M. MORALES Introuction to Crystallography magali.morales@ensicaen.fr
Classification of the matter in 3 states : Crystallise soli liqui or amorphous gaz soli Crystallise soli : unique arrangement of atoms + long-range orer an symmetry 3D perioic array of atoms (translational perioicity) : A n can be euce from A (arbitrary origin) by: r r r AA n ua+ vb+ wc r r r a,b,c r r uvw : irect lattice basis vectors u, v, w integers A n is calle "noe"
From the unit cell to the crystal : CaF Unit cell : The smallest set of atoms arrange in a particular way. perioically repeate in three imensions Macroscopic crystal
The unit cell is the parallelepipe built on the crystallographic basis vectors a, b, c of the irect lattice (or irect space DS). In 3D, the unit cell is etermine by 6 parameters: the a, b an c cell lengths an the α, β an γ angles. r V (a Unit cell volume V: r r r b).c (b r r r c).a (c r r a).b Arbitrary choice of DS basis vectors unit cell multiplicity: primitive cell contains only one lattice point (1 noe). Non-primitive cell (multiple cell) contains more than one noe. Multiplicity of the cell ratio of its volume to the volume of primitive cell.
Which choice for the unit cell? When several unit cells are possible, the choice correspons to the smallest unit cell exhibiting all the irect lattice symmetry. Example: Orthorombic system ( a b c an α β γ 90 ): Primitive, boy centere (I), a,b or face (F) centere unit cell P I F The unit cell oes not bring out all the irect lattice symmetry : Projection along c Conventionnal unit cell unit cell 1
Unit cell choice: T an T : simple translations T+T an T-T are simple translations Unit cell kept is rectangular ancentere(in green) T+T an T-T are not simple translations Unit cell kept is simple rectangular (in green)
Reuce coorinates x, y, z OA, OB an OC : coorinates corresponing respectively to the 3 crystallographic axes X, Y an Z (imension of a length) x, y an z : reuce coorinates (without any imension) x OA a y OB b z OC c M M reuce coorinates : OA a 3 OB b OC 3c OM r ua+ r r vb+ wc r r 0
Cristalline structures classification (function of their symmetry) 3D Bravais lattice systems (7) cubic tetragonal orthorhombic hexagonal monoclinic rhomboheric triclinic
Different noe positions for a given symmetry : ifferent types of lattices (A, B, C, P, I, F, R) 14 Bravais lattices Bravais system type symbol Unit cell multiplicity Primitive P 1 Cubic Boy centere I Face centere F 4 Tetragonal Primitive Boy centere P I 1 Primitive P 1 Orthorhombic Boy centere Face centere I F 4 Base centere A, B or C Hexagonal Primitive P 1 Monoclinic Primitive Base centere P B 1 Rhomboheric Primitive R 1 (with rhomboheric axes) Triclinic Primitive P 1
14 Bravais lattices Λ(r) Triclinic P Monoclinic P Monoclinic C Tetragonal P Tetragonal I Orthorhombic P Orthorhombic C Orthorhombic I Orthorhombic F Hexagonal P Rhomboheric P Cubic P Cubic F Cubic I
Motif an crystal lattice: The crystal structure C(r) can be consiere to arise from the convolution of a basis omain B(r) also calle the motif with the Bravais lattice Λ(r) : C(r) B(r) Λ(r). Motif B(r) The motif B(r) smallest possible unit of atoms which by application of all translational symmetry generates the whole crystal Lattice Λ(r) The lattice Λ(r) can be viewe as a set of perioically space Dirac istributions, i.e. a Dirac δ(r-r 0 ) istribution locate on + + + r r r r r each noe of the lattice: Λ(r) (r ua vb wc) u v w
Lattice rows set of noes that lie on a straight line that passes through the origin labelle by three integers inserte into square brackets ([... ]) without commas, corresponing to the first noe next to the origin on the row: i.e. if R uvw ua + vb + wc is the first noe next to the origin on the straight line, the lattice row is labelle [u v w] u, v an w have no common integer ivisor ifferent from one. Following noes : nu nv nw with n Є Z
Families of lattice planes: (001) Lattice plane (h k l) plane which passes through lattice noes that o not all lie on the same straight line an labelle by three prime integer numbers h, k, l name Miller inices. hkl P 4 P 3 P P 1 Family of lattice planes set of parallel lattice planes P i. Distance between two neighbouring lattice planes spacing hkl.
Families of lattice planes: Family lattice plane (hkl) equation: hx + ky+ lz m with m integer m 0 the plane passes from the origin O m 1 or -1 : first neighbour planes of the origin. (3,,4) c b 1/4 1/ O m 0 1/3 a m 1 m The plane with m 1, P 1, cuts the [1 0 0], [0 1 0] an [0 0 1] irections at integer fractions of the basis vectors: a/h, b/k, c/l. For m n, P n plane cuts at na/h, nb/k, nc/l. If the plane P 1 is parallel to one of the basis vectors, the corresponing inex is set to zero.
Example: Miller inices Plane Reuce Inverse values Miller inices coorinates A x y z 1/x 1/y 1/z ha /x ka /y la /z 1 1 1 1/1 1/ 1 1 4 3 1/ 1/4 1/3 1 6 3 4 3 3 7 1/3 1/7 0 1 7 3 0
Reciprocal space RS
Reciprocal space (RS): Geometric efinition Introuce by Bravais an use again by Ewal (1917) : an essential concept for the stuy of crystal lattices an their iffraction properties. Basic vector efinitions: r r r r r r r b c r c a r a b a*, b*, c* V V V with V(a,b,c) irect cell volume an V*(a*,b*,c*)1/V reciprocal cell volume irect space DS b a Equivalent efinitions (D, 3D...) r r r r r r a*.a 1 b*.a 0 c*.a 0 r r r r r r a*.b 0 b*.b 1 c*.b 0 r r r r r r a*.c 0 b*.c 0 c*.c 1 RS b* a* Reciprocal lattice point coorinates: integers Q r r r n( ha* + kb* lc*) with h,k,l an n hkl +
RS Properties Symmetry : RS has the same symmetry as the irect space DS : DS b a b* a* RS Duality : The reciprocal space of RS is the irect space DS Inee, RS of RS is constitute by noes verifying Qhkl Qhkl.R m, If RR uvw the equation is verifie (i. e. R is a noe of DS) reciprocally if Rx a+y b+z c an verify x u + y v + z w m, x, y an z must be integers
Relation between lattice planes an RS To every lattice planes family (h k l) in the DS correspons a lattice row [h k l] of the same inices in the RS. The reciprocal lattice row [h k l] is perpenicular to the irect lattice planes (h k l) an its spacing r hkl is the inverse of the irect lattice plane spacing hkl. [h k l] (h k l) an r hkl 1/ hkl Since a family of irect lattice planes (h k l) efines a reciprocal lattice row [h k l], the noes on this reciprocal lattice row are: nh nk nl with n integer. R uvw.n m R uvw So Q nh nk nl belonging to the RS n hkl is perpenicular to a irect family plane an verify : 1 Q n The smallest lattice row vector moulus : q hkl 1/ hkl an Q nh nk nl n q hkl
hkl spacings hkl spacing: hkl 1 q hkl with q hkl smallest vector of the lattice row General case: hkl h a* + k b* + l c* 1 + hka*b*cosγ * + klb*c*cosα* + lha*c*cosβ* Hexagonal system : 1 a* b*, c*, γ* 60 3a c hkl 4 (h 3 + k a + hk) + l a ( ) c Cubic system : a* b* c*, α* β* γ* 90 1 a hkl h a + k + l
Case of multiple cells R R uvw uvw Boy centere cubic unit cell ua + vb+ wc ( u+ 0.5) a + ( v+ 0.5) b+ ( w+ 0.5) c The conition implies 1) h, k,l integers ) h + k + l n Ruvw Qhkl.Ruvw n a a* DS I RS F RR of RD : face centere cubic cell Hexagonal unit cell A a-b; Ba+b; Cc h + k n DS P I F A Existence conitions no conitions h + k + l n h, k,l same parity k + l n B C (a+ b) c 1 A* (a* b*) V V C A c (a b) 1 B* (b* + a*) V V RS P F I A b b* a a* B B* A A*
* Bragg law Family lattice plane iffraction with hkl spacing hkl sin θ mλ hkl s i θ Q s s i an s * Diffusion vector Q sinθ λ RS an iffraction * The reciprocal lattice can therefore be viewe as being a representation of all those scattering vectors Q that can give rise to iffraction. * To each scattering vector Q correspons a reciprocal lattice noe. m Q s s i λ : unitary incient an iffuse vectors Q is perpenicular to the iffracting planes Diffraction Q is a RS vector (in a reciprocal lattice row to the iffracting planes) Q n
DS Fourier transform Définition Function or S(r) istribution TF(S(r)) F(q) S(r)e TF -1 iq.r (F(q)) S(r) F(q)e iq.r 3 r 3 q 1 T q i u e T u δ h (q ht) Direct lattice is escribe by the «noe ensity» istribution : TF(S(r)) F(q) δ(r R uvw iq.ruvw e uvw δ(q h x S(r) uvw h) δ(q k u )e δ(r iq.r uvw y e iq x u k) δ(q l R uvw ) 3 r v e z iq y l) v w e iq z w q q a* + q b* + q c* x y z F ( q ) v* δ( q Q ) hkl hkl «noeensity»ofrs (V*(a*,b*,c*)(1/V) RS TF (DS)
TF properties RS an DS uality TF 1 ( hkl δ(q Q hkl )) V uvw δ(r R uvw ) Direct an reciprocal space symmetry If O is a symmetry operator in irect space F(O(q)) S(r)e io(q).r S(O(r'))e iq.r' 3 r 3 r' S(r)e iq.o S(r')e 1 (r) iq.r' 3 3 r' r F(q) O is a symmetry operator in reciprocal space Convolution prouct : Convolution prouct of f an g is f * g (f g)(r) f (u)g(r u) 3 u TF(f g) TF(f )TF(g) TF(fg) TF(f ) TF(g)
Application to low imensional objects a π/a a* 1D : chain S(r) δ(r u F(q) h ua) δ(r // δ(qx h) ) q q x a* set of parallel plane q qx a* + q F(q) y b* hk D : plane S(r) δ(r uvw δ(qx h) δ(qy k) ua + vb) δ(r // Ro lattice ) b b* a a*