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Assignment Problem Set 2
by Nicki Agron-Olshina

Problem 1

Given (i) the equation for X-ray absorption, I(l)/Io(l) = e-(m / r) (l) r x, (ii) that the (m / r) for lead at 1.54 Å is 230 cm2/g, and (iii) that the density of lead is 11.3 g/cm3, 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 is 0.062 mm.

3 Marks

The main beam-stop for an unfiltered non-monochromated copper X-ray source is usually much thicker than this value because it must be able to completely absorbe not only all the copper Ka but also the copper Kb X-rays which are of a shorter wavelength than the copper Ka and therefore more energitic.

2 Marks

Problem 2

The laboratory powder diffraction patterns of organic compounds are often broadened due to sample transparency. This ocurrs because the x-ray beam is able to penetrate deeper into the organic material because it contains primarily lighter elements, H, N. O and C. The deeper penetration of the x-ray beam results in a longer path length to the detector than the X-ray beam diffracted from the sample surface. The beams with longer path length are seen by the detector at slightly smaller 2q angles than those diffracted from the surface. This results in very broad peaks which appear to be shifted to lower scattering angles.

This problem can be overcome with a diffractometer configured for Bragg-Brentano geometry by mounting a thin coating of the finely ground organic powder on a zero background plate. Sample thickness is minimized and the peak broadening reduced giving sharper peaks and better quality data. The intensities of the diffracted peaks are much weaker for a sample mounted on a zero background plate due to the much smaller volume of diffracting material.

2 Marks

The highly collimated monochromatic synchrotron radiation is particularly beneficial for powder diffraction studies of organic powder samples. This nearly parallel beam obviates the need for focusing geometry with a flat plate sample. Since focussing geometry is inaccurate at low angles and many organic materials have low angle diffraction peaks, the highly collimated synchrotron radiation eliminates a significant source of low angle peak broadening found in conventional laboratory diffractometer data.

The two design features of synchrotron angle-dispersive powder diffractometers that exploit the high collimation of the beam are the Mulltiple Foil (Hart-Parrish) Mode and the Flat Plate Mode with or without an Analyser Crystal.

3 Marks

Problem 3

Given that 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

The range of d-spacings that 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° is 9.998 Å to 2.000 Å respectively.

1 Mark

Given nl=2dsinq we can substitute 2dsinq for l in the above equation getting;

E = 12.398 / 2dsinq
and by factoring out the 2 and putting in the value for q we get;
E = 6.199/d(sin1.75)

and substituting the value of sin1.75 in the equation we get;
d = 6.199/E(0.031).

By substituting the values for E (20 to 100 keV) in the above equation we get;

For E = 20keV
d = 6.199 / 20(0.031)
d = 6.199 /0.620
d = 9.998 Å.

Note the loss of precision by using only 2 significant figures for sin(1.75)


For E = 100 keV
d = 6.199 / 100(0.031)
d = 6.199 /3.100
d = 2.000 Å.

Note the loss of precision by using only 2 significant figures for sin(1.75)

3 Marks

Problem 4

A double-bounce monochromator (using Si (111) planes) on a synchrotron powder diffractometer would have to be rotated 1.8386° 2q to change the wavelength from 0.75 Å to 0.65 Å.

Given the d-spacing for Si(111) planes is 3.136Å and l 1 = 0.75 Å and l 2 = 0.65 Å
Using the equation nl=2dsinq and substituting the appropriate values we get:

For l 1
0.75 Å = 2(3.236 Å) sinq1
0.75 Å = 6.272 Å sinq1
0.75 Å / 6.272 Å = sinq1
0.11958 = sinq1
6.8678 = q1

For l 2
0.65 Å = 2(3.236 Å) sinq2
0.65 Å = 6.272 Å sinq2
0.65 Å / 6.272 Å = sinq2
0.10364 = sinq2
5.9485 = q2

Then q1 - q2 = 6.8678° - 5.9485° = 0.9193°
2q = 2(0.9193°) = 1.8386°

Comment: Don't confuse the axis name (which may be called either 2qM or qM) with the angle through which it is rotated. See model answer here.
4 Marks

Problem 5

For each symmetry operator listed below the corresponding symmetry element is given.
  1. -x,y,-z - A twofold rotation axis parallel to the y axis and passing through the origin and reversing the signs on the x- and z-coordinates.

    2. 1 Mark

  2. 1/2+x,1/2+y,1/2+z - A glide plane x,y,z with 1/2 added to the x-, y-, and z-coordinates (due to translation).

    3. 0 Marks

  3. -z,-y,-z - A point of inversion at the origin reversing the signs on the x-, y-, and z-coordinates.

    4. 1 Mark

  4. -x,-y,1/2+z - A 21 screw axis parallel to the Z-direction which passes through the origin and reverses the sign on the x- and y-coordinates and adds 1/2 to the z-coordinate (due to translation).

    5. 1 Mark

  5. x,y,-z - A mirror plane parallel to the x- and y- directions and passing through the origin and reversing the sign on the z-coordinate.

    6. 1 Mark

  6. 1/2+x,-y,z - A glide plane with its normal parallel to the Y-direction and passing through the origin, so that the sign of the y-coordinate is reversed (due to reflection) and 1/2 is added to the x-coordinate (due to translation).

    7. 1 Mark

  7. 1/2+x,-y,-z - A 21 screw axis parallel to the X-direction and passing through the origin reversing the signs on the y- and z-coordinates (due to reflection) and adding 1/2 to the x-coordinates (due to translation).

    8. 1 Mark

  8. 1/2+x,1/2+y,z - A glide plane parallel to the z-direction in which 1/2 is added to both the x- and y-coordinates (due to translation).

    9. 0 Marks

  9. 1/2-x,1/2-y,z - A twofold rotation axis parallel to the Z-direction and passing through a point located at (1/4,1/4,z).

    10. 1 Mark

  10. x,1/2-y,z - A glide plane parallel to the X- and Z-directions intersecting the Y axis at (x,1/4,z).

    11. You were close - it is a mirror plane
    0 Marks

See model answers for those where you scored zero. Otherwise answers fine.
0 Marks