asgn2

Assignement 2 - Josef Buršík

1. The thickness of lead foil required to attenuate a Cu Ka X-ray beam of intensity 10⁸ photons/s down to 10 photons/s should be in my opinion 0.062 mm..

The main beam-stop for an unfiltered non-monochromated copper X-ray source is said to be usually much thicker than this value because it is necessery to filter off also the higher energy white radiation (l shorter then l of characteristic radiation) which is more penetraiting.

2. The laboratory powder diffraction patterns of organic compounds are often broadened due to sample transparency because the organic materials contain only light elements such as C, H, N, and O (these elements have lower number of electrons). For such transparent samples, the X-rays penetrate deeper into the sample than let say in case of ordinary ceramics powder. Their path out of the sample will be seen by the detector at a slightly smaller 2q angle and this contribute to the broadening of the refletion lines.

With a laboratory diffractometer configured for Bragg-Brentano geometry this problem may be overcome by measuring organic samples in form of thin layers.

The characteristic feature of synchrotron radiation that is particularly beneficial for powder diffraction studies of organic powder samples in my opinion is:

It is intense. 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 and highly monochromatic beam.This can enable measurements to be conducted at great speed and with superior statistics better counting statistics (better signal:noise ratio).
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.he nearly parallel beam, which obviates the need for focussing geometry with a flat plate sample. Since focussing geometry is inaccurate at low angles, this eliminates a significant source of low angle peak broadening found in conventional laboratory diffractometer data.
It has a smooth continuous spectrum. This gives a choice offering any single wavelength by the use of proper monochromators.

The two design features of synchrotron angle-dispersive powder diffractometers that enable this characteristic feature to be exploited in my opinion are

the so-called "double crystal" arrangement (the further improvement in resolution is a consequence of the second "analyser" crystal. This analyser crystal is set to diffract only X-ray photons of the same wavelength as those given by the monochromator (the "first crystal"), and will only diffract a narrow angular range of X-rays scattered by the sample.
Multiple Foil (Hart-Parrish) arrangement which enables choosing which part of the specimen (e.g. the surface layer of highly absorbing materials) we wish to study (and also in the case when the sample is in the form of the thin layer, we can ensure that the diffraction detected only comes from the sample and not the supporting medium).

3. Combining the equation E = 12.398 / l with the Bragg's law in terms of photon energy gives the energy-dispersive form: E d sinq = C

where C has the value 6.1993 keV. After that we can write

d = C/(E.sinq.

Putting the given values of E (20 and 100 keV) in this equation we obtain that d(1) = 5. 077 (A), and d(2) = 1.015 (A).

4. The basis for the calculation is the Bragg law 2 d sinq = l, (where q = q_M), which shall be fulfilled for difracted beam for both wavelengths (0.75 A, and 0.65A). When we calculate the angles q(1) andq(2) (using the d value for Si) we get values 6.8678° and 5.9485° for 0.75 A and 0.65 A wavelengths, respectively (e.g. q(1) = arcsin(0.75/2*3.136) = 6.86781°). The monochromator have to be rotated by 0.91925° so that the incident angle would decrease form 6.8678° to 5.9485°. (The difference is 0.91925° and the D(2q_M) = 1.8385°.)

5. The corresponding symmetry elements of symmetry operators given bellow in my opinion are:

-x, y, -z: a twofold rotation axes about the b-axis, i.e. about the line 0,y,0, going through the origin

1/2+x, 1/2+y, 1/2+z: point X(x,y,z) is moved to 1/2 + x in a-direction (x coordinate of a point X(x,y,z) changed to x+1/2), to 1/2 + y in b - direction (y-coordinate of a point X(x,y,z) changed to y+1/2), to 1/2 + z in c - direction (z-coordinate of a point X(x,y,z) changed to z+1/2). If the z - coordinate would changed to 1/2-z then the operator would be an n-glide plane perpendicular to the c-axis and passing through the point (0,0,1/2), i.e. the plane x,y,1/2 with translations 1/2 along both a and b. In the case 1/2+x, 1/2+y, 1/2+z the point is moved by the same distance (1/2 of given lattice parameter) in the all three dimensions without any reflection.

-x, -y, -z: a centre of inversion located at the origin (point (0, 0, 0))

-x, -y, 1/2+z: A 2₁ screw axis with a twofold rotation about the c-axis, passing through the origin, followed by a translation of 1/2 c along c

x, y, -z: a mirror plane perpendicular to the c-axis and passing through the origin, i.e. the plane
x,y,0

1/2+x, -y, z: an a-glide plane perpendicular to the b-axis and passing through the origin, i.e. the plane x,0,z with a translation 1/2 along a

1/2+x, -y, -z: a twofold rotation about the x-axis, followed by a 1/2 a translation along a, passing through the origin

1/2+x, 1/2+y, z: point X(x,y,z) is moved to 1/2 + x in a-direction (x coordinate of a point X(x,y,z) changed to x+1/2), to 1/2 + y in b - direction (y-coordinate of a point X(x,y,z) changed to y +1/2), z - coordinate remains unchanged. If the z - coordinate would changed to -z then the operator would be an n-glide plane perpendicular to the c-axis and passing through the origin, i.e. the plane x,y,0 with translations 1/2 along both a and b. But in the case 1/2+x,1/2+y,z it is done only the translation to 1/2+x,1/2+y along a and b, resp., without reflection.

1/2-x, 1/2-y, z: a twofold rotation about the z-axis, passing trough the point (1/4, 1/4, z)

x, 1/2-y, z: a mirror plane perpendicular to the b-axis and passing through the point (0, 1/2, 0), i.e. the plane x,1/2, z.