Che materia stai cercando?

Reticoli 3D - Gonzales Szwacki

Materiale didattico per il corso di Cristallografia della prof.ssa Elena Buonaccorsi. Trattasi del capitolo 2 del testo di Gonzalez Szwacki dal titolo "Basic Elements of Crystallography", all'interno del quale è trattato il tema dei reticoli tridimensionali dei cristalli.

Esame di Cristallografia docente Prof. E. Bonaccorsi

Anteprima

ESTRATTO DOCUMENTO

25

Three-Dimensional Crystal Lattice

Figure 30 Conventional cells, of the most general shape, for the 7 crystal systems in three

(a) (b) (c) (d) (e)

dimensions: triclinic, monoclinic, orthorhombic, tetragonal, cubic,

(f) (g)

trigonal, and hexagonal.

In the lattices belonging to the orthorhombic, tetragonal, and cubic

systems three mutually perpendicular symmetry axes coexist and the basis

26 Basic Elements of Crystallography

G G G

vectors , , are parallel to them. Figs. 30c and 30e show the

a a a

1 2 3

conventional cells and the three twofold and fourfold rotation axes for

the orthorhombic and cubic systems, respectively, while Fig. 30d shows

the conventional cell and a fourfold rotation axis present in lattices of the

tetragonal system. The remaining symmetry axes (not shown in Fig. 30d) are

The restrictions for the conventional cell parameters are

twofold.

summarized in Table 1. These constrains guarantee the presence of three

mutually perpendicular symmetry axes of the orders specified in

Figs. 30c-30e.

The lattices with only one threefold or sixfold symmetry axis belong

to the trigonal or hexagonal systems, respectively. A solid figure that

possesses a sixfold symmetry axis has the shape of a hexagonal prism shown

in Fig. 30g. In Fig. 30g is also shown a parallelepiped whose volume

represents of the volume of the hexagonal prism. This parallelepiped

1 3 G is

is a conventional cell for the hexagonal system. Its basis vector a 3

G G

parallel to the sixfold symmetry axis and the basis vectors and are

a a

1 2

lying in a plane orthogonal to this symmetry axis. The restrictions pointed

out in Table 1 guarantee the presence of a sixfold symmetry axis in the

lattices belonging to the hexagonal system. In Fig. 31 we show that the

conventional cell for the trigonal system and the hexagonal prism are

related. This will be explained in more details later. The restrictions on

the rhombohedral cell parameters given in Table 1 guarantee the presence

Table 1 Restrictions on conventional cell parameters for each crystal system. The following

G G G G G G

( ) ( ) ( )

α α α

= = =

abbreviations are used: , , .

a ,a a ,a a ,a

) ) )

1 2 12 1 3 13 2 3 23

Restrictions on conventional cell parameters

Crystal system , , , and , ,

α α α

a a a

1 2 3 12 13 23

Triclinic None

α α

= = °

90

Monoclinic 13 23

α α α

= = = °

90

Orthorhombic 12 13 23

=

a a

1 2

Tetragonal α α α

= = = °

90

12 13 23

= =

a a a

1 2 3

Cubic α α α

= = = °

90

12 13 23

= =

a a a

1 2 3

Trigonal α α α

= =

12 13 23

=

a a

1 2

Hexagonal α α α

= ° = = °

120 , 90

12 13 23 27

Three-Dimensional Crystal Lattice

of a threefold symmetry axis. Of course in the case when

G G G G G G

( ) ( ) ( )

= = = °

) ) )

a ,a a ,a a ,a 90

1 2 1 3 2 3

we are in the presence of a cube and this threefold symmetry axis coincides

with one of the four threefold axes of the cube.

To summarize, we can say that the order and the number of the highest

order symmetry axes characterize a crystal system. All the highest order

xes for each of the 7 crystal systems are shown in Fig. 30.

symmetry a

Conventional Cell for the Trigonal

6. System

The conventional cell for the trigonal system takes on the shape of a

rhombohedron. This rhombohedron can be constructed in a hexagonal

prism, what is shown in Fig. 31. We can see in that figure that two vertices

of the rhombohedron are located in centers of the hexagonal prism bases

the

and the other 6 form two groups with 3 vertices each. The plane defined by

the three vertices of one group is parallel to the prism bases, what means that

these vertices are at the same distance from tance between the

a base. The dis

three vertices which are closer to the top base and this base is the same as

the distance between the vertices from the other group and the bottom base,

of the prism height

and represents (see Fig. 31)

c .

1 3

The positions of the vertices belonging to each of the two groups can be

determined easily as their projections on the plane of the nearer prism base

Figure 31 Rhombohedron constructed inside a hexagonal prism.

28 Basic Elements of Crystallography

(bottom or top) coincide with the geometric centers of three equilateral

triangles, what is shown in Fig. 31. We can also see in figure that these

this

triangles are not next to each other and the three triangles of the bottom base

do not coincide with those of the top base.

Bravais Lattices

7. The 14

7.1. Introduction

In this section, we will describe all the three-

dimensional lattices or

more strictly speaking lattice types. If we place lattice points at the vertices

of each parallelepiped that represents the conventional cell of one of the

seven crystal systems, we obtain 7 different lattices. All the points

then

at the vertices of a cell contribute with 1 point to this cell. This is

placed

explained in Fig. 32 on the

th

e example of a cubic cell. A point placed in a

vertex of a cube belongs to 8 cubes (4 of which are shown in Fig. 32), so 1 8

of it belongs to each cube. Since there are 8 points at the vertices of the

ibute with 1

cube, they contr point to it and the cell is primitive.

A French scientist, Bravais (second half of the XIX century),

demonstrated that if we place an additional point in the geometric center

additional points of the parallelepipeds representing

or on the faces

conventional cells of the seven crystal systems (in such a way that the

points has the same symmetry

set of as the parallelepiped), then we will

lattices or strictly speaking lattice types. Therefore, we have a

obtain 7 new

total of 14 lattice types . It will be shown later that 11 of

in three dimensions

them belong to the monoclinic, orthorhombic, tetragonal, cubic

or crystal

ach of the remaining crystal systems (triclinic, trigonal, and

systems. E

hexagonal) has only one lattice type. In the case of each of the 7 new

parallelepiped, the same

lattices, the which represents the unit cell that has

Figure 32 1 8 of the point placed in each belongs to this cell

cell.

A vertex of a cubic cell 29

Three-Dimensional Crystal Lattice

symmetry as the infinite lattice, has more than one lattice point. Therefore,

this unit cell represents a non primitive cell of the lattice while a primitive

cell of such a lattice does not have its point symmetry.

Next, we will build the 7 new lattices mention above which are called

the centered Bravais lattices.

7.2. The Triclinic System

In the case of the triclinic system, there is only one lattice type. The

arguments are very simple. Since in the case of the triclinic system there are

no restrictions on its conventional cell parameters, a primitive cell of any

triclinic lattice represents a conventional cell of the triclinic system. By

placing additional points out of the vertices of the conventional cell, we

transform a primitive cell of one triclinic lattice onto a non primitive cell of

another triclinic lattice, but of course both lattices are of the same type since

a primitive cell of this new lattice represents another conventional cell for

the triclinic system.

7.3. The Monoclinic System

In Fig. 33a, we have placed lattice points at the vertices of the

conventional cell for the monoclinic system shown in Fig. 30b. This cell

can be centered in several different ways as shown in Figs. 33b-33d and

in Fig. 34. In all cases, the set of lattice points has the same point symmetry

as the conventional cell of the monoclinic system. Note that in Figs. 33 and

34 we have changed the notation for the cell parameters, and now we are

, , . The cells from Figs. 33 and 34 have their

using instead of

a, b, c a a a

1 2 3

unique symmetry axes parallel to the edges. Consequently, we speak of the

c

setting with unique axis (for short setting). In the case of the

c c-axis C-face

centered cell, shown in Fig. 33b, the centering lattice points are in the cell

bases (orthogonal to the Figs. 33c and 33d show the same cell, but

c-edge).

this time body and all-face centered, respectively. There are still two more

options for placing the additional lattice points within the conventional cell

of the monoclinic system. This is shown in Figs. 34a and 34b for the A-face

centered and centered cells, respectively. The symbols for the

B-face

centering types of the cells shown in Figs. 33 and 34 are listed in Table 2.

Let us now investigate to which monoclinic lattice types belong the

centered cells shown in Fig. 33. In Fig. 35, we demonstrate that in the lattice

shown in Fig. 33b we can find a primitive cell of the same type as the cell

from Fig. 33a, so this is, in fact, a primitive monoclinic lattice. It can be also

30 Basic Elements of Crystallography

Figure 33 (a) Conventional primitive cell for the simpl

simple

e monoclinic lattice. In the figure,

(b)

we have also drawn the cell from (a) centered in three different ways: face centered,

C-

(d)

(c) c

body centered, and all-face centered. The -axis setting is assumed.

Figure 34 (a) A

The cell from Fig. 33a centered in two different ways: face centered and

-face

(b) B c

face centered. The

-face -axis setting is assumed.

demonstrated (see Fig. 36) that in the lattice from Fig. 33d, there is a body

centered cell of the same type as the cell from Fig. 33c, so this is a body

centered monoclinic lattice. From all the above, we can conclude that

Three-Dimensional Crystal Lattice 31

Table 2 Symbols for the centering types of the cells shown in Figs. 33 and 34.

lattice points

Number of

Symbol Centering type of a cell per cell

P Primitive 1

A A-face centered 2

B B-face centered 2

C C-face centered 2

I Body centered 2

F All-face centered 4

Figure 35 unit located in

inside

A primitive cell of the same type as the cell shown in Fig. 33a,

the monoclinic lattice from Fig. 33b.

33 The c-axis setting is assumed.

Figure 36 A body centered e as the cell shown in Fig. 33c,

unit cell of the same type placed

the monoclinic lattice from Fig. 33d. The

inside c-axis setting is assumed.

we have four cells belonging to only two types of monoclinic

in Fig. 33

lattices, for which the arrangements of the lattice points

nts are shown in

Figs. 33a and 33c.

Next, we will check the cases shown in Fig. 34. In lattice type

each

this figure

figure, cell. This is

plotted in we can find a body centered monoclinic

in Fig. 37 Thus, in the

demonstrated for the lattice from Fig. 34b.

32 Basic Elements of Crystallography

Figure 37 A body centered of the same type as the cell shown in Fig. 33c

33c,

unit cell located

the monoclinic lattice from Fig. 34b. The

inside c-axis setting is assumed.

monoclinic lattice shown in this figure,

figure there are two different conventional

(body centered and contain the same

monoclinic cells B-face centered) that

number

ber of lattice points

points, face centered

so we may consider this lattice as a B-face

or a body centered. Similarly, the monoclinic lattice from Fig. 34a may be

A

A-face Therefore

Therefore,

considered as an centered or a body centered. if we

xis setting, then

the face centered, and the

assume the c-axis the A-face centered, B-face

body centered monoclinic lattices are In conclusion,

conclusion

mutually equivalent.

there are only two types of monoclinic lattices, the primitive one and one of

the following three lattices: face centered, or body

A-face centered, B-face

face centered lattice is selected to represent the centering

centered. The B-face

monoclinic lattice (if the axis setting is assumed). The symbols

type of the c-axis

of the two monoclinic lattice types are then However, in

mP and mB. the

literature we can find more often the case when the axis setting is

b-axis

and then the In this case,

assumed, mA, mC, and mI lattices are equivalent.

is selected to identify the center

centering

ing type of the monoclinic

the mC lattice

lattice. the axis setting) lattices the

mB (c-axis setting) or mC (b-axis

In the case of

smallest cell that has the point symmetry of the infinite lattice contains 2

points,

nts, while the primitive cells of these lattices do not have their

lattice

point symmetry.

The Orthorhombic System

7.4. In the same way,

way for the monoclinic system,

as it was done in Sec. II.7.3

we can place the lattice points within the conventional cell for the

orthorhombic system. The resulting set of points will have the same point

33

Three-Dimensional Crystal Lattice

Figure 38 (a) Conventional primitive orthorhombic lattice

lattice.

unit cell of the simple In the

(b)

figure, we have also drawn the cell from C-

(a) centered in three different ways: face

(c) (d) (e)

body centered, and shows two bases for two

centered, all-face centered. Figure

adjacent cells from (b) and (d). The base of a primitive (or body centered) cell of the lattice

shown in (b) (or (d)) is highlighted in (e).

symmetry as the cell. Since the three edges in the conventional cell for this

system are mutually orthogonal, only

onl y the cases described in Fig. 38 will be

considered. The rest of the cases, with face centered cells, do not

and

A- B-

lead to any new lattice types. his time,

Contrary to the monoclinic system, t

neither the case from Fig. 38b he cases

nor the case from Fig. 38d match t

. 38a a primitive

described in Figs and 38c, respectively, since neither nor a

are present in the

body centered unit cells with edges mutually orthogonal

lattices shown in Figs. 38b and 38d, respectively. Therefore, we can

conclude that in the case of the there are four types of

orthorhombic system,

lattices: primitive ( ), and all-

), face centered ( ), body centered (

oP C- oC oI

face centered ( ).

oF

34 Basic Elements of Crystallography

7.5. The Tetragonal System

The conventional unit cell of the tetragonal system, instead of having

a rectangle at the base (as it was the case of the orthorhombic system), has a

square. For this system, we have to analyze the same types of centering of

its conventional cell, as those shown in Figs. 38b-38d for the orthorhombic

system. Thus, the C-face centered, body centered, and all-face centered

tetragonal cells will be considered. Here, as before, the c edge is orthogonal

to the cell base. The presence of a fourfold axis parallel to the c edge

excludes the possibility of having A- and B-face centered cells in tetragonal

lattices. It is easy to demonstrate that now the lattice represented by the C-

face centered tetragonal unit cell is effectively a primitive lattice and the

lattice represented by the all-face centered tetragonal unit cell is just a body

centered lattice. This is shown in Fig. 39, where we have displayed two

bases of a C-face centered or all-face centered tetragonal unit cells. One of

those bases is labeled as I, whereas the square marked as II is the base of a

primitive or body centered tetragonal cell.

To conclude, we can say that in the case of the tetragonal system, there

are two types of lattices: tP and tI.

Figure 39 Two bases (labeled as I) of two adjacent C-face centered and all-face centered

tetragonal unit cells. The base labeled as II corresponds in one case to a primitive tetragonal

unit cell and in the other case to the body centered tetragonal cell.

7.6. The Cubic System

We now move to the case of a cubic system. The search for the possible

lattices belonging to this system will be held using again Fig. 38. This time,

the only relevant cases are those described in Figs. 38a, 38c, and 38d, since,

due to the point symmetry, those are the only cases that can be a priori

expected in the lattices of the cubic system (remember that in the present

consideration all the cells shown Fig. 38 are cubes). In the cubic lattice with

35

Three-Dimensional Crystal Lattice

Figure 40 bases (labeled as I) of two adjacent all

all-face

Two centered cubic unit cells. The

base labeled as II corresponds to a noncubic body centered unit cell.

the same arrangement of lattice is excluded

points as shown in Fig. 38d, it

cubic is explained in Fig. 40.

the presence of a body centered unit cell, what

Lastly, we can conclude that in the case of the there are

cubic system,

. The primitive cubic lattice is also called

3 types of lattices: , , and

cP cI cF

) and the body and face centered cubic lattices are

simple cubic ( all-

sc

commonly abbreviated (face centered

as (body centered cubic) and

bcc fcc

respectively.

cubic),

The Trigonal and Hexagonal Systems

7.7. There is only one type of lattices, namely, in the trigonal and

and

hR hP

hexagonal systems, respectively. Later we will explain the origin of the

for the trigonal lattice.

symbol used

hR

Finally, we may say that there are all together 14 types of lattices in

three dimensions, called the Bravais lattices. The 14 Bravais lattices are

shown in Fig. 41.

Symbols for Bravais

7.8. Lattices

In Table 3 are summarized the symbols for the 14 Bravais lattices. We

can observe in this table, that the lattices are classified in 6 crystal families,

that are symbolized by lower case letters (see column two

, , , , , and

a m o t h c

of Table 3). The second classification is according to the discussed by us 7

crystal systems. We can see in the table that in three dimensions the

classifications according to crystal families and crystal systems are the same

except for the hexagonal family, which collects two crystal

cr

ystal systems: trigonal

and hexagonal. The two parts of the Bravais lattice symbol are: first, the

symbol of the crystal family and second, a capital letter ( ,

, , , )

P S I F R

designating the Bravais lattice centering. As a reminder, the symbol is

P

the primitive lattices. The symbol face centered

given to denotes a one-

S symbols for the

lattice ( and are the standard, setting independent,

mS oS

36 Basic Elements of Crystallography

Table 3 Symbols for the 14 Bravais lattices.

Crystal Family Symbol Crystal System Bravais Lattice Symbol

Triclinic a Triclinic aP

(anorthic) mP

Monoclinic m Monoclinic mS (mA, mB, mC)

oP

oS (oA, oB, oC)

Orthorhombic o Orthorhombic oI

oF

tP

Tetragonal t Tetragonal tI

Trigonal (rhombohedral) hR

Hexagonal h Hexagonal hP

cP

Cubic c Cubic cI

cF

one-face centered monoclinic and orthorhombic Bravais lattices,

respectively). For the last case also the symbols A, B, or C are used,

describing lattices centered at the corresponding A, B, or C faces. The

symbols F and I are designated for all-face centered and body centered

Bravais lattices, respectively. Finally, the symbol R is used for a trigonal

lattice.

7.9. Conclusions

To conclude we can say that the carried out identification of the 14

Bravais lattices was nothing more than the classification of all the three-

dimensional lattices in 14 groups. The lattices belonging to a given group

have the same point symmetry. Besides that, they have the same number

and location of the lattice points within the smallest unit cell, which has the

point symmetry of the lattice. We could see that half of the Bravais lattices

appear as centered ones. This means, the smallest unit cells, that have the

same point symmetry as the infinite lattices, contain more than one lattice

point. However, it is important to point out that for each of the 14 Bravais

lattices it is possible to choose a unit cell that contains only one lattice point,

it means, a primitive unit cell. The basis vectors, , , , that define

G G G

a a a

1 2 3

such a cell are primitive translation vectors of the Bravais lattice. Finally, a

Bravais lattice represents a set of points whose positions are given by

G defined as

vectors R Three-Dimensional Crystal Lattice 37

Figure 41 The 14 Bravais lattices. The conventional cells of the crystal systems are that from

Fig. 30.

38 Basic Elements of Crystallography

G G G G

= + +

R a a a , (II.1)

n n n

1 1 2 2 3 3

∈ .

where ]

, ,

n n n

1 2 3

Coordination Number

8. Since all the lattice points in a Bravais lattice have equivalent positions

in space, they have identical each point has the

surroundings. Therefore,

NN

NNs

s (points that are the closest to it) and this number,

same number of , is a characteristic of a given Bravais lattice.

called the number

coordination number,

In the literature, we find more often an alternative definition in which the

coordination number is the number s of an atom in a crystal (or

of the NN

molecule). Our definition, however, is more general, since we may think of

substituting the lattice points with different objects like single atoms, or

groups of atoms, or even molecules, and the definition still remains

rem

ains valid,

objects will have identical surroundings.

since all these

On occasions the information about the next nearest

neares

t neighbors (NNNs)

and even the third nearest neighbors of a lattice point

(TNNs) is also

42 shows the s of a lattice point in

important. Figure NNs, NNNs, and TNN

lattice. In this figure, the s of a lattice point placed in the center of

the NN

sc of 8 smaller cubes) are placed of a regular

a large cube (built at the vertices

octahedron. Since the octahedron has 6 vertices, the coordination number

lattice is 6. The s are in the middle of the 12 edges of the

for the NNN

sc

large cube, so there are 12 lattice. The

NNNs of a lattice point in the sc

tice point in consideration are of the large

TNNs of the lat at the vertices

Figure 42 sc

The NNs, NNNs, and TNNs of a lattice point in a lattice. The 6 NNs of a lattice

point placed in the center of the large cube are of the regular octahedron. The

at the vertices

12 NNNs are in the middle of the large cube edges and the 8 TNNs are in its vertices. 39

Three-Dimensional Crystal Lattice

cube, so the number of them is 8. lattice is equal

The NN distance in the sc

to the lattice parameter , and the TNN distance

the NNN distance is

a, 2a

3a

is .

Body Centered Cubic Lattice

9. 43 shows three examples of a set of three primitive translation

Figure

vectors that define the primitive unit cell of the lattice. In these three

bcc

cases at least one of the vectors involves two of lattice points,

“types”

namely, those from cube vertices and those

t

hose from cube centers. This, of

course, is essential in the case of a primitive cell, since with this cell it is

possible to reproduce the entire lattice. The primitive cell defined by vectors

G G G in Fig. 43c is shown in Fig. 44. it is drawn a

, , In this figure,

a a a

1 2 3

rhombohedron which represents the most symmetric primitive unit cell of

lattice. We can figure that one diagonal of the

the also see in this

bcc

rhombohedron is lying along one of the diagonals of the cube. Those

diagonals represent a threefold axis . This is the unique threefold

of each cell

Figure 43 bcc

Three sets of three primitive translation vectors of the lattice.

40 Basic Elements of Crystallography

Figure 44 bcc

A primitive rhombohedral unit cell of the lattice.

axis of the rhombohedron, while the cube has still three more such axes. The

G G G

angles between the basis vectors 43c and 44

, , shown in Figs. are

a a a

1 2 3

the same: G G

G G G G

( ) ( ) ( ) ′

= = = °

a ,a a ,a a ,a 109 28

) ) )

.

1 2 1 3 2 3

Next, we will calculate the volume of the primitive unit cell and

0

compare it with the volume of the cube. The volume of given by

this cell is

ˆ ˆ ˆ

x y z § ·

1 1 1

G G G

( ) 1 1 1 ˆ ˆ ˆ

= × ⋅ = − ⋅ − + +

a a a x y z

a a a a a a

ȍ ¨ ¸

© ¹

0 1 2 3 2 2 2 2 2 2

1 1 1

a a a ,

2 2 2

§ · § ·

1 1 1 1 1 1 1 1

3 3 3

2 2

= + ⋅ − + = + =

+ +

ˆ ˆ ˆ ˆ ˆ ˆ

0x y z x y z

a a a a a a a a

¨ ¸ ¨ ¸

© ¹ © ¹

2 2 2 2 2 4 4 2 (II.2)

the volumes ratio is

and 3

V a

cube = = 2 (II.3)

.

1

ȍ 3

a

0 2 41

Three-Dimensional Crystal Lattice

A primitive unit cell of the lattice has one lattice point while the cubic

bcc

cell has two points. The ratio, given by Eq. (II.3), between the cell volumes

is equal to the ratio between the numbers of points belonging to them.

Therefore, the same volume corresponds to each lattice point.

0

us now demonstrate that the two points that are within the

Let cubic

bcc

cell have equivalent positions in the lattice. This is shown and

unit bcc

explained in Fig. 45. In s of a point of the

Fig. 46 we show the NN bcc

This lattice has a coordination number 8.

lattice.

We will now consider the lattice points within the cubic cell of the bcc

lattice. It is convenient being a sum of

sometimes to associate the point,

Figure 45 Demonstration of the equivalence of the two lattice points within the cubic unit cell

bcc

of the lattice.

Figure 46 bcc

The lattice points from the vertices of the cubic unit cell of the lattice represent

the NNs of the lattice point that is in the center of the cell.

42 Basic Elements of Crystallography

Figure 47 (a) A cubic cell of the lattice points located at the vertices

bcc lattice. Each of the

1 8 (b)

to the unit cell so Positions of t

the

contributes with the cell contains 2 lattice points.

2 points within the cubic cell.

cell placed in the

The point, which is a sum of eight fractions, is

vertex of the cube that coincides with the origin of the cell. The coordinates are expressed in

units of a. Fig.

eight fractions (see f the vertices of the cube.

47a), with only one o

shows

Figure 47b such point in the cube vertex that coincides with

a G ( )

G G

initial point of the basis vectors

the 0,0,0

, , . Its position is . The

b

a c

of the second lattice point within the cubic cell

position given with respect

,

G ( )

G G

to the , , axes, 1 2,1 2,1 2 , where the coordinates are expressed

is

b

a c G

G G G

vector

in units of a. The t 1 2a 1 2 b 1 2 c represents one of the

= + +

vectors of the bcc

shortest translation the

lattice and it coincides with vector

In Figs. 43b and 43c there are shown other examples of

from Fig. 43a.

G

a 3 shortest translation vectors in the bcc

the lattice. At least one of such vectors

to appear in each set of basis vectors that define a

has unit

primitive cell of

this lattice (see Fig. the lattice point located in the center of

43), since both,

cubic cell and the point

the represented by

from its vertices, are then the

vert

lattice point from ices of the primitive unit cell.

Face Centered Cubic Lattice

10. First, us consider

let unit

the lattice points within the cubic cell of

The two

the fcc A-

lattice. faces (orthogonal to

lattice points placed in the

G

the basis vector in half of the

Fig. 48a) contribute to the cubic cell with

a

will

point each. We represent these lattice

two fractions with one point

G

G G

that contains the origin of the basis vectors

placed in the A-face

face , ,

b

a c

is

(see Fig. 48b), that is,

, at the shortest distance from the origin. The position

( )

of this point is 0,1 2,1 2 expressed in units

, where the coordinates are

Three-Dimensional Crystal Lattice 43

Figure 48 (a) (b)

face centered cubic unit cell of the Positions of the four

All-face fcc lattice.

lattice points within the cubic cell. The coordinates are expressed in units of which is the

a,

length of the cube edge.

a which is the length of the basis vectors. ,

of , In the similar way we can

B- C-

the he positions of

place the points in and faces (see Fig. 48b). Finally, t

fcc

the four lattice points belonging to the cubic cell of the lattice are:

(0,0,0) , , , and .

(0,1 2,1 2) (1 2,0,1 2) (1 2,1 2,0) fcc

49 shows the most symmetric primitive unit cell

Figure of the lattice.

. Each of them represents one of

It is defined by the basis vectors , ,

G G G

a a a

1 2 3

fcc

the shortest translation vectors of the lattice. In the definition of the

four lattice points belonging to

vectors , , are involved all the the

G G G

a a a

1 2 3

and this primitive cell can reproduce

cubic unit cell guarantees that the the

entire lattice. The primitive unit cell shown in Fig. 49 takes on the shape of a

is inscribed in the cubic cell. The threefold axis of the

rhombohedron that

rhombohedron coincides

coin

cides with one of the threefold axis of the cube. The

define this axis are of the two cells, while

lattice points that at the vertices

the rest of the rhombohedron vertices coincide with the centers of the cube

faces. Figure 49 A primitive rhombohedral unit cell of the fcc lattice.

44 Basic Elements of Crystallography

For the cell form Fig. 49 it is easy to show that

G G G G G G (II.4)

(a , a ) (a , a ) (a , a ) 60

= = = ° .

) ) )

1 2 1 3 2 3

Indeed, since ( 2 2) we have that

= = =

a a a a

1 2 3 2

G G G G G G

2

a a cos (a ,a ) cos (a ,a )

⋅ = =

a a a (II.5)

) )

1 2 1 2 1 2 1 2

4

and using the vector coordinates we have also that 1

G G 2

a a

⋅ = + + =

a a a a a a a (II.6)

,

1 2 1 2 1 2 1 2

x x y y z z 4 G G

then comparing the two expressions for the scalar product , we obtain

a a

1 2

2 1 1

G G G G

Ÿ

2 2

c

os (a ,a ) cos (a ,a )

= =

a a (II.7)

.

) )

1 2 1 2

4 4 2

Repeating the same procedure as done in Eqs. (II.5-7), for all the vector

G G G

, , of a primitive

pairs, we finally get Eq. (II.4). So the basis vectors a a a

1 2 3

rhombohedral unit cell of the lattice are at angles of to each other.

fcc 60°

Let us now calculate the volume, , of the primitive unit cell of the fcc

0

lattice and compare it with the volume of the cubic cell. We have that

ˆ ˆ ˆ

x y z § ·

1 1

G G G

( ) 1 1 ˆ ˆ

a a 0 y z

a

= × ⋅ = ⋅ +

a a a a

ȍ ¨ ¸

© ¹

3

0 1 2 2 2 2 2

1 1 0

a a

2 2

§ · § ·

1 1 1 1 1 1 1 1

3 3

2 2 2 3

ˆ ˆ ˆ ˆ ˆ

y (II.8)

x y z z

a a a a a a a a

= − + + ⋅ + = + =

¨ ¸ ¨ ¸ .

© ¹ © ¹ 4

8 8

4 4 4 2 2

The primitive unit cell has one lattice point while the cubic cell contains four

lattice points, so the ratio between the volumes of these cells

3

V a

cube 4

= = (II.9)

1

ȍ 3

a

0 4

is equal to the ratio between the numbers of lattice points in them. 45

Three-Dimensional Crystal Lattice

Figure 50 Demonstration of the equivalence of all lattice points in the cubic cell of the fcc

lattice. we will demonstrate that lattice points within the

Next, different cubic

e.

unit cell of the lattice have equivalent positions in this lattic This is

fcc

In the explanation we are using two sets of cubes.

shown in Fig. 50. The

second set of cubes in Fig. 50)

(represented by a gray colored cube is

obtained by translating the first one in the direction of a diagonal from

half of the diagonal The correspondence of the lattice

its bases by length.

points in the two sets of cubes is Upon making a

explained in Fig. 50. side

similar translation, but now in a plane which coincides with the faces

sets of ult will be that the points that are in

of one of the two cubes, the res

the middle of the faces of the two types of cubes (from the two sets) will

y the positions of the points at ,

occup the vertices. In this manner we can

demonstrate the equivalence between the lattice points

positions of at the

vertices and faces of the cube.

Figure 51 fcc

Nearest neighbors of a point in the lattice.

46 Basic Elements of Crystallography

Since all the lattice points in the cubic lattice have equivalent

fcc

positions, the neighborhood of each lattice point is the same and therefore,

each point has the same number of s. We will consider the neighborhood

NN

from the face of a cube. shown in Fig. 51.

of a lattice point This is As

explained in this figure the coordination number of the lattice is 12.

fcc

Rhombohedral Unit Cell in a Cubic Lattice

11. We have already learned in Secs. II.9 and II.10 that a rhombohedron

represents a primitive unit cell of both the ices. A cube,

and the latt

bcc fcc

which is a primitive cell of the case of a

lattice, is also a particular

sc

rhombohedron. However,

However , a rhombohedron represents, at first, the

conventional unit cell the trigonal system, and now we know that when

of

G G G cell are at

the basis vectors , , of a primitive rhombohedral unit

a a a

1 2 3 ′

°

109 28 is a primitive

angles of , or , or to each other, then this cell

90°

90

60°

cell of a lattice belonging to the cubic system, it means, possesses a higher

symmetry than the symmetry of a trigonal lattice. Moreover, the

point

presence of a rhombohedral unit cell with its threefold symmetry axis in a

cubic lattice is not surprising, since this lattice possesses threefold symmetry

As next we will show a centered rhombohedral unit cell in the

axes. sc

lattice. ohedral Unit Cell

11.1. Rhombohedral of the Lattice

sc of course,

Besides of the primitive rhombohedral unit cells there are,

centered rhombohedral unit cells in belonging to the cubic system.

lattices

Figure 52 sc

A body centered rhombohedral unit cell of the lattice.


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AUTORE

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DETTAGLI
Corso di laurea: Corso di laurea in scienze geologiche
SSD:
Università: Pisa - Unipi
A.A.: 2011-2012

I contenuti di questa pagina costituiscono rielaborazioni personali del Publisher Atreyu di informazioni apprese con la frequenza delle lezioni di Cristallografia e studio autonomo di eventuali libri di riferimento in preparazione dell'esame finale o della tesi. Non devono intendersi come materiale ufficiale dell'università Pisa - Unipi o del prof Bonaccorsi Elena.

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