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Objectives

Objectives. By the end of this section you should: understand the concept of planes in crystals know that planes are identified by their Miller Index and their separation, d be able to calculate Miller Indices for planes know and be able to use the d-spacing equation for orthogonal crystals

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Objectives

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  1. Objectives By the end of this section you should: • understand the concept of planes in crystals • know that planes are identified by their Miller Index and their separation, d • be able to calculate Miller Indices for planes • know and be able to use the d-spacing equation for orthogonal crystals • understand the concept of diffraction in crystals • be able to derive and use Bragg’s law

  2. Lattice Planes and Miller Indices Imagine representing a crystal structure on a grid (lattice) which is a 3D array of points (lattice points). Can imagine dividing the grid into sets of “planes” in different orientations

  3. All planes in a set are identical • The planes are “imaginary” • The perpendicular distance between pairs of adjacent planes is the d-spacing Need to label planes to be able to identify them Find intercepts on a,b,c: 1/4, 2/3, 1/2 Take reciprocals 4, 3/2, 2 Multiply up to integers: (8 3 4)[if necessary]

  4. Exercise - What is the Miller index of the plane below? Find intercepts on a,b,c: Take reciprocals Multiply up to integers:

  5. General label is(h k l)which intersects at a/h, b/k, c/l (hkl) is the MILLER INDEX of that plane (round brackets, no commas). Plane perpendicular to y cuts at , 1,   (0 1 0) plane This diagonal cuts at 1, 1,   (1 1 0) plane NB an index 0 means that the plane is parallel to that axis

  6. Using the same set of axes draw the planes with the following Miller indices: (0 0 1) (1 1 1)

  7. Using the same set of axes draw the planes with the following Miller indices: (0 0 2) (2 2 2) NOW THINK!! What does this mean?

  8. Planes - conclusions 1 • Miller indices define the orientation of the plane within the unit cell • The Miller Index defines a set of planes parallel to one another (remember the unit cell is a subset of the “infinite” crystal • (002) planes are parallel to (001) planes, and so on

  9. d-spacing formula For orthogonal crystal systems (i.e. ===90) :- For cubic crystals (special case of orthogonal) a=b=c :- e.g. for (1 0 0) d = a (2 0 0) d = a/2 (1 1 0) d = a/2 etc.

  10. A cubic crystal has a=5.2 Å (=0.52nm). Calculate the d-spacing of the (1 1 0) plane A tetragonal crystal has a=4.7 Å, c=3.4 Å. Calculate the separation of the: (1 0 0) (0 0 1) (1 1 1) planes

  11. Question 2 in handout: If a = b = c = 8 Å, find d-spacings for planes with Miller indices (1 2 3) Calculate the d-spacings for the same planes in a crystal with unit cell a = b = 7 Å, c = 9 Å. Calculate the d-spacings for the same planes in a crystal with unit cell a = 7 Å, b = 8 Å, c = 9 Å.

  12. X-ray Diffraction

  13. Diffraction - an optical grating Path difference XY between diffracted beams 1 and 2: sin = XY/a  XY = a sin  For 1 and 2 to be in phase and give constructive interference, XY = , 2, 3, 4…..n so a sin  = n where n is the order of diffraction

  14. Consequences: maximum value of  for diffraction sin = 1  a =  Realistically, sin <1  a >  So separation must be same order as, but greater than, wavelength of light. Thus for diffraction from crystals: Interatomic distances 0.1 - 2 Å so  = 0.1 - 2 Å X-rays, electrons, neutrons suitable

  15. Diffraction from crystals X-ray Tube Detector ?

  16. Beam 2 lags beam 1 by XYZ = 2d sin  so 2d sin  = nBragg’s Law

  17. e.g. X-rays with wavelength 1.54Å are reflected from planes with d=1.2Å. Calculate the Bragg angle, , for constructive interference.  = 1.54 x 10-10 m, d = 1.2 x 10-10 m, =? n=1 :  = 39.9° n=2 : X (n/2d)>1 2d sin  = n We normally set n=1 and adjust Miller indices, to give 2dhkl sin  = 

  18. Example of equivalence of the two forms of Bragg’s law: Calculate  for =1.54 Å, cubic crystal, a=5Å 2d sin  = n (1 0 0) reflection, d=5 Å n=1, =8.86o n=2, =17.93o n=3, =27.52o n=4, =38.02o n=5, =50.35o n=6, =67.52o no reflection for n7 (2 0 0) reflection, d=2.5 Å n=1, =17.93o n=2, =38.02o n=3, =67.52o no reflection for n4

  19. Use Bragg’s law and the d-spacing equation to solve a wide variety of problems 2d sin  = n or 2dhkl sin  = 

  20. Combining Bragg and d-spacing equation X-rays with wavelength 1.54 Å are “reflected” from the (1 1 0) planes of a cubic crystal with unit cell a = 6 Å. Calculate the Bragg angle, , for all orders of reflection, n. d = 4.24 Å

  21. d = 4.24 Å n = 1 :  = 10.46° n = 2 :  = 21.30° n = 3 :  = 33.01° n = 4 :  = 46.59° n = 5 :  = 65.23° = (1 1 0) = (2 2 0) = (3 3 0) = (4 4 0) = (5 5 0) 2dhkl sin  = 

  22. Summary • We can imagine planes within a crystal • Each set of planes is uniquely identified by its Miller index (h k l) • We can calculate the separation, d, for each set of planes (h k l) • Crystals diffract radiation of a similar order of wavelength to the interatomic spacings • We model this diffraction by considering the “reflection” of radiation from planes - Bragg’s Law

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