The resip iprocal space and unit itcell Bragg ggs la law - PowerPoint PPT Presentation

In Introduction to cry rystallography The unit itcell ll The resip iprocal space and unit itcell Bragg ggs la law Structure factor F hkl and and atomic ic scattering factor f z θ

The specimen Introduction to crystallography We divide materials into two categories: • Amorphous materials • The atoms are ”randomly” distributed in space • Not quite true, there is short range order • Examples: glass, polystyrene (isopor) • Crystalline materials • The atoms are ordered • Short range and long range order • Deviations from the perfect order are of importance for the properties of the materials. Why do we see facets on the surface

The specimen Introduction to crystallography

The specimen Basic aspects of crystallography • Crystallography describes and characterise the structure of crystals • Basic concept is symmetry • Translational symmetry: if you are standing at one point in a crystal, and move a distance (vector) a the crystal will look exactly the same as where you started. 2D a a a a b 1D a a a a

The specimen The lattice • described as a set of mathematical points in space • each of these points represents one or a group of atoms, basis Basis + Lattice a a a = crystal structure a a a

The specimen Axial systems z The point lattices can be described by 7 axial systems (coordinate systems) c Axial system Axes Angles a ≠b≠c α≠β≠γ≠90 o Triclinic α β y b a ≠b≠c α = γ =90 o ≠ β Monoclinic γ a ≠b≠c α = β = γ =90 o Orthorombic a a =b≠c α = β = γ =90 o Tetragonal x α = β = γ =90 o Cubic a=b=c a 1=a2=a3≠c α = β =90 o Hexagonal γ =120 o α = β = γ ≠ 90 o Rhombohedral a=b=c

The specimen Bravais lattice The point lattices can be described by 14 different Bravais lattices Hermann and Mauguin symboler: P (primitiv) F (face centred) I (body centred) A, B, C (bace or end centred) R (rhombohedral)

The specimen Unit cell • The crystal structure is described by specifying a repeating element and its translational periodicity • The repeating element (usually consisting of many atoms) is replaced by a lattice point and all lattice points have the same atomic environments. • The unit cells are the smallest building blocks. • A primitive unit cell has only one lattice point in the unit cell. Repeating element, basis c α b β Lattice point γ a

The specimen Exaples of materials with a face centered cubic lattice Copper

The specimen Exaples of materials with a face centered cubic lattice Silicon

The specimen Exaples of materials with a face centered cubic lattice ZnS

The specimen What about other symmetry elements? • We have discussed translational symmetry , but there are also other important symmetry operations: • Mirror planes • Rotation axes • Inversion • Screw axes • Glide planes • The combination of these symmetry operations with the Bravais lattices give the 230 space groups

The specimen Space groups • Crystals can be classified • Structural data for known crystalline according to 230 space groups . phases are available in “books” like “Pearson’s handbook of crystallographic data….” but also • Details about crystal description electronically in databases like “Find it” can be found in International or f.ex . “Crystallography open Tables for Crystallography . database”: • Criteria for filling Bravais http://www.crystallography.net/cod/ . point lattice with atoms. • Pearson symbol like cF4 indicate the • A space group can be referred to axial system (cubic), centering of the by a number or the space group lattice (face) and number of atoms in symbol (ex. Fm-3m is nr. 225) the unit cell of a phase (like Cu).

The specimen Lattice planes • Miller indexing system z • Crystals are described in the axial system of their unit cell c/l • Miller indices (hkl) of a plane is found b/k a/h 0 from the interception of the plane with y the unit cell axis (a/h, b/k, c/l). x Z • The reciprocal of the interceptions are rationalized if necessary to avoid fraction (110) numbers of (h k l) and 1/ ∞ = 0 Y • Planes are often described by their normal Z Z (001) X (111) • (hkl) one single set of parallel planes (010) • {hkl} equivalent planes Y Y (100) X X

The specimen Directions z w c • The indices of directions (u, v and w) can be found from [uvw] the components of the vector in the axial system a , b , c . c b u a a v b • The indices are scaled so that all are integers and as small as possible y x • Notation Zone axis [uvw] • [uvw] one single direction or zone axis • <uvw> geometrical equivalent directions (hkl) • [hkl] is normal to the (hkl) plane in cubic axial systems uh+vk+wl= 0

The specimen Recip iprocal l vectors, pla lanar dis istances • The reciprocal lattice is defined by the vectors : – The normal of a plane is given by the vector:    * * * g hkl h a k b l c   * ( a b c ) / V   * b ( c a ) / V – Planar distance between the planes {hkl} is   given by: * c ( a b ) / V  d 1 / g hkl hkl a*=(bcsin α )/V b*=(casin β )/V – Planar distance (d-value) between planes {hkl} in a cubic crystal with lattice parameter c*=(absin γ )/V a: a  d hkl   2 2 2 h k l

Interaction with sample Elastic scattering Why do we want a monochromatic wave in diffraction studies? • X-rays are scattered by the electrons in a material • Electrons are scattered by both the electron and the nuclei in a material • The electrons are directly scattered and not by an field to field exchange as in the case for X-rays • The diffraction theory is the same for electrons, X-rays and neutrons. • Based on Braggs law

Interaction with sample Bragg ggs law Coherent incoming Gives the angle when wave the scattered wave is in phase. Elastic scattering

Interaction with sample Deduction of Braggs law: ψ i Sum ψ scattered in phase θ θ d x x In order for the scattered waves to be in phase (resulting in constructiv interference), the difference in the travelled distance of the waves must be a multiple of λ (i.e 2X = n λ ). X= d sin θ and hence 2d sin θ = n λ .

Interaction with sample Effect of adding a scattering plane with d/2 ψ i θ (001) d/2 (002) x’ x’ θ d x x Imaging that the red planes represents the (001) planes in a cubic structure with the same atom (=electron) density on the (002) plane (like in fcc and bcc structures). The (002) plane will now scatter the incoming wave and travel 1/2n λ shorter/longer than the waves scattered from (001). The amplitude of the sum of the waves in this situation is 0 (x’=d/2 sinθ, 2x’= d sinθ and this is equal to 1/2nλ. In this example “d“ represented the planar distance of the (001) planes and “d/2” the planar distance of the (002) plane. This is commonly written as d 001 and d 002 . For a cubic unitcell d 001 and d 002 is equivalent to d 100 and d 200 . The latter ones are the ones commonly tabulated in d-value tables if the intensity is not 0.

Interaction with sample Example: Bragg angle for (002) relative to (001) ψ i θ (001) d/2 (002) x’ x’ θ d x x Imaging that the wave length and the crystal are the same as in the previous examplered planes represents the (001) planes in a cubic structure with the same atom (=electron) density on the (002) plane (like in fcc and bcc structures). The (002) plane will now scatter the incoming wave and travel 1/2n λ shorter/longer than the waves scattered from (001). The amplitude of the sum of the waves in this situation is 0 (x’=d/2 sinθ, 2x’= d sinθ and this is equal to 1/2nλ. In this example “d“ represented the planar distance of the (001) planes and “d/2” the planar distance of the (002) plane. This is commonly written as d 001 and d 002 . For a cubic unitcell d 001 and d 002 is equivalent to d 100 and d 200 . The latter ones are the ones commonly tabulated in d-value tables if the intensity is not 0.

Atomic scattering factor (X-ray) Z Intensity of the scattered X-ray beam

Structure factors N       X-ray: ( x ) i hu kv lw F F f exp( 2 ( )) z j j j g hkl j w j c  j 1 r j c The coordinate of atom j within the crystal unit cell is given r j =u j a +v j b +w j c . a b v j b u j a h, k and l are the miller indices of the Bragg reflection g (and represents the normal y to the plane (hkl). x N is the number of atoms within the crystal unit cell. f j (n) is the x-ray scattering factor, or x-ray scattering amplitude, for atom j. The intensity of a reflection is The structure factors for x-ray, neutron and  F g F proportional to: electron diffraction are similar. For neutrons and g (n) or f j (e) . electrons we need only to replace by f j

Example: Cu, fcc N  • e iφ = cos φ + isin φ      F F f exp( 2 i hu kv lw ( )) g hkl j j j j  • e n π i = (-1) n j 1 • e ix + e -ix = 2cosx Atomic positions in the unit cell: [000], [½ ½ 0], [½ 0 ½ ], [0 ½ ½ ] F hkl = f (1+ e π i(h+k) + e π i(h+l) + e π i(k+l) ) What is the general condition If h, k, l are all odd then: for reflections for fcc? F hkl = f(1+1+1+1)=4f What is the general condition If h, k, l are mixed integers (exs 112) then for reflections for bcc? F hkl =f(1+1-1-1)=0 (forbidden)