{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Questions 33 - 38"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Q33 Volume of unit cell\n",
"Find the general formula for the volume of a unit cell using equation 30. Simplify the answer as far as possible.\n",
"\n",
"## Q34 Bravais lattice\n",
"Determine what Bravais lattices make $\\displaystyle n_2=\\frac{\\cos(\\alpha)-\\cos(\\beta)\\cos(\\gamma)}{\\sin(\\gamma}=0$, see eqn 29.\n",
"\n",
"## Q35 Unit cell\n",
"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.\n",
"\n",
"## Q36 Basis set for crystal\n",
"(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.\n",
"\n",
"(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 $2-1-5$.\n",
"\n",
"**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 $(2a, 4a)$,\n",
"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.\n",
"\n",
"![Drawing](vectors-fig24.png)\n",
"\n",
"figure 24. Geometry for a cube and hexagonal structure such as graphite.\n",
"______\n",
"\n",
"## Q37 Tetrazine bond lengths and angles\n",
"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.\n",
"\n",
"## Q38 Recalculate Q37 using matrices."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3 (ipykernel)",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.9.6"
}
},
"nbformat": 4,
"nbformat_minor": 2
}