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Essentials of Materials Science and Engineering 3rd edition Donald R. Askeland, Wendelin J. Wright - Solutions

Consider the CuAu crystal structure. It can b e described as a simple cubic lattice with a basis of Cu (0, 0, 0), Cu (1/2, 1/2, 0), Au (1/2, 0, 1/2), and Au (0, 1/2, 1/2). (a) How many atoms of each type are there per unit cell? (b) Draw the unit cell for CuAu. Use a different symbol for each type
A typical paper clip weighs 0.59 g and consists of BCC iron. Calculate (a) The number of unit cells; and (b) The number of atoms in the paper clip. (See Appendix A for required data.)
Calculate the atomic packing fraction for the hexagonal close-packed crystal structure for which c = (√8 / 3)a . Remember that the base of the unit cell is a parallelogram.
α-Mn has a cubic structure with a0 = 0.8931 nm and a density of 7.47 g/cm3. β-Mn has a different cubic structure with a0 = 0.6326 nm and a density of 7.26 g/cm3. The atomic weight of manganese is 54.938 g/mol and the atomic radius is 0.112 nm. Determine the percent volume change that would occur
Determine the Miller indices for the directions in the cubic unit cell shown in Figure 3-33.
Sketch the following planes and directions within a cubic unit cell: (a) [101] (b) [010] (c) [122] (d) [301] (e) [201] (f) [213] (g) [011] (h) [102] (i) [002] (j) [130] (k) [212] (l) [312]
What are the indices of the six directions of the form < 110 > that lie in the (111) plane of a cubic cell?
Determine the angle between the [110] direction and the (110) plane in a tetragonal unit cell; then determine the angle between the [011] direction and the (011) plane in a tetragonal cell. The lattice parameters are a0 = 4.0 A and c0 = 5.0 A. What is responsible for the difference?
Determine the crystal structure for the following: (a) A metal with a0 = 4.9489 Å, r = 1.75 Å and one atom per lattice point; and (b) A metal with a0 = 0.42906 nm, r = 0.1858 nm, and one atom per lattice point.
Determine the Miller indices of the plane that passes through three points having the following coordinates. (a) (0, 0, 1); (1, 0, 0); and (½, ½, 0) (b) (½, 0, 1); (½, 0, 0); and (0, 1, 0) (c) (1, 0, 0); (0, 1, ½); and (1, ½, ¼) (d) (1, 0, 0); (0, 0, ¼); and (½, 1, 0)
The lattice constant of BaTiO3, a ceramic material used to make capacitors, for the cubic crystal structure is 4 Å. This material is analyzed using copper K-α radiation of wavelength 1.54 Å. What will be the value of 2θ at which the (200) reflection from the diffracted x-rays can be expected?
A sample of cubic SiC was analyzed using XRD. It was found that the (111) peak was located at a 2θ of 16°. The wavelength (λ) of the x-ray radiation used in this experiment was 0.6975 Å. Determine the lattice constant (a0) of this form of SiC.
Calculate the theoretical density of α-Sn. Assume the diamond cubic structure and obtain the atomic radius information from Appendix B.
A particular unit cell is cubic with ions of type A located at the corners and facecenters of the unit cell and ions of type B located at the midpoint of each edge of the cube and at the body-centered position. The ions contribute to the unit cell in the usual way (1/8 ion contribution for each ion
The lattice constant of zinc selenide (ZnSe) is 0.567 nm. The crystal structure is that of zinc blende. Show that the theoretical density of ZnSe should be 5.26 g/cm3.
The theoretical density of germanium is 5.324 g/cm3 at 300 K. Germanium has the same crystal structure as diamond. What is the lattice constant of germanium at 300 K?
Recently, gallium nitride (GaN) has been used to make light-emitting diodes (LEDs) that emit a blue or ultraviolet light. Such LEDs are used in DVD players and other electronic devices. This material has two crystal structures. One form is the zinc-blende crystal structure (lattice constant a0 =
Would you expect CsBr to have the sodium chloride, zinc blende, fluorite, or cesium chloride structure? Based on your answer, determine (a) The lattice parameter; (b) The density; and (c) The packing factor.
Would you expect BeO to have the sodium chloride, zinc blende, or fluorite structure? Based on your answer, determine (a) The lattice parameter; (b) The density; and (c) The packing factor.
The density of potassium, which has the BCC structure, is 0.855 g/cm3. The atomic weight of potassium is 39.09 g/mol. Calculate (a) The lattice parameter; and (b) The atomic radius of potassium.
Would you expect UO2 to have the sodium chloride, zinc blende, or fluorite structure? Based on your answer, determine (a) The lattice parameter; (b) The density; and (c) The packing factor.
Would you expect NiO to have the cesium chloride, sodium chloride, or zinc blende structure? Based on your answer, determine (a) The lattice parameter; (b) The density; and (c) The packing factor.
What is the radius of an atom that will just fit into the octahedral site in FCC copper without disturbing the crystal structure?
Determine the minimum radius of an atom that will just fit into (a) The tetrahedral interstitial site in FCC nickel; and (b) The octahedral interstitial site in BCC lithium.
Suppose that FCC rhodium is produced as a 1-mm thick sheet, with the (111) plane parallel to the surface of the sheet. How many (111) interplanar spacings d111 thick is the sheet?
Determine the repeat distance, linear density, and packing fraction for BCC lithium, which has a lattice parameter of 0.35089 nm, in the [100], [110] and [111] directions. Which of these directions is close-packed?
Determine the planar density and packing fraction for FCC nickel in the (100), (110), and (111) planes. Which, if any, of these planes is close-packed?
The density of thorium, which has the FCC structure and one atom per lattice point, is 11.72 g/cm3. The atomic weight of thorium is 232 g/mol. Calculate (a) The lattice parameter; and (b) The atomic radius of thorium.
A metal has a cubic structure with a density of 2.6 g/cm3, an atomic weight of 87.62 g/mol, and a lattice parameter of 6.0849 Å. One atom is associated with each lattice point. Determine the crystal structure of the metal.
A metal has a cubic structure with a density of 1.892 g/cm3, an atomic weight of 132.91 g/mol, and a lattice parameter of 6.13 Å. One atom is associated with each lattice point. Determine the crystal structure of the metal.
Indium has a tetragonal structure with a0 = 0.32517 nm and c0 = 0.49459 nm. The density is 7.286 g/cm3, and the atomic weight is 114.82 g/mol. Does indium have the simple tetragonal or body-centered tetragonal structure?
Gallium has an orthorhombic structure with a0 = 0.45258 nm, b0 = 0.45186 nm, and c0 = 0.76570 nm. The atomic radius is 0.1218 nm. The density is 5.904 g/cm3, and the atomic weight is 69.72 g/mol. Determine (a) The number of atoms in each unit cell; and (b) The packing factor in the unit cell.
Beryllium has a hexagonal crystal structure with a0 = 0.22858 nm and c0 = 0.35842 nm. The atomic radius is 0.1143 nm, the density is 1.848 g/cm3, and the atomic weight is 9.01 g/mol. Determine (a) The number of atoms in each unit cell; and (b) The packing factor in the unit cell.
Gold has 5.82 × 108 vacancies/cm3 at equilibrium at 300 K. What fraction of the atomic sites is vacant at 600 K?
Tin atoms are introduced into an FCC copper crystal, producing an alloy with a lattice parameter of 3.7589 × 10-8 cm and a density of 8.772 g/cm3. Calculate the atomic percentage of tin present in the alloy.
We replace 7.5 atomic percent of the chromium atoms in its BCC crystal with tantalum. X-ray diffraction shows that the lattice parameter is 0.29158 nm. Calculate the density of the alloy.
Suppose we introduce one carbon atom for every 100 iron atoms in an interstitial position in BCC iron, giving a lattice parameter of 0.2867 nm. For the Fe-C alloy, find the density and the packing factor.
The density of BCC iron is 7.882 g/cm3, and the lattice parameter is 0.2866 nm when hydrogen atoms are introduced at interstitial positions. Calculate (a) The atomic fraction of hydrogen atoms; and (b) The number of unit cells required on average that contain hydrogen atoms.
Suppose one Schottky defect is present in every tenth unit cell of MgO. MgO has the sodium chloride crystal structure and a lattice parameter of 0.396 nm. Calculate(a) The number of anion vacancies per cm3; and(b) The density of the ceramic.
ZnS has the zinc blende structure. If the density is 302 g / cm3 and the lattice parameter is 0.59583 nm, determine the nuber of Schottky defects (a) Per unit cell; and (b) Per cubic centimeter.
Draw a B Burgers circuit around the dislocation shown in Figure 4-17. Clearly indicate the Burgers vectors that you find. What type of dislocation is this? In what direction will the dislocation move due to the applied shear stress r? Reference you answers to the coordinate axes shown.
The crystal shown in Figure 4-18 contains two dislocations A and B. if a shear stress is applied to the crystal as shown, what will happen to dislocations A and B?
What are the Miller indices of the slip directions (a) On the (111) plane in an FCC unit cell? (b) On the (011) plane in a BCC unit cell?
What are the Miller indices of the slip planes in FCC unit cells that include the [101] slip direction?
Calculate the number of vacancies per cm3 expected in copper at 1080°C (just below the melting temperature). The energy for vacancy formation is 20, 000 cal/mol.
A single crystal of silver is oriented so that the (111) slip plane is perpendicular to an applied stress of 50 MPa. List the slip systems composed of close-packed planes and directions that may be activated due to this applied stress.
What are the Miller indices of the {110} slip planes in BCC unit cells that include the [111] slip direction?
Calculate the length of the Burgers vector in the following materials: (a) BCC niobium and (b) FCC silver.
Determine the interplanar spacing and the length of the Burgers vector for slip on the expected slip systems in FCC aluminum. Repeat, assuming that the slip system is a (110) plane and a [11] direction. What is the ratio between the shear stresses required for slip for the two systems? Assume that
Determine the interplanar spacing and the length of the Burgers vector for slip on the (110)/[11] slip system in BCC tantalum. Repeat, assuming that the slip system is a (111)/[ 10] system. What is the ratio between the shear stresses required for slip for the two systems? Assume that k = 2 in
How many grams of aluminum, with a dislocation density of 1010 cm/cm3, are required to give a total dislocation length that would stretch from New York City to Los Angeles (3000 miles)?
A single crystal of an FCC metal is oriented so that the [001] direction is parallel to an applied stress of 5000 psi. Calculate the resolved shear stress acting on the (111) slip plane in the [10], [01], and [10] directions. Which slip system (s) will become active first?
A single crystal of a BCC metal is oriented so that the [001] direction is parallel to the applied stress. If the critical resolved shear stress required for slip is 12, 000 psi, calculate the magnitude of the applies stress required to cause slip to begin in the [11] direction on the (110),
Compare the c/a ratios for the following HCP metals, determine the likely slip processes in each, and estimate the approximate critical resolved shear stress. Explain. (a) Zinc (b) Magnesium (c) Titanium (d) Zirconium (e) Rhenium (f) Beryllium
The strength of titanium is found to be 65,000 psi when the grain size is 6.7 × 10-4 in. and 82,000 psi when the grain size is 3.15 × 10-5 in. Determine (a) The constants in the Hall-Petch equation; and (b) The strength of the titanium when the grain size is reduced to 8.00 × 10-6 in.
The fraction of lattice points occupied by vacancies in solid aluminum at 660 °C is 10-3. What is the energy required to create vacancies in aluminum?
A copper-zinc alloy has the following properties: Grain diameter (mm) Strength (MPa) 0.015 .................................. 170 MPa 0.025 .................................. 158 MPa 0.035 .................................. 151 MPa 0.050 .................................. 145 MPa Determine (a) The
The yield stresses of several samples of a steel containing 0.12% carbon with different grain sizes were measured. The data are shown here.Grain-Size Inverse(b) Which sample has the grain size of 27.7 Î¼m? (c) Fit these data to a straight line and calculate the constants Ïƒ0
A nanotechnology researcher develops a sample of 0.12% carbon steel such that the value of d-1/2 is 110 mm-1/2. What will be the grain size of this steel? Can she use the Hall-Petch relationship developed for this steel in the previous problem to predict the yield stress of this sample?
The density of a sample of FCC palladium is 11.98 g/cm3, and its lattice parameter is 3.8902 Å. Calculate (a) The fraction of the lattice points that contain vacancies; and (b) The total number of vacancies in a cubic centimeter of Pd.
The density of a sample of HCP beryllium is 1.844 g/cm3, and the lattice parameters are α0 = 0.22858 nm and c0 = 0.35842 nm. Calculate (a) The fraction of the lattice points that contain vacancies; and (b) The total number of vacancies in a cubic centimeter.
BCC lithium has a lattice parameter of 3.5089 × 10-8 cm and contains one vacancy per 200 unit cells. Calculate (a) The number of vacancies per cubic centimeter; and (b) The density of Li.
FCC lead has a lattice parameter of 0.4949 nm and contains one vacancy per 500 Pb atoms. Calculate (a) The density; and (b) The number of vacancies per gram of Pb.
Cu and Ni form a substitutional solid solution. This means that the crystal structure of a Cu-Ni alloy consists of Ni atoms substituting for Cu atoms in the regular atomic positions of the FCC structure. For a Cu-30% wt.% Ni alloy, what fraction of the atomic sites does Ni occupy?
A niobium alloy is produced by introducing tungsten substitutional atoms in the BCC structure; eventually an alloy is produced that has a lattice parameter of 0.32554 nm and a density of 11.95 g/cm3. Calculate the fraction of the atoms in the alloy that are tungsten.
Atoms are found to move from one lattice position to another at the rate of 5 × 105 jumps/s at 400 °C when the activation energy for their movement is 30,000 cal/mol. Calculate the jump rate at 750 °C?
A 4-cm-diameter, 0.5-mm-thick spherical container made from BCC iron holds nitrogen at 700 °C. The concentration at the inner surface is 0.05 atomic percent and at the outer surface is 0.002 atomic percent. Calculate the number of grams of nitrogen that are lost from the container per hour?
A BCC iron structure is to be manufactured that will allow no more than 50 g of hydrogen to be lost per year through each square centimeter of the iron at 400 °C. If the concentration of hydrogen at one surface is 0.05 H atom per unit cell and 0.001 H atom per unit cell at the second surface,
Determine the maximum allowable temperature that will produce a flux of less than 2000 H atoms/(cm2·s) through a BCC iron foil when the concentration gradient is -5 × 1016 atoms/(cm3 · cm). (Note the negative sign for the flux?)
Certain ceramic materials such as those based on oxides of yttrium, barium, and copper have been shown to be superconductors near liquid nitrogen temperature (~77 to 100 K). Since ceramics are brittle, it has been proposed to make long wires of these materials by encasing them in a silver tube. In
Diffusion of oxygen in YBa2Cu3O7 doped with silver was measured. It was seen that the diffusion of oxygen was slowed down by silver doping, as shown in the data here.Temperature T (˚C)Diffusion Coefficient D (cm2/s)650..........................................................2.89 ×
Zinc oxide (ZnO) ceramics are used in a variety of applications, such as surge protection devices. The diffusion of oxygen in single crystals of ZnO was studied by Tomlins and co-workers. These data are shown in the table here.Using these data, calculate the activation energy for the diffusion of
Use the diffusion data in the table below for atoms in iron to answer the questions that follow. Assume metastable equilibrium conditions and trace amounts of C in Fe. The gas constant in SI units is 8.314 J/(mol · K).(a) Plot the diffusion coefficient as a function of inverse temperature
The plot has three lines representing grain boundary, surface, and volume self-diffusion in a metal. Match the lines labe led A, B, and C with the type of diffusion. Justify you r answer by calculating the activation energy for diffusion for each case?
The diffusion of yttrium ions along grain boundaries in chromium oxide (Cr2CO3) has been studied by Lesage and co-workers. Their data for grain-boundary diffusivities are shown here.(a) From these data, show that the activation energy for grain-boundary diffusion of yttrium in Cr2CO3 is 190
The number of vacancies in a material is related to temperature by an Arrhenius equation. If the fraction of lattice points containing vacancies is 8 × 10-5 at 600 °C, determine the fraction of lattice points containing vacancies at 1000 °C?
Consider a 2-mm-thick silicon wafer to be doped (infused with impurities) using antimony. Assume that the dopant source (gas mixture of antimony chloride and other gases) provides a constant concentration of 1022 atoms/m3. We need a dopant profile such that the concentration of Sb at a depth of 1
Compare the diffusion coefficient of carbon in BCC and FCC iron at the allotropic transformation temperature of 912 oC and explain the difference.
A carburizing process is carried out on a 0.10% C steel by introducing 1.0% C at the surface at 980 ˚C, where the iron is FCC. Calculate the carbon content at 0.01 cm, 0.05 cm, and 0.10 cm beneath the surface after 1 h?
Iron containing 0.05% C is heated to 912 °C in an atmosphere that produces 1.20% C at the surface and is held for 24 h. Calculate the carbon content at 0.05 cm beneath the surface if (a) The iron is BCC; and (b) The iron is FCC. Explain the difference.
What temperature is required to obtain 0.50% C at a distance of 0.5 mm beneath the surface of a 0.20% C steel in 2 h, when 1.10% C is present at the surface? Assume that the iron is FCC?
A 0.15% C steel is to be carburized at 1100o C, giving 0.35% C at a distance of 1 mm beneath the surface. If the surface composition is maintained at 0.90% C, what time is required?
A 0.02% C steel is to be carburized at 1200 °C in 4 h, giving 0.45% C at a distance of 0.6 mm beneath the surface. Calculate the carbon content required at the surface of the steel?
A 1.2% C tool steel held at 1150 °C is exposed to oxygen for 48 h. The carbon content at the steel surface is zero. To what depth will the steel be decarburized to less than 0.20% C?
A 0.80% C steel must operate at 950 °C in an oxidizing environment for which the carbon content at the steel surface is zero. Only the outermost 0.02 cm of the steel part can fall below 0.75% C. What is the maximum time that the steel part can operate?
A steel with the BCC crystal structure containing 0.001% N is nitrided at 550 °C for 5 h. If the nitrogen content at the steel surface is 0.08%, determine the nitrogen content at 0.25 mm from the surface?
The diffusion coefficient for Cr+3 in Cr2O3 is 6 × 10-15 cm2/s at 727 °C and 1 × 10-9 cm2/s at 1400 °C. Calculate(a) The activation energy; and(b) The constant D0.
What time is required to nitride a 0.002% N steel to obtain 0.12% N at a distance of 0.002 in. beneath the surface at 625 °C? The nitrogen content at the surface is 0.15%?
We can successfully perform a carburizing heat treatment at 1200 °C in 1 h. In an effort to reduce the cost of the brick lining in our furnace, we propose to reduce the carburizing temperature to 950 °C. What time will be required to give us a similar carburizing treatment?
Calculate the constant surface concentrations of phosphorus and boron needed to achieve a concentration of 1018 atoms/cm3 at a depth of 0.1 μm from the surface of single crystal silicon using a thermal treatment at 1100 °C for 1 hour. Consider phosphorus and boron separately. The diffusion
When processing silicon single crystals for microelectronics applications, precise quantities of impurities are often introduced at relatively shallow depths by ion implantation and dif fused into the silicon substrate in a subsequent thermal treatment. This can be approximated as a finite source
The diffusion coefficient for O-2 in Cr2O3 is 4 × 10-15 cm2 / s at 1150 °C and 6 × 10-11 cm2/s at 1715 °C. Calculate (a) The activation energy; and (b) The constant D0.
A 1-mm-thick BCC iron foil is used to separate a region of high nitrogen gas concentration of 0.1 atomic percent from a region of low nitrogen gas concentration at 650 °C. If the flux of nitrogen through the foil is 1012 atoms / (cm2 · s), what is the nitrogen concentration in the low
A 0.2-mm-thick wafer of silicon is treated so that a uniform concentration gradient of antimony is produced. One surface contains 1 Sb atom per 108 Si atoms and the other surface contains 500 Sb atoms per 108 Si atoms. The lattice parameter for Si is 5.4307 Å (Appendix A). Calculate the
When a Cu-Zn alloy solidifies, one portion of the structure contains 25 atomic percent zinc and another portion 0.025 mm away contains 20 atomic percent zinc. The lattice parameter for the FCC alloy is about 3.63 × 10-8 cm. Determine the concentration gradient in(a) Atomic percent Zn per cm;(b)
A 0.001 in. BCC iron foil is used to separate a high hydrogen content gas from a low hydrogen gas at 650 °C. 5 × 108 H atoms/cm3 are in equilibrium on one side of the foil, and 2 × 103 H atoms/cm3 are in equilibrium with the other side.Determine(a) The concentration gradient of hydrogen; and(b)
A 1-mm-thick sheet of FCC iron is used to contain nitrogen in a heat exchanger at 1200 °C. The concentration of N at one surface is 0.04 atomic percent and the concentration at the second surface is 0.005 atomic percent. Determine the flux of nitrogen through the foil in N atoms/(cm2 · s)?
An 850-lb force is applied to a 0.15-in.-diameter nickel wire having a yield strength of 45,000 psi and a tensile strength of 55,000 psi. Determine (a) Whether the wire will plastically deform; and (b) Whether the wire will experience necking.
The following data were collected from a standard 0.505-in.-diameter test specimen of a copper alloy (l0 = 2.0 in.):After fracture, the total length was 3.014 in., and the diameter was 0.374 in. Plot the engineering stress-strain curve and calculate (a) The 0.2% offset yield strength; (b) The
The following data were collected from a 0.4-in.-deameter test specimen of polyvinyl chloride (l0 = 2.0 in.):After fracture, the total length was 2,09 in., and the diameter was 0.393 in. Plot the engineering stress-strain curve and calculate (a) The 0.2% offset yield strength; (b) The tensile

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