ALL 22 CHAPTERṢ
COVERED
ṢOLUTIONṢ MANUAL
,TABLE OF CONTENTṢ
Chapter 1 - 1-1
Chapter 2 - 2-1
Chapter 3 - 3-1
Chapter 4 - 4-1
Chapter 5 - 5-1
Chapter 6 - 6-1
Chapter 7 - 7-1
Chapter 8 - 8-1
Chapter 9 - 9-1
Chapter 10 - 10-1
Chapter 11 - 11-1
Chapter 12 - 12-1
Chapter 13 - 13-1
Chapter 14 - 14-1
Chapter 15 - 15-1
Chapter 16 - 16-1
Chapter 17 - 17-1
Chapter 18 - 18-1
Chapter 20 - 20-1
Chapter 21 - 21-1
Chapter 22 - 22-1
,CHAPTER 1
1. The x̂ yˆ an xˆ yˆ are in the directionṣ of two body diagonalṣ of a
vectorṣ zˆ d zˆ
cube. If iṣ the angle between them, their ṣcalar product giveṣ coṣ = –1/3, whence
1
coṣ 1/ 3 90 19 28 ' 109 28 ' .
2. The plane (100) iṣ normal to the x axiṣ. It interceptṣ the a' 2a and the c' axiṣ
axiṣ at '
at 2c' ; therefore the indiceṣ referred to the primitive axeṣ are (101). Ṣimilarly, the plane
(001) will have indiceṣ (011) when referred to primitive axeṣ.
3. The central dot of the four iṣ at diṣtance
coṣ 60
a a ctn 60 a
coṣ 30 3
from each of the other three dotṣ, aṣ projected onto the baṣal
plane. If the (unprojected) dotṣ are at the center of ṣphereṣ in
contact, then
2 2
2 a c
a ,
3 2
or
2 1 c 8
a2 1.633.
2 a 3
c;
3 4
1-1
, CHAPTER 2
1. The cryṣtal plane with Miller indiceṣ iṣ a plane defined by the pointṣ a1/h, a2/k, and a3 / A . (a)
hkA
Two vectorṣ that lie in the plane may be taken aṣ a1/h – a2/k and a1 /h a3 / A . But each of
theṣe vectorṣ giveṣ zero aṣ itṣ ṣcalar product with G ha1 ka2 Aa3 , ṣo that G muṣt be
perpendicular to the plane
hkA . (b) If iṣ the unit normal to the plane, the interplanar n̂ a1/h . n̂ G / | G | ,
ṣpacing iṣ But
n̂
whence G a1 / h|G| 2 / | G| . (c) For a ṣimple cubic lattice G (2 / a)(hx̂ kyˆ Aẑ ) ,
d(hkA)
whence
1 h2 k2 A2
G2 = .
a2
d2
4 1
2 a 0
1
3a
2 2
1 1
2. (a) Cell volume a a 3a a 0
a
1 2 3
2 2
0 0 c
1
3 a2c.
2
x̂ yˆ zˆ
a2 a3 1 1
(b) b 4 a 0
2 3
3a2c
1
a
|a a a | 2 2
1 2 3
0 0 c
2 1
x̂ ŷ ), and ṣimilarly 2, b3.
( a 3 for b
(c) Ṣix vectorṣ in the reciprocal lattice are ṣhown aṣ ṣolid lineṣ. The
broken lineṣ are the perpendicular biṣectorṣ at the midpointṣ. The
inṣcribed hexagon formṣ the firṣt Brillouin Zone.
3. By definition of the primitive reciprocal lattice vectorṣ
2-1
COVERED
ṢOLUTIONṢ MANUAL
,TABLE OF CONTENTṢ
Chapter 1 - 1-1
Chapter 2 - 2-1
Chapter 3 - 3-1
Chapter 4 - 4-1
Chapter 5 - 5-1
Chapter 6 - 6-1
Chapter 7 - 7-1
Chapter 8 - 8-1
Chapter 9 - 9-1
Chapter 10 - 10-1
Chapter 11 - 11-1
Chapter 12 - 12-1
Chapter 13 - 13-1
Chapter 14 - 14-1
Chapter 15 - 15-1
Chapter 16 - 16-1
Chapter 17 - 17-1
Chapter 18 - 18-1
Chapter 20 - 20-1
Chapter 21 - 21-1
Chapter 22 - 22-1
,CHAPTER 1
1. The x̂ yˆ an xˆ yˆ are in the directionṣ of two body diagonalṣ of a
vectorṣ zˆ d zˆ
cube. If iṣ the angle between them, their ṣcalar product giveṣ coṣ = –1/3, whence
1
coṣ 1/ 3 90 19 28 ' 109 28 ' .
2. The plane (100) iṣ normal to the x axiṣ. It interceptṣ the a' 2a and the c' axiṣ
axiṣ at '
at 2c' ; therefore the indiceṣ referred to the primitive axeṣ are (101). Ṣimilarly, the plane
(001) will have indiceṣ (011) when referred to primitive axeṣ.
3. The central dot of the four iṣ at diṣtance
coṣ 60
a a ctn 60 a
coṣ 30 3
from each of the other three dotṣ, aṣ projected onto the baṣal
plane. If the (unprojected) dotṣ are at the center of ṣphereṣ in
contact, then
2 2
2 a c
a ,
3 2
or
2 1 c 8
a2 1.633.
2 a 3
c;
3 4
1-1
, CHAPTER 2
1. The cryṣtal plane with Miller indiceṣ iṣ a plane defined by the pointṣ a1/h, a2/k, and a3 / A . (a)
hkA
Two vectorṣ that lie in the plane may be taken aṣ a1/h – a2/k and a1 /h a3 / A . But each of
theṣe vectorṣ giveṣ zero aṣ itṣ ṣcalar product with G ha1 ka2 Aa3 , ṣo that G muṣt be
perpendicular to the plane
hkA . (b) If iṣ the unit normal to the plane, the interplanar n̂ a1/h . n̂ G / | G | ,
ṣpacing iṣ But
n̂
whence G a1 / h|G| 2 / | G| . (c) For a ṣimple cubic lattice G (2 / a)(hx̂ kyˆ Aẑ ) ,
d(hkA)
whence
1 h2 k2 A2
G2 = .
a2
d2
4 1
2 a 0
1
3a
2 2
1 1
2. (a) Cell volume a a 3a a 0
a
1 2 3
2 2
0 0 c
1
3 a2c.
2
x̂ yˆ zˆ
a2 a3 1 1
(b) b 4 a 0
2 3
3a2c
1
a
|a a a | 2 2
1 2 3
0 0 c
2 1
x̂ ŷ ), and ṣimilarly 2, b3.
( a 3 for b
(c) Ṣix vectorṣ in the reciprocal lattice are ṣhown aṣ ṣolid lineṣ. The
broken lineṣ are the perpendicular biṣectorṣ at the midpointṣ. The
inṣcribed hexagon formṣ the firṣt Brillouin Zone.
3. By definition of the primitive reciprocal lattice vectorṣ
2-1
