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- A/The volume density of atoms for a simple cubic lattice is 3 x 1022 cm. Assume that the atoms are hard spheres with each atom touching its nearest neighbor. Determine the lattice constant and the radium of the atom and APF. 15 M B/A th 0 051 ig diff oli calt0.4 nm 0.3 nm +x -0.5 nm- Direction 2 Fig. Q3(b) c) Calculate the atomic packing factor for the face-centered cubic (FCC) crystal structure. Formula Sheet nA V.NA p= djkl Direction 1 APF = = V V nλ = 2d hk sin 0 S +y a √h² +k² +1²if radius of manganese ion is 66ppm nd fluoride ion is 134 ppm then what is the shortest distance between two fluoride ions
- 6. Write the expression that relates reciprocal lattice vector to crystal lattice.6. A reciprocal-lattice vector has a dimension equal to the reciprocal of length, for example, cm-'. Is it meaningful to compare the magnitudes of a direct-lattice vector R with a reciprocal-lattice vector G? Is it meaningful to compare their directions? If the latter answer is yes, find the angle between R and G in terms of their components in a cubic crystal. What is the angle between R = [111] and G = [110]? 7. Does a real lattice vector have a corresponding unique reciprocal vector?different colors-> different atoms (2D crystal) There is a square lattice that has 2 basis atoms A and B. Atomic erent form factor of each is fA and fB (a) Find the primitive unit cell of this lattice and write the coordinates of basis atoms (How many basis atoms in a primitive cell?) (b) Draw reciprocal lattice of this crystal and find reciprocal lattice vectors. (c) Find structure factor SG of a reciprocal lattice points about G. structure factor SG =zj fj exp(-iG-rj) * (j: primitive unit cell OFo| 25E basis A) (d) If A andB are the same atoms fA=fB=f. If so, find G, so that SG is not zero. Draw all G points. (e) in (d) find primitive lattice vector and unit cell in real space and k-space. Find lattice constant.
- 6. Assume that a cubic lattice constant a equals 0.40 nm. Find the volume of a primitive cell for simple cubic, face-centered-cubic, and body-centered-cubic crystal lattices.Q5 // Answer three branches: A-Draw [313] and (101) B- Calculate the number of atoms / unit area for (111), (110), (100) for the BCC crystal. Derive in terms of a C- Explain that sulfur has covalent bonds? With drawing? D-vanadium V has an atomic radius of 0.132nm and a density of 6.1g / cm. Does it have a FC C crystal structure or BC C? 50Cteatmat A2- Show that the length Tik of the reciprocal lattice vector īe and the interplanar distance dħke of direct lattice satisfy the relation: Fdhke = hke =1.
- 6. A reciprocal-lattice vector has a dimension equal to the reciprocal of length, for example, cm-'. Is it meaningful to compare the magnitudes of a direct-lattice vector R with a reciprocal-lattice vector G? Is it meaningful to compare their directions? If the latter answer is yes, find the angle between R and G in terms of their components in a cubic crystal. What is the angle between R = [111] and G = [110]?Date 1 Page 1 9. Ga As has lattice constant of 0.565 nm. Estimate the percent of atoms on surface for N = 2, 251, 100. where 'N' represents of Layers. number4. Liner ionic crystal. Consider a line of 2N ions of alternating charge ±q with a repulsive potential energy A/R" between nearest neighbors. (a) Show that at the equilibrium separation (CGS) (b) Let the crystal be compressed so that R. → R. (1-8). Show that the work done in a compressing a unit length of the crystal has the leading term C8²/2, where 2 (n-1)q² In 2 (CGS) C = U(R): = 2Nq² In 2 (₁-1). Ro n Ro To obtain the results in SI, replace q² by q²/4л. Note: We should not expect to obtain this result from the expression for U(R), but we must use the complete expression for U(R).