60, and each of its axes is l/\/2 times the length of the axes of the

cubic cell.

Each

cubic cell has four lattice points associated with_it, each

rhombohedral

cell has _one,

and

the

former

_has, correspondingly, four times the

volume

of the latter.

Never-theless, ^{it is} usually

more

convenient touse the cubic cell rather

than

the

rhombohedral

one because the former

immediately

suggests the cubic

symmetry ^which

^{the lattice} actually possesses. Similarly, the other cen-tered nonprimitive ^{cells listed in}

Table

2-1 are preferred to the primitive cells possible ⁱⁿtheir respective ^lattices.

Ifnonprimitive lattice cells are used, the vector

from

the _{origin to}

any

point ⁱⁿ the lattice will

now have components which

^are nonintegral

mul-tiples oftheunit-cellvectors _a,b, ^c.

The

_position of

any

^lattice point ⁱⁿ a

cell

may

be given ⁱⁿ

terms

of itscoordinates] ^if the vector

from

the _origin of the unit cell to the given point has

components

xa, yb, zc,

where

x, y,

and

z are _fractions, then the coordinates of the point ^are x_yz.

Thus,

point

A

inFig. 2-7,

taken

asthe_origin, has coordinates

000

while points Bj C,

and _D, when

referred to cubic _axes,

have

coordinates

Off,

f f^,

and _{f f}

0, respectively. Point

E

has coordinates

f \

¹

and

is equivalent FIG. 2-7. Face-centered cubicpoint lattice referred to cubic and rhombo-hedral cells.

2-6]

LATTICE DIRECTIONS AND PLANES

37 to point Z), being separated

from

it

by

^{the vector c.}

The

coordinates of equivalent points ⁱⁿ different unit cells

can always

^be

made

identical

by

the addition or subtraction of a set of integral coordinates; in this case, subtraction of 1

from f

^ ¹ (the coordinates of

E)

gives

^ f

^(the coordinates of D).

Note

that the coordinates of a body-centered point, for example, ^are

always

| ^

^ no matter whether

^{the unitcellis} cubic, tetragonal, or ortho-rhombic,

and whatever

its size.

The

coordinates ofa point position, such as

^ ^

^\,

may

^alsobe regarded^as

an

_{operator which,}

when

"applied" ^to a point ^at the _origin, ^will

move

or translate it to the _position \

\

\, the final position being obtained

by

simple addition ^of the operator

\ \ \ and

the _{original position}

000.

In this sense, the _positions

000, \ \ \

are called the "body-centering translations," since they ^will

produce

^the

two

_point positions characteristic ^of a body-centered ^cell

when

_applied to a point^atthe _{origin. Similarly,} the four pointpositions characteristic ofa face-centered _cell,

namely

0, \ ^, \ ^,

and

_\

_\

_0, are called the face-centering translations.

The

base-centering translations

depend on which

_pair of opposite ^{faces are} centered; ^if centered

on

the

C

_face, for example, they ^are 0, \ \^0.

2-6

Lattice directions

and

_planes.

The

direction of

any

^{line in a}

lat-tice

may

be described

by

^first

drawing

^{a line}

through

^the origin parallel to the given ^line

and

then _giving the coordinates of

any

point

on

the line

through

^the origin. Let the line pass

through

^the origin ^of the unit cell

and any

point

having

coordinates

u

vw,

where

these

numbers

arenot neces-sarily integral. (This ^{line will also} pass

through

^the points

2u

2v2w,

3u

3v3w, etc.)

Then

_[uvw], written in square brackets, are the indices of the direction of the line.

They

^{are also the indices of}

any

line parallel to the given line, since the lattice is infinite

and

the _origin

may

^be

^taken

at

any

point.

Whatever

the values of _{i/, v,} w, they^are

always

^converted toasetofsmallest integers

by

multi-plication or divisionthroughout:thus,

[||l],

[112],

and

_[224] all represent the

same

_direction, but _[112] is the preferred form.

Negative

indices are written with a bar over the

number,

e.g.,[uvw]. Directionindicesare illus-trated in Fig. 2-8.

Direction^^related

by symmetry

^are

called directions ofa form,

and

a set ofthese are|Pepresented

by

^theindices

of

one

of

them

enclosed in angular

bracHts;

^for example, the four

body

_Fib/^-8.

[100]

38

THE

[CHAP.

diagonals ^of a _{cube, [111],} [ill], [TTl],

and

_[Til],

may

^all be represented

by

^the

symbol

(111).

The

orientation of planes ⁱⁿ a lattice

may

^{also be} represented

sym-bolically,according^toa

system

popularized

by

the English crystallographer Miller. In the _{general case,} the given plane^will be tilted with _{respect to} the crystallographic axes, and, ^since these axes

form

a convenient

frame

of reference,

we _might

describe the orientation of the plane

by

giving the actual distances,

measured from

the _origin, at

which

^it intercepts the three axes. Better _still,

by

expressing these distances as fractions of the axial lengths,

we

can obtain

numbers which

are

independent

^of the par-ticular axial lengths involved ⁱⁿ the given ^lattice.

But

a _difficulty then arises

when

the given plane ^is parallel to a certain crystallographic axis, because such a plane does not intercept that _{axis, i.e.,} its "intercept" can only

be

described as "infinity."

To

avoid the introduction of _infinityinto the description ^of plane orientation,

we

can use the reciprocal of the frac-tional intercept, ^this reciprocal being ^zero

when

_{the plane}

^and

^{axis are}

parallel.

We

^{thusarrive at a}

workable symbolism

^{for the} orientation of a plane ⁱⁿ a_lattice, the Miller _indices,

which

are defined as ^thereciprocals of thefractional intercepts which the plane

makes

with the crystallographic ^axes.

For

_example, ^if the Miller indices of a plane ^are (AW), written in paren-theses, then the plane

makes

fractional intercepts of I/A, I/A*, \/l with the axes,and, ^ifthe axiallengths ^are a, 6, c, the plane

makes

actual intercepts ofa/A, b/k, c/l, as

shown

ⁱⁿ Fig. 2-9(a). Parallel to

any

plane ⁱⁿ

any

lat-tice, thereis a

whole

set of_parallelequidistant planes,

one

of

which

_passes

through

^the origin; the Miller indices (hkl) usually ^{refer to} that plane ⁱⁿ theset

which

^isnearest the _origin, although they

may

^{be taken} as referring to

any

other plane ⁱⁿ theset or to the

whole

set

taken

together.

We _may

determine the Miller indices of the plane

shown

ⁱⁿ Fig. 2-9_(b) as follows^:

1A 2A 3A 4A

(a) (b)

FIG. 2-9. Plane designationby^{Millerindices.}

2-6]

LATTICE DIRECTIONS AND PLANES 39

Miller indices are

always

^cleared of fractions, as

shown

above.

As

stated earlier, ^ifa plane ^isparallelto a given axis, ^its fractional intercept

on

that axis is

taken

as _infinity

and

the corresponding ^Miller index is zero. If a plane cuts a negative ^axis, the corresponding index^is negative

and

is writ-ten

with

a bar over it. Planes

whose

indices are the negatives ^of

one

another are _parallel

and

^lie

on

_oppositesides of the _{origin, e.g.,} (210)

and

(2lO).

The

_{planes (nh}

nk

_nl) are _parallel tothe planes (hkl)

and have 1/n

the_spacing.

The same

_plane

may

belong ^to

two

different _sets, the Miller indices ofonesetbeing multiples^ofthoseoftheother;thus the

same

_plane belongs ^to the _{(210) set}

and

the ₍₄₂₀₎ set, and, in fact, the planes ^of the (210) ^set

form

every second plane ⁱⁿ the (420) ^set. jjn the cubic system,

it is convenient ^to

remember

^{that a direction} [hkl] ^is

always

perpendicular to a plane ^{(hkl) of} the

same

_indices,

but

thisis not generally true ⁱⁿ other systems.

Further

familiarity with Miller indices can

be

_gained

from

a

study

of Fig. 2-10.

A

slightly different

system

^of plane indexing ^is

used

in the hexagonal system.

The

unit cell of a hexagonal ^{lattice is} defined

by two

equal

and

coplanar vectors ^ai

and

a2, at 120 to

one

_another,

and

a third axis c at right angles [Fig. 2-11_(a)].

The _complete

lattice ^is built up, as usual,

by

HfeocH

(110)

(110) (111)

FIG. 2-10. Millerindices of latticeplanes.

(102)

In document WITHIN THE (pagina 54-57)