X-Ray Diffraction (XRD)

1. X-Ray Diffraction (XRD)

2. Principle In XRD, an incident wave is directed into a material and a detector is typically moved about to record the directions and intensities of the outgoing diffracted waves.

3. X-ray diffraction has acted as the corner stone of the twentieth century science. Its development has catalyzed the develpments of all of the rest of solid state science and much of understanding of chemical bonding

4. What is X-rays • X-rays are high energy electromagnetic radiation having energies from ~200eV to 1 MeV • Between the γ-rays and ultraviolet (UV) in the electromagnetic spectrum. • Gamma rays and x-rays are essentially identical with γ –rays being more energetic and shorter in wavelength

6. Each quantum of of electromagnetic radiation or photon has an energy, E which is proportional to its frequency,  : • E=h h=Plank’s constant=4.136 x 10-15eV.s • λ=hc/E λ= wavelength ; c=2.998 x 108 m/s • The useful range of wavelength for x-ray diffraction studies is between 0.05 and 0.25 nm. • * Interatomic spacings in crystals are typically about 0.2 nm (2Å)

7. Production of X-rays X-rays are produced in an x-ray-tube consisting of two metals electrodes enclosed in a vacuum chamber

8. Electrons are produced by heating a tungsten filament cathode which is at a high negative potential The e are accelerated at high velocity towards the anode (water-cooled) Loss of energy due to the collision with the metal anode produced X-rays Only 1% of the e beam converted to x-rays – the rest as heat

9. A typical x-ray spectrum - A continuous radiationdue to electrons losing their energy in a series collisions with atoms of target anode.

10. Illustration of the origin of continuous radiation

11. Short Wavelength Limit (λSWL) When e loses all its energy in a single collision with a target atom x-rays photon with maximum energy or shortest wavelength is produced - short wavelength limit Characteristic lines When an e has sufficient energy to eject an inner shell e →atom will be in the xcited state with vacancy in the inner shell

13. An e from an outer shell will fill up the vacancy Energy equals to the difference in the e energy levels will b released in d form of x-ray photon. This is characteristic of the target metal producing a sharp peaks in the spectrum –known as characteristic lines It is this characteristic that are most useful in XRD

14. The accelerating potentials necessary to produce x-rays having comparable to interatomic spacings are about 10kV. Usually high accelerating potentials are used to produce higher intensity line spectrum of target metal Higher accelerating potentials changes λSWL but not the characteristic wavelengths. Intensity of characteristic line depends both on the applied potential and the tube current I I = Bi(V – VK)n B=proportional constant I = current VK = potential req. to eject an e from K shell V= applied potential n =a cosntant for a particular value of V

15. The fig indicate that there are more than 1 characteristic line This line correspond to electron transitions line between different energy levels Characteristic line classified as K, L, M (Bohr model)

16. Transition L → K Kά M → L Lα M → K Kβ

17. Due to the presence of subshells Kά or Kβ can be further resolved into Kά1 and Kά2 LIII→ K Kά1 LII → K Kά2 Energies of the K,LII ,and LIII Levels of Molybdenum

18. The most important radiations in diffraction work are those corresponding to the filling of the inner most K shell from adjacent shells giving Kά1,Kά2 dan Kβ lines Wide choice of characteristic Kά lines obtained by using different target metals as shown on Table2 but Cu Kά is the most common radiation used.

19. The Kάlines are used bcos they are more energetic than Lά, therefore less strongly absorbed by the material we want to examine. The wavelength spread of each line is extremely narrow and each wavelength is known with high precision

20. Interaction of X-rays with matter Any mechanism which causes a photon, in the collimated incident X-ray beam to miss the detector is called absorption. Most mechanism – conversion of photon energy to another form, while some simply change the direction-diffraction

21. There are various processes taking place when x-ray interact w matter: • No interaction • Conversion to heat • Photoelectric effect – flourescence and auger electron • Compton Scattering • Coherent Scattering – leads to the phenomena of diffraction **Note: please do ur own reading on d above topics

22. Just like an incident electron, X-ray photons can initiate electronic transitions Decrease in intensity  distance traversed by the X-ray beam X-ray absorption

23. weight fractions • Beer’s law where m is the linear absorption coefficient • Problem: m depends on the density of the absorbing material, but the ratio m/r does not (mass absorption coefficient) and for a mixture (or alloy):

24. Properties of the absorption coefficient The way μ varies with wavelength λ gives the indication to the interaction of X-rays and atom

25. Properties of the absorption coefficient E l • There is a sharp discontinuity in the dependence of the absorption coefficient on energy (wavelength) at the energy corresponding to the energy required to eject an inner-shell electron • The discontinuity is known as an absorption edge • Away from an absorption edge, each “branch” of the absorption curve is given by:

26. Values of / are tabulated in the International Tables for Crystallography as well as in most X-ray diffraction textbooks Note the discontinuities in the tabulated data at the absorption edges Tabulated values of mass absorption coefficients

27. We see many absorption edges…

28. An interesting application of absorption edge is when the edge of one element A is located between the Kά and Kβlines of another element B When this occurs the Kβfrom the B atoms will be very strongly absorbed while the longer wavelength Kά will be slightly absorbed. With suitable thickness ,element A can act as a beta-filter for characteristic radiation from element B

29. Using an absorber as an X-ray filter can reduce undesirable wavelength contamination in a diffraction experiment Absorption and X-ray filters

30. Fluorescence is the opposite of absorption -- when energy is absorbed, a vacancy is produced in an electron shell Other electrons fill that vacancy, producing radiation Absorption at an edge generates high fluorescence Fluorescence can be a source of background in a diffraction experiment Cu Ka -- l = 1.54Å Cu-radiation fluoresces Fe K-edge -- l = 1.74Å iron, but Cr-radiation Cr Ka -- l = 1.79Å does not Fluorescent radiation is characteristic to specific elements and is widely used for chemical analysis X-ray fluorescence

31. DIFFRACTION Diffraction is a general characteristic of all waves can be define as modification of the behaviour of light or other waves by its interaction with an object. When a beam of x-rays incident on an atom, ē’s in the atom oscillate about their mean positions The process of absorption and reemission of electromagnetic radiation by ē is known as scattering

32. When there is no change in energy bet. Incident and emitted photon- radiation is elastically scattered and is known as coherent scattering. The photon changes direction after colliding with the electron but transfer none of its energy to the electron.Thus this scattered photon leaves in a new direction but with the same phase and energy as the incident photon. ------ phenomenon of diffraction When there is a photon energy loss – inelastic scattering , also known as Compton scattering

33. The above Fig. shows an atom containing several electrons arranged as points around the nuclues Consider two waves that are incident on the atom

34. The upper wave is scattered by electron A in the forward direction Similarly, the lower wave is scattered in the forward direction by electron The two waves scattered in the forward direction are said to be in phase across wavefront XX’ bcos the waves have traveled the same distance before and after. No path difference → in phase

35. Hence adding these two waves will give twice the amplitude but same wavelength The other scattered waves in the above Fig will not be in phase across the wavefront YY’ when the path difference (CB-AD) is not an integral number of wavelength If we add these two waves across the wave front the resultant amplitude of the scattered waves is less than the wave scattered by the same electron in the forward direction.

37. The efficiency of how an atom scatter a beam of X-rays is define by the atomic scattering factor, f When scattering is in the forward direction (scattering angle =0o ) f =Z ,since the waves scattered by all the electrons in the atom are in phase and the amplitude sum up As θ increases, the waves become more and more out of phase bcos they travel different path length and therefore the amplitude,or f, decreases.

38. The atomic scattering factor also depends on the wavelength of the incident rays Take note that most of the scattering occurs in the forward direction θ=0o

39. DIFFRACTION FROM CRYSTALLINE MATERIALS-BRAGG’S LAW Atoms scatter x-rays and these scattered waves from all atoms can interfere. If scattered waves are in phase(coherent) , constructive interference and the beams are diffracted in specific directions. These directions are governed by the wavelength λ, and the nature of the crystalline sample – Braggs Law nλ = 2dsinθ

40. In deriving Bragg’s Law, it is convenient to think the x-ray as being reflected from plane of atoms (mind you x-rays are really not being reflected)

41. Consider diffracted waves in the above Fig and assumed that it make the same angle,θ with atomic plane as does the incident wave. Criterion for wave to be diffracted,the reflected x-rays should all be in phase across the wavefront such as BB’ To be in phase, the path lengths between wfs AA’ and BB’ must differ by exactly an integral number n of wavelength λ The path difference, δ=nλ , n is an integer Since lines CC’ and CD are also wavefronts, δ = DE +EC’ = 2EC’

42. But, δ = 2CE Sinθ Bcos CE is the interplanar spacing d’ But, δ = 2d’ Sinθ nλ = 2d’Sinθ Bragg’s Law This is extremely important equation in indexing x-ray diffraction pattern and hence crystal structure of the materials. n known as order of reflection, is the path difference,in terms of the number of wavelengths between waves scattered by adjacent planes.

45. Although Bragg's law was used to explain the interference pattern of X-rays scattered by crystals, diffraction has been developed to study the structure of all states of matter with any beam, e.g., ions, electrons, neutrons, and protons, with a wavelength similar to the distance between the atomic or molecular structures of interest.

46. What is X-ray Diffraction ? The atomic planes of a crystal cause an incident beam of X-rays to interfere with one another as they leave the crystal. The phenomenon is called X-ray diffraction.

47. Why XRD? • Measure the average spacings between layers or rows of atoms • Determine the orientation of a single crystal or grain • Find the crystal structure of an unknown material • Measure the size, shape and internal stress of small crystalline regions

49. Bragg’s Law and Diffraction: How waves reveal the atomic structure of crystals nλ = 2dsinθ n = integer Diffraction occurs only when Bragg’s Law is satisfied Condition for constructive interference (X-rays 1 & 2) from planes with spacing d