Assignment 2

Zhanhui Yuan

Answer:

1. From the mass-absorption law: I(l) / I_o(l) = exp{- (m / r)_l rx}
10 / 10⁸ = exp{- (230 x 11.3 x X)}, X = ln(10/10⁸) / (-230 x 11.3) = 0.0062cm = 0.062mm
So the thikness of lead foil for Cu is 0.062mm. According to the above result, when the thickness of lead foil is 0.062mm, the intensity of transmitted Cu Ka X-ray beam could be down to 10 photos/s. As the main beam-stop, it should reduce the transmitted X-ray beam as small as possible. According to the mass-absorption law, the only way of reducing transmitted X-ray intensity is increasing the thickness of the metal foil. Therefore, the main beam-stop for an unfiltered non-monochromated copper X-ray source is usually much thicker than above value.

2. I.
For transparent samples, such as organic compounds, the X-ray penetrate deeper into the sample than normals. The diffraction happens from the surface layer and below the surface. Although the Bragg angles of the scattered X-ray are the same in both cases, the diffractted X-ray from below the surface will be seen by the detector at a slightly smaller 2q angle than those from the surface layer. So the powder-diffraction patterns of organic compounds are often broadened due to sample transparancy.
This problem maybe overcome by coating the fine powder on a silicon single-crystal sample holder. However, the intensities of the diffracted peaks are much weaker due to the volume of diffracting material is much smaller although the peaks are much sharper. In a word, the best way to collect data is to exploit transmission geometry, which is ideal for organic materials with high transparency.
II.
There are five main features of synchrotron radiations:

• It is intense. This can enable measurements to be conducted at great speed and with superior statistics.
• It is highly collimated (divergence in the order of mrads). This results in less wastage of radiation in its passage through the optical components towards the sample, and greater eventual resolution in measurement due to its spatial precision.
• It has a smooth continuous spectrum. This gives a choice of conduction experiments with white radiation, or offering any single wavelength by the use of monochromators.
• It is horizontally polarised.
• It has a precise flashing time structure.

Synchrotron powder diffraction has superiority due to two propertiees of the synchrotron beam:

1. The great intensity of the beam arriving at the monochromator. This means that the monochromator can be made highly wavelength selective, i.e. we can afford to reject most of the incoming beam and still have a sufficiently intense monochromatic beam.
2. The nearly parallel beam, which obviates the need for focussing geometry with a flat plate sample. This is particularly beneficial for powder diffraction studies of organic powder samples since it eliminates a significant source of low angle peak broadening found in conventional laboratory diffractometer data.

For synchrotron angle-dispersive powder diffraction experiments, the wavelength is an experimental variable. Two design features need to be considered:
A synchrotron wavelength should not be equal to that of the standard copper X-ray tube used in the laboratory. Otherwise, it does not take full advantage of the range of wavelengths produce by the synchrotron.
Another point to consider is the possible existence of X-ray absorption edges in the sample. For most routine powder diffraction studies, it is strongly advisable to choose a wavelength that is either slightly above or well below any of the absorption edges of elements in the sample. In special cases, it may be necessary to choose a wavelengtn that is close to the absorption edge of a particular element in the sample so as to exploit the so-called anomalous scattering effect.

3. From Bragg's Law: nl = 2dsinq (for argument, n=1), d = l / 2sinq
When E = 12.398 / l ( E = 20 keV ), 2q = 3.5° , so d = 12.398 / 2Esin(q/2) = 12.398 / 2 x 20 x sin(3.5^o/2) = 10.15 Å
When E = 12.398 / l ( E = 100 keV ), 2q = 3.5° , so d = 12.398 / 2Esin(q/2) = 12.398 / 2 x 100 x sin(3.5^o/2) = 2.03 Å
So the range of d-spacing is between 2.03 - 10.15 Å.

4. From Bragg's Law: nl = 2dsinq (for argument, n=1),

sinq₁ = l/2d = 0.75/2x3.136  q₁ = 6.68  2q₁ = 13.72^o
sinq₂ = l/2d = 0.65/2x3.136  q₂ = 5.95  2q₂ = 11.90^o
2q = 2q₁ - 2q₂ = 13.72^o - 11.90^o = 1.82^o
Therefore, the monochromator has to be rotated by 1.82^{o if Si(111) planes are used.}

5.

 I. -x, y, -z

It is a twofold rotation axis parallel to the Y-direction and passing the origin (0,0,0).

 II. 1/2+x, 1/2+y, 1/2+z

It contains following operation: (1). Two-one screw axis operation with rotation by 180^o about the screw axis followed by thanslation one half unit along the a-axis direction, i.e. 1/2+x, -y, -z. (2). It is twofold rotation axis parallel to the Z-direction and passing the origin, i.e. -(1/2+x), y, -z. (3). A two-one screw axis operation with rotation by thanslation one half unit along the b-axis dirction, i.e. 1/2+x, 1/2+y, z. (4). Twofold rotation axis parallel to the Z-direction and passing the origin, i.e. -(1/2+x), -(1/2+y), z. (5). Two-one screw axis o peration with rotation by 180^o about the screw axis followed by thanslation one half unit along the c-axis direction, i.e. 1/2+x, 1/2+y, 1/2+z.

 III. -x, -y, -z

It is an inversion operation with combine a rotation with an inversion through a point on the axis of rotation.

 IV. -x, -y, 1/2+z

It is a two-one screw operation which is rotation by 180^o about the screw axis followed by a translation of one half of the repeat unit along the c-axis direction.

 V. x, y, -z

It is a mirror operation which is perpendicular to the conventional Cartesian Z-axis direction.

 VI. 1/2+x, -y, z

It contains two operations which are a two-one screw axis with rotation by 180^o about the screw axis followed by a translation one half unit along the a-axis direction and a mirror operation which is perpendicular to the conventional Cartesian Z-axis direction.

 VII. 1/2+x, -y, -z

It is two-one screw axis with rotation by 180^o about the screw axis followed by a translation of one half unit along the a-axis direction.

 VIII. 1/2+x, 1/2+y, z

It contains following operation: (1). Two-one screw axis operation with rotation by 180^o about the screw axis followed by thanslation one half unit along the a-axis direction, i.e. 1/2+x, -y, -z. (2). It is twofold rotation axis parallel to the conventional Cartesian Z-direction and passing the origin, i.e. -(1/2+x), y, -z. (3). A two-one screw axis operation with rotation by thanslati on one half unit along the b-axis direction, i.e. 1/2+x, 1/2+y, z.

 IX. 1/2-x, 1/2-y, z

It contains following operation:(1). It is twofold rotation axis parallel to the conventional Cartesian Y-direction and passing the origin, i.e. -x, y, -z. (2).Two-one screw axis operation wit h rotation by 180^o about the screw axis followed by thanslation one half unit along the a-axis direction, i.e. 1/2-x, -y, z. (3). It is twofold rotation axis parallel to the conventional Cartesian Y-direction and passing the origin, i.e. -(1/2-x), -y, -z. (4). A two-one screw axis with rotation by 180^o about the screw axis followed by a translation one half unit along the b-axis direction, i.e. 1/2-x, 1/2-y, z.

 X. x, 1/2-y, z

It includes two operations which are an inversion symmetry with combine a rotaiton with an inversion through a point on the axis of rotation and a two-one screw axis with rotation by 180^o about the screw axis followed by a translation of one half unit along the b-axis direction.