Solution to Midterm 1Material Science and Engineering: An Intduction,

MSE, National Chiao Tung University, 100A

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1. For a FCC structure: (a) Draw its unit cell. (b) What is the coordination number? (c) Calculate its atomic packing factor. (d) What is its closest-packing plane? Calculate the planar density of that plane. (3+2+2+3 points)

• C.N. = 12

• atomic packing factor = 0.74

• (1,1,1), 91%

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!

4R

R

2R 3

2. Draw the curve of potential energy v.s. interatomic distance for ionic materials. Label attractive, repulsive, and net force/energy. Label the position of equilibrium spacing. What are the three materials properties that could be obtained from the curve of potential energy v.s. interatomic distance? Discuss how the shape of the curve relates to the three properties. . (5+5 points)

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• 熔點、熱膨脹係數、楊氏模數

• E-R圖曲線越深越窄，熔點越高E-R圖曲線越對稱，熱膨脹係數越低E-R圖曲線越深越窄，楊氏模數越高

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3. Draw a BCC unit cell. How many tetrahedral sites in a BCC unit cell?Mark them on the cell. (10 points)

• 每個面有四個 x ½ x 六面 = 12個

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4. (a) Draw the unit cell for ZnS. (b) What is the coordination number? (c) Write down the translation locations for the atom positions. (4+2+4 points)

• Coordination number for cation is 4, and 4 for anion, too.4S: 000,1/2 1/2 0, 0 1/2 1/2, 1/2 0 1/24Zn: 1/4 1/4 3/4, 1/4 3/4 1/4, 3/4 1/4 1/4, 3/4 3/4 3/4

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5. What is an edge dislocation? Draw a Burgers circuit to find Burgers vector of the dislocations. Mark the dislocation line in the figure. (4+6 points)

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6. (a) What are the diffusion mechanisms? Draw schematic pictures for each mechanism? (b) Which one is faster? Why? (6+4 points)

• Vacancy Diffusion Interstitial Diffusion

• For most metal alloys, interstitial diffusion is faster than vacancy because interstitial atoms are smaller.

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7. We can observe the grain structures of Cu by optical microscopy, as shown in Figure 1 below. (a) Explain the procedures for the specimen preparation. (b) Why the grain boundaries appear in black under an optical microscope? Why grains have contrast? (c) What are the straight lines in the figure? (3+4+3)

(a) Prepare the Cu sample with gridding and polishing, and make it etched in chemical solution, normally acid, for seconds.

(b) The chemical reactivity of the grains of some single-phase materials depends on crystallographic orientation. Atoms along grain boundary regions are more chemically active.

(c) Those straight lines in figure are twin boundaries.

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8. Draw the following planes in your cubic unit cell; mark the intersections of the planes and the three axes: (a) (021); (b) (211); (c) (312) (9 points)

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(021)

(211)

(312)

9. What are the requirements for substitutional impurities? (6 points)

• The requirements to form substitutional impurities : (1) Atomic size difference ＜±15% (2) Same crystal structure (3) Electronegativity (電負性) must be similar.(4) A metal with higher valences can be dissolved easier than the lower one

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10. Determine the indices for the vectors shown in the unit cell in Fig. 2. (8 points)

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A: [331]

B: [403]

C: [361]

D: [111]

11. Why we can use X-ray to determine crystal structure? What information (parameters) we can obtain from X-ray diffraction? (4+3 points)

• 1. The wavelength of x-ray is several Å, which is identical to the distance of atoms in most materials.2. The spacing between planes of crystalline material must meet the Bragg’s law:

• (1) Lattice constant, (2) Crystal structure (bcc or fcc,…….), (3) Chemical composite (ex. AB or A2B)

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12. For the HCP crystal structure, show that the ideal c/a ratio is 1.633. (8 points)

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線段MH為c軸長度的一半利用正三角形的特殊性質與畢氏定理

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JH 2+ MH 2

= JM 2

a2"23

#

$ %

&

' ( 2

+c2#

$ % &

' ( 2

= a2

c 2

a2=83)

ca*1.633

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