Presentation on theme: "Objectives By the end of this section you should:"— Presentation transcript:
1 Objectives By the end of this section you should: understand the concept of planes in crystalsknow that planes are identified by their Miller Index and their separation, dbe able to calculate Miller Indices for planesknow and be able to use the d-spacing equation for orthogonal crystalsunderstand the concept of diffraction in crystalsbe 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-spacingNeed to label planes to be able to identify themFind intercepts on a,b,c: 1/4, 2/3, 1/2Take reciprocals 4, 3/2, 2Multiply 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 reciprocalsMultiply 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) planeThis diagonal cuts at 1, 1, (1 1 0) planeNB 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 1Miller indices define the orientation of the plane within the unit cellThe 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) A cubic crystal has a=5.2 Å (=0.52nm). Calculate the d-spacing of the (1 1 0) planeA 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.
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…..nso 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 Åso = ÅX-rays, electrons, neutrons suitable
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 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 m, d = 1.2 x m, =?n=1 : = 39.9°n=2 : X (n/2d)>12d sin = nWe normally set n=1 and adjust Miller indices, to give2dhkl sin =
18 Example of equivalence of the two forms of Bragg’s law: Calculate for =1.54 Å, cubic crystal, a=5Å2d sin = n(1 0 0) reflection, d=5 Ån=1, =8.86on=2, =17.93on=3, =27.52on=4, =38.02on=5, =50.35on=6, =67.52ono reflection for n7(2 0 0) reflection, d=2.5 Ån=1, =17.93on=2, =38.02on=3, =67.52ono reflection for n4
19 2dhkl sin = 2d sin = n or Use Bragg’s law and the d-spacing equation to solve a wide variety of problems2d sin = nor2dhkl sin =
20 X-rays with wavelength 1.54 Å are “reflected” from the Combining Bragg and d-spacing equationX-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° = (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 spacingsWe model this diffraction by considering the “reflection” of radiation from planes - Bragg’s Law