ +t2) (1 ,t2) (m,0 ½ ½ +t2) ...... (I,tj) (2,0 ½ ½ +tj) (1 ,tj) (m,0 ½ ½ +tj) ...... inversion centers (1 ,p q r): 1 at p/2,q/2,r/2 (2,0 ½+v ½) 21screw axes (2,u ½+v ½ +w) (2,u ½ ½ +w) Symmetry Operations Block
TYPE of the symmetry operation
SCREW/GLIDE component
ORIENTATION of the geometric element
LOCATION of the geometric element
GEOMETRIC INTERPRETATION OF THE MATRIX- COLUMN PRESENTATION OF THE SYMMETRY OPERATIONS Example: Cmm2 General position Diagram of symmetry elements 0 b
TG TG 2 TG my TG mx
(I,0) (2,0) (my,0) (mx,0)
(I,t1) (2,t1) (my,t1) (mx, t 1) a (I,t2) (2,t2) (my,t2) (mx,t2) Diagram of ...... general position points (I,tj) (2,tj) (my,tj) (mx, t j) Space group P21/c (No. 14)
EXAMPLE
Matrix-column presentation
Geometric interpretation
Bilbao Crystallographic Server
Problem: Matrix-column presentation Geometrical interpretation GENPOS
space group
14 Example GENPOS: Space group P21/c (14) Space-group symmetry operations short-hand notation matrix-column presentation
Geometric interpretation
Seitz symbols
General positions ITA data SEITZ SYMBOLS FOR SYMMETRY OPERATIONS
short-hand description of the matrix-column presentations of Seitz symbols { R | t } the symmetry operations of the space groups
- specify the type and the order of the symmetry rotation (or linear) operation; part R - orientation of the symmetry element by the direction of the axis for rotations and rotoinversions, or the direction of the normal to reflection planes.
1 and 1 identity and inversion m reflections 2, 3, 4 and 6 rotations 3, 4 and 6 rotoinversions
translation part t translation parts of the coordinate triplets of the General position blocks EXAMPLE Seitz symbols for symmetry operations of hexagonal and trigonal crystal systems
Glazer et al. Acta Cryst A 70, 300 (2014) EXAMPLE
Matrix-column presentation
Geometric interpretation
Seitz symbols (1) {1|0} (2) {2010|01/21/2 } (3) {1|0} (4) {m010|01/21/2} EXERCISES
Problem 2.16 (b)
1. Characterize geometrically the matrix-column pairs listed under General position of the space group P4mm in ITA. 2. Consider the diagram of the symmetry elements of P4mm. Try to determine the matrix-column pairs of the symmetry operations whose symmetry elements are indicated on the unit-cell diagram.
3. Compare your results with the results of the program SYMMETRY OPERATIONS SPACE-GROUPS DIAGRAMS Diagrams of symmetry elements
0 b c 0 three different settings a a permutations c of a,b,c
0 b
Diagram of general position points
Diagram of symmetry elements Space group Cmm2 (No. 35) 0 b
conventional setting
a
General Position
How many general position points per unit cell are there? Diagram of general position points EXAMPLE Space group Cmm2 (No. 35) Geometric interpretation
glide plane, t=1/2a glide plane, t=1/2b 0 b at y=1/4, ⊥b at x=1/4, ⊥a
Matrix-column General Position presentation of symmetry a operations
x+1/2,-y+1/2,z -x+1/2,y+1/2,z Example: P4mm
Diagram of symmetry Diagram of general elements position points ⎫ ⎭ ⎬ Symmetry elements
Geometric Fixed points Symmetry element } elements } + Symmetry operations Element set that share the same geometric element Examples
line All rotations and screw rotations with the same axis, the same Rotation axis st th } 1 , ..., (n-1) powers + angle and sense of rotation and all coaxial equivalents the same screw vector (zero for rotation) up to a lattice translation vector.
plane Glide plane All glide reflections with the same } defining operation+ reflection plane, with glide of d.o. all coplanar equivalents (taken to be zero for reflections) by a lattice translation vector. Symmetry operations and symmetry elements Geometric elements and Element sets
P. M. de Wolff et al. Acta Cryst (1992) A48 727 Diagram of symmetry elements Example: P4mm
Element set of (00z) line
A l l r o t a t i o n s a n d s c r e w Symmetry operations 1st, 2 nd, 3 rd powers + rotations with the same axis, that share (0,0,z) as all coaxial equivalents geometric element } the same angle and sense of rotation and the same screw vector (zero for rotation) up to Element set of (0,0,z) line a lattice translation vector.
2 -x,-y,z
4+ -y,x,z
4- y,-x,z
2(0,0,1) -x,- y,z+1 ......
ORIGINS AND
ASYMMETRIC UNITS Space group Cmm2 (No. 35): left-hand page ITA
Origin statement
The site symmetry of the origin is stated, if different from the identity. A further symbol indicates all symmetry elements (including glide planes and screw axes) that pass through the origin, if any.
Space groups with two origins For each of the two origins the location relative to the other origin is also given. Example: Different origins for Pnnn Example: Asymmetric unit Cmm2 (No. 35)
ITA:
(output cctbx: Ralf Grosse-Kustelve)
ITA: An asymmetric unit of a space group is a (simply connected) smallest closed part of space from which, by application of all symmetry operations of the space group, the whole of space is filled. Example: Asymmetric units for the space group P121 b
a
c
(output cctbx: Ralf Grosse-Kustelve) GENERAL AND SPECIAL WYCKOFF POSITIONS SITE-SYMMETRY Group Actions Group Actions
Orbit and Stabilizer
Equivalence classes General and special Wyckoff positions
Orbit of a point Xo under G: G(Xo)={(W,w)Xo,(W,w)∈G} Multiplicity
Site-symmetry group So={(W,w)} of a point Xo
(W,w)Xo = Xo
a b c w x0 x0 1 Multiplicity: |P|/|So| d e f w y0 = y0 2 g h i w z0 z0 ( 3 )
General position Xo Special position Xo S={(1,o)}≃ 1 S> 1 ={(1,o),...,} Multiplicity: |P| Multiplicity: |P|/|So| Site-symmetry groups: oriented symbols General position
~ (i) coordinate triplets of an image point X of the original point X= x under (W,w) of G y z -presentation of infinite image points X~ under the action of (W,w) of G: 0≤xi<1
(ii) short-hand notation of the matrix-column pairs (W,w) of the symmetry operations of G -presentation of infinite symmetry operations of G (W,w) = (I,tn)(W,w0), 0≤wi0<1 General Position of Space groups ~ As coordinate triplets of an image point X of the original point X= x under (W,w) of G y z General position
(I,0)X (W2,w2)X ... (Wm,wm)X ... (Wi,wi)X
(I,t1)X (W2,w2+t1)X ... (Wm,wm+t1)X ... (Wi,wi+t1)X
(I,t2)X (W2,w2+t2)X ... (Wm,wm+t2)X ... (Wi,wi+t2)X ...... (I,tj)X (W2,w2+tj)X ... (Wm,wm+tj)X ... (Wi,wi+tj)X ...... ~ -presentation of infinite image points X of X under the action of (W,w) of G: 0≤xi<1 Example: Calculation of the Site-symmetry groups
Group P-1
S={(W,w), (W,w)Xo = Xo}
-1 0 1/2 -1/ 2 -1 0 0 = 0 -1 0 1/2 -1/ ( ) 2
Sf={(1,0), (-1,101)Xf = Xf} Sf≃{1, -1} isomorphic Example Space group P4mm EXAMPLE Space group P4mm General and special Wyckoff positions of P4mm Bilbao Crystallographic Server Wyckoff positions Problem: Site-symmetry groups WYCKPOS Coordinate transformations space group
Standard basis ITA settings Transformation of the basis Bilbao Crystallographic Server Example WYCKPOS: Wyckoff Positions Ccce (68)
2 1/2,y,1/4
2 x,1/4,1/4 EXERCISES Problem 2.17
Consider the special Wyckoff positions of the the space group P4mm. Determine the site-symmetry groups of Wyckoff positions 1a and 1b. Compare the results with the listed ITA data
The coordinate triplets (x,1/2,z) and (1/2,x,z), belong to Wyckoff position 4f. Compare their site-symmetry groups.
Compare your results with the results of the program WYCKPOS. EXERCISES Problem 2.18
Consider the Wyckoff-positions data of the space group I41/amd (No. 141), origin choice 2. Determine the site-symmetry groups of Wyckoff positions 4a, 4c, 8d and 8e. Compare the results with the listed ITA data.
Compare your results with the results of the program WYCKPOS.
Characterize geometrically the isometries (3), (7), (12), (13) and (16) as listed under General Position. Compare the results with the corresponding geometric descriptions listed under Symmetry operations block in ITA. Comment on the differences between the corresponding symmetry operations listed under the sub-blocks (0, 0, 0) and (1/2, 1/2, 1/2). Compare your results with the results of the program SYMMETRY OPERATIONS.
How do the above results change if origin choice 1 setting of I41/amd is considered? CO-ORDINATE TRANSFORMATIONS IN CRYSTALLOGRAPHY 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.BY IntroductionH. ARNOLD 5.1.3. General transformation 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 transformations in crystallography. Here the crystal structure is considered to be at rest, whereas the when nonconventional descriptions of a crystal structure are of point X is given by (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. 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 0 1 structures, of phase transitions andcells’of aregroup–subgroupderived. relations. 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 while i.e.0the 1new basis vectors a0, b0, c0 and the new origin O0 (Fig. cells’ are derived. 5.1.3.1),z by the position vector keeping the coordinate system unchanged. Symmetry operations are B C (ii) Description of the symmetry operations (motions) of an @ A treated in Chapter 8.1 and Part 11. TheTheysaareme pointbrieflyXreviewedis given winith respect to a new coordinater0 xsystem,0a0 y0b0 z0c0: object (crystal structure). This involvesChapterthe5.2.transformation of the coordinates of a point or the components of a position vector while i.e. the new basis5.1. vectorsTRANSFORMATIONSa0, bCo-ordinateIn0, cthis0 andsection,OFtheTHEnewtheCOORDINATEo relationsrigintransformationO0 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 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 following 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 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@ A by the symbol As (1 3) row matrices: coordinate system consists of two(P, p).arts, a linearTheseparttransformatiand aonshiftrules apply(P,palso) 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 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:both of the basis(i) linearvectors part:a, b, cchange, i.e. of orientation or length:0 1 0 1   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 pare needed. They are @ A @ A 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 B C (u=v=w) the indices of a direction indirect space P11 P12 positionP13 vectorPr11ora the@Pcoordinates21b P31cof, a pointA X in direct space x, y, z depend 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, 1 0 1 x x=y=z the coordinates of a point in directoriginspaceO to the new originq OP0 À p: 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 The matrix qconsists 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 0 1 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 À 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 Thus, the transformation (Q, q) is the inverse Ptransformationa P b of P cthe origin O’ 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 : coordinate 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 : x x=y=z=1 the augmented 4 1 column matrix0 of0 0 left-handed one (orcoordinatesvice versa). If (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  are linearly dependentthe oldand coordinatedo notQ 31formx systemQa32ycompletQ33 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 À elements of Q are set as follows: Qsystem.33 1 and w w1=w2=w3 the translation part of a symmetry The determinant of P, det P , should 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 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 4 1 column matrixvectors ofa, b, c areleft-handedcalled covariantone (orquantitiesvice versandaare). Ifwrittenbasisdet Pasvectorsrow0, thea randnewof pointb,basiscanX isvectorsbetransformedconsideredby as a three-dimen sional  if matrices.P is appliedTheyareare:to alinearly1 3dependentrow matrix,and do not form 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 in three-dimensionalrs0pacea, bare, c PQ y a0, b0, c0 y0 : W the rotation part of a symmetry (ii) A shift of origin is defined by the0shiftz 1 vector 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 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 4 wtrans-ith invariant c axis. This is achievedIn this case,byr settingr0, i.e. the position vector is invariant, although Q must be Papplied to a 3 1 column  the basis vectors and the components are transformed. For a pure 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 0 13 23 31 32 shift of origin, i.e. P Q I, the transformed position vector r0 vectors are fixedbecomesat the new origin O0 which has the coordinates W the rotation part ofQa symmetryq theUsually,augmentedthe Milleraffineindices4are made4 trans-relative prime before and after Q (ii) A shift oforigin1 is definedp1b,yp2t,hep3 shiftin thevectorold coordinate system (Fig. 5.1.3.1). 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 covariant wpith respectp1a top2theb basisp3c: 0 1 2 3 P p the augmented affineW 4w 4 trans-the augmented 4 4 matrix of a For a pure origin shift, the basis 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. o 1 formation matrix, 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 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 the pureÀ shift 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: o 1 formation matrix, with Q PÀ 1 2 3 À À À 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  bÃ0 Q bà o 1 symmetry operation in direct space (cf. 0or orientations.1 0 1In this case, the transformation matrix P is the unit c c It is convenient to introduce the augmented 4 4 matrix Q Chapter 8.1Copyrightand Part ©11). 2006 International UnionBmatrixÃ0 Cof CrystallographyI andB ÃtheC symbol of the pure shift becomeswhich is(Icomposed, p). of the matrices Q and qin the followi ng manner @ A @ A Q11 Q12 Q13 aà (cf. Chapter 8.1): 78 0 Q21 Q22 Q23 10 bà 1 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 o 1 Q31 Q32 Q33 q3 Copyright © 2006 International Union of Crystallography Q21aà Q22bà Q23cà : B 0 0 0 1 C 0 1 B C Q31aà Q32bà Q33cà @ A B C with othe 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 EXAMPLE EXAMPLE EXAMPLE Transformation matrix-column pair (P,p)
1/2 1/2 0 1/2 1 -1 0 -1/4 -1 (P,p)= -1/2 1/2 0 1/4 (P,p) = 1 1 0 -3/4 ( 0 0 1 0 ) ( 0 0 1 0 ) a’=1/2a-1/2b a=a’+b’ b’=1/2a+1/2b b=-a’+b’ c’=c c=c’ -1/4 1/2 -3/4 1/4 O=O’+ O’=O+ 0 0 Short-hand notation for the description of transformation matrices
Transformation5.1. TRANSFORMATIONS matrix:OF THE COORDINATE(aSYSTEM,b,c), origin O 1 P QÀ :
aà aÃ0 P11 P12 P13 p1 bà P bÃ0 : 0 1(P,p 0)=1 P21 P22 P23 cà cÃ0 p2 @ 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 0 1 -written by columnsw0 w Also, the inverse matrices of P and pare needed. They are @ A @ A 1 In contrast to all quantities mentioned above, the components of a Q PÀ -coefficients 0, +1, -1 notation 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: À x x The matrix qconsists 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 0 1 q q1a0 q2b0 q3c0: z0 z 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 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 Example Q13 Q23 Q31 Q32 0. { The quantities which transform 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 matrices. They are: 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 : 0 1 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 vectors 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 becomes 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 vectors a, b, c are contravariant with respect to the basis vectors r q1a q2b q3c 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 À À À matrix and their transformation is achieved by the matrix Q: r p1a p2b p3c: À À À aÃ0 aà Here the transformed vector r0 is no longer identical with r. bÃ0 Q bà 0 1 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 qin the followi ng manner @ A @ A Q11 Q12 Q13 aà (cf. Chapter 8.1): 0 Q21 Q22 Q23 10 bà 1 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 o 1 Q31 Q32 Q33 q3 Q21aà Q22bà Q23cà : B 0 0 0 1 C 0 1 B C Q31aà Q32bà Q33cà @ A B C with othe 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 Transformation of the coordinates of a point X(x,y,z): -1 (X’)=(P,p)-1(X) x´ P11 P12 P13 p1 x y = P21 P22 P23 p2 y -1 -1 =(P , -P p)(X) z ( P31 P32 P33 p3 ) z
special cases -origin shift (P=I): -change of basis (p=o) :
EXAMPLE 1 -1 0 -1/4 3/4 1/4 X’=(P,p)-1X= 1 1 0 -3/4 1/4 = 1/4 ( 0 0 1 0 ) 0 0 Covariant and contravariant crystallographic quantities
direct or crystal basis P11 P12 P13 (a’,b’,c’)=(a, b, c)P =(a, b, c) P21 P22 P23 ( P31 P32 P33 ) reciprocal or dual basis a*’ a* -1 P11 P12 P13 a* -1 b*’ = P b* = P21 P22 P23 b* ( P31 P32 P33 ) c*’ c* c* covariant to crystal basis: Miller indices (h’,k’,l’)=(h, k, l)P contravariant to crystal basis: indices of a direction [u] -1 u´ P11 P12 P13 u v´ = P21 P22 P23 v w´ ( P31 P32 P33 ) w Transformation of symmetry operations (W,w)
original coordinates
new coordinates
(W’,w’)=(P,p)-1(W,w)(P,p) Transformation of the coordinates of a point X(x,y,z): -1 (X’)=(P,p)-1(X) x´ P11 P12 P13 p1 x y = P21 P22 P23 p2 y =(P-1, -P-1p)(X) 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 EXERCISES Problem 2.19 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 (i) 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. (ii) Determine the coordinates X’ of a point X= 0,70 with respect to the new basis (a’,b’,c’). 0,31 0,95
Hints (W’,w’)=(P,p)-1(W,w)(P,p)
(X’)=(P,p)-1(X) Problem: SYMMETRY DATA ITA SETTINGS
530 ITA settings of orthorhombic and monoclinic groups SYMMETRY DATA: ITA SETTINGS
Monoclinic descriptions
abc cba¯ Monoclinic axis b Transf. abc ba¯c Monoclinic axis c abc ¯a c b Monoclinic axis a C 12/c1 A12/a1 A112/a B112/b B2/b11 C 2/c11 Cell type 1 HM C 2/c A12/n1 C 12/n1 B112/n A112/n C 2/n11 B2/n11 Cell type 2 I 12/a1 I 12/c1 I 112/b I 112/a I 2/c11 I 2/b11 Cell type 3
Orthorhombic descriptions No. HM abc ba¯c cab ¯c b a bca a¯cb 33 Pna21 Pna21 Pbn21 P21nb P21cn Pc21n Pn21a Bilbao Crystallographic Server
Problem: Coordinate transformations Generators GENPOS General positions
space group
Transformation ITA-settings of the basis symmetry data Example GENPOS:
default setting C12/c1
(W,w)A112/a= (P,p)-1(W,w)C12/c1(P,p)
final setting A112/a Example GENPOS: ITA settings of C2/c(15)
default setting A112/a setting Bilbao Crystallographic Server
Problem: Coordinate transformations Wyckoff positions WYCKPOS
space group
ITA settings Transformation of the basis EXERCISES Problem 2.23
Consider the space group P21/c (No. 14). Show that the relation between the General and Special position data of P1121/a (setting unique axis c ) can be obtained from the data P121/c1(setting unique axis b ) applying the transformation (a’,b’,c’)c = (a,b,c)bP, with P= c,a,b.
Use the retrieval tools GENPOS (generators and general positions) and WYCKPOS (Wyckoff positions) for accessing the space-group data. Get the data on general and special positions in different settings either by specifying transformation matrices to new bases, or by selecting one of the 530 settings of the monoclinic and orthorhombic groups listed in ITA. EXERCISES Problem 2.24 Use the retrieval tools GENPOS or Generators and General positions, WYCKPOS (or Wyckoff positions) for accessing the space-group data on the Bilbao Crystallographic Server or Symmetry Database server. Get the data on general and special positions in different settings either by specifying transformation matrices to new bases, or by selecting one of the 530 settings of the monoclinic and orthorhombic groups listed in ITA. Consider the General position data of the space group Im-3m (No. 229). Using the option Non-conventional setting obtain the matrix-column pairs of the symmetry operations with respect to a primitive basis, applying the transformation (a’,b’,c’) = 1/2(-a+b+c,a-b+c,a+b-c) METRIC TENSOR 3D-unit cell and lattice parameters lattice basis: {a, b, c} C B unit cell: the parallelepiped A defined by the basis vectors primitive P and centred unit cells: A,B,C,F, I, R
number of lattice points per unit cell Lattice parameters (3D) An alternative way to define the metric properties of a lattice L
Given a lattice L of V3 with a lattice basis: {a1, a2, a3}
Remark: the lengths of basis vectors are measured in nm (1nm=10-9 m) Å (1Å=10-10 m) pm (1pm=10-12 m) Metric tensor G in terms of lattice parameters METRIC TENSOR (FUNDAMENTAL MATRIX) Given a lattice L of V3 with a lattice basis: {a1, a2, a3} The lattice L inherits the metric properties of the Euclidean space and they are conveniently expressed with respect to a lattice basis (right-handed coordinate system) Metric tensor G of L a1 G11 G12 G13 T .{a1, a2, a3}= G21 G22 G23 G ={a1, a2, a3} . {a1, a2, a3}= a2 G31 G32 G33 a3
Metric tensor G is symmetric:Gik=Gki Gik=(ai,ak)=aiakcosαj, Scalar product of arbitrary vectors: (r,t)=rTGt Transformation properties of G under basis transformation
{a’1, a’2, a’3}= {a1, a2, a3} P
G´={a’1, a’2, a’3}T. {a’1, a’2, a’3}= PT{a1, a2, a3}T. {a1, a2, a3} P G´=PT G P Crystallographic calculations: Volume of the unit cell Volume of the unit cell in terms of lattice parameters (Buerger,1941)
Basis vectors with respect to Cartesian basis
det (A)=det(AT)
= det (G) EXERCISE (Problem 2.20)
Write down the metric tensors of the seven crystal systems in parametric form using the general expressions for their lattice parameters. For each of the cases, express the volume of the unit cell as a function of the lattice parameters.
For example: tetragonal crystal system: a=b, c, α=β=γ=90
a2 0 0 G = 0 a2 0 V=? 0 0 c2 EXERCISES Problem 2.21
A body-centred cubic lattice (cI) has as its conventional basis the conventional basis (aP,bP,cP) of a primitive cubic lattice, but the lattice also contains the centring vector 1/2aP+1/2bP+1/2cP which points to the centre of the conventional cell.
Calculate the coefficients of the metric tensor for the body-centred cubic lattice: (i) for the conventional basis (aP,bP,cP); (ii) for the primitive basis: aI=1/2(-aP+bP+cP), bI=1/2(aP-bP+cP), cI=1/2(aP+bP-cP) (iii) determine the lattice parameters of the primitive cell if aP=4 Å metric tensor Hint transformation G´=Pt G P EXERCISES Problem 2.22 A face-centred cubic lattice (cF) has as its conventional basis the conventional basis (aP,bP,cP) of a primitive cubic lattice, but the lattice also contains the centring vectors 1/2bP+1/2cP, 1/2aP+1/2cP, 1/2aP+1/2bP, which point to the centres of the faces of the conventional cell.
Calculate the coefficients of the metric tensor for the face-centred cubic lattice: (i) for the conventional basis (aP,bP,cP); (ii) for the primitive basis:
aF=1/2(bP+cP), bF=1/2(aP+cP), cF=1/2(aP+bP) (iii) determine the lattice parameters of the primitive cell if aP=4 Å