Powder Diffraction Assignments

Answers to Problem Set 1

  1. Lead foil is often used for the absorption of X-rays. Given (i) the equation for X-ray absorption, (ii) that m/r for lead at 1.54 Å is 230 cm2/g, and (iii) that the density of lead is 11.3 g/cm3, determine the thickness (in mm) of lead foil required to attenuate a Cu Ka X-ray beam of intensity 108 photons/s down to 10 photons/s. Explain why the main beam-stop for an unfiltered non-monochromated copper X-ray source would be much thicker than this value?

    Answer:

    I(l) / Io(l) = exp{- (m / r)l rx}
      =>   (10 photons/s) / (108 photons/s) = exp{- 230 cm2/g · 11.3 g/cm3 · x}
      =>   10-7 = exp{- 259.9 mm-1 · x}
      =>   x = 0.062 mm

    The main beam-stop for an unfiltered non-monochromated copper X-ray source would have to significantly attenuate not only Cu Ka line, but the whole spectrum of radiation. m/r for lead at other wavelengths may be much smaller than that for Cu Ka, so that one would have to rely exclusively on thickness of the filter as a factor of attenuation.♦

  2. Powder diffraction patterns of organic compounds are often broadened due to sample transparency. Briefly discuss how narrow peaks might be obtained (a) from a laboratory diffractometer configured for Bragg-Brentano geometry, and (b) from an angle-dispersive synchrotron powder diffractometer.

    Answer:

    Improvement of fineness of the sample is essential for obtaining narrow peaks. Then, the sample should be as thin as possible, in order to avoid unwanted transparency effects (this is especially valid for the case (a), Bragg-Brentano geometry). The things one generally should take care of are alignment of diffractometer axis, the size of the beam, and its parallelism.

    On the other hand, narrower peaks can be achieved also using transmission geometry, which is favourable for organic compounds due to their high sample transparency.

    In order to obtain as narrow peaks as possible, in the case of angle-scanning synchrotron powder diffractometer, apart from general remarks, one should work on improvement of monochromaticity of the beam. ♦

  3. Crystal monochromators exploit the fact that only planes in the Bragg condition diffract X-rays in the required direction. On synchrotron powder diffractometers, double-bounce monochromators are rotated about a 2q axis so as to tune the instrument to a particular wavelength. If the wavelength on the instrument has to be changed from 0.70 to 0.65 Å (e.g. to avoid the high absorption of a zirconium compound), by what angle would the monochromator have to be rotated if Si (111) planes are used? (The d-spacing for Si (111) planes is 3.136 Å.) Show your workings clearly in your answer.

    Answer:

    Monochromators diffract only the X-ray wavelength satisfying Bragg's Law (l = 2d sin q, for n = 1).

    For the first wavelength the equation will be

    0.70 Å = 2 · 3.136 Å sin q,

    and for the second

    0.65 Å = 2 · 3.136 Å sin (q+Dq),

    where q denotes the first angle position, and Dq wanted angle of rotation.

    This is equivalent to the next two equations:

    0.1116 = sin q
    0.1036 = sin (q+Dq)
    which give q = 6.408° and Dq = -0.461°.     ♦

  4. In energy dispersive diffraction, the equation relating an X-ray photon energy (E in keV) to its wavelength (l in Å) is:
    E = 12.398 / l
    What range of d-spacings can be measured for a detector designed to record X-ray photons with energies in the range 20-100 keV when the detector angle (2q) is set to 3.5°? Show your workings clearly in your answer.

    Answer:

    From Bragg's Law (l = 2d sin q, for n = 1) we have that

    d[Å] = 12.398 / (2 · E[keV] · sin 3.5°).
    Therefore, for E = 100[keV] is
    dmin = 1.015 Å,
    and for E = 20[keV] we have
    dmax = 5.077 Å.
    ♦

  5. For each symmetry operator listed below, state the corresponding symmetry element giving reasons for your answer:
    1. -x, y, -z
    2. 1/2+x, 1/2+y, 1/2+z
    3. -x, -y, -z
    4. -x, -y, 1/2+z
    5. x, y, -z
    6. 1/2+x, -y, z
    7. 1/2+x, -y, -z
    8. 1/2+x, 1/2+y, z
    9. 1/2-x, 1/2-y, z
    10. x, 1/2-y, z
    (You should note that none of the answers involves high-order, i.e. n > 2, symmetry axes.)

    Answer:

    No.Symmetry operatorSymmetry elementExplanation
    i.-x, y, -z 2 Rotation by 180° about Y-axis
    ii.1/2+x, 1/2+y, 1/2+z translation Vector of translation: t = 1/2a + 1/2b + 1/2c
    iii.-x, -y, -z -1 Inversion (by definition)
    iv.-x, -y, 1/2+z 21 Screw axis: Z
    v.x, y, -z m Mirror plane: XY
    vi.1/2+x, -y, z a Glide symmetry; mirror plane XZ & translation along X-axis
    vii.1/2+x, -y, -z 21 Screw axis: X
    viii.1/2+x, 1/2+y, z translation Vector of translation: t = 1/2a + 1/2b
    ix.1/2-x, 1/2-y, z 2 Rotation by 180° about (1/4,1/4,z)
    x.x, 1/2-y, z m Mirror plane which is parallel to the XZ-plane and passes through y=1/4
    ♦

Student: Siniša Prokic