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§2.3 Indices of crystal planes & directions Ⅰ. What are crystal planes and directions ? The atomic planes and directions passing through the crystal are called (crystal) planes and directions respectively.
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材料科学基础 Fundamentals of Materials ScienceFundamentals of Materials Science
Chapter 2Fundamentals of CrystallologyLan YuFaculty of Material Science and EngineeringKunming University of Science and technology
ReferencesReferences 1. E. J.Mittemeijer, Fundamentals of materials science. Springer-Verlag
Berlin Heideiberg 2010. 2. W.D. Callister, J.r, Foundations of materials science and engineering.
USA.5th-ed. John Wiley & Sons, Inc. 2001. 3. W.F. Smith, Foundations of materials science and engineering. New York,
McGraw-Hill book Co.1992. 4. C. Kittel, Introduction to solid state physics. USA. 8 th-ed. John Wiley &
Sons, Inc. 2005. 5. 周公度，结构与物性，第三版，高等教育出版社， 2009. 6. 潘金生等，材料科学基础，修订版，北京，清华大学出版社， 2011. 7. 胡赓祥，蔡珣等 材料科学基础，第三版，上海，上海交通大学出版社， 2010.
§2.3 Indices of crystal planes & §2.3 Indices of crystal planes & directionsdirections
ⅠⅠ. What are crystal planes and directions ?. What are crystal planes and directions ?
The atomic planes and directions passing through the crystal The atomic planes and directions passing through the crystal are called (crystal) planes and directions respectively.are called (crystal) planes and directions respectively.
1. Steps to determinate the plane indices: ① Establish a set of coordinate axes ② Find the intercepts of the planes to be indexed on a, b
and c axes (x, y, z).
a
c
b
x
y
z
ⅡⅡ. Plane indices. Plane indices
③ Take the reciprocals of the intercepts 1/x, 1/y, 1/z.④ Clear fractions but do not reduce to lowest integers.⑤ Enclose them in parentheses, (h k l) Example: 1/2,1,2/3 2,1,3/2 (423)
Plane indices referred to three axes a, b and c are also called Miller Indices.
Several important points for the Miller indices of planes :
Planes and their negatives are identical. Therefore .
Planes and their multiples are not identical.
In cubic systems, a direction that has the same indices as a plane is perpendicular to that plane.
)020()020(
2. The important planes in cubic crystals2. The important planes in cubic crystals
(110(110))
(112(112))
(111(111))
(001(001))
3. A family of planes consists of equivalent planes so far as the atom arrangement is concerned.
)110()011()110(
)101()101()110(}110{
Total: 6Total: 6
)111(
)111()111()111(}111{
Total: 4Total: 4
)121()112()112()211(
)112()121()211()121(
)211()211()121()112(}112{
)132()123()213()321()231()213()123()312()321(
)312()132()213()123()132()312()231()213()231()321(
)132()312()321()231()123(}123{
Total: 12Total: 12
Total: 4×3Total: 4×3 ！！ =24=24
晶面族 {hkl}
晶体中具有相同条件（原子排列和晶面间距完全相同），空间位向不同的各组晶面。用 {hkl} 表示。 如在立方晶胞中 同属 {111}晶面族。 11 111
-111 1 111（ ），（ ），（ ），（ ）
ⅢⅢ. Direction Indices. Direction Indices
1. Derivation for the crystallographic direction1. Derivation for the crystallographic direction① Firstly, set a point on the indexed direction as the origi
n point of coordinate axes.
② Find the coordinates of another point on the indexed direction ： x ， y ， z.
③ Reduce x ， y ， z to three smallest integers: u, v, w.
④ Enclose in square brackets [u v w].
* 指数看特征，正负看走向
（ x1,y1,z1 ） ,(x2,y2,z2) 二点连线的晶向指数： [x2-x1,y2-y1,z2-z1]
晶向族 <uvw> 晶体中原子排列情况相同但空间位向不同的一组晶向，用 <uvw> 表示。
在立方晶系里，数字相同，但排列顺序不同或正负号不同的晶向属于同一晶向族。
如 <100>=[100]+[001]+[010]
晶向族 <u v w>具有等同性能的晶向归并而成
2. The important direction in cubic crystals: <100> : crystal axes <110> : face diagonal <111> : body diagonal <112> : apices to opposite face-centers [’eipisi:z] apex[’eipeks]
3. Family of directions consists of crystallographically equivalent directions, denoted <u v w>
e.g.
]100[]010[]001[
]001[]010[]100[100
§2.4 §2.4 Hexagonal axes for hexagonal Hexagonal axes for hexagonal crystalscrystals
ⅠⅠ. Why choose four-axis system?. Why choose four-axis system?
Four indices has been devised for hexagonal unit cells because of the unique symmetry of the system.
aa
cc
)100(
]110[]100[
)011(
bb
ⅡⅡ. Plane indices (. Plane indices (hkilhkil)) It can be proved: i ≡ － (h ＋ k)
)0110()100(
)0011()011(
Important planes :
a1
a2
a3
c
)0110(
)0211(
)2110(
)0001(
)1110(
ⅢⅢ. Direction indices [ . Direction indices [ u v t wu v t w ] ] To make the indices unique, an additional condition is
imposed. ---- Let t＝－ (u＋ v)
Important directions
]0001[
]0101[
]0111[
]0112[
晶面指数：在四个轴上的截距，求倒数，整数化 （ h k i l ） h+k+i=0晶向指数：行走法， [u v t w] ， u+v+t=0
Transformation of indicesTransformation of indices Transformation of 3 to 4 indices, or vice versa. Suppose we have a vector, whose 3 indices [u v w], and 4 indices [u v t w].
We haveWe have cwatavauL 321
cWaVaU 21
SinceSince )()( 213 vutaaa
or:
cWaVaU
cwaavuavau
21
2121 ))((
cWaVaUcwavuavu
21
21 )2()2(
wWuvVvuU 22
)2(31 VUu )2(31 UVv
Ww )( vut
1. Quick way for indexing the directions in cubic crystals: The value of a direction depends on its feature while the
sign on direction.
Examples and DiscussionsExamples and Discussions
2. The coordinate origin can be set arbitrarily (for example on apices, body-center, face-centers etc.), but never on plane in questions, otherwise the intercepts would be 0,0,0 .
3. The coordinate system can be transferred arbitrarily, but rotation is forbidden.
4. The atomic arrangement and planar density of the important direction in cubic crystal.
plane indices
BCC FCCatomic
arrangement planar density atomic arrangement planar density
{100}
{110}
{111}
2214
14
aa
224.1
2
1414
aa
22
58.0
23
613
aa
2221
414
aa
224.1
2212
414
aa
22
3.2
23
213
613
aa
a
a
a
a2
a2a2
a2
a
a
a
a2
a2 a2
a2
5. The atomic arrangement and linear density of the important direction in cubic crystal.
linear indices
BCC FCCatomic
arrangement linear density atomic arrangement linear density
<100>
<110>
<111>
aa12
12
aa7.0
2212
aa16.1
3
1212
aa12
12
aa4.1
2
1212
aa58.0
3212
a a
a2 a2
a3 a3
Exercise1. Calculate the planar density and planar packing fraction for the
(010) and (020) planes cubic polonium, which has a lattice parameter of 0.334nm.
214
2
atoms/cm1096.8
334.01
faceofareafaceperatom
)010(densityplanar
Solution
79.0)2(
)(1
faceofareafaceperatomsofarea)010(fractionpacking
2
2
rr
However, no atoms are centered on the (020) planes. There fore, the planar density and the planar packing fraction are both zero.
Thanks !
