Crystal Structure

PERIODIC ARRAYS OF ATOMS

Lattice translation vectors
Basis and the crystal structure
Primitive lattice cell
FUNDAMENTAL TYPES OF LATTICES
Two-dimensional lattice types
Three-dimensional lattice types
INDEX SYSTEM FOR CRYSTAL PLANES
SIMPLE CRYSTAL STRUCTURES
Sodium chloride structure
Cesium chloride structure
Hexagonal close-packed structure
Diamond structure
Cubic zinc s d d e structure
DIRECT IMAGING OF ATOMIC STRUCTURE 18
NONIDEAL CRYSTAL STRUCTURES 18
Random stacking and polytypism 19
CRYSTAL STRUCTURE DATA 19
SUMMARY 22
PROBLEMS 22

1. Tetrahedral angles
2. Indices of planes
3. Hcp structure

UNITS: 1 A = 1 angstrom = 10-'cm = O.lnm = 10-"m.

 (c)
Figure 1 Relation of the external form of crystals to the form of tlre elementary building hlackr.
The building blocks are identical in (a) and (b), but Merent c ~ s t a lfaces ;Ire developed.
( c ) Cleaving a crystal of rocksalt.
 CHAPTER 1: CRYSTAL STRUCTURE

PERIODIC ARRAYS O F ATOMS

The serious study of solid state physics began with the discovery of x-ray
diffraction by crystals and the publication of a series of simple calculations of
the properties of crystals and of electrons in crystals. Why crystalline solids
rather than nonclystalline solids? The important electronic properties of solids
are best expressed in crystals. Thus the properties of the most important semi-
conductors depend on the crystalline structure of the host, essentially because
electrons have short wavelength components that respond dramatically to the
regular periodic atomic order of the specimen. Noncrystalline materials, no-
tably glasses, are important for optical propagation because light waves have a
longer wavelength than electrons and see an average over the order, and not
the less regular local order itself.
We start the book with crystals. A crystal is formed by adding atoms in a
constant environment, usually in a solution. Possibly the first crystal you ever
saw was a natural quartz crystal grown in a slow geological process from a sili-
cate solution in hot water under pressure. The crystal form develops as identical
building blocks are added continuously. Figure 1 shows an idealized picture of
the growth process, as imagined two centuries ago. The building blocks here
are atoms or groups of atoms. The crystal thus formed is a three-dimensional
periodic array of identical building blocks, apart from any imperfections and
impurities that may accidentally be included or built into the structure.
The original experimental evidence for the periodicity of the structure
rests on the discovery by mineralogists that the index numbers that define the
orientations of the faces of a crystal are exact integers. This evidence was sup-
ported by the discovery in 1912 of x-ray diffraction by crystals, when Laue de-
veloped the theory of x-ray diffraction by a periodic array, and his coworkers
reported the first experimental observation of x-ray diffraction by crystals.
The importance of x-rays for this task is that they are waves and have a wave-
length comparable with the length of a building block of the structnre. Such
analysis can also be done with neutron diffraction and with electron diffraction,
hut x-rays are usually the tool of choice.
The diffraction work proved decisively that crystals are built of a periodic
array of atoms or groups of atoms. With an established atomic model of a crys-
tal, physicists could think much further, and the development of quantum the-
ory was of great importance to the birth of solid state physics. Related studies
have been extended to noncrystalline solids and to quantum fluids. The wider
field is h o w n as condensed matter physics and is one of the largest and most
vigorous areas of physics.
 An ideal crystal is constn~ctedby the infinite repetition of idenbcal groups
of atoms (Fig 2) A group is called the basis. The set of mathematical points to
which the basls is attached is called the lattice The lattice in three dimensions
may be defined by three translabon vectors a,, a,, a,, such that the arrange-
ment of atoms in the crystal looks the same when viewed from the point r as
when viewed from every polnt r' translated by an mtegral multiple of the a's:

Here u,, u,, u , are arhitraryintegers. The set of points r' defined by (1)for all
u,, u,, u,defines the lattice.
The lattice is said to be primitive if any two points from which the atomic
arrangement looks the same always satisfy (1)with a suitable choice of the in-
tegers ui.This statement defines the primitive translation vectors a,. There
is no cell of smaller volume than a, .a, x a, that can serve as a building block
for the crystal structure. We often use the primitive translation vectors to de-
fine the crystal axes, which form three adjacent edges of the primitive paral-
lelepiped. Nonprimitive axes are often used as crystal axes when they have a
simple relation to the symmetry of the structure.

Figure 2 The crystal srmchre is formed by

the addition af the basis (b) to evely lattice
point of the space laisice (a). By looking at
( c ) , one oan recognize the basis and then one
can abstract the space lattice. It does not
matter where the basis is put in relation to a
lattice point.
 1 Crystal Struchrre 5

Basis and the Crystal Structure

The basis of the crystal structure can be idendled once the crystal mes
have been chosen Figure 2 shows how a crystal is made by adding a basis to
every lamce pomnt-of course the lattice points are just mathematical con-
structions. Every bas~sin a given crystal is dentical to every other ~n composi-
tmn, arrangement, and orientation
The number of atoms in the basis may be one, or it may be more than one.
The position of the center of an atom3 of the basis relahve to the associated
lattice point is

We may arrange the origin, wl~ichwe have called the associated lattice point,
so that 0 5 x,, yj, zj 5 1.

(b) (c)
Fieure 3a Latttce nomts of a soam lnmce m two &mensrons AU ~ a r of
s vector? a,,a, are trans-

twice the area of a primitive cell.

Figare 3b Primitive ccll "fa space lattice in three dimensions

Figure 3c Suppose these points are identical atoms: Sketchin on the figure a set of lattice points,
s choice ofprimitive axes, aprimitivc cell, and the basis of atoms associatedwith alattice point.
F i p e 4 A primitive cell may also be chosen fol-
lowing this procedure: (1) draw lines to connect a
given lattice point to all nearby lattice points: (2) at
the midpoint and normal to these lines, draw new
lines or planes. The smallest volume enclosed in this
way is the Wigner-Seitz primitive cell. All space may
be filled by these cells, just as by the c e h of Fig. 3.

Primitive Lattice Cell

The parallelepiped defined by primitive axes al,a,, a, is called a primitive
cell (Fig. 3b). A primitive cell is a t).pe of cell or unit cell. (The adjective unit is
superfluous and not needed ) A cell will fill all space by the repetition of suit-
able crystal b-anslatlon operationq. A primitive cell is a m~nimum-volumecell.
There are many ways of choosing the prnnitive axes and primitive cell for a
given lattice. The number of atoms m a primitive cell or primtive basis is
alwap the same for a given crystal smllcture
There is always one lattice point per primitive cell. If the primitwe cell is a
parallelepiped with lattice po~utsat each of the eight corners, each lattice
point is shared among e ~ g hcells,
t so that the total number of lattice points in
the cell is one: 8 X = 1.The volume of a parallelepiped wrth axes a,, %, a3is

-
V, = (a, as X a, 1 , (3)
by elementary vector analysis. The basis associated wrth a primitive cellis cdued
a prim~tivebans. No basis contans fewer atoms than a pnmibve basis contains.
Another way of choosing a primitive cell is shown in Fig. 4. This is h o w n to
physicists as a Wigner-Seitz cell.

FUNDAMENTAL TYPES OF LATTICES

Crystal lattices can be carried or mapped into themselves by the lattice

translations T and by vanous other symmetzy operattons. A typical symmetry
operation is that of rotation about an axis that passes through a lattice point.
Lattices can be found such that one., two-, three., four., and sixfold rotation
axes cany the lattice into itself, corresponhng to rotatrons by ZT, 2 ~ 1 22, d 3 ,
2 ~ 1 4and
, 2 ~ 1 6radians and by integral multiples of these rotations. The rota-
tion axes are denoted by the symbols 1,2, 3, 4, and 6.
We cannot find a lattice that goes into itself under other rotations, such as
by 2.d7 rachans or 2 ~ / rachans.
5 A single molecule properly designed can have
any degree of rotational symmetry, but an Infinite periodic lattice cannot. We
can make a crystal from molecules that individually have a fivefold rotation ms,
but we should not expect the latbce to have a fivefold rotation m s . In Fig. 5 we
show what happens if we try to construct a penodic latbce havlng fivefold
 Figure 5 A fivefold t u i s of symmetry can-
not exist in a periodic lattice because it is
not ~ossibleto fa thc area of a plane with
a mmected m a y of pentagons. We can,
however, fill all the area oTa plane with just
two distina designs of "tiles" or elemeutary
potysms.

(c) (e)
Figure 6 (a) A plane of symmetry to the faces of a cube. (b) A diagonal plane of s)rmmehy
in a cube. (c) The three tetrad xes of a cube. (d)Thc four tridaxes of a cube. (e) The six diad axes
of a cube.

symmetry: the pentagons do not fit together to fdl all space, shoulng that we can-
not combmne fwefold point symmetrywith the r e q u e d translational penodicity.
By lattice point group we mean the collection of symmetry operations
which, apphed about a lathce pomt, carry the lattice into itself The possible ro-
tations have been listed. We can have mirror reflecbons m about a plane through
by -r The symmetry axes and symmetry planes of a cube are shown in Fig 6.

Two-DimensionalLattice Types
The lattice in Fig. 3a was d r a m for arbitrary al and as. A general lattice
such as this is known as an oblique lattice and is invariant only under rotation
of .rr and 27r ahout any lattice point. But special lattices of the oblique type can
, 2 d 6 , or under mirror reflection.
he invariant under rotation of 2 ~ 1 32, ~ 1 4or
We m i s t impose restrictive conditions on a, and a% if we want to construct a lat-
tice that will he invariant under one or more of these new operations. There are
four distinct types of restriction, and each leads to what we may call a special
lattice type. Thus there are five distinct lattice types in two dimensions, the
oblique lattice and the four special lattices shown in Fig. 7. Bravais lattice is
the common phrase for a distinct lattice.%e; we say that there are five Bravdrs
lattices in two dimensions.

Figure 7 Four special lattices in twodirnmsions,

 1 Crystal St-ture

Three-Dimensional Lattice Types

The point symmetry groups in three dimensions require the 14 different
lattice types listed in Table 1. The general lattice is triclinic, and there are
13 special lattices. These are grouped for convenience into systems classified
according to seven types of cells, which are triclinic, monoclinic, orthorbom-
bic, tetragonal, cubic, higonal, and hexagonal. The division into systems is
expressed in the table in terms of the adal relations that describe the cells.
The cells in Fig. 8 are conventional cells: of these only the sc is a primitive cell.
Often a nonprimitive cell has a more obvious relation with the point symmetry
operations than has a primitive cell.
There are three lattices in the cubic system: the simple cubic (sc) lattice,
the body-centered cubic (hcc) lattice, and the face-centered cubic (fcc) lattice.
Table 1 The 14 latlioe types in three dimensions

Numhrr of Restr~chonson cunvenhonal

System lathces cell axes and angles

Triclimc

Monoclinic

Orthorhomb~c

Cubic

Trigonal

Hexagonal

Figure 8 The cubic space lattices. The cells s h m are the conventional cells.
 a
Lattice points p e r cell 1 2 4
Volume. primitive cell a3 ZQ
1 3 xa
1 3
Lattice points per unit volume l/a3 2/aR 4/a3
Numher of nearest neighbors 6 8 12
Nearest-neighbor distance a 3u2a/2 = 0.866a a/2'" 0.707a
Number of second neighbors 12 6 6
Second neighbor distance 2'% a a
Packing fractionn ZW &V5 id5
=0.524 =O.B80 =0.740

"The packing fraction is the manirnum proportion of the available volume that can be filled
with hard spheres.

Figure 10 Pnmibve translation vectors of the body-

~i~,,9 , . ~~ ~ d c,lbic~ lattice,
. shouing
~ a~ centered
~ ~ cubic
~ lattice;
~ these
~ vectors
d connect the lattice
prilnitive ~h~ rimifive she%,,,, is a rhonl+,o point at the origin to lattice points at the body centers.
2
hedron ,f edge ; 3 a, and the angle behen adja.
cent edges is 109"ZR'.
The primitive ceU is obtained on completing the rhody-
huhedron. In terms of the cube edge a, the primitive
translation vectors are
al=ia(i+i-2) ; as=$~(-i+j+Li).
a3=;a(%-f+i)
Ifere i , j , i are the Cartesian untt vectors

The charactenstics of the three cubic lattxes are summanzed in Table 2. A

prirmhve cell of the bcc lattice is shown in Fig. 9, and the primtive tran~lation
vectors are shown in Fig. 10 The primitive translabon vectors of the fcc lathce
are shown In Fig. 11. Primitive cells by definition contain only one lattice
point, but the conventional bcc cell contains two lattice points, and the fcc cell
contains four lattice points.
 I Crystal Stwctun 11

Figore 11 The rhombohedra1 primitive cell of the hce-centered Figore 12 Relation of the plimitive cell
cubic clystal. The primitive translation vectors a,, a,, connen in the hexagonal system (heavy lines) to
the lattice point at the origin with lattice points at the face centers, a prism of hexagonal symmew. Here
As drawn, the primitive vectors are: rr,=o,#a,.

The angles between& axes are 60'

The pos~honof a point in a cell is spec~fiedby (2) in terms of the atomic

coordinates x , y, z. Here each coordinate is a fraction of the axla1length a,, a,,
a, in the direction of the coordinate axis, with the origin taken at one corner of
the cell Thus the coorhnate? of the body center of a cell are ;$$, and the face
centers include iiO, 0,s;
11 1 1
5%. In the hexagonal system the primitive cell is a
right prism based on a rhomhu3 with an included angle of 120". F~gure12
shows the relabonship of the rhombic cell to a hexagonal prism.

INDEX SYSTEM FOR CRYSTAL PLANES

The orientahon of a crystal plane is determined by three points in the

plane, provided they are not collinear. If each point lay on a different crystal
axis, the plane could be specdied by giving the coordinates of the points in
terms of the latbce constants a,, a,, a3.However, it turns out to b e more useful
for structure analysis to specify the orientation of a plane by the indices deter-
mined by the followlug rules (Fig. 13).

Find the intercepts on the axes in terms of the lattice constants a,, a,, a,.
The axes may be those of a primitive or nonprimibve cell.
Figure I3 This plane intercepts
the a,, +, a, axes at 3a,, Za,, Za,

.,.
The recrprocals of these numbers
, I &
are 5, The smallest three mte-
gers havlng the same rabo are 2, 3,
3, and thus the m&ces of the plane
are (233) I

Figure 14 Inlces oElmportant lanes in a cubx ctystal l (100) and

The plane (200)is ~ a r d eto
to (100)

Take the reciprocals of these numbers and then reduce to three integers
having the same ratio, usually the smallest three integers. The result, en-
closed in parentheses (hkl),is called the index of the plane.
For the plane whose intercepts are 4 , 1 , 2 ,the reciprocals are $, 1,and $: the
smallest three integers having the same ratio are (142).For an intercept at infin-
ity, the corresponding index is zero. The indices of some important plades in a
cubic crystal are illustrated by Fig. 14. The indices (hkl) may denote a single
plane or a set of parallel planes. If a plane cuts an axis on the negative side of the
origin, the corresponding index is negative, indicated hy placing a minus sign
 1 Crystal Structure 13

above the index: (hkl). The cube faces of a cubic crystal are (100). (OlO), (OOl),
(TOO),( o ~ o )and
, (001). Planes equivalent by s y m m e q may he denoted by curly
brackets (braces) around indices; the set of cube faces is {100}.When we speak
of the (200) plane we mean a plane parallel to (100) but cutting the a, axis at i n .
The indices [uvw] of a direction in a clystal are the set of the smallest inte-
gers that have the ratio of the components of a vector in the desired direction,
referred to the axes. The a, axis is the [loo] direction; the -a, axis is the [ o ~ o ]
direction. In cubic crystals the direction [hkl] is perpendicular to a plane (hkl)
having the same indices, but this is not generally true in other crystal systems.

SIMPLE CRYSTAL STRUCTURES

We di~cusssimple crystal structures of general interest the sohum chlo-

nde, cesium chloride, hexagonal close-packed, h a m o n d and c u h ~ czinc sulfide
structures.

Sodium Chloride Structure

The sohum chloride, NaCI, structure 1s shown in Figs. 1.5 and 16. The
lattice is face-centered cublc: tile basis conslsts of one Na+ Ion and one C 1 ion

Figure 15 We may construct the soditzrn chloride

cvstal shuchlre by arranging Naf and C 1 ions alter-
nately at the lattice points o f a simple cubic lattice. I n
the ctysral each ion is surrounded by six nearest neigh-
bors of the opposite charge. The space lattice is fcc,
...
.
and the basis has one C I ion at 000 w d one Na'
L,? ,L. The 'figre shows one conventional cubic cell.
The ionic diameters here are reduced in relation tothe
Fiip r e 16 M
?md e r than th
ulll cirlnride The sodium iuns are
ns. (Courtesy of A. N. Holden and
cell in order to darify the spatial arrangement. P. :Singer)
Pigure 17 Na1ui.d c~?stals of lead snlfide. PbS. whir11lias the Figure 18 The cesium chloride cqstal
l ~ R. Burl?aon.)
NaCl crystal stmcturc. ( P h u t a g ~ a pby struehne. The space lattice is silnple
cubic, and &the basis has one Cst ion at
000 and one C 1 ion at i.

separated by one-half the body diagonal of a umt cube There are four units of
NaCl ~neach unit cube, with atoms in tlie pos~tions

Each atom has as nearest neighbors six atoms of the opposire kind. Represen-
tative crystals having the ~ karrangement
l include those in the following
table. The cube edge a is given in angstroms; 1if --cm lo-'' m 0.1
nm. Figure 17 is a photograph of crystals of lead sulfide (PbS) from Joplin,
Missouri. The Joplin specimens form in beautiful cuhes.

Cesium Chloride Stnccture

The cesium chloride structure is shown in Fig. 18. There is one molecule

.
per primitive cell, with atoms at the comers 000 and body-centered positions
,l -,i -l of the simple cubic space lattice. Each atom may he viewed as at the center
-
Figure 19 A close-packed layer of spheres is shown, with centers at points marked A. A second
and identical layer of spheres can he placed on top of this, above and patallel to the plane of the
drawing, with centen over the points marked B. There are two choices for a third layer. It can go
in over A or over C . Ifit goes in over A, the sequence is AEABAB . . . and the structure is hexagonal
close-packed. If the third layer goes in over C, the sequence is ABCABCABC . . . and the Struchlre
is face-centered cubic.

B Figure 20 The hexagonal close-packed structure.

c The atom positions in t h i s smcture do not constitute
a soace lattice. The mace lattice is s i m ~ l ehexa~onal

A I w i k a basis of hvo ;dentical atoms asiociatedwith

each latttce nomt. The lamce parameters o and c are
inhcated, where o is in the basal plane and c is the
rnagmtude of the an?a, of R g 12

of a cube of atoms of the opposite kind, $0 that the number of nearest neigh-
bors or coordination number is eight.

Hexagonal Close-Packed Structure (hcp)

There are an infinite number of ways of arranging identical spheres in a
regular array that maximuzes the packing fraction (Fig. 19) One 1s the face-
centered cub^ structure; another is the hexagonal close-packed structure
(Fig. 20). The fraction of the total volume occupled by the spheres is 0.74 for
both structures. No structure, regilar or not, has denserpaclung.
Figure 21 The primitive cell has a, = h,with an
included angle of 120". The c axis (or a,) is normal
to the plane of a, and a,. The ideal hcp stmohre has
c = 1.633 a, The two atoms of one basis are shown
as solid circles. One atom of the bais is at the ori-
gin; the other atom is at $Qb, which means at the
position r = $a, + +a, + $a,.

Spheres are arranged in a single closest-packed layer A by placmg each

sphere in contact with SIX others in a plane. This layer may serve as either the
basal plane of an hcp structure or the (111)plane of the fcc s h c t u r e . A sec-
ond s~milarlayer B may be added by placlng each sphere of B in contact with
three spheres of the bottom layer, as in Figs. 19-21. A third layer C may be
added In two ways. We obtam the fcc structure if the spheres of the third layer
are added over the holes in the first layer that are not occupied by B We
obtain the hcp structure when the spheres in the third layer are placed directly
over the centers of the spheres in the first layer.
The number of nearest-nelghbor atoms is 12 for both hcp and fcc stnlc-
tures. If the b~ndingenergy (or free energy) depended only on the number of
nearest-neighbor bonds per atom, there would be no difference in energy
between the fcc and hcp structures.

Diamond Structure
The diamond structure is the structwe of the semiconductors silicon and
germanium and is related to the structure of several important semicondnctor
binary compouncLs. The space lattice of damond 1s face-centered cubic. The
primitive basis of the diamond structure has two identical atoms at coordinates
000 and 2;; assoc~atedmth each point of the fcc latt~ce,as shown in Fig. 22.
Because the convenbonal unit cube of the fcc lattice contains 4 latbce points,
i t follows that the conven~onalunit cube of the dlamond structure contams
2 X 4 = 8 atoms. There is no way to choose a primitive cell such that the basis
of diamond contains only one atom.
Figure 22 Atomic positions in the cubic cell uf the diamond Figure 23 Crystal structure of diamond,
s t ~ u c h ~projected
rr on a cub? face; fiacticms denote height showingthetetrahedralbondarrangement.
above the hasp in units of a cubc edge. The paints at 0 and $
are on the fcc lattice those at and are on a similar lattice
displvcerl along the body diagonal by one-fourth of its lengh.
With a fcc space lattice, the basis consists of mia identical
atoms at 000 0 d i i ; .

The tetrahedral bonding characteristic of the diamond structure is shown

in Fig. 23. Each atom has 4 nearest neighbors and 12 next nearest neighbors.
The diamond structure is relatively empty: the maximum proportion of the
available volume which may he filled by hard spheres is only 0.34, which is 46
percent of the f11ling factor for a closest-packed stmcture such as fcc or hcp.
The diamond structure is an example of the directional covalent bonding
found in column IV of the periodic table of elements. Carbon, silicon, germa-
nium, and tin can crystallize in the diamond structure, with lattice constants
n = 3.567, 5.430, 5.658, and 6.49 A, respectively. Here a is the edge of the
conventional cubic cell.

Cubic Zinc Sulfide Structure

The diamond structure may be viewed as twn fcc structures displaced
from each other by one-quarter of a body diagonal. The cubic zinc sulfide
(zinc blende) structure results when Zn atoms are placed on one fcc lattice and
S atoms on the other fcc lattice, as in Fig. 24. The conventional cell is a cube.
The coordinates of the Zn atoms are 000; 0;;; $0;; $ $0; the coordinates of the
,
1 1 1 1 3 3 313 351
S atoms are 444; 44; 2 jj;4 4 4. The lattice is fcc. There are four molecules
'5-
ZnS per conventional cell. About each atom there are four equally distafft
atoms of the opposite kind arranged at the comers of a regular t e t r a h e d ~ w ..-
 Figure 24 Crystal structure of cubic zinc
sulfide

The diamond structure allows a center-of-inversion symmetry operation

at the midpoint of every hne between nearest-neighbor atoms. The inversion
operation carries an atom at r into an atom at -r. The cubic ZnS struc-
ture does not have inversion symmetry. Examples of the cubic zinc sulfide
structure are

The close equality of the lattice constants of several pairs, notably (Al, Ga)P
and (Al, Ga)As, makes possible the construction of sem~conductorhetemjunc-
tions (Chapter 19).

DIRECT IMAGING OF ATOMIC STRUCTURE

Direct images of crystal structure have been produced by transmission

electron microscopy. Perhaps the most beaubful Images are produced by scan-
ning tunneling microscopy; in STM (Chapter 19) one exploits the large vana-
tions in quantum tunneling as a function of the height of a fine metal tip above
the surface of a crystal. The image of Fig 25 was produced m t h way. ~ ~ An
STM method has been developed that will assemble single atoms Into an orga-
nized layer nanometer structure on a crystal substrate.

NONIDEAL CRYSTAL STRUCTURES

l",v
;
.
:
t
-
..,.*
.,
>;
. ,.
The ideal crystal of classical crystallographers is formed by the periodic
~ 7 '
repetition of identical units in space. But no general proof bas been given that
 I Crystal Structure 19

Figure 25 A scanning tunneling microscope

image of atorns on a (111)surface of fcc plat-
inum at 4 K.The nearest-neighbor spacing is
2.78 A. (Photo courtesy of D. M. Eigler, IHM
R r e r a r c h Divi~irn.)

the ideal crystal is the state of minimum energy of identical atoms at the tem-
perature of absolute zero. At finite temperatures this is likely not to be true. We
give a further example here.

Random Stacking and Polytypism

The fcc and hcp structures are made up of close-pack<:d planes I3f atoms.
- .
The structures differ in the stacking sequence of the planes, fcc having the se-
quence ABCABC . . . and hcp having the sequence ABABAB . . . . Structures
are h o w n in which the stacking sequence of close-packed planes is random.
This is known as random stacking and may be thought of as crystalline in two
dimensions and noncrystalline or glasslike in the third.
Polytypism is characterized by a stacking sequence with a long repeat
unit along the s t a c h g axis. The hest known example is zinc sulfide, ZnS, in
which more than 150 polytypes have been identified, with the lnngest period-
icity being 360 layers. Another example is silicon carbide, Sic, which occurs
with more than 45 stacking sequences of the close-packed layers. The polytype
of SiC known as 393R has a primitive cell with a = 3.079 A and c = 989.6 A.
The longest primitive cell observed for Sic has a repeat distance of 594 layers.
A given sequence is repeated many times within a single crystal. The mecha-
nism that induces such long-range crystallographic order is not a la~ng-range
force, but arises from spiral steps due to dislocations in the growtlI nucleus
(Chapter 20).

CRYSTAL STRUCTURE DATA

In Table 3 we hst the more common crystal stmctureq and lattlce structures
of the elements Values of the atomic concentration and the density are glven in
Table 4. Many dements occur m several crystal structures and transform from
 Table 3 Crystal structures of the elements
.rhe data given are at morn temperature for the most common form, Y.
the stated temperature in deg K. (Inorganic Cr)-stal Stlucture Database
 Table 4 Density and atomic concentration
The data are given at atmosphericpressure and mom temperature, or at the

mzg~mr~~~m~eam~s~m~~exsm~rs~vv
more stable.

SUMMARY

A lattice is an array of points related by the lattice translation operator

T = ula, f uza2 f u,a3, where u,, us, u3 are integers and a,, a,, a, are the

. crystal axes.
To form a crystal we attach to every lattice point an identical basis composed
+
of s atoms at the positions r, = xja, y,a, + zja3,withj = 1,2, . . . , s. Here
x, y, z may be selected to have values between 0 and 1.
. The axes a,, a*, a3 are primitive for the minimum cell volume la,. as X a,(
for which the crystal can be constructed from a lattice translation operator T
and a basis at every lattice point.

Problems
1. Tetrahedral angles. The angles behveen the tetrahedral bonds of diamond are the
same as the angles between the body diagonals of a cube, as in Fig. 10. Use elemen-
tary vector analysis to find the value of the angle.
2. Indices of planes. Consider the planes with indices (100) and (001);the lattice is
fcc, and the indices refer to the conventional cubic cell. What are the indices of
these planes when referred to the primitive axes of Fig. II?
3. Hcp structum. Show that the c/a ratio for an ideal hexagonal close-packed struc-
ture is (:)IR = 1.633. If c/a is significantly larger than this value, the crystal structure
may be thought of as composed of planes of closely packed atoms, the planes being
loosely stacked.