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\date{}\begin{document}
NATURAL SCIENCES TRIPOS Part IB\\
NATURAL SCIENCES TRIPOS Part II (General)
\vskip 6.5mm
\hrule\vskip 4mm
Monday 1 June 2009~~~~~~~~09.00 to 12.00
\vskip 4mm\hrule \vskip 11mm
{\bf MATERIALS SCIENCE AND METALLURGY (2)}\\

\begin{description}

\item [\rm ] {\em Answer {\bf five} questions, not more than {\bf two} being chosen from any one section.

\item [\rm ] Write on {\bf one} side of the paper only, and begin your answer to each question at the top of a new sheet.

\item [\rm ] Where questions are divided into a number of parts, the {\bf approximate} fraction of credit allocated to each part is indicated by the percentage in the right hand margin.

\item [\rm ] Candidates using electronic calculators are advised to indicate clearly the sequence of steps in their working.  Appropriate credit can then be given for the intermediate stages, even if the final stage is incorrect.

\item [\rm ] The answer to {\bf each question} must be tied up {\bf separately}, with its own cover--sheet.

\item [\rm ] Write the relevant {\bf question number} in the square labelled  `Section'.  Also, on {\bf each} cover-sheet, list the numbers of {\bf all} questions attempted.}
\end{description}
\vskip 0.5cm
\begin{table}[htp]
\begin{center}
\begin{tabular*}{16cm}{lcl}
{\bf STATIONERY REQUIREMENTS} & ~~~~~& {\bf SPECIAL REQUIREMENTS} \\[3pt]
{\em Metric graph--paper} &  ~~~~~&{\em Data Book (supplied by Department)}\\[3pt]
{\em Blue cover sheet $\times$ 5 }& ~~~~~& {\em Approved calculator allowed }\\[3pt]
{\em A4 script paper} &  ~~~~~&\\[3pt]
{\em Rough work pad} & ~~~~~& \\[3pt]
{\em Tags} &  ~~~~~&\\[3pt]
\end{tabular*}
\end{center}
\end{table}%


\centerline {\boxit{\boxit{\vskip 0.1cm \parindent=0pt \begin{center}{\bf You may not start to read the questions\\ printed on the subsequent pages of this\\ question paper until instructed that you\\ may do so by the Invigilator}\end{center}\vskip 0.1cm}}}

\newpage \pagestyle{fancy}\fancyhf{}\fancyhead[C]{Page \thepage\ of 7}\renewcommand{\headrulewidth}{0pt}

~\vskip 1cm \centerline{\bf SECTION D}\vskip 1cm
{\bf 10.}
\begin{enumerate}[label=(\roman*),align=left]
\item   Explain why two samples fabricated from the same brittle material using an identical process can fail at different tensile loads. 	\hfill[15\%]

\item  Fifty samples were tested to failure in uniaxial tension by loading them in increments of 100\,g. The accumulated mass at which each sample failed is listed below.  Calculate the mean failure load.													\hfill[10\%]
\begin{table}[htp]
\begin{center}
\begin{tabular}{cc}
Accumulated mass / g & 	Number of samples which failed \\
700 & 0\\
800 & 	4\\
900	& 5\\
1000	& 7\\
1100	& 10\\
1200	& 11\\
1300	& 8\\
1400	& 4\\
1500	& 1\\
\end{tabular}
\end{center}
\label{default}
\end{table}%

\item  Make a Weibull plot of the data and calculate the Weibull modulus.	\hfill [40\%]

\item  Estimate the load for which the probability of failure is a) 0.1 and b) 0.001. Comment on the relative confidence in these estimates.		\hfill	[15\%]

\item  A new batch of samples is fabricated using the same processing method. They have the same cross sectional area,  but are three times longer. Explain whether the average tensile strength is likely to increase, decrease or remain the same. Estimate the median tensile failure load of these longer samples.		\hfill [20\%]

\end{enumerate}

\newpage\fancyfoot[R]{\bf TURN OVER}
{\bf 11.}

\begin{enumerate}[label=(\roman*),align=left]
\item   Explain what is meant by the {\em principal stresses} of a stress state.  Sketch the three Mohr's circles for a stress state with principal stresses $\sigma_1,~\sigma_2$ and $\sigma_3$, and indicate how these can be used to establish the maximum shear stress and the orientation of the plane(s) on which it acts.   \hfill[25\%]

\item  	An aluminium alloy drive shaft of an aircraft is designed as a solid cylinder of 20\,mm diameter. It is expected to be subjected to a torque of 100\,N\,m and an axial tensile force of 10\,kN.  Using a Mohr's circle, or otherwise, evaluate the two principal stresses acting in the surface of the shaft, and the angle between the larger of these and the axis of the shaft. \hfill[50\%]

\item   Stating any assumptions, calculate the minimum uniaxial yield stress required of the alloy if plastic deformation is to be avoided.\hfill[25\%]
 
\end{enumerate}
[The shear stress at the free surface of a solid cylinder, of radius $R$, generated by the application of a torque, $T$, is given by $\sigma_{\theta z}=RT/I_p$, where $I_p$ is the polar second moment of area given by $\pi R^4/2$.]
\vskip 1cm
{\bf 12.}
\begin{enumerate}[label=(\roman*),align=left]
\item  Explain what is meant by a bending moment, $M$, and show that the moment acting in the section of a beam bent to a curvature $\kappa$ is given by $M=\kappa EI$, where $E$ is the Young's modulus of the material and $I$ is the second moment of inertia of the section of the beam.\hfill[25\%]

\item  Show that the second moment of area of a rectangular section beam, of width $w$ and thickness $h$, is given by $I=wh^3/12$. \hfill[15\%]
\item  Derive an expression for the deflection of an end-loaded cantilever beam, as a function of distance along it.  Hence show that the vertical displacement half way along the beam is 5/16 of the deflection at the end. \hfill[30\%]
\item The component of an atomic force microscope which contacts the specimen is a cantilever made of silicon ($E = 150$\,GPa), with a width of 50\,$\mu$m, a thickness of 5\,$\mu$m and a length of 300\,$\mu$m.  The maximum deflection of the end of the cantilever is 20\,$\mu$m.  Calculate the force that must be applied to the end of the cantilever in order to generate this height change.  \hfill[30\%]
\end{enumerate}

\newpage\fancyfoot[R]{}~\vskip 1cm \centerline{\bf SECTION E}\vskip 1cm
{\bf 13.}
\begin{enumerate}[label=(\roman*),align=left]
\item  The first Brillouin zone for the cubic close--packed crystal structure is shown  below. How many faces does the shape have, and what are their  $\{$hkl$\}$ indices? Indicate the number of mirror planes, four--fold, three--fold and two--fold 
rotation axes. \hfill[20\%] 
\item  Label the wavevectors of the  positions L, X, W and K relative to the coordinates $k_x$, $k_y$ and $k_z$.\hfill[35\%] 
\item  Show that the distances $\Gamma$--L, L--K, K--W and W--$\Gamma$ have magnitudes in the  ratios $2\sqrt3:\sqrt6:\sqrt2:2\sqrt5$ . Sketch and account for the $E$--$k$ relationship for free--electron 
wavevectors in the first Brillouin zone for the circuit $\Gamma$--L--K--W--$\Gamma$, plotting the  circuit as a single horizontal axis, so that the magnitudes of the distances involved are to  scale, with $E(k)$ as ordinate. \hfill[45\%]
\end{enumerate}

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=9cm]{brillouin2.eps}
\end{center}
\end{figure} 
\newpage\fancyfoot[R]{\bf TURN OVER}
{\bf 14.}
\begin{enumerate}[label=(\roman*),align=left]
\item  In terms of flat band diagrams, differentiate between {\em simple metals}, 
 {\em  intrinsic semiconductors, extrinsic semiconductors} ($n$--type and $p$--type) and {\em insulators. } Indicate the Fermi level in each case. \hfill[30\%]

\item  The Fermi level of an extrinsic semiconductor is coincident with the donor levels  at 240\,K. Raising the temperature to 300\,K changes the occupation probability of the donor levels by 0.3. Determine where the Fermi level lies relative to the donor level at 300\,K.  	\hfill[30\%]

\item  The electron energy levels in a particular semiconductor are free--electron--like  with respect to the bottom of the band at $E = 0$, so that for $ E > 0$, the number of  occupied states takes the form:
\begin{equation}
n(E)=\frac{1}{2\pi^2}\biggl(\frac{2m}{\hbar}\biggr)^{\frac{3}{2}} E^{\frac{1}{2}}\exp\biggl(-\frac{E-E_F}{k_B T}\biggr) \nonumber
\end{equation}
Determine the most highly populated electron energy in the conduction 
band relative to $E =0$. Find the average electron energy. 
\hfill[40\%]
\end{enumerate}\vskip 1cm

{\bf 15.}
\begin{enumerate}[label=(\roman*),align=left]
\item  Explain what is meant by a {\em reciprocal lattice} and interpret the physical meaning of a reciprocal lattice vector. Express the reciprocal lattice vectors {\bf a$^*$}, {\bf b$^*$} and {\bf c$^*$} in  terms of the real space vectors {\bf a}, {\bf b} and {\bf c}. 	\hfill[15\%]

\item  Show that when a direction with real space indices [{\em uvw}] lies in a plane ({\em hkl}), 
\begin{equation}hu + kv+lw = 0 \nonumber\end{equation}
irrespective of the symmetry of the lattice. 	\hfill[20\%]
\item What are the angles in a hexagonal lattice between:  {\bf a$^*$} and {\bf b$^*$}; {\bf a$^*$} and {\bf c$^*$}; and  {\bf b$^*$} and {\bf c$^*$}? \hfill[15\%]
\item What are the angles   between:  {\bf a$^*$} and {\bf b$^*$}; {\bf a$^*$} and {\bf c$^*$}; and {\bf b$^*$} and {\bf c$^*$}; in a monoclinic lattice in which the obtuse angle is $\beta$? \hfill[15\%]
\item Determine the spacings of the (002), (110) and (012) planes in a 
monoclinic crystal with $a = 2.885$\,\AA, $b = 4.120$\,\AA, $c = 4.622$\,\AA\ and $\beta = 96.8^\circ$.\hfill[35\%] 
\end{enumerate}

\newpage\fancyfoot[R]{}~\vskip 1cm \centerline{\bf SECTION F}\vskip 1cm
{\bf 16.}

An alloy of composition 95Al--5Cu\,wt\%\ is solidified in a sand mould. Representative microstructures are shown below.

\begin{enumerate}[label=(\roman*),align=left]
\item Sketch the important features of the microstructures illustrated, indicating the phases present.			\hfill [25\%]

\item Explain the development of the microstructure during cooling from the liquid to room temperature.	\hfill [25\%]

\item Calculate the fraction of the alloy expected to have the eutectic composition if the casting conditions permit perfect mixing in the liquid but allow negligible diffusion in the solid. Explain the relevance of the result to the micrographs shown below. \vskip0.1mm \hfill[25\%]	
\item Explain the annealing treatment needed to homogenise the microstructures illustrated below. \hfill	[25\%]	
\item[][Inter-diffusivity of Al and Cu: $D = 5 \times 10^{-6}\exp(-Q/RT)\,{\rm m^2\, s^{-1}},~ Q = 120\, {\rm kJ\, mol^{-1}}$]
\end{enumerate}


\begin{figure}[htp]
\centering 
\subfigure{
\includegraphics[width=7.5cm]{micrographa.eps}}
\subfigure{
\includegraphics[width=7.5cm]{micrographb.eps}}
\end{figure}



\newpage ~\vskip 0.5cm
{\bf 17.} Answer {\bf ALL} of the following questions. [All questions carry equal marks]

\begin{enumerate}[label=(\alph*),align=left]
\item Using  energy level diagrams, show how an Ohmic contact can be formed 
between a metal and an $n$--type semiconductor. 

\item Why is the separation of the anodic and the cathodic reactions in the aqueous corrosion of steel larger than during its oxidation?

\item Explain why an interfacial polycondensation process would not be a good method for producing high strength nylon fibres.

\item Explain the basis of energy dispersive X--ray analysis as implemented in a scanning electron microscope.

\end{enumerate}

~\vskip 1cm
{\bf 18.} Answer {\bf ALL} of the following questions. [All questions carry equal marks]

\begin{enumerate}[label=(\alph*),align=left]
\item Explain the appearance of polymer spherulites under polarised light, in particular the ÒMaltese CrossÓ pattern and banding that can occur, in terms of the molecular structure of the polymer chains.

\item Explain the physical basis of infrared spectroscopy, commenting on the types of materials for which it is most suited.

\item Stating any assumptions, show that during elastic deformation the fractional change in volume is equal to the sum of the linear strains in three perpendicular directions. Hence prove that volume is conserved during elastic uniaxial--tension only when the Poisson's ratio is 0.5.

\item It is proposed to build a house on top of the trunk of a spruce tree which is 4\,m high and 20\,cm in diameter. Calculate the maximum load which could in theory be borne by the tree. Comment on your answer and explain why the real failure load is likely to be different.
\end{enumerate}


\vskip 2cm\centerline{\bf END OF PAPER}
\end{document}
