The optimum thickness, x of the filter can be determined from the mass-absorption law:

where (m/r) is the mass absorption coefficient at the wavelength l, r is the density of the material, which for nickel metal is 8.92 g/cm³, and I(l) and I_o(l) are the transmitted and incident X-ray intensities, respectively.

An unfiltered, non-monochromated copper X-ray source would contain a much higher amount of high energy background radiation. This radiation would be much more penetrating and so the main beam stop would need to be much thicker than 0.062mm.

(A)
For organic materials the X-rays penetrate deeper into the sample than normal (they are less absorbing) . Each reflection peak will be broadened since reflections from the bottom of the sample will be seen by the detector at a slightly smaller 2q angle. In Bragg-Brentano geometry, the sample is rotated so that the incident X-ray angle is always the same as the angle of the detector to the sample surface. As 2q angle increases, the X-ray penetration increases and there is an increased broadening of organic reflections.

The situation can be improved by depositing the organic material as a fine powder coating on a silicon single-crystal sample holder. The 'thin' nature of the sample produces sharper peaks since the thickness effect is reduced. However, with this set-up there could be more preferred orientation due to the way the organic crystals lie on the substrate, and the overall peak intensities will be lower due to greater sampling of the substrate.

(B)
The synchrotron concentrates the X-ray intensity into a well defined near-parallel X-ray beam (the how). The monochromator can, therefore, be highly wavelength selective and still produce an intense beam (even though most of the incoming beam is rejected). With a reduced spread of wavelength hitting the sample, the range of angle representing a given d-spacing will also be reduced (from 2dsinq =nq)(the why). The result is that peaks are sharper i.e. better resolution).

The organic sample could be put into a glass capillary and the entire 2q range scanned using the Debye-Scherrer set-up (the how). Since the X-rays bathe all the sample there is no sampling of the substrate - although some scattering from the glass capillary is seen in the diffractogram.

20KeV photons l=12.398/20 = 0.6199Å

100KeV photons l=12.398/100 = 0.1240Å

Now 2dsinq = nl

(assume n=1), also 2q=2.7° therefore q =1.35°

For l = 0.6199Å  d = 0.6199 / 2sin1.35 = 13.2Å
For l = 0.1240Å  d = 0.1240 / 2sin1.35 = 2.6Å

The range of d-spacings is 2.6 to 13.2Å

(You should note that none of the answers involves high-order, i.e. n > 2, symmetry axes.)

[10 Marks]

1. –x, y, –z

Symmetry Element: Two-fold rotation axis parallel to the Y direction

Reason: The 180° rotation about the origin (o) in the plane of the screen causes the x and z values to invert. The y value is unaffected.



The rotation causes A to go to B and C to go to D.

2. 1/2+x, 1/2+y, z

Symmetry Element: Translation (diagonal)

Reason: A translation of half of a repeat unit in both the X and Y directions.


3. –x, –y, –z

Symmetry Element: Point of inversion at the origin of the X,Y,Z

Reason: All points are inverted

New x = – (old x)
New y = – (old y)
New z = – (old z)
4. –x, –y, 1/2+z

Symmetry Element: A two-one screw axis parallel to the Z-direction

Reason: The two-fold (180°) about the Z-axis inverts the x and y values (similar to (i)). This is followed by a translation in the Z-direction of one half the repeat unit.

5. x, y, –z

Symmetry Element: Mirror plane perpendicular to the Z-axis and cutting the Z-axis at z=0

Reason: Similar to a mirror plane facing this page. z=0 is assigned to the position of the plane above the page. The x and y values remain unchanged and the Z values are inverted

6. 1/2+x, –y, z

Symmetry Element: Glide plane that is parallel to the X-direction and perpendicular to the Y-direction (passing through y=0).

Reason: The Y co-ordinate is reversed (due to reflection) and one half of the repeat unit is added to the X co-ordinate (due to translation). The Z values are unaffected.

7. 1/2+x, –y, –z

Symmetry Element: A two-one screw axis parallel to the X-direction (similar to (iv)).

Reason: The two-fold (180°) rotation about the X-axis inverts the y and z values (similar to (i)). This is followed by a translation in the X-direction of one half of the repeat unit.

8. 1/2–x, 1/2–y, –z

Symmetry Element: A glide plane parallel to the Z-direction due to a two-fold rotation axis at ¼, ¼, z and a mirror plane perpendicular to the rotation axis, passing through Z=0 (but without translation after the reflection).

Reason: An atom at x,y,z goes to 2T_x – x, 2T_Y – y, z

Therefore

2T_x – x = ½ – x   which gives   T_x = ¼ 2T_Y – y = ½ – y  which gives  T_Y = ¼

A two-fold rotation axis at ¼, ¼, z will have the symmetry operator ½ – x, ½ – y, z.

A mirror plane perpendicular to the Z-direction and passing through z=0 will invert the z values to produce the symmetry operator ½ – x, ½ – y, – z.

9. –x, 1/2–y, z

Symmetry Element: Two-fold rotation axis at 0, ¼, z parallel to the Z-direction.

Reason:  From viii), for a rotation axis with a non-zero origin, an atom at x, y, z goes to 2T_x – x, 2T_y – y, z

Therefore,
2T_x – x = – x which gives T_x=0
2T_y – y = ½ – y which gives T_y=¼

A two-fold rotation axis at 0, ¼, z has the symmetry operator – x, ½ – y, z

10. x, 1/2–y, z

Symmetry Element: A glide plane due to a two-fold rotation at 0, ¼, z and reflection from a mirror plane perpendicular to the X-direction (No translation along the Z-direction).

Reason: The two-fold rotation is the same as that in (ix) with a symmetry operator of – x, ½–y, z. In this case the mirror plane in perpendicular to the X-direction (passing through x=0). The x-values are inverted due to reflection.