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EXAFS Phase Shifts

EXAFS Phase Shifts

This short presentation motivates why the Fourier transform of the EXAFS chi(k) function always displays the near-neighbor peak at a value about 1/2 Angstrom shorter than the known structure.

Bruce Ravel

May 02, 2018
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  1. EXAFS Phase Shifts
    Bruce Ravel
    Synchrotron Science Group
    National Institute of Standards and Technology
    &
    Beamline for Materials Measurements
    National Synchrotron Light Source II
    May 1, 2018
    1 / 6
    EXAFS Phase Shifts

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  2. The EXAFS Phase Shift in Hematite
    Here is ˜
    χ(R) for hematite, Fe2
    O3
    .
    Hematite has a known crystal structure∗ with Fe in a six-coordinated oxygen
    octahedron. There are 3 near neighbor oxygen atoms at 1.95 ˚
    A and 3 others
    2.12 ˚
    A.
    Why is the first peak in ˜
    χ(R) at about 1.4 ˚
    A when the nearest
    neighbor is at 1.95 ˚
    A?
    2 / 6
    EXAFS Phase Shifts
    ∗R.L. Blake, R.E. Hessevick, T. Zoltai, L.W. Finger American
    Mineralogist 51 (1966) 123-129, Refinement of the hemtatite structure

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  3. EXAFS Equation
    Here’s the EXAFS equation:
    χ(k, Γ) =
    (NΓ
    S2
    0
    )FΓ
    (k)e−2σ2
    Γ
    k2
    e−2RΓ/λ(k)
    2 kR2
    Γ
    sin (2kRΓ
    + ΦΓ
    (k)) (1)
    χtheory
    (k) =
    Γ
    χ(k, Γ) (2)

    = R0,Γ
    + ∆RΓ
    (3)
    k = 2me
    (E0 − ∆E0
    )/ 2 ≈ (E0 − ∆E0
    )/3.81 (4)
    The oscillatory term is a function not of 2kR, but of 2kR + Φ(k).
    The integral that makes ˜
    χ(R) is usually done over 2k, i.e.
    ˜
    χ(R) = d(2k) kkw · χ(k). This makes ˜
    χ(R) look somewhat like a radial
    distribution function with peaks near sensible values of R (half-path-length),
    rather than 2R (full-path-length).
    3 / 6
    EXAFS Phase Shifts

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  4. Scattering Amplitues and Phase Shifts
    Remember that the complex scattering function (for which F(k) is the amplitude
    and Φ(k) is the phase) is structured and Z-dependent. Here are some
    representative examples for elements from different rows of the periodic table.

    (k) ΦΓ
    (k)
    Very heavy elements have a discontinuity in Φ(k), like Pb at about 5.5 ˚
    A−1.
    Lighter scatterers, like O and Fe, have fairly smooth phase functions.
    4 / 6
    EXAFS Phase Shifts
    These figures are from Matt Newville’s Fundamentals of X-ray
    Absorption Fine Structure

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  5. Examining the Phase Function
    The phase functions for the lighter elements are valued near 0 at k = 0 ˚
    A−1 and
    decrease to almost 20 at k = 20 ˚
    A−1. To some level of approximation, these
    phase functions can be described by a line of slope -1, i.e. Φ(k) ≈ −1 · k
    Using that crude approximation, the oscillatory term of the EXAFS equation is
    sin(2kR − k) = sin 2k · (R − 1
    2
    ) . When the integral is done over d(2k), the
    first peak in the resulting ˜
    χ(R) shows up around (R-1
    2
    ) ˚
    A.
    5 / 6
    EXAFS Phase Shifts

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  6. This is why the first peak is shifted inward
    Obviously, the approximation of Φ(k) as a straight line is inaccurate.
    The peak shift is not exactly 1
    2
    ˚
    A. And for heavier scatterers, the
    approximation is even worse.
    But this explains in a hand-waving sense why the peaks are shifted to
    lower R in ˜
    χ(R).
    This is yet another reason why ˜
    χ(R) is NOT a radial distribution
    function.
    6 / 6
    EXAFS Phase Shifts

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