Powder Diffraction at SSRL

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					Structural Analysis


      Apurva Mehta
Physics of Diffraction


                      X-ray Lens not
                        very good



                      Mathematically



                    Intersection of
         Ewald sphere with Reciprocal Lattice
outline

Information in a Diffraction pattern

Structure Solution

Refinement Methods

Pointers for Refinement quality data
      What does a diffraction
         pattern tell us?

 Peak Shape & Width:
         crystallite size
         Strain gradient
 Peak Positions:
         Phase identification
         Lattice symmetry
         Lattice expansion
 Peak Intensity:
         Structure solution
         Crystallite orientation
                                      niasin(a)
                         A(a) = S e                = FT(a)
Sample Diffraction            A = S cos(nf)
                                 + i sin(nf)

                    a
                        A=   Feiwt
                                  S    e
                                            i nf



                        A = S F(a)    e
                                       iwt + inf




                                     iwt + inf
                             F(a) e
                a
       iwt
                             F(a) eiwt + i3f
   e
                             F(a) eiwt + i2f
                                      iwt + if
 asin (a) = f                F(a) e
                                      iwt
  Laue’s Eq.                 F(a)e
Sample  Diffraction
 Diffraction Pattern ~ {FT(sample) } {FT(sample) }




         =                 x                         *

                                                      M
                                                      o
                                                      t
                                                      i
             Sample size                              f
                (S)              Infinite Periodic
                                    Lattice (P)      (M)
Sample  Diffraction
     FT(Sample) = FT((S x P)*M)
         Convolution theorem
   FT(Sample) = FT(S x P) x FT(M)
 FT(Sample) = (FT(S) * FT(P)) x FT(M)
FT(S)


        X

  


        Y
FT(P)





FT (S x P) = FT(S) * FT(P)




    *                  x


    =



                  y
     FT(M)




FT(sample) = FT(S x P) x FT(M)
           Along X direction




       x
                 
X
      What does a diffraction
         pattern tell us?

 Peak Shape & Width:
         crystallite size
         Strain gradient
 Peak Positions:
         Phase identification
         Lattice symmetry
         Lattice expansion
 Peak Intensity:
         Structure solution
         Crystallite orientation
Structure Solution

 Single Crystal                 Powder
                                    Due to small crystallite size
    Protein Structure               kinematic equations valid

    Sample with heavy Z            Many small molecule
     problems Due to                 structures obtained via
                                     synchrotron diffraction
       Absorption/extinction
        effects
                                    Peak overlap a problem – high
                                     resolution setup helps
    Mostly used in Resonance
     mode                           Much lower intensity – loss on
                                     super lattice peaks from small
       Site specific valence
                                     symmetry breaks. (Fourier
       Orbital ordering.            difference helps)
       Diffraction from Crystalline Solid

 Long range order ----> diffraction pattern periodic
 crystal rotates ----> diffraction pattern rotates




                                       Pink beam laue pattern
                                      Or intersection of a large
                                       Ewald Sphere with RL
From 4 crystallites
From Powder
Powder Pattern

         Loss of angular information
            Not a problem as peak
             position = fn(a, b & a )



         Peak Overlap :: A problem
            But can be useful for precise
             lattice parameter
             measurements
Peak Broadening

 ~ (invers.) “size” of the sample
  Crystallite size
  Domain size

  Strain & strain gradient


Diffractometer resolution should be better than
 Peak broadening But not much better.
  Diffractometer Resolution



Wd2 = M2 x fb2 + fs2
M= (2 tan q/tan qm -tan qa/ tan qm -1)

Where
    f = divergence of the incident beam,
      b
    f = cumulative divergences due to slits and apertures
      s
q, q and q = Bragg angle for the sample, analyzer and the monochromator
    a      m
                Powder Average
                 Single crystal – no intensity
Fixed 2q
                  Even if Bragg angle right,
                                                 Q +/- d(Q) = q +/- d(q)
                 But the incident angle wrong

 d(q)                 d(q) = Mosaic width ~ 0.001 – 0.01 deg
                      d(Q) = beam dvg ~ >0.1 deg for sealed tubes
                                      ~ 0.01- 0.001 deg for
                      synchrotron
   q                                         For Powder Avg
           2q            Need <3600 rnd crystallites – sealed tube
                         Need ~ 30000 rnd crystallites - synchrotron
                Q
                        Powder samples must be prepared carefully
                    And data must be collected while rocking the sample
Physics of Diffraction


             No X-ray Lens




             Mathematically
         Phase Problem

rxyz = Shkl Fhkl exp(-2pi{hx + ky + lz})
  Fhkl is a Complex quantity
  Fhkl(fi, ri): (Fhkl)2 = Ihkl/(K*Lp*Abs)
     = Shkl CIhkl exp(-(f + Df))
 rxyz
  Df = phase unknown
          Hence Inverse Modeling
Solution to Phase Problem

 Must be guessed
  And then refined.


How to guess?
  Heavy atom substitution, SAD or MAD
  Similarity to homologous compounds

  Patterson function or pair distribution analysis.
Procedure for
Refinement/Inverse Modeling

Measure peak positions:
  Obtain lattice symmetry and point group
     Guess the space group.
         Use all and compare via F-factor analysis
Guess the motif and its placement
  Phases for each hkl

Measure the peak widths
  Use an appropriate profile shape function

Construct a full diff. pattern and compare with
 measurements
Inverse Modeling Method 1
                                        7000


                                        6000




Reitveld Method
                                        5000




                     Data               4000




                            Intensity
                                        3000


                                        2000


                                        1000


                                           0

                                               40   60   80    100   120   140   160

                                                              q




          Profile
          shape                                                                  Refined Structure
Model

        Background
Inverse Modeling Method 2
                                                 7000


                                                 6000




Fourier Method
                                                 5000




                              Data               4000




                                     Intensity
                                                 3000


                                                 2000


                                                 1000


                                                    0

                                                        40   60   80    100   120   140   160

                                                                       q


                   subtract
                   Background

                    Profile
                    shape

             Integrated
             Intensities
                                                                                          Refined Structure
Model
          phases
Inverse Modeling Methods

 Rietveld Method                     Fourier Method
    More precise                        Picture of the real space
    Yields Statistically reliable       Shows “missing” atoms,
     uncertainties                        broken symmetry,
                                          positional disorder


Should iterate between Rietveld and Fourier.
   Be skeptical about the Fourier picture if Rietveld
    refinement does not significantly improve the fit with
    the “new” model.
           Need for High Q




            Many more reflections at higher Q.
Therefore, most of the structural information is at higher Q
Profile Shape function

Empirical
  Voigt function modified for axial divergence
    (Finger, Jephcoat, Cox)
    Refinable parameters – for crystallite size, strain
     gradient, etc…


From Fundamental Principles
   Collect data on Calibrant
  under the same conditions

Obtain accurate wavelength and
 diffractometer misalignment parameters
Obtain the initial values for the profile
 function (instrumental only parameters)
Refine polarization factor

Tells of other misalignment and problems
Selected list of Programs

CCP14 for a more complete list
  http://www.ccp14.ac.uk/mirror/want_to_do.html

  GSAS
  Fullprof
  DBW
  MAUD

  Topaz – not free - Bruker – fundamental
   approach
                    Structure of MnO
                         MnO @ 6530eV
            7000
                                                                                                  Scattering
            6000                                                                                  density
            5000
                                                                          structural fit
                                        2000

                                                                                               fMn(x,y,z,T,E)
                         In te n sity




            4000
Intensity




            3000                           0                                                   fO(x,y,z,T,E)
            2000
                                               72              74                   76

                                                                2q
            1000


               0


            -1000
                    40   60                         80   100        120       140        160

                                                         2q
                             Resonance Scattering
            Fhkl = Sxyz fxyz exp(2pi{hx + ky + lz})


            7000


            6000


            5000


            4000
Intensity




            3000


            2000


            1000


               0

                   40   60   80    100   120   140   160




                                                                fxyz = scattering density
                                  q




                                                           Away from absorption edge
                                                                   a electron density
  Anomalous Scattering Factors

fxyz = fe{fiexyzT} f = Thomson scattering for an electron
                                   e



fi = fi0(q) + fi’(E) + i fi”(E)
m(E) = E * fi”(E)
Kramers -Kronig :: fi’(E) <-> fi”(E)
                              
                          2
           f ' ( E0 ) =
                          p  0
                                  f " ( E ) E 2 E E 2 dE
                                                - 0
                             Resonance Scattering vs Xanes
                                                        7000


                                                        6000


                                                        5000


                                                        4000
                                            Intensity




                                                        3000


                                                        2000
                                                                                                                                   -2          MnO2
                             5000
                                                        1000
                                                                                                                                   -3
                             4000                            0
                                                                                                                                                      from resonance scattering
                                                                    40            60     80         100          120   140
                                                                                                                                   -4
                             3000                                                              2q
                 Intensity




                             2000                                                                                                  -5
                             1000
                                                                                                                                   -6
                                                                                                                                                                                                     9000

                               0                                                                                                                                                                     8000

                                                                                                                                   -7                                                                7000

                                    40                  60              80         100        120         140                                                                                        6000
                                                                                                                             f'
                                                                             2q
                                                                                                                                   -8                                                                5000




                                                                                                                                                                                         Intensity
                                                                                                                                                                                                     4000

                                                                                                                                                                                                     3000


                                                                                                                                   -9                                                                2000

                                                                                                                                                                                                     1000

                                                                                                                                                                                                       0
           1.0
                                                                                                                                  -10                                                                       40   60   80        100   120   140


                                                                                                                                           from KK transform of XANES                                                      2q

           0.8
                                                                                                                                  -11
           0.6
                                                                                                                                  -12
m = f"/E




           0.4
                                                                                                                                  -13
           0.2                                                                                                                          6440   6460   6480   6500   6520   6540   6560               6580         6600
           0.0                                                                                                                                                 Energy(eV)
                             6500        6520                    6540        6560        6580             6600

                                                                 Energy (eV)
XANE Spectra of Mn Oxides


                           0.8
                                                                                          Mn Valence
                                                                   MnO
                                                                                        MnO2    Mn
               bsorption




0.8
              A




                           0.4




                                                                   Mn3O4                        Mn(II)?
                           0.0
                                                                                        Mn2O3
                                 6540

                                        Energy(eV)
                                                     6570   6600

                                                                                                  Mn(II)?
0.4                                                                 Mn2O3
                                                                                                Mn(I)?
 Absorption




                                                                                        Mn3O4
                                                                      MnO2
                                                                                                  Mn(I)?
0.0                                                                                     MnO     Mn
                                               6540                              6560
                                                                                         Avg.    Actual
                                                                   Energy (eV)
                                                                      F’ for Mn Oxides
                                                                      -3
                                                     f' (electrons)




                                                                      -6
                                                                                                            M n 2O 3:1
                                                                                                            M n 2O 3:2
                            -6
f ' ( e le c t r o n s )




                                                                                                                         M nO   2

                            -8                                        M nO
                                                                      -9
                                                                                               M n 3O 4:2
                                                                                  M n 3O 4:1
                           -1 0
                                  C r o m e r -L ib e r m a n M n
                                           6530                            6540                  6550                6560
                                                                           6450                                  6500               6550
                                                                                  E n e rg y (e V )

                                                                                                                     Energy (eV)
Why Resonance Scattering?

Sensitive to a specific crystallographic
 phase. (e.g., can investigate FeO layer growing on
  metallic Fe.)
Sensitive to a specific crystallographic site
 in a phase. (e.g., can investigate the tetrahedral
  and the octahedral site of Mn3O4)
  Mn valences in Mn Oxides

•Mn valence of the two sites in
      Mn2O3 very similar
•Valence of the two Mn sites in
       Mn3O4 different but not as                                              CL
                                                                          0




                                                               d(f')/dE
       different as expected.
          -3
                                   Mn3O4
          -4


          -5


          -6
     f'




          -7

                           Mn1                                                6540   6544      6548        6552   6556
          -8               Mn2
                                                                                            X Axis Title
          -9   A. Mehta, A. Lawson, and J. Arthur

                   6460     6480      6500       6520   6540
                                   Energy (eV)

				
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posted:11/6/2012
language:English
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