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# Reciprocal Lattice & Ewald Sphere Construction

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1. MATERIALS SCIENCE & ENGINEERING Part of A Learner’s Guide AN INTRODUCTORY E-BOOK Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur- 208016 Email:anandh@iitk.ac.in, URL:home.iitk.ac.in/~anandh http://home.iitk.ac.in/~anandh/E-book.htm Reciprocal Lattice & Ewald Sphere Construction

2. Reciprocal Lattice and Reciprocal Crystals • A crystal resides in real space. The diffraction pattern of the crystal in Fraunhofer diffraction geometry resides in Reciprocal Space. In a diffraction experiment (powder diffraction using X-rays, selected area diffraction in a TEM), a part of this reciprocal space is usually sampled. • From the real lattice the reciprocal lattice can be geometrically constructed. The properties of the reciprocal lattice are ‘inverse’ of the real lattice → planes ‘far away’ in the real crystal are closer to the origin in the reciprocal lattice. • As a real crystal can be thought of as decoration of a lattice with motif; a reciprocal crystal can be visualized as a Reciprocal Lattice decorated with a motif* of Intensities.Reciprocal Crystal = Reciprocal Lattice + Intensities as Motif* • The reciprocal of the ‘reciprocal lattice’ is nothing but the real lattice! • Planes in real lattice become points in reciprocal lattice and vice-versa. * Clearly, this is not the crystal motif- but a motif consisting of “Intensities”.

3. Motivation for constructing reciprocal lattices • In diffraction patterns (Fraunhofer geometry) (e.g. SAD), planes are mapped as spots (ideally points). Hence, we would like to have a construction which maps planes in a real crystal as points. • Apart from the use in ‘diffraction studies’ we will see that it makes sense to use reciprocal lattice when we are dealing with planes. • As the index of the plane increases → the interplanar spacing decreases → and ‘planes start to crowd’ around the origin in the real lattice (refer figure). Hence, we work in reciprocal lattice when dealing with planes. As seen in the figure the diagonal is divided into (h + k) parts.

4. We will construct reciprocal lattices in 1D and 2D before taking up a formal definition in 3D

5. Let us start with a one dimensional lattice and construct the reciprocal lattice Real Lattice Reciprocal Lattice How is this reciprocal lattice constructed One unit cell • The plane (2) has intercept at ½, plane (3) has intercept at 1/3 etc. • As the index of the plane increases it gets closer to the origin (there is a crowding towards the origin) Real Lattice Note in 1D planes are points and have Miller indices of single digit (they have been extended into the second dimension (as lines) for better visibility Each one of these points correspond to a set of ‘planes’ in real space Reciprocal Lattice Note that in reciprocal space index has NO brackets

6. Reciprocal Lattice Now let us construct some 2D reciprocal lattices Each one of these points correspond to a set of ‘planes’ in real space Example-1 Real Lattice Reciprocal Lattice g vectors connect origin to reciprocal lattice points The reciprocal lattice has an origin! Overlay of real and reciprocal lattices Note that vectors in reciprocal space are perpendicular to planes in real space (as constructed!) But do not measure distances from the figure!

7. The real lattice Example-2 Reciprocal Lattice Real Lattice The reciprocal lattice Note that vectors in reciprocal space are perpendicular to planes in real space (as constructed!) But do not measure distances from the figure!

8. Reciprocal Lattice Properties are reciprocal to the crystal lattice The basis vectors of a reciprocal lattice are defined using the basis vectors of the crystal as below BASIS VECTORS B The reciprocal lattice is created by interplanar spacings

9. A reciprocal lattice vector is  to the corresponding real lattice plane • The length of a reciprocal lattice vector is the reciprocal of the spacing of the corresponding real lattice plane • Planes in the crystal become lattice points in the reciprocal lattice  ALTERNATE CONSTRUCTION OF THE REAL LATTICE • Reciprocal lattice point represents the orientation and spacing of a set of planes

10. Going from the reciprocal lattice to diffraction spots in an experiment • A Selected Area Diffraction (SAD) pattern in a TEM is similar to a section through the reciprocal lattice (or more precisely the reciprocal crystal, wherein each reciprocal lattice point has been decorated with a certain intensity). • The reciprocal crystal has all the information about the atomic positions and the atomic species. • Reciprocal lattice* is the reciprocal of a primitivelattice and is purely geometrical  does not deal with the intensities decorating the points Physics comes in from the following: • For non-primitive cells ( lattices with additional points) and for crystals having motifs ( crystal = lattice + motif) the Reciprocal lattice points have to be weighed in with the corresponding scattering power (|Fhkl|2) Some of the Reciprocal lattice points go missing (or may be scaled up or down in intensity) Making of Reciprocal Crystal: Reciprocal lattice decorated with a motif of scattering power (as intensities) • The Ewald sphere construction further can select those points which are actually observed in a diffraction experiment * as considered here

11. To summarize: Real Lattice Decoration of the lattice with motif Real Crystal Purely Geometrical Construction Reciprocal Lattice Decoration of the lattice with Intensities Structure factor calculation Reciprocal Crystal Ewald Sphere construction Selection of some spots/intensities from the reciprocal crystal Diffraction Pattern

12. Crystal = Lattice + Motif • In crystals based on a particular lattice the intensities of particular reflections are modified  they may even go missing Lattice Position of the diffraction spots Is determined by Motif Intensity of the diffraction spots Diffraction Pattern Position of the diffraction spots  RECIPROCAL LATTICE Intensity of the diffraction spots  MOTIF’ OF INTENSITIES

13. Making of a Reciprocal Crystal • There are two ways of constructing the Reciprocal Crystal: 1) Construct the lattice and decorate each lattice point with appropriate intensity* 2) Use the concept as that for the real crystal** • The above two approaches are equivalent for simple crystals (SC, BCC, FCC lattices decorated with monoatomic motifs), but for ordered crystals the two approaches are different (E.g. ordered CuZn, Ordered Ni3Al etc.) (as shown soon). * Point #1 has been considered to be consistent with literature ** Point #2 makes reciprocal crystals equivalent in definition to real crystals

14. Examples of 3D Reciprocal Lattices weighed in with scattering power (|F|2) SC 001 011 101 111 Lattice = SC 000 010 100 110 No missing reflections Reciprocal Crystal = SC SC lattice with Intensities as the motif Figures NOT to Scale

15. 002 022 BCC 202 222 011 101 020 000 Lattice = BCC 110 200 100 missing reflection (F = 0) 220 Reciprocal Crystal = FCC Weighing factor for each point “motif” FCC lattice with Intensities as the motif Figures NOT to Scale

16. 002 022 FCC 202 222 111 020 000 Lattice = FCC 200 220 100 missing reflection (F = 0) 110 missing reflection (F = 0) Weighing factor for each point “motif” Reciprocal Crystal = BCC BCC lattice with Intensities as the motif Figures NOT to Scale

17. Order-disorder transformation and its effect on diffraction pattern • When a disordered structure becomes an ordered structure (at lower temperature), the symmetry of the structure is lowered and certain superlattice spots appear in the Reciprocal Lattice/crystal (and correspondingly in the appropriate diffraction patterns). Superlattice spots are weaker in intensity than the spots in the disordered structure. • An example of an order-disorder transformation is in the Cu-Zn system: the high temperature structure can be referred to the BCC lattice and the low temperature structure to the SC lattice (as shown next). Another examples are as below. Click here to know more about Ordered Structures Click here to know more about Superlattices & Sublattices

18. Diagrams not to scale Positional Order In a strict sense this is not a crystal !! High T disordered BCC Probabilistic occupation of each BCC lattice site: 50% by Cu, 50% by Zn 470ºC G = H  TS Sublattice-1 (SL-1) Sublattice-2 (SL-2) SC Low T ordered SL-1 occupied by Cu and SL-2 occupied by Zn. Origin of SL-2 at (½, ½, ½)

19. SC Click here to see structure factor calculation for NiAl (to see why some spots have weak intensity) →Slide 27 Click here to see XRD powder pattern of NiAl → Slide 5 Diffraction pattern from the ordered structure (3D) Ordered FCC BCC Reciprocal Crystal = FCC FCC lattice with Intensities as the motif Reciprocal crystal This is like the NaCl structure in Reciprocal Space!

20. SC Click here to see structure factor calculation for Ni3Al (to see why some spots have weak intensity) →Slide 29 Click here to see XRD powder pattern of Ni3Al → Slide 6 Diffraction pattern from the ordered structure (3D) Ordered BCC FCC Reciprocal Crystal = BCC BCC lattice with Intensities as the motif Reciprocal crystal

21. There are two ways of constructing the Reciprocal Crystal: 1) Construct the lattice and decorate each lattice point with appropriate intensity 2) Use the concept as that for the real crystal (lattice + Motif) 1) SC + two kinds of Intensities decorating the lattice 2) (FCC) + (Motif = 1FR + 1SLR) Motif  FR  Fundamental Reflection SLR  Superlattice Reflection 1) SC + two kinds of Intensities decorating the lattice 2) (BCC) + (Motif = 1FR + 3SLR) Motif

22. Example of superlattice spots in a TEM diffraction pattern The spots are periodically arranged    Superlattice spots SAD patterns from a BCC phase (a = 10.7 Å) in as-cast Mg4Zn94Y2 alloy showing important zones

23. Example of superlattice peaks in XRD pattern NiAl pattern from 0-160 (2) Superlattice reflections (weak)

24. The Ewald Sphere * Paul Peter Ewald (German physicist and crystallographer; 1888-1985) • Reciprocal lattice/crystal is a map of the crystal in reciprocal space → but it does not tell us which spots/reflections would be observed in an actual experiment. • The Ewald sphere construction selects those points which are actually observed in a diffraction experiment

25. Circular of a Colloquium held at Max-Planck-Institut für Metallforschung 7. Paul-Peter-Ewald-Kolloquium Freitag, 17. Juli 2008 organisiert von: Max-Planck-Institut für MetallforschungInstitut für Theoretische und Angewandte Physik,Institut für Metallkunde,Institut für Nichtmetallische Anorganische Materialiender Universität Stuttgart Programm

26. The Ewald Sphere • The reciprocal lattice points are the values of momentum transfer for which the Bragg’s equation is satisfied. • For diffraction to occur the scattering vector must be equal to a reciprocal lattice vector. • Geometrically  if the origin of reciprocal space is placed at the tip of ki then diffraction will occur only for those reciprocal lattice points that lie on the surface of the Ewald sphere. See Cullity’s book: A15-4

27. Bragg’s equation revisited Rewrite • Draw a circle with diameter 2/ • Construct a triangle with the diameter as the hypotenuse and 1/dhklas a side (any triangle inscribed in a circle with the diameter as the hypotenuse is a right angle triangle: APO = 90): AOP • The angle opposite the 1/d side is hkl (from the rewritten Bragg’s equation)

28. Now if we overlay ‘real space’ information on this: • Assume the incident ray along AC and the diffracted ray along CP. Then automatically the crystal will have to be considered to be located at C with an orientation such that the dhkl planes bisect the angle OCP (OCP = 2). • OP becomes the reciprocal space vector ghkl(often reciprocal space vectors are written without the ‘*’).

29. The Ewald Sphere construction Which leads to spheres for various hkl reflections Crystal related information is present in the reciprocal crystal The Ewald sphere construction generates the diffraction pattern Chooses part of the reciprocal crystal which is observed in an experiment Radiation related information is present in the Ewald Sphere

30. Ewald Sphere • When the Ewald Sphere (shown as circle in 2D below) touches the reciprocal lattice point  that reflection is observed in an experiment (41 reflection in the figure below). The Ewald Sphere touches the reciprocal lattice (for point 41)  Bragg’s equation is satisfied for 41 K = K =g = Diffraction Vector

31. Ewald sphere  X-rays Diffraction from Al using Cu K radiation Row of reciprocal lattice points Rows of reciprocal lattice points The 111 reflection is observed at a smaller angle 111 as compared to the 222 reflection (Cu K) = 1.54 Å, 1/ = 0.65 Å−1 (2/ = 1.3 Å−1), aAl = 4.05 Å, d111 = 2.34 Å, 1/d111 = 0.43 Å−1

32. Ewald sphere  X-rays Now consider Ewald sphere construction for two different crystals of the same phase in a polycrystal/powder (considered next). Click to compare them (Cu K) = 1.54 Å, 1/ = 0.65 Å−1 (2/ = 1.3 Å−1), aAl = 4.05 Å, d111 = 2.34 Å, 1/d111 = 0.43 Å−1

33. POWDER METHOD Diffraction cones and the Diffractometer geometry • In the powder method  is fixed but  is variable (the sample consists of crystallites in various orientations). • A cone of ‘diffraction beams’ are produced from each set of planes (e.g. (111), (120) etc.)(As to how these cones arise is shown in an upcoming slide). • The diffractometer moving in an arc can intersect these cones and give rise to peaks in a ‘powder diffraction pattern’. Click here for more details regarding the powder method

34. ‘3D’ view of the ‘diffraction cones’ Different cones for different reflections Diffractometer moves in a semi-circle to capture the intensity of the diffracted beams

35. THE POWDER METHOD Understanding the formation of the cones Cone of diffracted rays In a power sample the point P can lie on a sphere centered around O due all possible orientations of the crystals The distance PO = 1/dhkl

36. Circular Section through the spheres made by the hkl reflections The 440 reflection is not observed(as the Ewald sphere does not intersect the reciprocal lattice point sphere) Ewald sphere construction for Al Allowed reflections are those for h, k and l unmixed

37. The 331 reflection is not observed Ewald sphere construction for Cu Allowed reflections are those for h, k and l unmixed