1 Copper (Cu) is a metal that crystallises in the face centred cubic (fcc) crystal structure with a lattice constant of 3.61 . It is one of the most important elements on the planet due to its high electrical conductivity and excellent mechanical properties. a) Imagine that you are doing an x-ray diffraction experiment using a monochromatic beam of x-rays with a photon energy of 5 keV. Calculate the Bragg angles of all the diffraction peaks that you will observe from a copper crystal in this experiment. [5] b) Calculate the 'distance' from the centre of the first Brillouin zone to the closest Brillouin zone bounding plane, and the next closest bounding plane (i.e. the Brillouin zone faces) for copper. What photon energies (in eV) would these two distances correspond to if they were wavevectors, and to what two diffraction planes (Miller indexes) do these two photon energies correspond? [5] c) If each copper atom in the crystal contributes a single free electron, calculate the Fermi energy (in eV) of copper and show that the Fermi sphere fits entirely within the first Brillouin zone. [5] d) Imagine that you make an ultrathin film of copper such that it can be considered a 2D metal crystal. Show that the density of electron states g(?) in such a 2D metal is independent of the electron energy ? . [5] e) Imagine that you cleave or polish a single crystal of copper to expose the Cu(110) surface. Using an Ewald sphere construction, work out the number and arrangement of diffraction spots that will be observed in a low energy electron diffraction (LEED) experiment performed on this surface with an electron energy of 37 eV. [5]
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The energy of the X-rays is given by E = hc/λ, where h is Planck's constant and c is the speed of light. From this, we can calculate the wavelength of the X-rays as λ = hc/E = 6.626 x 10^-34 Js x 3 x 10^8 m/s / 5 x 10^3 eV = 2.48 x 10^-10 m. The interplanar Show more…
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Nickel and copper both have FCC crystal structures. According to the equilibrium phase diagram, Ni and Cu form a solid solution over the entire concentration range (that is, they are completely miscible), and the lattice parameter of the solid solution changes with concentration as shown in the first figure below. An X-ray diffraction pattern from an unknown Cu-Ni alloy, acquired using Zn Ka1 radiation (Η = 1.436 ), is presented in the second figure. To solve this problem, please use the following data: MNi = 58.7 g/mol MCu = 63.5 g/mol ρalloy = 8.925 g/cm³ (μ/ρ)Ni = 325 cm²/g (μ/ρ)Cu = 42 cm²/g Please also use the following approximation: (μ/ρ)Alloy = xNi(μ/ρ)Ni + xCu (μ/ρ)Cu, where x is mole fraction. a. Solve the diffraction pattern below. b. Determine the lattice parameter of the alloy using the relationship between the lattice parameter and (cos2θ)/(sinθ) (hint: plot a vs. (cos2θ)/(sinθ), and then extraction to -1.0) c. Estimate the concentration of the alloy. d. Calculate the mass absorption coefficient of the alloy from the data given above for λ = 1.436 .
Sri K.
Neutron diffraction. Consider a beam of neutrons scattering from a crystal (Figure 10.14 ). The interaction between neutrons and the nuclei in the crystal is short ranged, and can be approximated as $$V(mathbf{r})=frac{2 pi hbar^{2} b}{m} sum_{i} delta^{3}left(mathbf{r}-mathbf{r}_{i} ight)$$ where the $mathbf{r}_{i}$ are the locations of the nuclei and the strength of the potential is expressed in terms of the nuclear scattering length $b$ (a) In the first Born approximation, show that $$frac{d sigma}{d Omega}=b^{2}left|sum_{i} e^{-i mathbf{q} cdot mathbf{r}_{i}} ight|^{2}$$ where $mathbf{q} equiv k-k^{prime}$ (b) Now consider the case where the nuclei are arranged on a cubic lattice with spacing $a .$ Take the positions to be $$mathbf{r}_{i}=l a hat{i}+m a hat{j}+n a hat{k}$$ where $l, m,$ and $n$ all range from 0 to $N-1,$ so there are a total of $N^{3}$ nuclei. Show that $$frac{d sigma}{d Omega}=b^{2} frac{sin ^{2}left(N q_{x} a / 2 ight)}{sin ^{2}left(q_{x} a / 2 ight)} frac{sin ^{2}left(N q_{y} a / 2 ight)}{sin ^{2}left(q_{y} a / 2 ight)} frac{sin ^{2}left(N q_{z} a / 2 ight)}{sin ^{2}left(q_{z} a / 2 ight)}$$ (c) Plot $$frac{1}{N} frac{sin ^{2}left(N q_{x} a / 2 ight)}{sin ^{2}left(q_{x} a / 2 ight)}$$ as a function of $q_{x} a$ for several values of $N(N=1,5,10)$ to show that the function describes a series of peaks that become progressively sharper as $N$ increases. (d) In light of $(mathrm{c}),$ in the limit of large $N$ the differential scattering cross section is negligibly small except at one of these peaks: $$mathbf{q}=mathbf{G}_{ell m n}=frac{2 pi}{a}(l hat{imath}+m hat{jmath}+n hat{k})$$ for integer $l, m,$ and $n .$ The vectors $mathbf{G}_{l m n}$ are called reciprocal lattice vectors. Find the scattering angles $( heta)$ at which peaks occur. If the neutron's wavelength is equal to the crystal spacing $a$, what are the three smallest (nonzero) angles? Comment: Neutron diffraction is one method used, to determine crystal structures (electrons and x-rays can also be used and the same expression for the locations of the peaks holds). In this problem we looked at a cubic arrangement of atoms, but a different arrangement (hexagonal for example) would produce peaks at a different set of angles. Thus from the scattering data one can infer the underlying crystal structure.
Adi S.
Walter White who lives in Simeranya (a utopic city in Anarres) prepares blue crystals, named as myth, which adopt a side-centered cubic (scc) cell, where atoms occupy the center of sides of a cubic cell. (The myth molecule is assumed to have a perfect spherical shape.) a) Find out how many atoms are present in each unit cell. b) Calculate the atomic packing factor (APF) of scc unit cell. c) Given the density of the material as 6.184 g/cm³ and a molar mass of 132 g/mol, determine the radius (in nm) of myth molecule. d) Find the linear density (in nm⁻¹) of the following vectors in scc; [111], [110], and [100]. (You can choose the origin as one of the corners of scc cell.) e) Determine the number of holes with coordination numbers 6, 8, and 12 per unit cell.
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