{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Questions 1 - 3\n",
"\n",
"## Q1 Graphite lattice\n",
"In a teaching lab experiment to measure the lattice spacing of graphite, the angle at which the electrons diffract is measured as their accelerating voltage $V$ is changed. By the de-Broglie relationship, the electron's wavelength is $\\lambda = h/p$ where $p$ is the momentum. The relationship between kinetic energy and accelerating potential is $eV = p^2/(2m)$, hence, $p = 2meV$ where $e$ is the charge on the electron and $m$ its mass. Combining the wavelength and Bragg's law $n\\lambda = 2d\\sin(\\theta)$, produces the equation for the (first-order diffraction) inter-planar spacing $d$,\n",
"\n",
"$$\\displaystyle d=\\frac{h}{\\sin(\\theta)\\sqrt{8meV}}$$\n",
"\n",
"In the experiment, the angle $\\theta$ and the voltage are measured; all the other terms are constants. If the error on the angle is $\\sigma_\\theta$ and on the voltage $\\sigma_V$V, what is the variance on $d$? Express the result in terms of $d^2$.\n",
"\n",
"**Strategy:** Use equation 20, and differentiate in terms of $\\theta$ and $V$. Simplify and substitute for $d$.\n",
"\n",
"## Q2 Electrode potential\n",
"The electrode potential $E$ of a reaction $A_{ox} + ne^- \\to A^-_{red}$ is measured to obtain the ratio of activities of oxidized to reduced species, $Q = a_{red} /a_{ox}$ via the Nernst equation,\n",
"\n",
"$$\\displaystyle E=E^\\ominus -\\frac{RT}{nF}\\ln\\left(\\frac{a_{red}}{a_{ox}}\\right)$$\n",
"\n",
"(a) If the standard deviation of the measured potential is $\\sigma_E$ and $\\sigma_T$ on the temperature, calculate how these affect the measured ratio of activities. Simplify the answer.\n",
"\n",
"(b) At a given temperature, where will the standard deviation be smallest?\n",
"\n",
"## Q3 Molecular beam\n",
"Barlow (1989) gives an example of errors in a tracking chamber measured in cylindrical polar coordinates, and how they are related to errors in $x, y$, and $z$. This question examines a similar situation. In a molecular beam experiment, molecules are photo-dissociated and the image of the fragments, ions, or electrons is observed on a CCD camera at angle $\\theta$ and distance $r$ from the origin at the centre of the detector, figure 6. This is also in the $x-y$ plane. There is an error in both $\\theta$ and $r$ because the laser source has a certain size, and dissociation occurs from a finite volume in space, and because the molecules are themselves moving.\n",
"\n",
"Calculate the error in $x$ and $y$ on the surface of the CCD camera.\n",
"\n",
"The cylindrical polar coordinates have to be converted to Cartesians and the conversions are $x = r\\cos(\\theta), y = r\\sin(\\theta)$, and $z = z$. As the distance to the detector is large and fixed, assume that there is no error in $z$, and ignore this in the calculation.\n",
"\n",
"![Drawing](analysis-fig6.png)\n",
"\n",
"Figure 6. Geometry of a molecular beam photo-dissociation experiment and a sketch of an image as it might appear on a CCD camera. The oval shape is a consequence of the polarization of the absorption and rate of dissociation.\n",
"______\n",
"\n",
"**Strategy:** Write down the $G$ matrix, which is also the Jacobian transformation between Cartesian and polar coordinates (see Chapter 4.11), and multiply out the matrices."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
},
{
"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
}