# Questions 33 - 38#

## Q33 Volume of unit cell#

Find the general formula for the volume of a unit cell using equation 30. Simplify the answer as far as possible.

## Q34 Bravais lattice#

Determine what Bravais lattices make $$\displaystyle n_2=\frac{\cos(\alpha)-\cos(\beta)\cos(\gamma)}{\sin(\gamma}=0$$, see eqn 29.

## Q35 Unit cell#

If a space group is monoclinic then $$\alpha = \gamma = 90^\text{o}$$ and $$\beta \ne 90^\text{o}$$ and the unit cell dimensions are $$a, b, c$$. By convention, $$\beta$$ is the angle between sides $$a$$ and $$c$$, see figure 20. Show that the bond distance between two atoms is the same using equation 21 as equation 34.

## Q36 Basis set for crystal#

(a) Write down a basis set to define the position of atoms in a tetragonal, orthorhombic or cubic crystal with sides $$a, b, c$$, and calculate the angle $$\theta$$ if point 4 is $$a/3$$ along the side.

(b) For a two-dimensional hexagonal structure such as graphite, as shown in the figure, the unit cell axis are at $$60^\text{o}$$. The axes can be defined with unit vectors $$\vec{u}$$ and $$\vec{v}$$. Calculate the lengths $$1-2, 1-3, 1-4, 1-5$$, and angles $$2-1-3, 2-1-4$$, and $$2-1-5$$.

Strategy: (b) The natural basis set should lie along the sides of the hexagonal unit cell and then this is labelled with vectors $$\vec u$$ and $$\vec v$$. If the basis set is written as $$(u, 0), (0, t)$$ then this would be an orthogonal set, but the angle between the vectors is $$60^\text{o}$$ not $$90^\text{o}$$ so this is cannot be right. It is better to transform the vectors into an orthogonal $$x-y$$ set using the transformation matrix described in the text; equation 31. As the structure is two dimensional, then the axis $$c$$ is zero and the matrix becomes two dimensional. Taking point 1 to be the origin, point 2 is at $$(3a, 3a)$$, 3 at $$(2a, 4a)$$, 4 at $$(1a, 4a)$$, and 5 at $$(4a, 1a)$$ in $$\vec u$$ and $$\vec v$$ unit vectors. This can be seen by counting the number of diamond shapes defined by the $$u-v$$ basis set needed to cross the hexagons to a given point.

figure 24. Geometry for a cube and hexagonal structure such as graphite.

## Q37 Tetrazine bond lengths and angles#

Using Python, repeat the tetrazine example in the text then calculate the $$\mathrm{C-N_2, N_1-N_3}$$ bond lengths and $$\mathrm{CN_1N_3}$$ bond angle.