Part 1 - Structures of crystalline materials Flashcards
What is the definition of a lattice?
A lattice is a spatial pattern of points with equal and equally oriented surroundings. It is an abstract mathematical concept.
What is a motif?
A motif could be a small group of atoms, a molecule or a collection of several molecules. A motif corresponds to one lattice point. Alternatively called a basis.
What is a unit cell, and what are the rules for choice of unit cell?
A unit cell is a small unit that fully represents the whole structure upon periodic repition.
The rules that apply for choice of unit cell are:
1) Its rotation symmetry is the same as that of the lattice.
2) The edges and angles are as similar to each other as possible.
3) The number of right angles is maximised
4) The volume is minimsied.
If present, an inversion centre will be located at the origin.
What are fractional coordinates in a unit cell?
A fraction of the parameter lengths. Varies from 0 to 1. E.g. in a triclinic cell with a ≠ b ≠ c, fractional coordinates (0.2, 0.4, 0.6) would mean r = 0.2a + 0.4b + 0.6c.
How are directions in a lattice denoted?
Using [uvw]-notation.
u, v and w are points a line goes through from the origin in the [uvw]-direction. u, v and w have integer values, so a line going through (1/2,1,1/2) would be [121]. Also, [242] and [363] would also indicate the exact same direction.
How are a set of symmetrically equivalent lattice directions denoted?
In angle brackets, e.g. <100>. This would correspond to [100], [010], [001], [-100], [0-10] and [00-1] in a cubic lattice.
How are sets of parallel equidistant planes in a lattice denoted?
Using hkl-notation.
Starting with the plane that contains the origin, a hkl-set of equidistant planes divides the unit cell into h parts along the a-direction, k parts along the b-direction and l parts along the c-direction.
If the plane that faces the origin crosses a, b or c in the negative directions, the indices will have negative signs (denoted with a bar above).
How is a single plane in a crystal (such as a crystal face) denoted ?
Using Miller-indices, or (hkl)-notation.
Similar to set of parallell equidistant planes. A set of all equivalent orientations of such a plane is denoted by curly brackets, {hkl}.
What is a point group?
A set of symmetry operations where at least one point is fixed. The set forms a mathematical group.
What is the order of a point group?
Equal to the number of symmetry operations.
What are the elements of point symmetry in 3D?
Identiy, inversion centre, mirror plane, rotation axis and rotoinversion axis.
How can we mathematically describe a point symmetry operation?
For a point (x,y,z) going to (x’,y’,z’), we can describe this be:
r’ = Rr,
where r’ is the vector describing the new poistion, r is the vector describing the old position and R is a matrix representing the symmetry operation.
What is a crystal class?
A crystal class is equivalent to a crystallographic point group.
What are the seven crystal systems?
Cubic, tetreagonal, hexagonal, trigonal, orthorhombic, monoclinic and triclinic.
What is the minimum point group symmetry of the cubic system?
23.
How many Bravais lattices are there?
14
What are the symbols for the centerings of the different Bravais lattices?
P - primitive I - body centered (Innenzentrierung) F - face centered A/B/C - side-centered R - rhomohedral (only 7 trigonal space groups)
What does the Bravais lattice specify?
Translational symmetry
Does the terms trigonal, tetragonal and hexagonal refer to symmetry or to the lattice?
Symmetry.
Does rhombohedral refer to symmetry or the lattice?
Lattice
What is the difference between a rhombohedral structure expressed in a hexagonal lattice and a rhombohedral lattice?
In the rhombohedral lattice, it is a primitive unit cell formed as a rhombohedron. In the hexagonal lattice, the volume is three times as large, and contains three lattice points: (000), (1/3, 2/3, 2/3) and (2/3, 1/3, 1/3).
What is a Seitz operator, and how is it denoted?
A Seitz operator describes both the rotational and translational symmetry of a given symmetry element, e.g.:
(R|t) = Rr + t,
where R is the rotational matrix and t is the translation vector, which can describe glide planes or a screw axes.
What is a double glide plane?
A double glide plane is a glide plane in which the centering causes a situation where two alternative shift directions along a glide plane result in the same point.
Denoted by the symbol e, and only present in 5 of the space groups.
What is a diamond glide plane?
A diamond glide plane is a glide plane where the shift is 1/2 of the F- or I-centering vector.
For F-centered orthorhombic or cubic lattices this means 1/4a+1/4b for a plane in the c-direction.
For I-centered tetragonal and cubic lattice this means 1/4a+1/4b+1/4c.
In a crystal structure, what is the asymmetric unit?
The asymmetric unit contains only the atomic coordinates in the unit cell that are crystallographic unique.
For Wyckoff site labels, what is characteristic of the Wyckoff site with letter a?
The Wyckoff site with the letter a is the highest symmetry site.
What determines whether ccp or hcp is more stable?
Orbital symmetry.
What is the Jagodzinski-Wyckoff notation for stacking sequences?
It is denoting the stacking sequence between adjacent closest-packed layers according to whether the shift is hexagonal (spheres in layers above and below lie directly above each other) or cubic (spheres in layers above and below are not directly above each other).
Each layer that has a hexagonal shift is denoted h and for each layer that has a cubic shift is denoted c.
This is then summed up for the smallest unit cell to fully describe the structure, such as hhc for a structure with three layers needed to describe this.
What is the Ramsdell symbol?
The Ramsdell symbol gives the number of close-packed layers per unit cell specified with a letter H (hexagonal), R (rhombohedral), T (trigonal with P Bravais lattice) or C (cubic).
What type of voids are there in hcp and ccp structures?
Tetrahedral (2 per sphere) and octahedral (1 per sphere) holes
How many octahedral holes can be filled in hcp and ccp?
All holes can be filled.
How many tetrahedral holes can be filled in hcp and ccp?
In ccp both tetrahedral holes can be filled. In hcp only one can be filled (if two were filled they would be too close).
Describe NaCl in terms of closest-packing and filling of holes.
NaCl is ccp of Cl- with Na+ in all octahedral holes.
Could also be described the other way around.
Describe NiAs in terms of closest-packing and filling of holes.
NiAs is hcp of As2- with Ni2+ in all octahedral holes.
As is coordinated by six Ni in a trigonal prism.
