%%%%%%%%%% START: DO NOT EDIT THIS SECTION %%%%%%%%%%%%%%%%%%%%
\include{preamble}
%%%%%%%%%% END: DO NOT EDIT THIS SECTION %%%%%%%%%%%%%%%%%%%%

% Anything starting with a % is a comment, and ignored by LaTeX so it won't appear in your exam paper.

% Tell LaTeX this is the start of the document
\begin{document}

%%% START OF TITLE PAGE
\FrontPage{PHY00001I}
{
	UNIVERSITY OF YORK

	DEPARTMENT OF PHYSICS

	Second Year Examinations 2012

        LAGRANGIAN MECHANICS AND SOLID STATE PHYSICS

        % \textbf means "put text in bold face". Everything inside the curly braces will be emboldened.
        % Generally speaking LaTeX uses curly braces for all mandatory arguments to commands.
	\textbf{For all second year Physics students}

	Time allowed: THREE HOURS

        % a double backslash \\ means "insert linebreak" (i.e. go onto a new line)
	FOUR questions are to be answered\\
	(TWO questions from Section A and TWO questions from Section B)
        % there isn't enough vertical spacing here, so let's add 10mm vertical space:
        \vspace{10mm}

	\textbf{ALL STUDENTS}\\
        % \underline underlines everything in the curly braces.
	START EVERY QUESTION USING A \underline{\textbf{NEW}} ANSWER BOOKLET
}
{
\textbf{Materials supplied:}\\
Calculator\\
Tables of Constants and Mathematical Formulae
}
%% END OF TITLE PAGE


\Notice{Section A: Introduction to Solid State}

\Notice{Answer TWO questions (out of three)}

%% Each question starts with the command \Question
%% Sections (a), (b) etc. are called sub-questions, and start with the command \SubQuestion
%% Finer-grained sections, e.g. (a)(i), (a)(ii), are sub-sub-questions, and start with the command \SubSubQuestion
%% Thus the following starts question 1 (a) (i)
\Question \SubQuestion \SubSubQuestion Indicate the (200) and (111) planes on a sketch of a cubic crystal. 

% Marks for the present part of the question, regardless of whether it is a sub- or sub-sub-question. These marks are for Threshold
% material, so we use the \ThresholdMarks command. This command appends a star to the marks automatically when they are typeset.
\ThresholdMarks{2}

% Now move on to the next sub-sub-question, i.e. 1 (a) (ii)
\SubSubQuestion Determine the Miller indices for the plane in the figure below.\ThresholdMarks{2}

% Images are included using the "figure" environment, and may be PDF, JPG or PNG. PDF is preferred for
% diagrams as it allows a proper vector format. The option [h] tells LaTeX to put the figure right here; 
% if you don't specify it LaTeX will probably move it to a more aesthetically pleasing place!
%
% If your raw image is too large, you can use the "trim" option to select the region you want
% by specifying the left, bottom, right and top distances. In the example below the PDF image
% is quite large, and so the leftmost 1cm, lower 19cm, rightmost 6cm and top 1cm are trimmed away
% and the final trimmed image scaled to fill 0.4 of the width of a text column.
% You can rotate the image using a nonzero angle.
\begin{figure}[h]
\begin{center}
	\includegraphics[clip=true,trim=1cm 19cm 6cm 1cm,angle=0,width=0.4\columnwidth]{diagram.pdf}
\end{center}
\end{figure}

% A pair of dollar signs $ blah blah $ is used when including snippets of maths in the middle of ordinary text. LaTeX understands 
% about maths typesetting and has sensible default rules about how variables and numbers should appear, and everything in between 
% the dollar signs will be typeset using these rules. In particular variables should always be typeset in maths mode so that they 
% are appropriately italicised, but units are not supposed to be italicised -- in this example LaTeX doesn't know that "nm" is a 
% unit, not a variable, so we use \mathrm to tell it to typeset nm as part of the maths but in an ordinary Roman font (not italicised).
\SubSubQuestion Calculate the separation between the (111) planes of a cube with a lattice constant $a=0.4\mathrm{nm}$.\ThresholdMarks{2}

% Greek letters are available in maths mode as \alpha, \beta etc. for lowercase and \Alpha, \Beta etc. for uppercase.
% Common mathematical functions are available as e.g. \cos, \sec, \sin, \csc (for cosec), \ln, \log (base-10 log)
\SubQuestion \SubSubQuestion Show that Bragg's Law for diffraction is given by: $n \lambda = 2 d \sin \theta$ and indicate on a diagram what the different parameters represent. 

% Marks for the present part of the question, regardless of whether it is a sub- or sub-sub-question. These marks are *not* for Threshold material, 
% so we use the \Marks command.
\Marks{4}

% Remember the \mathrm here means typeset everything in the curly braces in a Roman font. The reason it's \mathrm not \textrm is because it's in a maths part,
% denoted by the pair of dollar signs.
%
% In maths mode you can use a caret ^{} to make everything inside the curly braces a superscript of whatever was before the underscore. If you don't use
% curly braces, only the following character will be treated as a superscript so x^{32} would appear as the 32nd power of x, but x^32 would appear as
% x cubed followed by a 2, probably not what you wanted...
%
% The special character \circ is a small circle, useful as a superscript for "degrees". 
% \times means "the multiplication symbol".
\SubSubQuestion An X-ray diffraction experiment is carried out on a body centred cubic crystal and the first reflection found at $21.66^\circ$. Calculate its lattice constant. The wavelength of the X-rays used is $1.54\times 10^{-10}\mathrm{m}$. \Marks{4}

\SubSubQuestionNote Here is a note that is a \textbf{sub-sub-question-note}. Notice that it has the same indentation as a sub-sub-question.

\SubQuestionNote Here is a note that is a \textbf{sub-question-note}. Notice that it has the same indentation as a sub-question.

% Normally LaTeX would split the next part of the question over two pages which isn't very good, so we tell it to start a new page right here
% using the \newpage command.
\newpage

% NB instead of using \\ to start a new line, you can use the \newline command
\SubQuestion Using the X-ray structure factor as given below, show that for a face centred cubic structure the conditions for allowed reflections are that $h$, $k$, and $l$ must be either all odd or all even.\newline
% In maths mode you can use an underscore, _{} to make everything inside the curly braces a subscript of whatever was before the underscore.
The X-ray structure factor $F_{hkl}(\bm{\Delta k})$ is given by
% To insert a whole line of maths we can use \[ to start maths mode, and \] to turn it off -- everything in between is treated as maths. 
% It doesn't matter how long the equation you type here is or how many lines it takes up in this LaTeX file, in the final PDF it will 
% all be put on a single line. This does mean it might not fit, so be careful... See later for how to do a multi-line equation.
%
% Note here we use \bm{...} to embolden the vectors, not \mathbf. This is because \mathbf{} switches off the italics and doesn't understand 
% all of the symbols you might want in an equation.
\[
	F_{hkl}(\bm{\Delta k}) = \sum_{j=1}^N f_j(\bm{\Delta k})\,e^{-i\bm{\Delta k}.{\bm r}_j}
\]
where
% You can put a hat on any section of maths using \hat{...}, useful for operators, unit vectors etc.
\[
	\bm{\Delta k} = \frac{2\pi n}{a}\left(h\hat{\bm x} + k \hat{\bm y} + l \hat{\bm z} \right)
\]
The face centred structure can be treated as a simple cubic unit cell with a 4 atom basis as follows:
% There's a bit too much space left here by default, so we tell LaTeX to move back up the page by 5mm.
\vspace{-5mm}
% Here we need lots of equations and we want to align all the equals signs. We do this using the equation array environment,
% which starts \begin{eqnarray*} and ends \end{eqnarray*}. Everything in between is typeset as maths. The \\ has the usual
% meaning of "start a new line here", and the pair of ampersands & ... & tells LaTeX what should be aligned between the 
% different lines, i.e. here all the = signs are aligned by LaTeX so they are directly under each other.
%
% Fractions are specified with the \frac{}{} command, the first argument is the numerator and the second the denominator,
% so here \frac{a}{2} means a/2.
%
% When you bracket a section of maths sometimes the brackets are smaller than the contents. You can use \left( and \right) 
% instead of just ( and ) to ensure LaTeX used appropriately sized brackets.
\begin{eqnarray*}
	\bm r_1 & = & 0 \\
	\bm r_2 & = & \frac{a}{2}\left(\hat{\bm x}+\hat{\bm y}\right) \\
	\bm r_3 & = & \frac{a}{2}\left(\hat{\bm y}+\hat{\bm z}\right) \\
	\bm r_4 & = & \frac{a}{2}\left(\hat{\bm z}+\hat{\bm x}\right)
\end{eqnarray*}
\Marks{6}

% Here's a sample table
\begin{table}[h]
% Switch to everything being centred
\begin{center}
% Specify a 3-column table with a left vertical line as a border, a left-aligned column, a separating line, a centred column,
% a separating line, a right-aligned column and finally a right vertical line. Removing a | removes a vertical line, and you
% add or subtract columns just by adding or removing a l, c or r as appropriate.
\begin{tabular}{|l|c|r|}
% Draw a horizontal line across the table, which will form the top border
\hline
% We want the title to go across the whole table, i.e. to span all 3 columns, and we want this to also have the left and
% right vertical lines. The \multicolumn{}{}{} command handles this, the 1st argument is the number of columns to span
% (doesn't have to be all of them), the 2nd is how to format it (just like the main table) and the 3rd argument is the
% actual text you want to appear.
\multicolumn{3}{|c|}{A sample table with three columns}\\
\multicolumn{3}{|c|}{(This text spans 3 columns and is centred)}\\
% Draw a horizontal line across the table
\hline
% An ampersand & tells LaTeX where the current column stops and the next one starts. When you want a new row just use
% the new line command \\
this column is  & this column is & and this column is\\
left-aligned    & centred        & right-aligned\\
% Draw a horizontal line across the table, which will form the bottom border
\hline
\end{tabular}
% Switch back to normal (left-aligned)
\end{center}
\end{table}

%% At the end of a question use the \EndOfQuestion command. This typesets the "Turn Over" instruction as well as the page number.
\EndOfQuestion

\newpage

\Question In an insulated metal rod, the temperature $T(x,t)$ obeys the diffusion equation below in which $x$ is the displacement along the bar, $D$ is the thermal diffusivity coefficient and $t$ is time.
% Here we use \partial to create the "curly d" of the partial derivative, and \frac{}{} to typeset the derivative properly as a fraction.
\[
	\frac{\partial^2 T}{\partial x^2} = \frac{1}{D} \frac{\partial T}{\partial t}
\]
Consider a rod of length $L$. The whole rod is heated so that a uniform temperature gradient exists along the bar increasing from 0$^\circ$C at $x=0$ to 200$^\circ$C at $x=L$. Then at time $t=0$ the rod is positioned in an insulating jacket and both its ends placed in contact with ice baths so that they are held at 0$^\circ$C.

\SubQuestion Prove that the partial differential equation has special solutions of the form 
\[
	T_n(x,t)=P_n \sin \left(\frac{n\pi x}{L}\right)\exp\left(-\frac{t}{\tau_n}\right)
\]
% When you use $ ... $ to put a piece of maths inside a paragraph, LaTeX typesets it slightly differently to ordinary text so as not
% to make the line too tall. Sometimes this is a bit ugly, usually when you're using fractions or summations e.g. \sum_{i=1}{i=N}.
% You can tell LaTeX not to worry about the line's height and typeset it as ordinary maths using \displaystyle
where $P_n$ is a constant, $n$ is an integer and $\tau_n$ is given by $\displaystyle \tau_n = \frac{1}{D}\left(\frac{L}{n\pi}\right)^2$
\Marks{8}

\SubQuestion Given that the temperature distribution along the bar at $t=0$ can be represented by the Fourier series
% Sometimes you want LaTeX to make maths just a little bit bigger -- use the \mathlarger{} command to enlarge everything in the
% curly braces. There's also a \mathsmaller{} command for when you want the opposite.
\[
	T(x,0) = -\frac{400}{x}\mathlarger{\mathlarger{\sum_n}
	} \frac{(-1)^n}{n} \sin \left(\frac{n\pi x}{L}\right)
\]
show that the full solution found via comparison with the general solution is
\[
	T(x,t) = -\frac{400}{x}\sum_{n=1} \frac{(-1)^n}{n} \sin \left(\frac{n\pi x}{L}\right) \exp\left(-\frac{t}{\tau_n}\right)
\]
\Marks{4}

\SubQuestion Using the first three terms in the series, find the approximate temperature of the rod at $\displaystyle x=\frac{L}{2}$ at 
time $\displaystyle t_m = \frac{L^2}{\pi^2D}$.
\Marks{4}

\SubQuestion If the rod has a length of 1m and is made from steel ($D = 1 \times 10^{-4}\mathrm{m^2 s^{-1}}$) use only the first non-zero term in the full solution to estimate the time taken for the middle of the rod to fall to 40$^\circ$C.
\Marks{4}

%% It is important to note the end of the examination paper so that the correct rubric can be inserted. There is no need to use \EndOfQuestion for the final question.
\EndOfPaper

% Tell LaTeX this is the end of the document
\end{document}

% Other useful stuff not illustrated so far...
%
% Maths symbols:
%
% \sum     a large capital sigma used for summations; upper and/or lower limits can be specified as super- and sub-scripts.
% \int     the integral sign; limits may be specified using super- and sub-scripts, just like \sum
% \iint    a double integral sign (looks better than two \int)
% \iiint   a triple integral sign (looks better than two \int)
% \oint    a loop integral sign
%
% \infty   the infinity symbol
% \neq     the "not equal to" symbol
% \approx  the "approximately equal to" symbol
% \ge      the "greater than or equal to" symbol
% \le      the "less than or equal to" symbol
