Ewald Sphere 721r3a

Reciprocal Lattice & Ewald sphere Presented by- Mohammad Rameez & Santhosh Kumar

Unit Cells Crystals consist of repeating asymmetric units which may be atoms, ions or molecules. The space lattice is the pattern formed by the points that represent these repeating structural units.

Unit Cells A unit cell of the crystal is an imaginary parallel-sided region from which the entire crystal can be built up. Usually the smallest unit cell which exhibits the greatest symmetry is chosen. If repeated (translated) in 3 dimensions, the entire crystal is recreated.

Crystal = Lattice + Motif Motif or Basis: typically an atom or a group of atoms associated with each lattice point Lattice  the underlying periodicity of the crystal Basis  Entity associated with each lattice points Lattice  how to repeat Motif  what to repeat

Lattice Translationally periodic arrangement of points

Crystal Translationally periodic arrangement of motifs

Crystal = Lattice (Where to repeat) + Motif (What to repeat) Crystal

a

=

Lattice

a

+ Motif

Note: all parts of the motif do not sit on the lattice point

a 2 Motifs are associated with lattice points  they need NOT sit physically at the lattice point

Space Lattice

A lattice is also called a Space Lattice

An array of points such that every point has identical surroundings  In Euclidean space  infinite array

 We can have 1D, 2D or 3D arrays (lattices) or

Translationally periodic arrangement of points in space is called a lattice

Crystal Planes

Family of planes 7

Miller index Reciprocal a1 intercept is 2 1/2 a2 intercept is 3 1/3 a3 intercept is 3 1/3

3

a3

a2 a1

2

Hence Miller indices are 3,2,2 and are depicted by 3

( hkl ) = (322)

Calculate reciprocal of these intercepts and reduce them to smallest three integers having same ratio. 8

Inter-planer Distance (hkl) represent a family of planes. All parallel crystal planes have the same Miller index. These planes are equally spaced at distance dhkl . This distance is defined as:

dhkl =

a

√h 2 + k 2 + l 2

9

b

a

Green a

b

c

3

2

∞

Inverse

1/3

1/2

1/ ∞

Multiply through

2

3

0

Intercept

Crystal planes in a cubic unit cell a3 a1

a2

(100)

(110)

(111)

dhkl = a

dhkl = a/√2

dhkl = a/√3 12

What is Real Space and Reciprocal Space? Real Space (i.e. spacing of surface atoms in nm)

a Reciprocal-Space (i.e. spacing of diffraction spots in nm–1)

2 G a

larger real-space

smaller reciprocal-space Diffraction Techniques

•

• •

•

•

Reciprocal Space A Bravais lattice is an infinite array of discrete points with an arrangement an orientation which appears exactly the same, from whichever of the points the array is viewed. There are 14 Bravais lattices with primitive vectors a1, a2, and a3. The set of all wave vectors k that yield plane waves with the periodicity of a given Bravais lattice is known as the reciprocal lattice. The primitive vectors of the reciprocal lattice are found from:  

 a j  ak bi  2    ai  a j  ak 

a2 a1

REAL b2=2/a2 b1=2/a1

Where cyclic permutations of i, j, and k generate the three primitive vector components.

Ref: Ashcroft and Mermin, Solid State Physics (1976).

RECIPROCAL

The Reciprocal “Lattice” of a 3D Array of Scattering Centres Reciprocal Lattice

Scattering centres in a real space crystal lattice

Fourier transform

Scattering centres in a random group (e.g. amorphous material)

The reciprocal of any repeated length scales give reciprocal “lattice” features

Fourier transform 1/L L

The reciprocal lattice • A diffraction pattern is not a direct representation of the crystal lattice • The diffraction pattern is a representation of the reciprocal lattice We have already considered some reciprocal features Miller indices were derived as the reciprocal (or inverse) of unit cell intercepts.

Reciprocal Lattice vectors Any set of planes can be defined by: (1) their orientation in the crystal (hkl) (2) their d-spacing

The orientation of a plane is defined by the direction of a normal (vector product)

Defining the reciprocal Take two sets of planes: Draw directions normal: These lines define the orientation but not the length G1

1 We use to define the lengths d These are called reciprocal lattice vectors G1 and G2 Dimensions = 1/length

G2

Properties Of Reciprocal Lattice Direct lattice is a lattice in ordinary space whereas the reciprocal lattice is a lattice in the Fourier space.  The vectors in reciprocal lattice has the dimensions of (length)-1 whereas the primitive vectors of the direct lattice have the dimensions of length  A diffraction pattern of a crystal is a map of the reciprocal lattice of the crystal whereas a microscopic image is a map of direct lattice  When we rotate a crystal, both direct and reciprocal lattice rotates  Each point in the reciprocal lattice represents a set of parallel planes of the crystal lattice  If the coordinates of reciprocal vector G have no common factor, then G is inversely proportional to the spacing of the lattice planes normal to G  The volume of unit cell of the reciprocal lattice is inversely proportional to the volume of unit cell of the direct lattice  The direct lattice is the reciprocal of its own reciprocal lattice  The unit cell of the reciprocal lattice need not be parallelopiped 

Reciprocal Lattice In Two Dimensions

Construction of Reciprocal Lattice

1. Identify the basic planes in the direct space lattice, i.e. (001), (010), and (001). 2. Draw normals to these planes from the origin. 3. Mark distances from the origin along these normals proportional to the inverse of the distance from the origin to the direct space planes

Above a monoclinic direct space lattice is transformed (the baxis is perpendicular to the page). Note that the reciprocal lattice in the last is also monoclinic with * equal to 180°−. The symmetry system of the reciprocal lattice is the same as the direct lattice.

Reciprocal Lattices 

BCC Lattice

ˆ a1  12 a(xˆ  yˆ  z) ˆ a3  12 a(xˆ  yˆ  z) 2 G1  yˆ  zˆ a

ˆ a 2  12 a(xˆ  yˆ  z) 0  a1 a2  a3  12 a3

2 G2  xˆ  zˆ a

2 G3  xˆ  yˆ a

The reciprocal lattice is represented by the primitive vectors of an FCC lattice.

23

Reciprocal Lattices 

FCC Lattice

2 a1  yˆ  zˆ a

2 a2  xˆ  zˆ a

2 a3  xˆ  yˆ a

0  a1 a2  a3  a3

ˆ G1  12 a (  xˆ  yˆ  z)

ˆ G 2  12 a (xˆ  yˆ  z)

ˆ G 3  12 a (xˆ  yˆ  z) The reciprocal lattice is represented by the primitive vectors of an BCC lattice. 24

Drawing Brillouin Zones

Wigner–Seitz cell

The BZ is the fundamental unit cell in the space defined by reciprocal lattice vectors.

25

Drawing Brillouin Zones

26

Direct Space to Reciprocal Space

Reciprocal Space to Direct Space

Reciprocal Lattice for Aluminium

Diffraction from a particle and solid Single particle • To understand diffraction we also have to consider what happens when a wave interacts with a single particle. The particle scatters the incident beam uniformly in all directions Solid material • What happens if the beam is incident on solid material? If we consider a crystalline material, the scattered beams may add together in a few directions and reinforce each other to give diffracted beams 30

Theories: Common Names •

Maxwell – Equations describing interactions of electromagnetic waves (X-rays)

•

Laue – Interference from 3D array of scattering centres in crystals

•

Ewald – Worked with on general solution for 3D array

•

Bragg – Simplified equation based on planes of scattering centres

•

Fresnel – Optical model based on uniform refractive index of a material (reflectivity)

•

Dynamical Diffraction (Darwin, Prins, James) – Theory for scattering from perfect single crystals including multiple scattering events within the crystal

•

Kinematical Theory – Simplified theory for “small crystals” not including multiple scattering

Scattering by a crystals A crystal contains a large number of unit cells

•For a general unit cell in a crystal, the structure factor is:

•The total wave scattered by the crystal is:

Scattering by a crystals

•A crystal does not diffract X-ray unless

Not constructive interference unless all waves have the same phase

This is the famous Laue condition

Laue conditions specify scattering conditions What is S???

S is perpendicular to the imaginary reflecting plane

What does Laue condition mean???

a, b, c are unit vectors of crystal unit cell axes For a particular crystal unit, X-ray will only be scattered along discrete directions

What does this mean?! Laue assumed that each set of atoms could radiate the incident radiation in all directions Constructive interference only occurs when the scattering vector, K, coincides with a reciprocal lattice vector, G

This naturally leads to the Ewald Sphere construction

The Ewald Sphere We have constructed the reciprocal lattice (RL) in of the reciprocal dspacings, 1/dhkl, another utility of this lattice in of crystallography is made apparent by the Ewald sphere, which tells us the angle at which each family of planes will diffract! Consider a circle of radius r, with points X and Y lying on the circumference.

If the angle XAY is defined as q, then the angle XOY will be 2q by geometry and sin(q = XY/2r If this geometry is constructed in reciprocal space, then it has some important implications. The radius can be set to 1/l, where l is the wavelength of the X-ray beam. If Y is the 000 reciprocal lattice point, and X is a general point hkl, then the distance XY is 1/dhkl Thus: sin(q) = (1/dhkl)/(2/l) or, rearranged: l = 2 dhkl sin(q) , and Bragg’s law is satisfied!

Ewald Sphere We superimpose the imaginary “sphere” of radiated radiation upon the reciprocal lattice For a fixed direction and wavelength of incident radiation, we draw -ko (=1/l) e.g. along a* Draw sphere of radius 1/l centred on end of ko Reflection is only observed if sphere intersects a point i.e. where K=G

Laue Condition Satisfied • In this case, the incident wavevector angle is adjusted such that two points lie on the surface of the sphere. • The diffraction angle is then half the angle between the incident and diffracted wavevectors. • Generally, only a few angles will yield the proper conditions for diffraction.

k'

k

2Q K

O

What does this actually mean?! Relate to a real diffraction experiment with crystal at O K=G so scattered beam at angle 2q Geometry:

x tan 2q  R

but so

x  2q  tan R 1

2sinqhkl K  l

2 1 1 x  K  sin tan  l 2 R

Practicalities This allows us to convert distances on the film to lengths of reciprocal lattice vectors. Indexing the pattern (I.e. asg (hkl) values to each spot) allows us to deduce the dimensions of the reciprocal lattice (and hence real lattice) In single crystal methods, the crystal is rotated or moved so that each G is brought to the surface of the Ewald sphere In powder methods, we assume that the random orientation of the crystallites means that G takes up all orientations at once.

The Ewald Sphere is defined as the sphere of radius 1/l that is the locus of possible scattering vectors (S) when the beam (s0) is fixed.

y s0

x z

This sphere is in scattering space (reciprocal space). If a scattered wave has S on the Ewald sphere, it is visible on the film/detector.

Crystal position defines the coordinate axes.

Moving the beam moves the Ewald Sphere y s0

x z

y x z

For a given direction of the incoming xrays, the set of possible scattering vectors S is the surface of a sphere of radius 1/l, ing through the crystallographic origin.

Keeping the crystal fixed, we rotate the X-ray source. The Ewald Sphere moves in parallel with the X-ray source. The new set of S vectors describe the phase vs. direction of scatter for that position of the source.

Moving the beam. Crystal fixed.

By moving the X-ray source relative to the crystal, we can sample every possible S

limit = 2/l q180

The visible part of reciprocal space The set of all vectors S (red), given all possible directions of the beam (black arrows), is called reciprocal space.

: in this view, the crystal is fixed (center of image, where the X-rays are pointed). In real life, we find it easier to move the crystal, not the source. It doesn’t matter which one you move, the crystal or the source. The results are the same.

The Ewald Sphere If one rotates the Ewald sphere completely about the (000) reciprocal lattice point in all three dimensions, the larger sphere (of radius 2/l) contains all of the reflections that it is possible to collect using that wavelength of X-rays. This construction is known as the “Limiting sphere” and it defines the complete data set. Any reciprocal lattice points outside of this sphere can not be observed. Note that the shorter the wavelength of the X-radiation, the larger the Ewald sphere and the more reflections may be seen (in theory).

The limiting sphere will hold roughly (4/3r3/ V*) lattice points. Since r = 2/l, this equates to around (33.5/ V*l3) or (33.5 V/l3) reflections. For an orthorhombic cell with a volume of 1600Å3, this means CuKa can give around 14,700 reflections while MoKa would give 152,000 reflections.

The Ewald Sphere construction Which leads to spheres for various hkl reflections

Crystal related information is present in the reciprocal crystal

Sinq hkl 

l 2 d hkl

1 d hkl  The Ewald sphere construction generates the diffraction pattern 2l

Radiation related information is present in the Ewald Sphere

Ewald Sphere for LEED

k'

K k

• In low energy electron diffraction, the electrons interact mainly with the surface atoms so the reciprocal lattice becomes regularly spaced rods perpendicular to the surface plane. • The Ewald construction gives the spot locations on the screen and hence the in-plane lattice vectors. • Intensity of the spots versus electron energy gives the dspacing.

Ewald Construction for RHEED • In RHEED, High energy electrons are incident at grazing incidence. • The Ewald construction illustrated why straks are observed rather than spots--the reciprocal lattice rods intersect the Ewald sphere nearly tangentially.

Ewald Sphere Construction for RHEED

l = 2a/(h2+k2)1/2 sinq

As q is very small in REED, a = (h2+k2)1/2 l L/t t is the distance between the streaks

Intensity Measurement by LEED

Interpretation of LEED patterns a) Sharpness of LEED patterns: Well ordered surfaces exhibit sharp bright spots and low background intensity. The presence of surface defects and crystallographic imperfection results in broadening and wakening of the spots and increased background b) LEED spot geometry: This yields information on the surface geometry, i.e. symmetry and lattice constants. Furthermore one can deduce information on possible reconstructions or superstructures caused by adsorbates To produce diffraction patterns

the surface area has to be at least the length of the coherence length (typically several 100 Å). Therefore sometimes superposition of several domains leads to new diffraction patterns eg. 2x2 superstructure on hexagonal lattice leads to the same pattern as three domains of (2x1) superstructures

c) LEED spot profile: The spot profile is determined by the perfectness of the surface. Any imperfections broaden the spot. Reducing the domain size broadens the spot too. Even for a perfect crystal surface there is some finite spot widths due to the finite coherence length, determined by the energy distribution and the angular spread of the electron beam. Regularly stepped surfaces lead to split spots. In this case the diffraction conditions are given by two regularities, the terraces and the atomic arrangement in the terraces. d) LEED I-V analysis: The spot geometry gives only information on the regular arrangement on a surface. No information can be obtained for the local arrangement of the surface atoms (adsorbates) to the underlying array. However, due to multiple scattering the local arrangement of the scatterer within the surface unit cell influences the scattering. This shows up in special modulations of the spot intensities as function of the electron beam: Therefore I-V curves have to be measured. On the other hand, I-V curves can be calculated by assuming a special atomic arrangement. Usually, by a trial and error method the best fit between experimental and theoretical I-V curves yields then the most probable atomic positions within the unit cell.

Substrate & over layer LEED pattern Only Substrate

Substrate +Adsorbate

Thank you!