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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 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 • 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] Exercise - What is the Miller index of the plane below? Find intercepts on a,b,c: Take reciprocals Multiply up to integers: 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 Using the same set of axes draw the planes with the following Miller indices: (0 0 1) (1 1 1) 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? 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 d-spacing formula 2 2 2 For orthogonal crystal systems (i.e. ===90) :- 1 h k l 2 2 2 2 d a b c For cubic crystals (special case of orthogonal) a=b=c :- 1 h2 k 2 l 2 2 d a2 e.g. for (1 0 0) (2 0 0) (1 1 0) d=a d = a/2 d = a/2 etc. 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 1 h 2 k 2 l2 2 2 2 d a c [a b] 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. X-ray Diffraction Diffraction - an optical grating Path difference XY between diffracted beams 1 and 2: 2 a sin = XY/a 1 X XY = a sin Y Coherent incident light Diffracted light 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 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 X-ray Tube Diffraction from crystals Detector “Reflected” radiation Incident radiation 1 2 X d ? Z Y Transmitted radiation “Reflected” radiation Incident radiation 1 2 X d Z Y Transmitted radiation Beam 2 lags beam 1 by XYZ = 2d sin so 2d sin = n Bragg’s Law 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, 2d sin n n sin 2d 1 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 = 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 n7 (2 0 0) reflection, d=2.5 Å n=1, =17.93o n=2, =38.02o n=3, =67.52o no reflection for n4 Use Bragg’s law and the d-spacing equation to solve a wide variety of problems 2d sin = n or 2dhkl sin = 2 2 2 1 h k l 2 2 2 2 d a b c 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. 1 h k l 2 d a2 2 2 d 18 2 2 11 0 0.056 2 6 d = 4.24 Å d = 4.24 Å n sin 2d 1 n=1: = 10.46° = (1 1 0) n=2: = 21.30° = (2 2 0) n=3: = 33.01° = (3 3 0) n=4: = 46.59° = (4 4 0) n=5: = 65.23° = (5 5 0) 2dhkl sin = 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