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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