Space Group: translations and point Group

Transcription

1 Space Group: translations and point Group Fyodorov and Schönflies in 1891 listed the 230 space Groups in 3d G= space Group with elements: r' = αr + a a = traslation, α = rotation matrix ( α = 1 No rotation). The operation denoted by Useful faithful matrix representation (different matrices for different operations): r' α a r α r + a = = Translations T =(0, a) are a subgroup H; a ( α a): A traslation observed from a rotated-translated reference is still a translation. Therefore, the subgroup H is invariant (more about thet below). 1

2 { } Hg = hg, h H right coset space group operation followed by a translation { } gh = gh, h H left coset (laterale sinistro) Cosets are labelled by α, and a multiplication of cosets may be defined. Point Group is a quotient Group: Point Group = Space Group Translation Group? Point Group Space Group Point Group*Translation Group = = G is not a direct product of H and the point Group!. Bad! ( The direct product would be β b)( α a) = ( βα b + a) 2

3 r' α a r α r + a = = β b αr + a β( αr + a) + b βαr + βa+ b = = Multiplication rule is ( β b)( α a) = ( βα b + βa) It is called semidirect product. (never Abelian) Inverse of ( α a) must be ( β b) such that ( β b)( α a) = ( βα b + βa) = (1 0) where 1 = no rotation. β = α b 1 1 (1 b + α a) = (1 0) b = α a ( α a) = ( α α a) 1 Evidently,. Then must be such that 3

4 ( α a) r = αr + a :first rotate then translate, but means: ( α a) r = α r α a = α ( r a): first translate back then rotate back ( a) = ( α α α 1 Classes: conjugation is ( β b) ( α a) ( β b)( α a) 1 = ( α a) ( βα b+ βa). Inserting the inverse, ( β b) ( α α a) ( βα b + βa) = ( α βα α ( b + βa a)). Th e rotations must be conjugated, i.e. same angle. If β=1, ( β b ) is a translation and the conjugate is a translation. This confirms that translations make an invariant subgroup. That is, if β=1 (translation) conjugation with any element gives a translation: translations are an invariant subgroup a) (rotations are non-invariant subgroup (conjugation does not change angle but may add translation) 4

5 Pure rotations Consider the Space Group operation ( α a), r r': r' = αr+ a The subgroup of Rotations is not invariant. When is it a pure rotation around some center site? Shift of origin Consider the Space Group operation r r' with r' = αr+ a. Shifting the origin to -b, the operation r r' must be rewritten s s' : r = s b r' = αr+ a becomes s' b= α( s b) + a : r' = s' b same rotation,but a a+ b αb. Shifting the origin changes the translation.

6 Consider the Space Group operation ( α a), r r': r' = αr+ a When is it a pure rotation around some center site? One wants b such that a+ b αb=0. O ( α 1) ( α 1) 1 a+ b= αb a = b a= b The shift of the origin is obtained by rotating the old translation, if a solution exists. Then one can consider the Space Group operation as a pure rotation around some origin.

7 A space Group generated by the Bravais translations and the point Group is said symmorphic The symmorphic Groups have only the rotations of the point Group and the translations of the Bravais lattice; nonsymmorphic Groups have extra symmetry elements are called screw axes and glide planes. screw: ( α, a ) glide: ( σ, a ) 7

8 Binary compounds with Hexagonal structure (CdS) 8

9 Example: Binary compounds with Hexagonal structure (CdS) c 2 Screw axis: C 6 operation and C/2 translation screw : ( α, a ) 9

10 c 2 glide plane:reflection and c/2 translation glide: ( σ, a ) 10

11 Graphite is elemental but nonsymmorphic C6 rotation screw axis and glide plane screw axes and glide planes depend on special relations between the dimensions of the basis (that is, of the unit which is periodically repeated) and of the Bravais translations. 11

12 screw operation: ( α, a ) Doubt: When is a screw axis really needed? c It is natural to ask whether one can eliminate the translation 2 from the screw axis operation by shifting the origin to some -b. Recall the condition for pure rotation around -b ( α ) 1 1 a = b If solution exists, the Space Group operation is a pure rotation around -b, the c/2 translation is not needed, and the Group is symmorphic.. but if αa=a there is no solution since (1- α ) -1 a=(1+α+α 2 +α 3 + )a blows up The translation cannot be removed when it is along the rotation axis, Then, it is a real screw axis. 12

13 Let the translation can be taken parallel to the rotation axis, αa=a screw operation: ( α, a), with a not a lattice translation t. How arbitrary is the choice of a? Now we show that for some integer n (t= Bravais lattice translation) na = t Proof: Since α belongs to the point group α n = 1 for some n; let us iterate ( α,a) recalling multiplication: ( β b)( α a) = ( βα b + βa) ( α a) = ( α a+ αa) = ( α 2a). n 1 n n k n ( α a ) = ( α α a ) = ( α na ), and k = 0 n for some n,( α a) = (1 na) 13

14 n 1 n n k n ( α a ) = ( α α a ) = ( α na ), and k = 0 n for some n,( α a) = (1 na) must eventually give a pure translation na = t Example screw-axis with an angle α = π/2, n=4 can have a translation a equal to 1/4, 2/4 o 3/4 of a Bravais vector. Example: for glide plane n=2 14

15 Glide plane: α is a reflection, n=2 a=1/2 t C 6 ½ c σ 15

16 Kinds of lattices in 3d Primitive (P): lattice points on the cell corners only. Body (I): one additional lattice point at the center of the cell. Face (F): one additional lattice point at the center of each of the faces of the cell. Base (A, B or C): one additional lattice point at the center of each of one pair of the cell faces.

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18 International notation (International Tables for X-Ray Crystallography (1952) The international notation for a Space Group starts with a letter ( P for primitive, I for body-centered, F for face centered, R per rombohedric) followed by a list of Group classes Screw axis with translation ¼ Bravais vector 4 1 Screw axis with translation 2/4 Bravais vector 4 2 Screw axis with translation ¾ Bravais vector 4 3 Screw axis with translation = Bravais vector

19 International Notation: Group symbols are lists of elements Example and comparison with Schoenflies notation: 4 2 F 3 F Face centered Cubic O m m h 4 C4 m 2 C2 m 3 C axis+orthogonal axis+orthogonal axis+inversion plane plane 3 this is symmorphic,while the diamond Group is not 41 2 F 3 F Face centered Cubic O d m h 41 1 C screw axis with t 4 d 4 translation+glide plane 2 C2 m 3 C axis+orthogonal axis+inversion plane 3 Tables readily available for purchase on internet 19

20 CdS in Wurtzite crystal structure P6 3 mc along c axis) group (P=primitive, c means glide translation CdS also has a cubic form with space group F 43m 20

21 Representations of the Translation Group and of the Space Group Recall Group elements: ( α a) denotes r' = αr + a inverse: ( α a) = ( α α a) ( α a) r = αr + a :first rotate then translate ( α a) r = α r α a = α ( r a):first translate back then rotate back 1 1 ( α a) ( α a) r = α ( αr + a a) = r Since f(r), Rf(r)=f(R r), ( α a)f(r) = f(( α a) r)=f( α (r-a)) Consider first the effect on plane waves, which are eigenfunctions of all the translations. 1 ik. r ik.( α, a) r 1 ( α, a) e = e = exp[ ik. α ( r a)] 21

22 Rotating two vectors by the same angle the scalar product does not change; so we may write. e ( α = ).( ) ( α, ae ) ik r i k r a In terms of Bloch functions, (α,a )ψ n (k,r ) yields a linear combination of ψ n (αk,r ), where n n because in general Point Group operations mix degenerate bands. k labels a representation of translation Group, basis=plane waves. Such representations are mixed by the Space Group. 22

23 Star of k is the set { αk, α point Group }. High symmetry k have smaller sets Star of k: subspace which is a basis set for a representation of all T and R in the Space Group. However some operations may mix k points at border of BZ with other k points differing by reciprocal lattice vectors G; these are equivalent and not distinct basis elements. The star of some special k may comprise just that k. 23

24 Example: square lattice Special Points: Γ, M, X Special Lines:,Z, Σ Γ is invariant under C X is invariant under 4v M is invariant under C C 4v 2v since theother corners areconnected to M by G Σ Z is invariant under σ, is invariant under σ, d y is invariant under σ,since it takes to equivalent points. x

25 this is C 4v this is C 2v just a reflection

26 Special Points: Γ, M, X, R Special Lines:,Z, Σ, ST, Γ, R are invariant under O h X, M are invariant under 4 / mmm Σ, S are invariant under 2 mm,, T are invariant under 4 mm, Z has 2 mirror planes and C. 2

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30 Define: Group of the wave vector or Little Group is the Subgroup G G which consists of the operations (a, α ) k such that : αk = k + G. In general one may have a set of wave functions at each k, so a basis for the Space Group must comprise all of them The set of the basis functions of a representation of the Little Group for all the points of a star provide a basis for a representation of the Space Group G. Such representations can be analyzed in the irreducible representations of the Space Group in the usual way. 30

31 The Magnetic Groups Magnetic Groups are obtained from the space groups by adding a new generator: time reversal T. They were studied by Lev Vasilyevich Shubnikov and refered to as color Groups. Лев Васи льевич Шу бников T flips spins as well as currents. It makes a difference in magnetic materials where equilibrium currents and magnetic moments exist. In this chain T is no symmetry, but T times a one-step translation is: T can only be a symmetry if there are no spins and no currents. Hamermesh (chapter 2) proves some theorems. Magnetic point Groups can be obtained from the non-magnetic ones in most cases the following way. 31

32 G point group, H subgroup having index 2, that is, G=H+aH, with a H, a G. Then the magnetic Group is G =H+TaH. T = iσ yk Along with C 3v which has index 2, there is a magnetic Group where the reflections are multipied by T. The rotation C 3 cannot be multiplied by T because otherwise the third power would give T itself as a symmetry. This is excluded because it would reverse spins. In this way one finds 58 new, magnetic Groups. Including 32 point Groups the total is 90 according to Hamermesh, 122 according to Tinkham. These can be combined with the translation ones to form generalized space Groups. The Magnetic Groups are 1651 in 3d 32

33 From Tinkham