CHEM-E5225 :Electron Microscopy. Diffraction 1

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1 CHEM-E5225 :Electron Microscopy Diffraction Yanling Ge Text book: Transmission electron microscopy by David B Williams & C. Barry Carter. 2009, Springer

2 Outline Diffraction in TEM Thinking in reciprocal space Diffraction from crystals (your calculation) Diffraction from small volumes

3 Diffraction in TEM

4 Why use Diffraction in TEM Is the specimen crystalline or amorphous? If it is crystalline, then what are the crystallographic characteristics (Lattice parameter, symmetry, etc.) of the specimen. If the specimen single crystalline? If not, what is the grain morphology, how large are the grains, what is the grain-size distribution, etc.? What is the orientation of the specimen or lf individual grains with respect to the electron beam? Is more than one phase present in the specimen? If so, how are they orientation relationship? Diffraction contrast in the image for defects.

5 Scattering from a plane of atoms = 2d sin B Wave-propagation vectors, or simply as wave vectors, k-vectors, which is normal to the wavefront. k I = k D = 1/ K = k D k I

6 Scattering from a Crystal When equals the Bragg angle, B : AC+CD = 2dsin B =n K B = 1/d, K B = g Scattering from a crystal: It does not matter how the atoms (scattering centers) are distributed on these two planes; the scattering from any two points on planes P 1 and P 2 will produce the same path difference 2dsin.

7 Bragg s law in vector

8 Meaning of n in the Bragg s law n = 2d sin B The physical meaning of n is high order of diffraction. Here we can consider n in the Bragg s law as indicating that electrons are diffracting from a set of planes with spacing d/n rather than d.

9 Forming DPs and images: the TEM imaging system

10 Caution need for using SAD aperture Two key points in forming an SAD pattern: (i) be sure specimen at the eucentric focus position (ii) remember to focus the DP with the intermediate lens (diffraction focus)

11 Thinking in Reciprocal Space The properties and its relation to real lattice The Mathematical definition

12 Construction of reciprocal space (100) planes n 010 d 010 (010) (100) (020) planes n 020 (110) planes d 110 n 110 (010) (020) d 020 (010) (020) (100) (100) (110)

13 Mathematical definition of the reciprocal lattice Real space: r n = n 1 a + n 2 b + n 3 c Reciprocal space: r* = m 1 a* + m 2 b* + m 3 c* a* b = a* c = b* c = b* a = c* a = c* b = 0 direction The vector a* is orthogonal to the vectors b and c. a* a = 1; b* b = 1; c* c = 1 length Definition: a* b V c c ; b* a V c c ; c* a b V c The vector, a*, is always perpendicular to the plane (100) even when a is not.

14 g hkl * = ha* + kb* + lc* The vector g The definition of the (hkl) indices is OA = a/h; OB = b/k: OC = c/l. the plane ABC can then be represented as (hkl). g = 1/d hkl Vector g hkl is normal to the plane (hkl) and its length is (1/d hkl ).

15 Diffraction equations n = 2d sin B Bragg s Law Laue equations

16 Laue equations

17 The Laue Equations and Their Relation To Bragg s Law Laue diffraction conditions K r n = N Setting r n equal to the three unit vectors in turn: K a = h K b = k Bragg s Law n = 2dsin K = g g hkl r n = N K c = l Bragg s law is a special form of the Laue equations.

18 The Ewald Sphere of Reflection Construction of Ewald sphere, start from the origin O of reciprocal-lattice and from O to have incident vector length 1/ as a radius to get the center of Ewald sphere C. The key point is that when the sphere cuts through the reciprocal-lattice point the Bragg condition is satisfied. k D could be any vector which begins at C and ends on the sphere. The origin O of reciprocallattice is fixed, not the center of sphere C, which moves with incident beam.

19 The Excitation Error Excitation error or deviation parameter is measured by a vector, s, in reciprocal space: K = g + s It is a measure of how far we deviate from the exact Bragg condition. s c : parallel vector CG; s z : parallel incident beam; s m : perpendicular to the surface of specimen or perpendicular to OG; s g : defined for a particular g; We define the sign of s to be negative when G is outside the sphere, while s is positive when G is inside the Ewald sphere.

20 The Excitation Error The change the value of s: First, if we tilt the specimen, the row of spots moves but the Ewald sphere does not. Second, if we tilt the beam above the specimen, the Ewald sphere moves, because k I tilts, because C moves.

$However, as the kv increases, the radius of the Ewald sphere increases and the diffraction spots appear to move together since the camera length changed.$

21 Thin-Foil Effect and the Effect of Accelerating Voltage The specimen is unchanged so the reciprocal lattice is the same. However, as the kv increases, the radius of the Ewald sphere increases and the diffraction spots appear to move together since the camera length changed. What is very important for TEM is that because is small, the radius of the Ewald sphere, 1/, is large and hence the Ewald sphere is quite flat.

22 Diffraction from Crystals

$for diffraction from crystals: The crystal is perfect and infinite. Scattering angle is small Fixed value of atom scattering factor f.$

23 Structure factors F amplitudeof thewavescattered by all theatoms of a unit cell amplitudeof thewavescattered by an electron Assumption for diffraction from crystals: The crystal is perfect and infinite. Scattering angle is small Fixed value of atom scattering factor f.

24 Scattering from a Unit Cell x/a = u y/b = v z/c = w Phase difference between 1 and 3 : 3 1 = 2 hu Phase difference between the wave scattered by atom B and scattered by atom A at the origin, for hkl reflection: = 2 (hu+kv+lw) B.D. Cullity, Elements of X-ray Diffraction, Addison-Wesley Publishing Company, 1978.

25 Scattering from An Unit Cell B.D. Cullity, Elements of X-ray Diffraction, Addison-Wesley Publishing Company, 1978.

26 Body-Centered Cubic

27 Face-centered Cubic

28 Hexagonal close-packed

29 Simple Cubic NiAl (B2) Ni 3 Al (L1 2 ) All the possible reciprocal-lattice points will give rise to Bragg reflections.

30 Supperlattice Reflections and Imaging NiAl Ni 3 Al

31 The forbidden reflections with F = 0 can sometimes actually be present due to dynamical scattering events. Forbidden Reflections

32 Diffraction from Small Volumes

There are also subsidiary maxima or minima when: Each point in the reciprocal lattice can

33 The Summation Approach Shape Effect If K a is an integer, then Bragg condition and the intensity is then a maximum. There are also subsidiary maxima or minima when: Each point in the reciprocal lattice can actually be associated with a rod, socalled relrod, the reason is that we have a thin specimen: a small thickness in real space gives a larger length in reciprocal space.

34 The Thin-Foil Effect

35 The Thin-Foil Effect The errors in SADP if s is not zero!

$Diffraction From Wedge-Shaped Specimens The relrod will always be normal to the surface. We will see two spots which lie along line which is normal to the edge of the wedge.$

36 Diffraction From Wedge-Shaped Specimens The relrod will always be normal to the surface. We will see two spots which lie along line which is normal to the edge of the wedge. Notice that all the pairs of spots are aligned in the same direction as we expected and that their separation is larger for larger values of s. This simple relrod model predicts that we would see only one spots if s=0 because the relrod model fails when we are in a strongly dynamical diffraction condition.

$Diffraction From Planar Defects The platelet is itself a thin$ Two spots in the DP, the separation of the spots increases with

37 Diffraction From Planar Defects The platelet is itself a thin parallelepiped which is inclined to the specimen parallelepiped. Two spots in the DP, the separation of the spots increases with increasing s. The line MN lies normal to the trace of the platelet.

$Diffraction From Particles small becomes$ the beam.

38 Diffraction From Particles small becomes large Platelets are orientated parallel to the beam. Short-range ordering of vanadium carbide gives rise to diffuse scattering in the DPs.

$Calculate the intensity of diffracted beam of NaCl using structure$

39 Homework 5 rd assignment: reading and presentation Homework: Calculate the intensity of diffracted beam of NaCl using structure factor

SOLID STATE 18. Reciprocal Space

SOLID STATE 18. Reciprocal Space SOLID STATE 8 Reciprocal Space Wave vectors and the concept of K-space can simplify the explanation of several properties of the solid state. They will be introduced to provide more information on diffraction