Crystal structures Visualization of atomic...
Transcript of Crystal structures Visualization of atomic...
Crystal structures
1. Visualization of atomic structures (Tool task)For this task, we use the following geometry file format to set up atomic structures:
1 l a t t i c e v e c t o r 10 .0 0 .0 0 .0 # l a t t i c e vec to r one ( dimension i s [A])2 l a t t i c e v e c t o r 0 . 0 10 .0 0 .0 # l a t t i c e vec to r two3 l a t t i c e v e c t o r 0 . 0 0 .0 10 .0 # l a t t i c e vec to r three4 atom frac 0 .0 0 .0 0 .0 S i # r e l a t i v e coord ina te s with r e spec t to5 atom frac 0 .0 0 .5 0 .5 S i # l a t t i c e v e c t o r s and sp e c i e s type6 atom frac 0 .5 0 .0 0 .5 S i7 atom 5 .0 5 .0 0 .0 S i # . . or a b so l u t e coord ina te s in [A] and sp e c i e s type8 . . .
A typical program for visualizing atomic structures is VESTA, which is available on Windows,Linux and Mac OS X. It can read the above structure file directly, if it is stored as a file named“geometry.in”. Create files for the following structures and visualize them with VESTA. Each“geometry.in” file consists of three lattice vectors and at least one atom.
• simple cubic (sc) α−Polonium (alat = 3.35 A)
• face centered cubic (fcc) γ−Iron (Austenite) (alat = 3.68 A)Can you spot the primitive cell (unit cell with one atom in the basis) as noted in the class?
• body centered cubic (bcc) α−Iron (Ferrite) (alat = 2.86 A)Is there a way to write a primitive cell as was possible for fcc?
• diamond Carbon (Diamond) (alat = 3.57 A)Can you find a primitive cell consisting of only a two atoms basis?
• hexagonal closed packed (hcp) Gallium Nitride (alat = 3.19 A, clat = 5.19 A)
A link to VESTA and a HandsOn manual can be found on our website. For a more detaileddescription use the User’s Manual on the VESTA website. Crystallographic information for manyelements can be found online, e.g. www.webelements.com.
2. Ideal c over a ratio in the hexagonal closed packed (hcp) structure (Analytical task)By assuming hard spheres in a hcp structure, calculate the ideal ratio between the in-plane spacinga and the stacking distance of identical planes c. Is this ratio fulfilled for hcp crystals such as GaN(lattice parameters as given above)?
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• Crystal structures:In the following, the lattice constant for cubic systems is denoted by a. In case of the hexagonalclosed packed structure, there are two lattice parameters denoted by a and c.
1. The primitive cell of the simple cubic (sc) lattice is given by the three lattice vectors
~a1 = (a, 0, 0) ~a2 = (0, a, 0) ~a3 = (0, 0, a) (1)
with one atom in the cell located at the fractional coordinates ~b1 = (0, 0, 0).
2. The conventional cell of the face centered cubic (fcc) lattice is given by the three lattice vectors
~a1 = (a, 0, 0) ~a2 = (0, a, 0) ~a3 = (0, 0, a) (2)
with four atoms in the cell located at the fractional coordinates ~b1 = (0, 0, 0), ~b2 = (12, 1
2, 0),
~b3 = (12, 0, 1
2), and ~b4 = (0, 1
2, 1
2).
3. The conventional cell of the body centered cubic (bcc) lattice is given by the three latticevectors
~a1 = (a, 0, 0) ~a2 = (0, a, 0) ~a3 = (0, 0, a) (3)
with two atoms in the cell located at the fractional coordinates ~b1 = (0, 0, 0) and ~b2 = (12, 1
2, 1
2).
4. The conventional cell of the diamond lattice is given by the three lattice vectors
~a1 = (a, 0, 0) ~a2 = (0, a, 0) ~a3 = (0, 0, a) (4)
with eight atoms in the cell located at the fractional coordinates ~b1 = (0, 0, 0), ~b2 = (12, 1
2, 0),
~b3 = (12, 0, 1
2), ~b4 = (0, 1
2, 1
2), ~b5 = (1
4, 1
4, 1
4), ~b6 = (3
4, 3
4, 1
4), ~b7 = (3
4, 1
4, 3
4), and ~b8 = (1
4, 3
4, 3
4).
5. The primitive cell of the hexagonal closed packed (hcp) lattice is given by the three latticevectors
~a1 =a
2(1,−
√3, 0) ~a2 =
a
2(1,√
3, 0) ~a3 = (0, 0, c) (5)
with two atoms in the cell located at the fractional coordinates ~b1 = (0, 0, 0) and ~b2 = (13, 2
3, 1
2).
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