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# Pend Fisika Zat Padat (4) indexing

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• 1. Pertemuan 4CRYSTAL INDEXING<br />IwanSugihartono, M.Si<br />JurusanFisika, FMIPA<br />UniversitasNegeri Jakarta<br />1<br />
• 2. Crystals<br />06/01/2011<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />2<br /><ul><li>Crystal structure basics
• 3. unit cells
• 4. symmetry
• 5. lattices
• 6. Diffraction
• 7. how and why - derivation
• 8. Some important crystal structures and properties
• 9. close packed structures
• 10. octahedral and tetrahedral holes
• 11. basic structures
• 12. ferroelectricity</li></li></ul><li>Objectives<br />By the end of this section you should:<br /><ul><li>understand the concept of planes in crystals
• 13. know that planes are identified by their Miller Index and their separation, d
• 14. be able to calculate Miller Indices for planes
• 15. know and be able to use the d-spacing equation for orthogonal crystals
• 16. understand the concept of diffraction in crystals
• 17. be able to derive and use Bragg’s law</li></ul>06/01/2011<br />3<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
• 18. Lattice Planes and Miller Indices<br />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<br />06/01/2011<br />4<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
• 19. All planes in a set are identical<br />The planes are “imaginary”<br />The perpendicular distance between pairs of adjacent planes is the d-spacing<br />Need to label planes to be able to identify them<br />Find intercepts on a,b,c: 1/4, 2/3, 1/2<br />Take reciprocals 4, 3/2, 2<br />Multiply up to integers: (8 3 4) [if necessary]<br />06/01/2011<br />5<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
• 20. Exercise - What is the Miller index of the plane below?<br />Find intercepts on a,b,c:<br />Take reciprocals<br />Multiply up to integers:<br />06/01/2011<br />6<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
• 21. General label is(h k l)which intersects at a/h, b/k, c/l <br />(hkl) is the MILLER INDEX of that plane (round brackets, no commas).<br />Plane perpendicular to y cuts at , 1, <br /> (0 1 0) plane<br />This diagonal cuts at 1, 1, <br /> (1 1 0) plane<br />NB an index 0 means that the plane is parallel to that axis<br />06/01/2011<br />7<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
• 22. Using the same set of axes draw the planes with the following Miller indices:<br />(0 0 1) <br />(1 1 1)<br />06/01/2011<br />8<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
• 23. Using the same set of axes draw the planes with the following Miller indices:<br />(0 0 2) <br />(2 2 2)<br />NOW THINK!! What does this mean?<br />06/01/2011<br />9<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
• 24. Planes - conclusions 1<br />Miller indices define the orientation of the plane within the unit cell<br />The Miller Index defines a set of planes parallel to one another (remember the unit cell is a subset of the “infinite” crystal<br />(002) planes are parallel to (001) planes, and so on<br />06/01/2011<br />10<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
• 25. d-spacing formula<br />For orthogonal crystal systems (i.e. ===90) :-<br />For cubic crystals (special case of orthogonal) a=b=c :-<br />e.g. for (1 0 0) d = a<br /> (2 0 0) d = a/2<br /> (1 1 0) d = a/2 etc.<br />06/01/2011<br />11<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
• 26. A cubic crystal has a=5.2 Å (=0.52nm). Calculate the d-spacing of the (1 1 0) plane<br />A tetragonal crystal has a=4.7 Å, c=3.4 Å. Calculate the separation of the:<br />(1 0 0)<br />(0 0 1)<br />(1 1 1) planes<br />06/01/2011<br />12<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
• 27. Question 2 in handout:<br />If a = b = c = 8 Å, find d-spacings for planes with Miller indices (1 2 3)<br />Calculate the d-spacings for the same planes in a crystal with unit cell a = b = 7 Å, c = 9 Å.<br />Calculate the d-spacings for the same planes in a crystal with unit cell a = 7 Å, b = 8 Å, c = 9 Å.<br />06/01/2011<br />13<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
• 28. X-ray Diffraction<br />06/01/2011<br />14<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
• 29. Diffraction - an optical grating<br />Path difference XY between diffracted beams 1 and 2:<br />sin = XY/a<br /> XY = a sin <br />For 1 and 2 to be in phase and give constructive interference, XY = , 2, 3, 4…..n<br />so a sin  = n where n is the order of diffraction<br />06/01/2011<br />15<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
• 30. Consequences: maximum value of  for diffraction<br />sin = 1  a = <br />Realistically, sin &lt;1  a &gt; <br />So separation must be same order as, but greater than, wavelength of light.<br />Thus for diffraction from crystals:<br />Interatomic distances 0.1 - 2 Å<br />so  = 0.1 - 2 Å<br />X-rays, electrons, neutrons suitable<br />06/01/2011<br />16<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
• 31. Diffraction from crystals<br />X-ray Tube<br />Detector<br />?<br />06/01/2011<br />17<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
• 32. Beam 2 lags beam 1 by XYZ = 2d sin <br />so 2d sin  = nBragg’s Law<br />06/01/2011<br />18<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
• 33. e.g. X-rays with wavelength 1.54Å are reflected from planes with d=1.2Å. Calculate the Bragg angle, , for constructive interference.<br /> = 1.54 x 10-10 m, d = 1.2 x 10-10 m, =?<br />n=1 :  = 39.9°<br />n=2 : X (n/2d)&gt;1<br />2d sin  = n<br />We normally set n=1 and adjust Miller indices, to give <br />2dhkl sin  = <br />06/01/2011<br />19<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
• 34. Example of equivalence of the two forms of Bragg’s law:<br />Calculate  for =1.54 Å, cubic crystal, a=5Å<br />2d sin  = n<br />(1 0 0) reflection, d=5 Å<br />n=1, =8.86o<br />n=2, =17.93o<br />n=3, =27.52o<br />n=4, =38.02o<br />n=5, =50.35o<br />n=6, =67.52o<br />no reflection for n7<br />(2 0 0) reflection, d=2.5 Å<br />n=1, =17.93o<br />n=2, =38.02o<br />n=3, =67.52o<br />no reflection for n4<br />06/01/2011<br />20<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
• 35. Use Bragg’s law and the d-spacing equation to solve a wide variety of problems<br />2d sin  = n<br />or<br />2dhkl sin  = <br />06/01/2011<br />21<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
• 36. Combining Bragg and d-spacing equation<br />X-rays with wavelength 1.54 Å are “reflected” from the <br />(1 1 0) planes of a cubic crystal with unit cell a = 6 Å. Calculate the Bragg angle, , for all orders of reflection, n.<br />d = 4.24 Å<br />06/01/2011<br />22<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
• 37. d = 4.24 Å<br />n = 1 :  = 10.46°<br />n = 2 :  = 21.30°<br />n = 3 :  = 33.01°<br />n = 4 :  = 46.59°<br />n = 5 :  = 65.23°<br />= (1 1 0)<br />= (2 2 0)<br />= (3 3 0)<br />= (4 4 0)<br />= (5 5 0)<br />2dhkl sin  = <br />06/01/2011<br />23<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
• 38. Summary<br /><ul><li>We can imagine planes within a crystal
• 39. Each set of planes is uniquely identified by its Miller index (h k l)
• 40. We can calculate the separation, d, for each set of planes (h k l)
• 41. Crystals diffract radiation of a similar order of wavelength to the interatomicspacings
• 42. We model this diffraction by considering the “reflection” of radiation from planes - Bragg’s Law</li></ul>06/01/2011<br />24<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
• 43. THANK YOU<br />06/01/2011<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />25<br />