347 Determine the Miller indices for the planes shown in the following unit cell:
347 Determine the Miller indices for the planes shown in the following unit cell:
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Machine elements are subjected to varieties of loads, some components are subjected to static loads, while some machine components are subjected to fluctuating loads, whose load magnitude tends to fluctuate. The components of a machine, when rotating at a high speed, are subjected to a high degree of load, which fluctuates from a high value to a low value. For the machine elements under the action of static loads, static failure theories are applied to know the safe and hazardous working conditions and regions. However, most of the machine elements are subjected to variable or fluctuating stresses, due to the nature of load that fluctuates from high magnitude to low magnitude. Also, the nature of the loads is repetitive. For instance, shafts, bearings, cams and followers, and so on.
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3.58
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102 · Chapter 3 I The Structure of Crystalline Solids
15 Camaet tha (111) and (O12) planes into the four- 3.61 (a) Derive planar density expressions for BCC
130 of 990
eme for hexagonal unit
(100) and (110) planes in terms of the atomic
radius R.
or the planes shown in
(b) Compute and compare planar density values
for these same two planes for molybdenum (Mo).
S ARUWE IAagUa unit cells:
3.62 (a) Derive the planar density expression for the
O HCP (0001) plane in terms of the atomic radius R.
(b) Compute the planar density value for this
same plane for titanium (Ti).
Polycrystalline Materials
3.63 Explain why the properties of polycrystalline
materials are most often isotropic.
X-Ray Diffraction: Determination
of Crystal Structures
3.64 The interplanar spacing da for planes in a unit
cell having orthorhombic geometry is given by
(a)
(c)
1
where a, b, and e are the lattice parameters.
(a) To what equation does this expression reduce
for crystals having cubic symmetry?
(b) For erystals having tetragonal symmetry?
3.65 Using the data for aluminum in Table 3.1, com-
• pute the interplanar spacing for the (110) set of
planes.
3.66 Using the data for a-iron in Table 3.1, compute
• the interplanar spacings for the (111) and (211)
sets of planes.
(b)
(d)
3.57 Sketch the (0111) and (21 10) planes in a hexago- 3.67 Determine the expected diffraction angle for the
O first-order reflection from the (310) set of planes
for BCC chromium (Cr) when monochromatic
radiation of wavelength 0.0711 nm is used.
3.58 (a) Derive linear density expressions for FCC 3,68 Determine the expected diffraction angle for the
first-order reflection from the (111) set of planes
for FCC nickel (Ni) when monochromatic radia-
nal unit cell.
Linear and Planar Densities
O (100] and [111] directions in terms of the atomic
radius R.
tion of wavelength 0.1937 nm is used.
(b) Compute and compare linear density values
for these same two directions for copper (Cu).
3.69 The metal rhodium (Rh) has an FCC crystal struc-
3.59 (a) Derive linear density expressions for BCC O ture. If the angle of diffraction for the (311) set of
(110] and [111] directions in temms of the atomic
radius R.
planes occurs at 36.12" (first-order reflection) when
monochromatic x-radiation having a wavelength of
0,0711 nm is used, compute the following:
(b) Compute and compare linear density values
for these same two directions for iron (Fe).
(a) The interplanar spacing for this set of planes
(b) The atomic radius for a Rh atom
3.60 (a) Derive planar density expressions for FCC
• (100) and (111) planes in terms of the atomic 3.70 The metal niobium (Nb) has a BCC crystal
radius R.
(b) Compute and compare planar density values
for these same two planes for aluminum (Al).
structure. If the angle of diffraction for the (211)
set of planes occurs at 75.99 (first-order reflec-
tion) when monochromatic x-radiation having
Questions and Problems. 103
a wavelength of 0,1659 nm is used, compute the
following:
(a) Determine the interplanar spacing for each of
the peaks.
(b) For each peak, determine the atomic radius
for Pt, and compare these with the value pre-
sented in Table 3.1.
(a) The interplanar spacing for this set of planes
(b) The atomic radius for the Nb atom
3.71 For which set of crystallographic planes will a first-
O order diffraction peak occur at a diffraction angle 3.75 The following table lists diffraction angles for the
of 44,53' for FCC nickel (Ni) when monochromatic
first three peaks (first-order) of the x-ray diffrac-
tion pattern for some metal. Monochromatic x-ra-
diation having a wavelength of 0.1397 nm was used.
radiation having a wavelength of 0.1542 nm is used?
3.72 For which set of crystallographic planes will a
O first-order diffraction peak occur at a diffraction
angle of 44.53 for BCC tantalum (Ta) when
monochromatic radiation having a wavelength of
0.1937 nm is used?
(a) Determine whether this metal's crystal struc-
explain the reason for your choice.
(b) If the crystal structure is either BCC or FCC,
identify which of the metals in Table 3.1 gives this
diffraction pattern. Justify your decision.
3.73 Figure 3.26 shows the first five peaks of the x-ray
O diffraction pattern for tungsten (W), which has a
BCC crystal structure; monochromatic x-radia-
tion having a wavelength of 0.1542 nm was used.
(a) Index (i.e., give h, k, and / indices) each of
these peaks.
(b) Determine the interplanar spacing for each of
the peaks.
Peak Number
Diffraction Angle (20)
34.51"
40.06
3
57.95
(c) For each peak, determine the atomic radius
for W, and compare these with the value pre-
sented in Table 3.1.
3.76 The following table lists diffraction angles for the
first three peaks (first-order) of the x-ray diffrac-
tion pattern for some metal. Monochromatic x-ra-
diation having a wavelength of 0.0711 nm was used.
(a) Delermine winether this metal's crystal strue-
3.74 The following table lists dmaeog Tor te
L diffeu](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9715461-f5a2-4dc1-935c-0b02c448b3b6%2Fdecbc7b8-9ca0-42bf-bde0-a1ec41e915e9%2F07mfymk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:5:20 1
Done
102 · Chapter 3 I The Structure of Crystalline Solids
15 Camaet tha (111) and (O12) planes into the four- 3.61 (a) Derive planar density expressions for BCC
130 of 990
eme for hexagonal unit
(100) and (110) planes in terms of the atomic
radius R.
or the planes shown in
(b) Compute and compare planar density values
for these same two planes for molybdenum (Mo).
S ARUWE IAagUa unit cells:
3.62 (a) Derive the planar density expression for the
O HCP (0001) plane in terms of the atomic radius R.
(b) Compute the planar density value for this
same plane for titanium (Ti).
Polycrystalline Materials
3.63 Explain why the properties of polycrystalline
materials are most often isotropic.
X-Ray Diffraction: Determination
of Crystal Structures
3.64 The interplanar spacing da for planes in a unit
cell having orthorhombic geometry is given by
(a)
(c)
1
where a, b, and e are the lattice parameters.
(a) To what equation does this expression reduce
for crystals having cubic symmetry?
(b) For erystals having tetragonal symmetry?
3.65 Using the data for aluminum in Table 3.1, com-
• pute the interplanar spacing for the (110) set of
planes.
3.66 Using the data for a-iron in Table 3.1, compute
• the interplanar spacings for the (111) and (211)
sets of planes.
(b)
(d)
3.57 Sketch the (0111) and (21 10) planes in a hexago- 3.67 Determine the expected diffraction angle for the
O first-order reflection from the (310) set of planes
for BCC chromium (Cr) when monochromatic
radiation of wavelength 0.0711 nm is used.
3.58 (a) Derive linear density expressions for FCC 3,68 Determine the expected diffraction angle for the
first-order reflection from the (111) set of planes
for FCC nickel (Ni) when monochromatic radia-
nal unit cell.
Linear and Planar Densities
O (100] and [111] directions in terms of the atomic
radius R.
tion of wavelength 0.1937 nm is used.
(b) Compute and compare linear density values
for these same two directions for copper (Cu).
3.69 The metal rhodium (Rh) has an FCC crystal struc-
3.59 (a) Derive linear density expressions for BCC O ture. If the angle of diffraction for the (311) set of
(110] and [111] directions in temms of the atomic
radius R.
planes occurs at 36.12" (first-order reflection) when
monochromatic x-radiation having a wavelength of
0,0711 nm is used, compute the following:
(b) Compute and compare linear density values
for these same two directions for iron (Fe).
(a) The interplanar spacing for this set of planes
(b) The atomic radius for a Rh atom
3.60 (a) Derive planar density expressions for FCC
• (100) and (111) planes in terms of the atomic 3.70 The metal niobium (Nb) has a BCC crystal
radius R.
(b) Compute and compare planar density values
for these same two planes for aluminum (Al).
structure. If the angle of diffraction for the (211)
set of planes occurs at 75.99 (first-order reflec-
tion) when monochromatic x-radiation having
Questions and Problems. 103
a wavelength of 0,1659 nm is used, compute the
following:
(a) Determine the interplanar spacing for each of
the peaks.
(b) For each peak, determine the atomic radius
for Pt, and compare these with the value pre-
sented in Table 3.1.
(a) The interplanar spacing for this set of planes
(b) The atomic radius for the Nb atom
3.71 For which set of crystallographic planes will a first-
O order diffraction peak occur at a diffraction angle 3.75 The following table lists diffraction angles for the
of 44,53' for FCC nickel (Ni) when monochromatic
first three peaks (first-order) of the x-ray diffrac-
tion pattern for some metal. Monochromatic x-ra-
diation having a wavelength of 0.1397 nm was used.
radiation having a wavelength of 0.1542 nm is used?
3.72 For which set of crystallographic planes will a
O first-order diffraction peak occur at a diffraction
angle of 44.53 for BCC tantalum (Ta) when
monochromatic radiation having a wavelength of
0.1937 nm is used?
(a) Determine whether this metal's crystal struc-
explain the reason for your choice.
(b) If the crystal structure is either BCC or FCC,
identify which of the metals in Table 3.1 gives this
diffraction pattern. Justify your decision.
3.73 Figure 3.26 shows the first five peaks of the x-ray
O diffraction pattern for tungsten (W), which has a
BCC crystal structure; monochromatic x-radia-
tion having a wavelength of 0.1542 nm was used.
(a) Index (i.e., give h, k, and / indices) each of
these peaks.
(b) Determine the interplanar spacing for each of
the peaks.
Peak Number
Diffraction Angle (20)
34.51"
40.06
3
57.95
(c) For each peak, determine the atomic radius
for W, and compare these with the value pre-
sented in Table 3.1.
3.76 The following table lists diffraction angles for the
first three peaks (first-order) of the x-ray diffrac-
tion pattern for some metal. Monochromatic x-ra-
diation having a wavelength of 0.0711 nm was used.
(a) Delermine winether this metal's crystal strue-
3.74 The following table lists dmaeog Tor te
L diffeu
![5:20 1
Done
(a) (011]
(b) [100]
Plane 2
Plane 1
3.40 Convert the [110] and [001] directions into the
four-index Miller-Bravais scheme for hexagonal
unit cells.
3.41 Determine the indices for the directions shown
al unit cells:
129 of 990
.04 mm
-0.4 nm-
3.46 Sketch within a cubie unit cell the following
planes:
(a) (101)
(e) (111)
(b) (211)
(n (212)
(c) (012)
(g) (312)
(a)
(b)
(d) (313)
(h) (301)
Questions and Problems · 101
347 Determine the Miller indices for the planes 3.50 Cite the indices of the direction that results from
S shown in the following unit cel:
the intersection of each of the following pairs of
planes within a cubic crystal:
(a) The (110) and (111) planes
(b) The (110) and (110) planes
(e) The (111) and (001) planes
3.51 Sketch the atomic packing of the following:
O (a) The (100) plane for the FCC crystal structure
(b) The (111) plane for the BCC crystal structure
(similar to Figures 3.125 and 3.13b).
3.52 Consider the reduced-sphere unit cell shown in
Problem 3.23, having an origin of the coordinate
system positioned at the atom labeled O. For the
following sets of planes, determine which are
equivalent:
(a) (100), (O10), and (001)
(b) (110). (101), (011), and (101)
3.48 Determine the Miller indices for the planes
3 shown in the following unit cell:
(c) (111). (111). (1II), and (11)
3.53 The accompanying figure shows three different
crystallographic planes for a unit cell of a hypo-
thetical metal. The circles represent atoms.
0.4 am
0.4 m
+y
1001)
(100)
(110)
(a) To what crystal system does the unit cell be-
long?
(b) What would this crystal structure be called?
3.49 Determine the Miller indices for the planes 3.54 The accompanying figure shows three different
crystallographic planes for a unit cell of some
S shown in the following unit cell:
hypothetical metal. The circles represent atoms.
le0.30 nm
0.20 nm
0.25 nm
(110)
(101)
(011)
(a) To what crystal system does the unit cell belong?
(b) What would this crystal structure be called?
(e) If the density of this metal is 18.91 g/em', de-
termine its atomic weight.
102· Chapter 3 I The Structure of Crystalline Solids
3.55 Convert the (111) and (01 me Eo
indey Miller Bravais scheme for bexagonal unit (100) and (110) planes in terms of the atomic
density expressions for BCC](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9715461-f5a2-4dc1-935c-0b02c448b3b6%2Fdecbc7b8-9ca0-42bf-bde0-a1ec41e915e9%2Ftwfo169_processed.jpeg&w=3840&q=75)
Transcribed Image Text:5:20 1
Done
(a) (011]
(b) [100]
Plane 2
Plane 1
3.40 Convert the [110] and [001] directions into the
four-index Miller-Bravais scheme for hexagonal
unit cells.
3.41 Determine the indices for the directions shown
al unit cells:
129 of 990
.04 mm
-0.4 nm-
3.46 Sketch within a cubie unit cell the following
planes:
(a) (101)
(e) (111)
(b) (211)
(n (212)
(c) (012)
(g) (312)
(a)
(b)
(d) (313)
(h) (301)
Questions and Problems · 101
347 Determine the Miller indices for the planes 3.50 Cite the indices of the direction that results from
S shown in the following unit cel:
the intersection of each of the following pairs of
planes within a cubic crystal:
(a) The (110) and (111) planes
(b) The (110) and (110) planes
(e) The (111) and (001) planes
3.51 Sketch the atomic packing of the following:
O (a) The (100) plane for the FCC crystal structure
(b) The (111) plane for the BCC crystal structure
(similar to Figures 3.125 and 3.13b).
3.52 Consider the reduced-sphere unit cell shown in
Problem 3.23, having an origin of the coordinate
system positioned at the atom labeled O. For the
following sets of planes, determine which are
equivalent:
(a) (100), (O10), and (001)
(b) (110). (101), (011), and (101)
3.48 Determine the Miller indices for the planes
3 shown in the following unit cell:
(c) (111). (111). (1II), and (11)
3.53 The accompanying figure shows three different
crystallographic planes for a unit cell of a hypo-
thetical metal. The circles represent atoms.
0.4 am
0.4 m
+y
1001)
(100)
(110)
(a) To what crystal system does the unit cell be-
long?
(b) What would this crystal structure be called?
3.49 Determine the Miller indices for the planes 3.54 The accompanying figure shows three different
crystallographic planes for a unit cell of some
S shown in the following unit cell:
hypothetical metal. The circles represent atoms.
le0.30 nm
0.20 nm
0.25 nm
(110)
(101)
(011)
(a) To what crystal system does the unit cell belong?
(b) What would this crystal structure be called?
(e) If the density of this metal is 18.91 g/em', de-
termine its atomic weight.
102· Chapter 3 I The Structure of Crystalline Solids
3.55 Convert the (111) and (01 me Eo
indey Miller Bravais scheme for bexagonal unit (100) and (110) planes in terms of the atomic
density expressions for BCC
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