Part-5+XRD(2)+theory(2018)+CP.ppt

Amplitude scattered by a single electron f hkl

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amplitude scattered by a single electron F hkl = Lattice F |F| 2 hkl SC f f 2 all BCC 0 0  h+k+l = odd number 2f 4f h+k+l = even number FCC 0 0  h, k, l   mixed 4f 16f 2 h, k, l   unmixed NaCl 4(f Na  - f Cl )  16(f Na  - f Cl 2                          h+k+l = odd number 4(f Na  + f Cl )  16(f Na  + f Cl 2   h+k+l = even number Diamond 0 0  h, k, l   mixed 4f(1- i ) 32f 2 h, k, l   unmixed   and  h+k+l = odd 0 0  h, k, l   unmixed   and (h+k+l)/2 = odd 8f 64f 2 h, k, l   unmixed   and (h+k+l)/2 = even Zinc-blende 0 0                        h, k, l   mixed 4(f Zn - i f S 16{(f Zn ) 2 + (f S ) 2 }                        h, k, l   unmixed  and  h+k+l = odd 4(f Zn -f S 16{(f Zn ) - 2f Zn f + (f S ) 2 }                               h, k, l   unmixed   and (h+k+l)/2 = odd 4(f Z +f S 16{(f Zn ) +2f Zn f + (f S ) 2 }                              h, k, l   unmixed   and (h+k+l)/2 = even (1) Structure Factors          j=1 N P. 131-144 Review 
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How does the x-ray scattering phenomena occur?  d This part considers superpositions of electric field induced by each e - . Z amplitude scattered by Z electrons in an atom amplitude scattered by a single electron f = f sin
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amplitude scattered by Z electrons in an atom amplitude scattered by a single electron f = The atomic scattering factor  f  = Z  for any atom in the forward direction (2  =  0)       As   increases   f  decreases   functional dependence of the decrease depends on  the details of the distribution of electrons around an atom (sometimes called the  form factor) f   is calculated from quantum mechanics Appendix 10 (page 634) f sin Cu 0  (29) Cu +1  (28) Cu +2  (27)
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Appendix 10 (page 634) 1.5 1.4 1.3 1. 1.1 1. .9 .8 .7 .6 .5 .4 .3 . .1 . 1 3 4 5 6 7 8 9 1 11 1 13 14 15 16 17 18 19  Sin   Atomic Scattering Factors C Si Na+ Cl- Ag(scale 1/3) Atomic scattering factor   YES: f~Z       for   ~ 0 or  forward scattering.   No : f< or << Z                  for   large angle scattering.
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2 d hkl sin = n (phase difference) Path-difference= 2 1 d h00 Miller integer 2 d hkl sin = n Scattering by a Unit Cell
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Need to sum up all waves scattered by  each plane of which phase is shifted by  Two waves case: E 1 =A 1 sin (2  t - 1 ) E 2 =A 2 sin (2  t - 2 ) ( x y z ) ( h k l ) a b c (h k l)  Plane u v w  Atom position
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Introducing Euler s notation e i cos i sin Mathematical procedure to add to two waves  (vector) E 1 =A 1 sin (2  t - 1 ) E 2 =A 2 sin (2  t - 2 ) A Ae i Re Im Ae i A cos i sin
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  • Fall '12
  • p
  • Diffraction, Neutron diffraction, X-ray scattering techniques

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