Chapter 3 Basic Crystallography and Electron Diffraction from Crystals. Lecture 9. Chapter 3 CHEM Fall, L. Ma

Transcription

1 Chapter 3 Basic Crystallography and Electron Diffraction from Crystals Lecture 9

2 Outline The geometry of electron diffraction Crystallography Kinetic Theory of Electron diffraction Diffraction from crystals For physical issues of diffraction, please refer to: Diffraction physics, by John M. Cowley, North-Holland Pub. Co, 1984.

3 Why use diffraction in the TEM the first event occurring between electrons and specimen measure the average spacing between layers or rows of atoms; determine the orientation of a single crystal or grain; find the crystal structure of an unknown material; identify unknown phase measure the size, shape and internal stress of small crystalline regions. relate the crystallography to the image, giving TEM its great advantage over SEM and visible-light microscope. The geometry of electron diffraction

4 E-gun Schematic lens configuration of a TEM system

5 Objective lens and specimen stage Heart of TEM The objective lens forms an inverted initial image, which is subsequently magnified. In the back focal plane of the objective lens a diffraction pattern is formed. The objective aperture can be inserted here. The effect of inserting the aperture is shown on the next page. The objective lens would not usually provide a magnification of more than 50 and a TEM is routinely used to view regions of the specimen which are only a mm or so across. What does this imply about the diameter of a typical aperture if it were placed at the back focal plane as shown in this diagram? Diffraction (prior knowledge) Diffraction is an interference effect which leads to the scattering of strong beams of radiation in specific directions. Diffraction from crystals is described by the Bragg Law n λ = 2 d sin θ where n is an integer (the order of scattering), λ is the wavelength of the radiation, d is the spacing between the scattering entities (e.g. planes of atoms in the crystal) and θ is the angle of scattering. Electron and X-ray diffraction are both particularly powerful because their wavelengths are smaller than the typical spacings of atoms in crystals and strong, easily measurable, diffraction occurs.

6 Example : Identification of material structure

7 Example

8 Diffracted Wave

9 Electron diffraction in a double-slit experiment: Electrons act like waves. In this double-slit experiment, you can shoot an electron at a double-slit with different speeds, and observe the diffraction patterns.

10 In analogy, for a 3-D crystal, a similar path difference arguments shows that the diffraction of monochromatic electron beam by the regularly spaced 3-D array gives an interference patterns of beams. Kinematical theory of electrons diffraction and assumption 1. The e-beam is monochromatic, that is the electrons all have the same energy and wavelength. 2. The crystal is free from distortion. 3. Only a negligible fraction of the incident beam is scattered by the crystal, that is every atom in the crystal receives an incident wave of same the amplitude. 4. The incident and scattered waves may be treated as plane waves. 5. There is no interaction between the incident beam and scattered wavelets, that is the refractive index of crystal is unity. 6. There is no attenuation of e-beam with increasing depth in the crystal, that is no absorption. 7. There is no re-scattering of scattered waves. 8. The kinematical approach is satisfactory for general description of diffraction patterns. 9. Dynamic theory of electron diffraction is more realistic.

11 Top of thin foil Crystal plane (hkl) Bottom of thin foil The path difference is PO 2 d ' + O Q sin θ For in-phase arrival at observation, or constructive interference, this path length difference must be equal to an integral number of wavelength, nλ or This is called Bragg Law. n labels the various diffraction orders for a given set of planes. high order of n in Bragg Law usually is for spectrometry n equal 1 is for diffraction = = ' hkl = d hkl sin θ + d hkl sin θ 2d sinθ = nλ hkl

12 Bragg s Law is widely applied to the diffraction of X-rays as well as electrons. It tells us that we can expect very few elastically scattered electrons to emerge from our specimen unless they are at an angle ө. n is the order of diffraction. Usually only n=1 is considered for convenience. ө is Bragg angle. E.g. Also, ө is very small in electron diffraction 2d 2d Q θ λ hkl hkl 2d θ θ is samll sin sin θ = = n λ λ In practice, an electron beam will only be strongly diffracted from planes of atoms which are almost parallel to the electron beam. This fact makes the geometry of electron diffraction patterns much simpler than that of X- ray diffraction patterns for which ө can be very large. Diffracted beam ө Incident beam

13 How to get diffraction operation mode in TEM? By lowering the focal strength of the intermediate lens

14 Objective aperture Change focal strength SAD aperture Intermediate lens The first intermediate lens magnifies the initial image that is formed by the objective lens. The lens can be focused on initial image formed by the objective lens, or Diffraction pattern formed in the back focal plane of the objective lens. This determines whether the viewing screen of Chapter the microscope 3 CHEM shows 793 a diffraction 2011 Fall, pattern or an image.

15 Connecting Electron Diffraction Pattern and Image (a) A single perfect crystal (b) A small number of grains- note that even with three grains the spots begin to form circle (c) A large number pf randomly oriented grainsthe spots have now merged into rings

16 Connecting Electron Diffraction Pattern and Image (a) Electron diffraction pattern from a thin film of amorphous carbon. (b) (c) The variation of intensity with scattering angle obtained from a. Diffraction pattern from a fine grained polycrystalline gold specimen. (d) Diffraction from a single crystal of aluminum. Unlike TEM image, diffraction pattern is somewhat difficult to understand and describe. So, we need find some effective ways to characterize it. Firstly, we need know basics of crystallography, since Bragg s Law describes the crystal diffraction.

17 d-spacing of crystal planes, only a few of planes satisfy Bragg s Law, i.e. diffraction occurs If we consider the lattice of a crystal, then we can see that many different planes with different spacings exist within that one lattice. It is difficult to imagine how a beam of waves entering a crystal might be diffracted with so many different planes set at different angles to the beam, all with different spacings. Which planes will satisfy the Bragg law to diffract incident beam?

18 Crystallography Basics (Further reading: Crystallography, W. Borchardt-Ott, 2 nd edition, Springer, Berlin, 1995) Crystal Greek: χρυσταλλοζ (ice) 晶 Chinese: Ideal-Crystal Infinite extended 3-dimentional periodic arrangement of atoms Anisotropism (properties are dependent upon direction e.g. pleochroism for optical properties) Real-Crystal Regular faces constant angles between faces Habitus (e.g prismatic, fibric; depends on growth conditions: faster growing faces become smaller)

19 Crystal properties Crystal: homogeneous solid possessing long-range, three-dimensional internal order. Crystals are far-ordered (> 20 Å) 1 Ångström= m = 0.1 nm = 100 pm The principles that control the crystal s atomic structure also affect the shapes of the crystal faces and the angles between them. Unit cell: smallest unit of the structure (or lattice) that can be indefinitely repeated to generate the whole structure (lattice). amorphous: solid material which lack any ordered internal atomic arrangement, the term amorphous might be replaced by: Short-range ordered (2-5 Å e.g. Si-O coordination polyhedral) Medium-range ordered (5-20 Å e.g. borosilicate glass structure

20 Constructing crystal: there are two most compact ways close-packed hexagonal (honeycomb) layers of atoms square arrangement Hexagonal layer Square layer

21 Mathematical definition Crystal structure: a regular arrangement of atoms decorating a periodic, 3-D lattice. Lattice: the set of points which is created by all integer linear combinations of three basis vectors, a, b, and c. Six numbers of lattice parameters of unit cell ( a,b,c, α,β,γ) Lattice vector : t = ua + vb + wc there are seven primitive lattices Lattice Property all lattice site are equivalent there is only one lattice site per unit set

22 Lattice vector : t = ua + vb + wc Centering vectors we can place additional lattice sites at the endpoints of so-called centering vectors. the possible centers are (0,1/2,1/2), (1/2,0,1/2),(1/2,1/2,0),(1/2,1/2,1/2), body center ( I-center) and face center (I-center)

23 Lattice vector : t = ua + vb + wc Combining primitive lattice and centered lattice, there are 14 types of lattice, derived by August Bravais in 1850, called Bravais lattice Detailed crystal system is illustrated in the International Tables for crystallography, Volume A. In this class, we only use the simple structures: simple cubic, face-centered cubic, body-centered cubic, and the close-packed hexagonal

24 The simple cubic structure (SC) Ex: α-polonium Simple Cubic 2r = a r = Radius of atom a = Lattice Parameter Concept of Coordination Number For Example : In a Simple Cubic system a Central Atom is surrounded by 6 atoms that are called First nearest neighbors Coordination Numbers (CN =6) a 2r To illustrate this we need to put together 8 unit cells of Simple as shown and mark one central atom and show CN

25 Constructing simple cubic structure Simple cubic structure is obtained by layers directly stacking on the top of each other. This is an uncommon structure, ex. α-polonium Practice to construct crystal structure: go home and use tennis or pen pang ball.

26 The body-centered cubic structure (BCC): ex. Cr, Mo, and Fe a 2r Hard Ball Model a In BCC: A Central Atom is surrounded by 8 atoms that are the First nearest neighbors CN=8 for BCC Stick and Ball Model Pl. Note: CN can be shown with one unit cell

27 Notice No class at Oct. 5, 2011