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Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
Pend Fisika Zat Padat (4) indexing
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Pend Fisika Zat Padat (4) indexing

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  • 1. Pertemuan 4CRYSTAL INDEXING
    IwanSugihartono, M.Si
    JurusanFisika, FMIPA
    UniversitasNegeri Jakarta
    1
  • 2. Crystals
    06/01/2011
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
    2
    • 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
  • Objectives
    By the end of this section you should:
    • 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
    06/01/2011
    3
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 18. 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
    06/01/2011
    4
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 19. 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]
    06/01/2011
    5
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 20. Exercise - What is the Miller index of the plane below?
    Find intercepts on a,b,c:
    Take reciprocals
    Multiply up to integers:
    06/01/2011
    6
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 21. 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
    06/01/2011
    7
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 22. Using the same set of axes draw the planes with the following Miller indices:
    (0 0 1)
    (1 1 1)
    06/01/2011
    8
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 23. 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?
    06/01/2011
    9
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 24. 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
    06/01/2011
    10
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 25. 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.
    06/01/2011
    11
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 26. 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
    06/01/2011
    12
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 27. 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.
    06/01/2011
    13
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 28. X-ray Diffraction
    06/01/2011
    14
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 29. 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
    06/01/2011
    15
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 30. 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
    06/01/2011
    16
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 31. Diffraction from crystals
    X-ray Tube
    Detector
    ?
    06/01/2011
    17
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 32. Beam 2 lags beam 1 by XYZ = 2d sin 
    so 2d sin  = nBragg’s Law
    06/01/2011
    18
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 33. 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  = 
    06/01/2011
    19
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 34. 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
    06/01/2011
    20
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 35. Use Bragg’s law and the d-spacing equation to solve a wide variety of problems
    2d sin  = n
    or
    2dhkl sin  = 
    06/01/2011
    21
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 36. 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
    06/01/2011
    22
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 37. 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  = 
    06/01/2011
    23
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 38. Summary
    • 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
    06/01/2011
    24
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 43. THANK YOU
    06/01/2011
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
    25

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