Karteikarten
ICMP Week 2 LNs
Karten in diesem Stapel (42)
- 1What is the aim of crystallography?to determine the crystal structure of a material of interest
under the relevant conditions of temperature, pressure or other parameters - 2State the following; Translation vectors in rec space & its kronecker delta identity rec lattice vectors and real space lattice planes relation interplanar spacing relations
- 3What is a reciprocal lattice vector with coord hkl perp to?perpendicular to the set of real-space lattice planes with
Miller indices (hkl) - 4For x ray diffraction explain.. local scattering strength how strongly the crystal diffracts xrays & what needs to be taken into accountIn an x-ray diffraction experiment, the x-rays are scattered by the electrons in the
crystal. The local scattering “strength” at position 𝒓 is thus given by the electron
density 𝑛(𝒓). We will calculate how strongly the crystal as a whole diffracts x-rays
in a particular direction by integrating over all volume elements in the crystal,
taking into account the local electron density and the phase differences between
the radiation scattered at different locations of the crystal. - 5Explain X ray diffraction in the classical sense
- 6Why doesnt wavelength change in x ray diffraction?Diffraction is a process that does not
change the energy of the x-ray photons, so their wavelength is also not changed
in the diffraction process. Hence |𝒌| = |𝒌|, but the outgoing wave will generally
be in a direction different from that of the incoming wave — 𝒌 and 𝒌 are not
parallel. With that, we can write down an expression for the scattering amplitude - 7What does the scattering amplitude take into account?takes into account the electron density in every volume element and the
phase differences. The scattering amplitude 𝐹 describes the ratio of the diffracted
and incoming electric field strength. - 8Do the following... Sketch X ray diffraction State the scattering amplitude relation State the change in wave vector relation
- 9Explain the terms in the relation for the scattering amplitude
- 10State the following; FT of electron density Fourier expansion of charge density Simplified FT
- 11State the following for Fourier Transforms; Boundary conditions (if FT is 0 or not) Laue Diffraction conditions Diffraction intensity and Fg relation
- 12Using the FT of electron density explain in detail how diffraction occurs only in directions for which the scattering vector is equal to the reciprocal lattice vectorThis yields the condition that diffraction occurs only in directions for which the
scattering vector equals a reciprocal lattice vector
In this way, one can associate every observed diffraction spot with a reciprocal
lattice vector 𝑮ℎ𝑘𝑙. The integer coordinates ℎ, 𝑘, 𝑙 of this vector are also the Miller
indices of a real-space lattice plane that is perpendicular to 𝑮, and from here it
can be shown that the Laue diffraction condition is equivalent to Bragg’s law
- 13State the fundamental limitation of x ray diffraction experimentsA fundamental limitation of x-ray diffraction experiments is that one measures
the diffraction intensities, not field amplitudes. It is therefore not possible to
reconstruct the electron density directly from the measured intensities. - 14How is the electron density distribution approximated?by a superposition of free atom (or ion) electron densities.
- 15Define the structure factorThe scattering strength or amplitude from
a single unit cell of the crystal is called structure factor - 16What is the advantage of the electron density distribution being periodic?it suffices to evaluate the
integral for the scattering from a single unit cell rather than integrating over the
whole crystal. The result simply scales with the number 𝑁 of
unit cells, or the volume of the crystal. - 17Describe the method in finding structure factors for a structure to test a model against experimental data. What is neglected here?require knowledge of the
electron density distribution. A pragmatic and in most cases sufficiently accurate
approximation is to use the superposition of the electron distributions of the free
atoms that constitute the crystal (➠ Appendix 3.C). The structure factor can now
be calculated using only the reduced coordinates for the atoms (labelled
with index 𝑗) in the unit cell. In this approximation, we neglect the redistribution
of the valence electrons due to chemical bonding, but it works sufficiently well in
practical crystal structure determinations - 18State; Structure Factor in terms of hkl Measured intensity in terms of structure factor
- 19For Lau Diffraction experiment; What was used? Modern uses Sketch single crystal x ray diffraction Bullet point the method
- 20For x ray diffraction experiments; What is used? What is required? Facility methodX-ray diffraction experiments are conducted routinely on diffractometers equipped
with conventional x-ray tubes. The Schools of Physics and Chemistry own and
operate several such machines. For some applications, such as protein crystallography or diffraction experiments at extreme conditions of pressure (hundreds
of kilobar), high-intensity and tightly-focussed x-ray beams are required. These
are available at synchrotron radiation sources. These facilities operate an electron storage ring, where electrons with kinetic energies of several GeV travel
in an approximate circle at nearly the speed of light. When they pass a section
with a rapidly oscillating magnetic field (an undulator or wiggler), the electrons
are forced into an oscillating, accelerated motion that causes the emission of
electromagnetic radiation, typically in the x-ray range. - 21For Bismuth; Describe the bright spots Finding positions of atoms in unit cell Missing hk combinationsThe are just rows of sharp, bright spots, which form essentially
a map of the reciprocal lattice points 𝑮ℎ𝑘𝑙 that were accessible in the experiment.
The assignment of the Miller indices (ℎ𝑘𝑙) to the rows of reflections is based on a
(somewhat sophisticated and computerised) trial-and-error procedure. This gives
the crystal lattice and lattice parameters. The positions of the atoms in the unit
cell can then be deduced (with substantial effort) from the relative intensities of
the reflections.
Careful inspection of the assigned Miller indices, (ℎ 𝑘 𝑙), shows that there are
various ℎ𝑘 combinations systematically missing. For example, there are no (1 4 𝑙)
reflections for any value of 𝑙. Such systematically missing reflections are known
as systematic absences - 22For Powder Diffraction experiments; Systematic Absences Method Source of allowed diffraction anglesSystematic absences can also be observed in powder diffraction experiments.
Here, the crystalline material is ground to a fine powder, with grain sizes in the
order of 1–10 µm. A monochromatic x-ray beam is directed at the sample, and
circular diffraction rings are observed on an area detector (now electronic, photographic plates were used originally). The rings originate from the superposition of
thousands or even millions of diffraction spots, each of them originating from an
individual crystal grain. Because the grains are randomly oriented, the equivalent
reflections from different grains fall on different positions along the same ring.
The “allowed” diffraction angles 2Θℎ𝑘𝑙 come from the discrete set of interplanar spacings 𝑑ℎ𝑘𝑙, via Bragg’s law. For further analysis, the diffraction intensity is
recorded as a function of diffraction angle 2Θ. - 23Sketch Monochromatic Powder X ray diffraction
- 24For Powder diffraction patterns; Why is there a fall off in intensity with increasing diffraction angle? State the relation and definition of the atomic form factor, what does it depend on?Powder diffraction patterns show an overall fall-off in intensity with increasing
diffraction angle. This is partially a geometric effect: With increasing diffraction
angle, the total diffracted intensity is spread out over a larger ring, thus reducing
the intensity measured at any point. A second factor is partially destructive
interference of x-rays scattered from different parts of the same atom. Going
back to the derivation of the structure factor, we can write down a definition
for the atomic form factor that shows its dependence on the scattering vector
𝑮, which relates to the diffraction angle and wavelength via Bragg’s law, 𝐺ℎ𝑘𝑙 =
4𝜋 sin(Θℎ𝑘𝑙)/𝜆. Note that the angle Θ used here is only half of the diffraction angle
2Θ measured in the diffraction experiment. As the free atoms considered here are
spherically symmetric, the atomic form factor depends only on the magnitude 𝐺
of the scattering vector. The expression for 𝑓 (𝐺) is clearly the Fourier transform
of the electron density distribution in the atom. The atomic form factor is also
known as the atomic scattering factor.
- 25State a difference from x ray and neutron diffractionFrom the point of view of data analysis, x-ray and neutron diffraction data are
very similar. An important difference is that neutrons are not scattered by the
extended electron clouds of the atoms, but instead by the effectively point-like
nuclei. This results in an atomic form factor that is independent of the scattering
angle, called neutron scattering length. As a result, neutron diffraction patterns
generally show less intensity fall-off at high diffraction angles.
- 26Describe the production of neutrons from a spallation neutron sourceIn a spallation neutron source, protons are accelerated to high particle energy and
then directed at a target made from a heavy element such as tungsten or mercury.
The collisions between protons and nuclei release neutrons that can be used for
diffraction and other experiments. Alternatively, neutrons can be produced in
nuclear reactors. The principle is the same as that of a nuclear power station, but
with operation at much lower power levels. - 27For an x ray diffraction experiment what defines the scattering vectorThe direction of the diffracted beam in space,
measured by two angles, and the wavelength of the diffracted radiation - 28For an x ray diffraction experiment, what defines the direction of the scattering vector?the line between the crystal and the illuminated point on
the detector - 29For conducting an x ray diffraction experiment on a crystal, explain in detail how the reciprocal lattice is found
- 30How is a reciprocal lattice generated in real space How are observed reflections labelled? State the relations for the reciprocal lattice translational vectors
- 31What do systematic absences tell us?something about the lattice centring and
potentially other symmetry properties of the crystal structure - 32Describe how systematic absences arise from BCC and FCC
- 33When considering the FCC reciprocal lattice points, why is the primitive translation vectors used?Otherwise, we would have more than one lattice
point per unit cell, which one cannot describe with the conventional translation
vectors alone - 34Derive the rec space translation vectors for FCC
- 35What is a similar structure to the rec translation vectors of an fcc lattice? Sketch this
- 36Describe the introduction of the concept of small angle scattering (x ray and neutron)
- 37What does the atomic form factor describe for a single atom?the
strength and angular dependence of the diffraction signal from a single atom. - 38What does the form factor depend on for microscopic objects?depends on the shape, size and material of the diffraction object. This make it
possible, for example, to probe the average size and shape of the nanoparticles
in a catalyst sample, so as to correlate these with their functional properties as
catalysts - 39Why was fivefold symmetry considered impossible in periodic crystals?There is no 3D crystal system that features a fivefold rotation axis. One can argue
that it is not possible to fill a plane with regular pentagons without leaving gaps.
As a crystal structure cannot have gaps, there should be no periodic crystals with
fivefold symmetry - 40Define the characteristics of a QuasicrystalQuasicrystals are characterised by a an ordered arrangement of atoms that is not
periodic. Penrose tilings are an example how a plane can be covered with two
types of building block in an ordered but non-periodic fashion. One can imagine
that each of these building blocks contains a set of atoms, and that 3D space can
be filled in a similar fashion. Note how the Penrose tiling contains local features
with fivefold rotation symmetry, but that this is neither the case for the tiles
themselves nor for the mosaic as a whole. - 41What is the origin of crystal binding?originates from electrostatic interactions between the atoms (or
ions) and their electrons. - 42Describe in detail the 4 idealised types of crystal binding Describe the cohesive energy of a crystal
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- Year 4 ICMP Karteikarten am ICMP Week 2 LNs, erstellt von R S C am 19/06/2024.
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