Dr. Wiezorek Suggested Solutions HW_3

Dr. Wiezorek Suggested Solutions HW_3 - M Homework...

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Unformatted text preview: M Homework Assignment #3 MEMS 1054 ‘ NAME: Return by next Wednesday! 7 Questions in total! This is an important homework for you in MEMS 1054. Get Started early, especially since I uploaded it only today, Friday! 1) Many concentrated solid solutions of metal atom species form chemically ordered phases at low temperatures. The chemical ordering involves changes from a statistically random distribution of the atoms among the atom sites to a regular distribution in which designated sites are occupied by one kind of atom. Thus, atomic sites, which are equivalent in the disordered solid solution are no longer equivalent in the ordered structures. These phenomena can be monitored and studied with diffraction techniques using X—rays, neutrons and electrons. fir'spr-dzred {Tut-c- Svoh'q'r 5¢Futian~ cu” . Ate-mt. in. A“? A” Figure 1 As shown schematically in the section of the binary phase—diagram (Figure 1), both Cu and Au have cubic close—packed structures and at temperatures above 410°C a complete solid solution exists between them across the entire composition range. Below critical temperatures of 390°C and 410°C ordered phases occur at the compositions of about CuAu and Cu3Au. Just as the disordered Cu and Au phases unit cells with lattice parameters 0t=[3=y=90° and a=b=c can be used HERE to good approximation to describe the ordered phases. Continued next page a) In the AB—compound CuAu the atomic ordering a crystal structure where Cu atoms occupy the positions 0&0 and IA, 1/2, 0 and the Au atoms occupy the sites 1/2 , 0, 1A and 0, r 1/6, 1/2 in the unit cell. .__.——- \ .___¢-——~ Draw a unit cell of ordered CuAu and determine the lattice type (P, C, LP) and crystal system of this crystal structure. [oat] :i ‘ ‘9 '0 “7‘ 'W WWW” my“; my am 2/ > / ‘5 ’5 W14 #3:? W m.) a; 44 W! mazezm, / ‘ 3 m #22!“ “252/: M to w r a M w“ M" Z: m>la 0/ Mil ‘ #6:: WW affaixwmf [a1 a/LNC)\ I l I m I o ‘ ‘ ‘ vita/w?” [7,,” f _ WW b) n the A3B—compound Cu3Au the a omic ordering a crystal structure where Au atoms occupy the positions 0,0,0 and Cu atoms occupy 1/2, 1A, 0 and 1A , 0, 1/5 and O, 1/2, IA in the unit cell. Draw a unit cell of ordered CuAu and determine the lattice type (P, C, I F) and crystal system of this crystal structure. 2) Consider the ionically bonded crystal structure of rock salt, NaCl. A conventional unit cell representation of this cubic structure is shown on the next page inclusive of some information regarding the atomic and ionic radii. Size effect in ionically bonded structures: e.g. NaCl Na Cl —-> Na+ Cl- electron transfer [Ne3s1] [Ne3823p5] [Ne] [Ar] 0.099 0.095 0.181 r [nm] 0.192 /7 a) How many lattice points are contained in this conventional unit cell? How many chemical units of NaCl are contained in this unit cell? % and? b) Determine by inspection of the unit cell the coordination (represents the number of nearest neighbors) for each of the two ion species in the NaCl structure. How many cations (anions) are nearest neighbors to the anion (cation) sites? Mn‘ W [Z /V£+Wm)£rém My]. 44. m“ a” a'M/f 42‘“ «ce'« 4 ,~5¢’/Mé/m‘/M< c) Is the cation coordination characteristic of NaCl structures called tetrahedral, cubic octahedral or polyhedral? arm/m fl/ 7%} f; oer/WfEDML eawamg; ’ {44/1 MVLMM 1'; AWM/ 5/ “A’Ké” d) For the sketched unit cell of NaCl define a coordinate system and indicate the direction [-1 2 0] by an arrow, draw into it the (101) plane and list all the planes contained in the form {111} using their Miller indices. 3) Compare the three cubic cystal structures shown below as unit cell representations. What are the atom positions and species? What are the Bravais lattices for these structures (cP, cI, CF)? How many atoms of each species (where applicable) are contained in the unit cell? 4) Find descriptions of the archetype close-packing related structure of fluorite (a) and for perovskite (b). The are in the notes I used in lecture and you can also find them per book and/or intemet research. Identify for each the Bravais LATTICE (there are 14 of these) and describe the MOTIF for each structure. . -r 111 FJEZ" WwJeaA a)CaF2(fluor1te) 6F; Japalg‘ FM‘ ’1” 2‘ 116; V,“ q 11 d) CaTi03 (perovskite) C CL, «)‘l M” M y 4“ if?!) g 21/”. 5' 0’7' 0 —- M5” 0v MW 5) The unit cell of a tetragonal crystal has three orthogonal axes (0L=[3=y=90°) with lattice constants a, b, and c equal to 3A, 3A, and 7A, respectively. a) Make a perspective sketch of one or more unit cells of this crystal. Outline planes with Miller indices (110) and (0-11). b) List six different orientations of planes, which have Miller indices that are permutations of i1 , :lzl , and O. For this tetragonal crystal, group these six planes into forms of crystallographically equivalent planes {hkl}. '_ / n . 101°), 1070) ,- 1‘ (la/)1 i (m7), i (911)]: (WI) 100/} #Jfl’j Mt / {In} i101} Jag/“ML c) Find the angles between each of the pairs of the [100] direction with the [101], [011], [1-10] and [111] in this tetragonal crystal. :1 I \ p z . (14/[[Ioo3itl00)=> 2‘3 : com, : 3m : “31.1 a d! ' {i at (Mm): o , =6...» 0114/ L4. I ) q 33 0 / «ms-mm»? W. WW7? ' a 1/ a ' . “(9 ¥ZEWJIEW3>$ 5050/9: W 3 {7 ’W d) Is the normal of the (10-1) plane parallel to the [10-1] direction in this tetragonal crystal? I M I . / 6) Draw the atomic arrangements of the close—packed planes in the elemental cF crystal of Cu and use it as a representation of the crystal in the projection along the normal to this close—packed plane. What are the Miller indices, (hkl), of the close packed plane you drew? ( n |) Now mark the three close—packed directions in this close—packed plane (hkl). m iElTojl iftwjfi i [0713 / 7) Miller indices problems from the textbook by Callister. .. 3.33, 3 .34 3.33 Determine the Miller indices for the planes shown in the following unit cell: a ‘3 3.34, Determine the Miller indices for the planes , Shown in the following unit cell: w v +2 ...
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