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?

Given parameters

I(l) = 10 photons/sec

I₀(l)= 10⁸ photons/sec

(m/r)_1.54 = 230 cm²/g

r = 11.3 g/cc

Substituting these values in the equation,

I(l) / I_o(l) = exp{- (m / r)_l rx}

The final value obtained was 53 mm.

The much higher thickness than the above value for a beam stop in an unfiltered non-monochromated copper X-ray source could be due to absorb the high energy background radiation.

2. Powder diffraction patterns of organic compounds are often broadened due to sample transparency. Briefly discuss how narrow peaks might be obtained from a laboratory diffractometer configured for Bragg-Brentano geometry. Briefly explain how and why narrow peaks can be obtained with organic samples on synchrotron angle-dispersive powder diffractometers.

The best way to obtain the narrow peak in Bragg-Brentano geometry is to grind the sample into fine powder. Other alternative way, for organic compounds in particular, is to apply a fine coating of the powder on a Silicon single crystal.

However, the diffraction pattern produced by synchroton radiation is much narrower due to the two important properties of such radiations. Firstly, the synchroton radiation is highly intense leading to better signal:noise ratio. Secondly, the near parallel beam obtained with the synchroton radiations eliminates the low angle peak broadening found in conventional laboratory diffractometer.

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 0.75 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.

Given parameters

l = 0.65 Å

d = 3.136 Å

Substituting these values in the equation l = 2d sinq one can calculate q. From this the 2q can be calculated and found to be 11.9°. Thus the double bounce monochromator is to be rotated about an able 11.9° in order to use the Si (111) plane.

4. 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 2.7°. Show your workings clearly in your answer.

Given parameter,

Energy = 20 – 100 keV

2q = 2.7°

Using the relation

Ed sin q = C

Where C= 6.1993 keV Å. Substituting values of 20 keV and 100 keV in the above equation and 1.35 for q, the range of d-spacings have been found to be ranging from 2.631Å to 13.156 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, 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. -x, 1/2-y, z
10. x, 1/2-y, z

Ans:

1. two fold rotation along the y-axis
2. two fold rotation along the z-axis in a clock wise direction that does not pass through the origin
3. inversion
4. 2₁ screw axis along z-axis
5. reflection along xy plane
6. a glide plane along xz plane
7. 2₁ screw axis along X-axis
9. b glide plane along yz