Introduction to The Solid State

CmPT R

II

I n t r o d u c t i o n to The Solid S t a t e This t r e a t i s e is w r i t t e n to elucidate a n d explain the c h a r a c t e r i s t i c s of p h o s p h o r s a n d the p h e n o m e n o n of l u m i n e s c e n c e . T h a t is, the n a t u r e of materials which

absorb e n e r g y a n d recruit

it as light.

We e n c o u n t e r

p h o s p h o r s every day w h e n we w a t c h television. The television t u b e (or p l a s m a p a n e l as it m a y be) h a s dots (or lines) c o m p o s e d of red, g r e e n a n d b l u e - e m i t t i n g p h o s p h o r s . T h e s e p r o d u c e the active p i c t u r e s t h a t we v i e w from Cable or T V - N e t w o r k b r o a d c a s t s s u c h as ABC, CBS a n d NBC. We also u s e p h o s p h o r s in f l u o r e s c e n t l a m p s as inside (and outside) lighting in o u r homes

a n d offices. However,

in o r d e r

to u n d e r s t a n d

how p h o s p h o r s

p e r f o r m , we n e e d to k n o w s o m e t h i n g about the n a t u r e of solids in o r d e r to c o m p r e h e n d t h e l u m i n e s c e n t state. This c h a p t e r will s u m m a r i z e t h e n a t u r e of solids as r e l a t e d

to p h o s p h o r s

c a n n o t be c o m p l e t e l y c o m p r e h e n s i v e .

in general.

Many scientific

Of necessity, w e texts have b e e n

w r i t t e n on e a c h of t h e s e individual s u b j e c t s a n d you c a n refer to t h e m to get m o r e specific a n s w e r s to c e r t a i n q u e s t i o n s t h a t c a n n o t be c o v e r e d here. This c h a p t e r will cover the basics of: I. A c o m p a r i s o n of the t h r e e s t a t e s of m a t t e r 2. How one d e t e r m i n e s the s t r u c t u r e of solids 3. An i n t r o d u c t i o n to the defect solid s t a t e Because of the i m p o r t a n c e of the defect solid (all p h o s p h o r s fall into t h i s category), we will devote a c h a p t e r to the defect-state. Since we n e e d to k n o w h o w p h o s p h o r s are m a d e , a c h a p t e r covering solid state r e a c t i o n s is also m a n d a t o r y . Finally, we will cover b o t h single crystal f o r m a t i o n a n d particle size d e t e r m i n a t i o n s . S o m e p h o s p h o r s are single crystal. A g o o d e x a m p l e is t h a t of the laser. The first laser i n v e n t e d by M a i m a n u s e d a single crystal of ruby, i.e.-

a l u m i n u m oxide

or s a p p h i r e

(corundum)

activated by trivalent c h r o m i u m (Cra). A great variety of laser crystals are n o w in u s e i n c l u d i n g YAG:Nd a*, i.e.- Y3A15011:N d a+. Since m o s t p h o s p h o r s are u s e d as p o w d e r s , we will also i n c l u d e a c h a p t e r s h o w i n g h o w o n e determines

particle

size a n d particle

size

distributions.

We will also

Introduction to the solidstate

include a d e s c r i p t i o n of the n a t u r e of light a n d p h o t o n s in general since p h o s p h o r s emit p h o t o n s once t h e y are excited by a b s o r p t i o n of energy. Color is also an i m p o r t a n t c o m p o n e n t of p h o t o n s as we perceive t h e m . B e c a u s e I believe t h a t one n e e d s to be aware of the h i s t o r y of science in o r d e r to t h o r o u g h l y u n d e r s t a n d any p a r t of it, I will try to include as m u c h b a c k g r o u n d a n d r e s u l t s of investigations as t h e y p e r t a i n to p h o s p h o r s . For example, the m e t h o d s of d e t e r m i n i n g s t r u c t u r e s of solids did not o c c u r o v e r n i g h t b u t were developed slowly over a period of time once R 6 n t g e n d i s c o v e r e d x-rays. Nowadays, if you are in this field, you probably use a c o m p u t e r - c o n t r o l l e d p r o g r a m to collect diffraction d a t a automatically and to do the calculations for you. Yet, you m u s t to k n o w how the calculations n e e d to be a c c o m p l i s h e d in o r d e r to a s c e r t a i n w h e t h e r you have obtained an a c c u r a t e depiction of the s t r u c t u r e of any material, or not. W h e n a detailed s t u d y of solids is u n d e r t a k e n , one quickly d e t e r m i n e s t h a t solids have p r o p e r t i e s w h i c h differ profoundly from t h o s e of t h e o t h e r s t a t e s of m a t t e r . We will be c o n c e r n e d mainly w i t h inorganic solids, a l t h o u g h the s a m e p r i n c i p l e s apply to organic solids as well. It is well to note here t h a t OLEDs, or "Organic Light E m i t t i n g Diodes" are t h e p r o m i s e of the future to replace inorganic p h o s p h o r s as l i g h t - e m i t t i n g displays in c e r t a i n applications. Inorganic solids are the p r i m a r y materials of

construction

for

use

in

electronics,

lighting,

communications,

i n f o r m a t i o n storage a n d display, s u p e r c o n d u c t o r s , a n d o t h e r s at this time, b u t have b e e n s u p p l a n t e d by organic b a s e d m a t e r i a l s in s o m e applications. In c o m p a r i n g the t h r e e s t a t e s of m a t t e r , o u r a p p r o a c h will be as follows. First, we will c o m p a r e the t h r e e (3) basic s t a t e s of m a t t e r , n a m e l y gases, liquids a n d solids. We will t h e n c o n t r a s t t h e s e states energetically and atomisticaUy. Next, we will d i s c u s s s t r u c t u r e s of solids a n d the factors involved in d e t e r m i n a t i o n of crystal s t r u c t u r e . Finally, we will i n t r o d u c e the c o n c e p t of t h e defect solid a n d how s u c h defects affect t h e m a c r o s c o p i c p r o p e r t i e s of the solid state. We c a n s u m m a r i z e differences b e t w e e n follows:

the three

s t a t e s of m a t t e r

as

1.1 Changes of state a. T h e solid is t h e m o s t c o n d e n s e d

3

p h a s e of the t h r e e

normally

possible, a n d c o n t a i n s t h e l e a s t e n e r g y . b. T h e solid h a s b u t t h r e e (3) d e g r e e s t h o s e usually p r e s e n t present

of f r e e d o m in c o n t r a s t to

in t h e g a s e o u s state. (The n u m b e r

in a gas d e p e n d s

upon

the

number

of a t o m s

actually in

the

molecule). c. All solids c o n t a i n defects. T h e r e is no s u c h t h i n g as a p e r f e c t solid. 1.1. - CHANGES OF STATE Of the

three

difference

s t a t e s of m a t t e r ,

gaseous,

liquid

a n d solid,

the

major

involves t h e i r a t o m i s t i c f r e e d o m of m o v e m e n t . For e x a m p l e , a

gas m o l e c u l e c a n move in t h r e e (3) d i m e n s i o n s w i t h o u t r e s t r i c t i o n a n d h a s full v i b r a t i o n a l a n d r o t a t i o n a l m o d e s w i t h i n t h e m o l e c u l e . We k n o w t h a t it is possible to c o n d e n s e g a s e s to liquids to solids by r e m o v a l of energy.

Each

condensation,

state

represents

a

and has a characteristic

succeedingly temperature

lower

degree

of

range

at w h i c h

it

exists, t h e gas b e i n g t h e "hottest", a n d solids the "coldest". T h u s , s o l i d s c o a t a i n t h e learnt e n e r g y . The actual t e m p e r a t u r e r a n g e for t h e gaseous, liquid a n d solid s t a t e of a given m a t e r i a l d e p e n d s ,

of course, u p o n t h e

n a t u r e of t h e a t o m s involved. Even e l e m e n t s differ quite widely. Gallium, for e x a m p l e , is a m e t a l t h a t is liquid slightly above r o o m t e m p e r a t u r e , b u t does not vaporize until above 1100 ~

T u n g s t e n , also a m e t a l , does n o t

m e l t u n t i l t h e t e m p e r a t u r e is above 3 8 5 0 ~ An e l e m e n t a r y w a y to u n d e r s t a n d h o w t h e solid differs from t h e g a s e o u s or liquid s t a t e s involves the following e x a m p l e . T a k e t h e MX2 m o l e c u l e . In t h e g a s e o u s state, t h e r e will be a m a x i m u m of 3 x 3, or n i n e d e g r e e s of freedom,

for t h e m o l e c u l e c o m p o s e d

of t h e s e t h r e e

a t o m s (there

are

t h r e e d i m e n s i o n s , x, y, & z, in w h i c h to operate). T h e s e c a n be d i v i d e d into 3 - t r a n s l a t i o n a l , 3-vibrational a n d 3 - r o t a t i o n a l d e g r e e s of f r e e d o m . But as we c h a n g e t h e s t a t e of m a t t e r , we k n o w t h a t t h e t r a n s l a t i o n a l d e g r e e s of f r e e d o m p r e s e n t in g a s e s d i s a p p e a r in solids (and are r e s t r i c t e d

in

liquids). For g a s e o u s molecule, t h e 3 vibrational d e g r e e s of f r e e d o m will have (2J+1 = 7) r o t a t i o n a l s t a t e s s u p e r i m p o s e d u p o n t h e m (J h e r e is t h e

Introduction to the solid state

n u m b e r of a t o m s in the molecule, a s s u m i n g q u a n t i z e d vibrational states). If we m e a s u r e the a b s o r p t i o n s p e c t r a of o u r molecule, MX2 , in the infrar e d region of the s p e c t r u m , we obtain r e s u l t s similar to t h o s e s h o w n as follows: 1.1.1.Energy States of a T r i a t o m i c Molecule as a Function of States of Matter

,, F r e q u e n c ~

The g a s e o u s p h a s e s h o w s seven well s e p a r a t e d rotational a n d vibrational states, the liquid p h a s e b r o a d e n e d vibrational plus rotational states ( t h e y are no longer distinct), while the solid exhibits only one b r o a d featureless b a n d in this region of the s p e c t r u m . We c o n c l u d e t h a t we are limited to t h r e e degrees of f r e e d o m in the solid state. The 3 - t r a n s l a t i o n a l a n d 3rotational d e g r e e s of f r e e d o m have d i s a p p e a r e d b e c a u s e of the i m p o s i t i o n of l o n g r a n g e o r d e r i n g of MX2 m o l e c u l e s in the solid. This point c a n n o t be o v e r - e m p h a s i z e d since long range o r d e r i n g is t h e major difference b e t w e e n solids a n d the o t h e r states of m a t t e r .

1.2 Energetics of changes of state 1.2.- ENERGETICS OF CHANGES OF STATE W h e n h e a t energy is a d d e d to a solid, one of two c h a n g e s will occur. Either the i n t e r n a l h e a t c o n t e n t ( t e m p e r a t u r e ) of the material will c h a n g e or a c h ~ of ~ will occur. On an atomistic level, this involves an increase in vibrational energy w h i c h m a n i f e s t s itself either as an i n c r e a s e in i n t e r n a l t e m p e r a t u r e , or the b r e a k i n g of b o n d s in the solid to f o r m either a liquid or gas. The m o s t c o m m o n example u s e d to illustrate energy involved in c h a n g e s of state is t h a t of water. We k n o w t h a t w a t e r , a liquid, will c h a n g e to a solid (ice) if its i n t e r n a l t e m p e r a t u r e falls b e l o w a certain t e m p e r a t u r e . Likewise, if its t e m p e r a t u r e rises above a c e r t a i n point, w a t e r c h a n g e s to a gas (steam). Because w a t e r is so plentiful on t h e Earth, it w a s u s e d in the p a s t to define C h a n g e s of State and even to define Temperature Scales. The invention of the t h e r m o m e t e r is generally c r e d i t e d to Galileo. In his i n s t r u m e n t , built about 1592, the c h a n g i n g t e m p e r a t u r e of an i n v e r t e d glass vessel p r o d u c e d the e x p a n s i o n or c o n t r a c t i o n of the air within it, w h i c h in t u r n c h a n g e d the level of the liquid with w h i c h the vessel's long, o p e n - m o u t h e d n e c k was partially f'flled. This general principle was p e r f e c t e d in s u c c e e d i n g years by e x p e r i m e n t i n g with liquids s u c h as alcohol or m e r c u r y a n d by providing a scale to m e a s u r e the e x p a n s i o n a n d c o n t r a c t i o n b r o u g h t about in s u c h liquids by rising and falling temperatures. By the early 18th c e n t u r y as m a n y as 35 different t e m p e r a t u r e scales h a d b e e n devised. The G e r m a n physicist, Daniel Gabriel F a h r e n h e i t , in t h e period of 1 7 0 0 - 3 0 , p r o d u c e d accurate m e r c u r y t h e r m o m e t e r s c a l i b r a t e d to a s t a n d a r d scale t h a t r a n g e d from 32, the m e l t i n g point of ice, to 9 6 for b o d y t e m p e r a t u r e . The u n i t of t e m p e r a t u r e (degree) on t h e F a h r e n h e i t scale is 1 / 1 8 0 of the difference b e t w e e n the boiling (212) a n d freezing p o i n t s of water. The first c e n t i g r a d e scale (made up of 1 0 0 degrees) is a t t r i b u t e d to the Swedish a s t r o n o m e r Anders Celsius, w h o developed it in 1742. Celsius u s e d 0 for the boiling point of w a t e r a n d 100 for the melting point of snow. This w a s later inverted to p u t 0 on t h e cold e n d a n d 100 on the hot end, a n d in t h a t form it gained w i d e s p r e a d

Introduction to the solid state

use. It w a s k n o w n s i m p l y as the c e n t i g r a d e scale u n t i l in 1948 the n a m e was changed

to h o n o r

Celsius. In

1848

t h e British p h y s i c i s t William

T h o m p s o n (later Lord Kelvin) p r o p o s e d a s y s t e m t h a t u s e d t h e d e g r e e s t h a t Celsius u s e d , b u t w a s k e y e d to a b s o l u t e zero ( - 2 7 3 . 1 5 ~ this scale is n o w k n o w n as t h e Kelvin, i.e.- ~ the Fahrenheit degree

The u n i t of

The R a n k i n e scale e m p l o y s

k e y e d to a b s o l u t e zero

(-459.67

~

, i.e.- ~

T h e s e are t h e four t e m p e r a t u r e scales t h a t we e m p l o y today. T h e factors involved in e n e r g y c h a n g e , i.e.- t e m p e r a t u r e c h a n g e , i n c l u d e : "heat capacity", i.e.- Cp or Cv, a n d "heat of t r a n s f o r m a t i o n " , H. The f o r m e r is c o n n e c t e d

with internal

temperature

change whereas

the latter is

involved in c h a n g e s of state. The actual n a m e s we u s e to d e s c r i b e

H

d e p e n d u p o n t h e d i r e c t i o n in w h i c h t h e t e m p e r a t u r e c h a n g e o c c u r s , vis: I. 2. I. -

CHANGE OF S T A T E ice to w a t e r

~ , HEAT OF: Fusion

TEMPERATURE oC. ~ 0

32

w a t e r to i c e

Solidification

0

32

w a t e r to s t e a m

Vaporization

100

212

s t e a m to w a t e r

Condensation

100

212

Note t h a t w h e n a c h a n g e of state, i.e.- w a t e r to ice or ice to w a t e r , o c c u r s , there

is no c h a n g e

in t e m p e r a t u r e

while

this is o c c u r r i n g

(this is a

c o n c e p t t h a t is s o m e t i m e s difficult for b e g i n n i n g s c h o l a r s to grasp). H e a t c a p a c i t y w a s originally d e f i n e d in t e r m s of water. T h a t is, h e a t c a p a c i t y w a s d e f i n e d as t h e a m o u n t of h e a t r e q u i r e d to raise t h e t e m p e r a t u r e of one c u b i c c e n t i m e t e r ~

by o n e ( l )

degree.

of w a t e r (whose d e n s i t y is d e f i n e d as 1.0000 @ 4 "Heat" itself is defined

as t h e a m o u n t of e n e r g y

r e q u i r e d to raise 1 cc. of w a t e r by one degree, i.e.- o n e (1.0) calorie. T h e calorie in t u r n w a s originally defined as t h e a m o u n t of h e a t r e q u i r e d to raise t h e t e m p e r a t u r e of o n e g r a m of w a t e r from 14.5 ~

to 15.5 ~

at a

c o n s t a n t p r e s s u r e of one (1) a t m o s p h e r e . Heat c a p a c i t y w a s also k n o w n as t h e r m a l capacity. We t h u s label h e a t c a p a c i t y as Cp, m e a n i n g the t h e r m a l c a p a c i t y at c o m e , a t constant

pressure.

Originally, Cv, the

thermal

c a p a c i t y at

v o l u m e w a s also u s e d , b u t its u s e is r a r e n o w a d a y s . This is d u e to

t h e fact t h a t s o m e m a t e r i a l s do not have a linear t e m p e r a t u r e e x p a n s i o n .

1.2 Energetics of changes of state T h e s e are two t y p e s of "heat" involved in t h e t h e r m a l c h a n g e s of a n y g i v e n m a t e r i a l . We specify t h e "heat" of a m a t e r i a l , i.e.- its i n t e r n a l energy, by HS,L,G. w h e r e S,L, or G refer to solid, liquid or gas. T h u s , t h e r e l a t i o n b e t w e e n HS,L,G a n d Cp is: 1.2.2.-

{H = C p - T } S , L , G ,

or

AHS,L,G

= Cp(S,L,G}

ATS,L,G

w h e r e t h e s c t t m l s t a t e of m a t t e r is e i t h e r S, L, o r G.

Note that: T h e internal t e m p e r a t u r e of a material d o e s n o t c h a n g e as t h e material u n d e r g o e s a c h a n g e of state. All of t h e e n e r g y g o e s into f o r m i n g a n e w state of matter. For a c h a n g e of s t a t e b e t w e e n solid a n d liquid, we w o u l d have: {HS -HL} = C p ( T ( S ) - T ( L ) )

1.2.3-

or A H = C p

AT

AH h e r e is a h e a t of t r a n s f o r m a t i o n involved in a c h a n g e of s t a t e w h e r e a s AHS,L,G { s o m e t i m e s w r i t t e n AES,L,G} r e f e r s to c h a n g e i n h a t e m a l heat for a given s t a t e of m a t t e r . We c a n n o w c a l c u l a t e t h e a m o u n t of e n e r g y r e q u i r e d to r a i s e o n e g r m n of

i c e at - 10 ~ 1.2.4.-

to f o r m o n e g r a m of s t e a m at 110 ~

CAIX)RIES R E Q U I R E D TO CHANGE .1 G m OF ICE TO S T E A M

FORM

AT

__Cl~

__Cl~A T

AH_X

Ice (Ice

-10 to 0 0

0.5 5.0 cal. . . . . . .

--80 cal.

5.0 cal. 8 0 . 0 cal.

{fusion} --5 4 0 cal.

1 0 0 . 0 cal. 5 4 0 . 0 cal.

to w a t e r } Water {Water

100 0

1.0 100 cal. . . . . . .

10

0.5

To s t e a m } Steam

TOTAL

{vaporization} 5.0 cal.

5.0 cal. 7 3 0 . 0 cal.

Introduction to the solid state

Note t h a t in this case AH x is specified in t e r m s of the type of change of state occurring, while Cp AT (= AH) is the c h a z ~ e in Luteraal heat w h i c h o c c u r s as the t e m p e r a t u r e

rises. At a given c h a n g e of state, all of t h e

e n e r g y goes to the c h a n g e of state a n d the t e m p e r a t u r e does not c h a n g e until the t r a n s f o r m a t i o n is c o m p l e t e . 1.3. - PROPAGATION MODELS AND T H E CLOSE-PACKED SOLID We have already said t h a t the solid differs from the o t h e r s t a t e s of m a t t e r in t h a t long range o r d e r i n g of a t o m s or m o l e c u l e s has appeared. To achieve long range o r d e r in any solid, one m u s t s t a c k a t o m s in a s y m m e t r i c a l way to completely, fill space. If the a t o m s are all of one kind, i.e.- one of the e l e m e n t s , the p r o b l e m is straight forward. Sets of eight a t o m s , each set a r r a n g e d as a cube, will g e n e r a t e a cubic s t r u c t u r e . T w o s e t s of t h r e e atoms, each set of three a r r a n g e d in a triangle, will p r o p a g a t e a hexagonal p a t t e r n with three d i m e n s i o n a l s y m m e t r y . T h e following d i a g r a m illustrates this point: 1.3.1.

TETRAHEBRON (4)

H E X A G O N A L [8) CLOSE-PACKING

OCTAHEDRON

CUBIC (8] CLOSE-PACKING

Propagation Units in the Solid State

1.3 Propagation models and the close-packed solid This d i a g r a m s h o w s a p e r s p e c t i v e of cubic a n d h e x a g o n a l c l o s e - p a c k i n g propagation

units

in

their

space-fiUing

aspects.

W h a t we

mean

by

p r o p a g a t i o n u n i t s are solid s t a t e b u i l d i n g b l o c k s t h a t we c a n s t a c k in a s y m m e t r i c a l form to infinity. We do n o t c o n s i d e r

1-, 2- or 3 - a t o m u n i t s

s i n c e t h e y are trivial. T h a t is, t h e y are only 1- or 2 - d i m e n s i o n a l at b e s t . But, 4 - a t o m s will form a 3 - d i m e n s i o n a l t e t r a h e d r o n (half a c u b e is only 2d i m e n s i o n a l ) w h i c h is a valid p r o p a g a t i o n unit. This m e a n s is t h a t we c a n take

tetrahedrons

a n d fit t h e m

together

3-dimensionally

s y m m e t r i c a l s t r u c t u r e w h i c h e x t e n d s to infinity. However, n o t t r u e for 5 - a t o m s , w h i c h

forms

a four-sided

to

form

a

the s a m e is

p y r a m i d . This

shape

c a n n o t be c o m p l e t e l y fitted t o g e t h e r in a s y m m e t r i c a l a n d space-ffiling m a n n e r . Even if we s t a c k t h e s e p y r a m i d s , we find t h a t t h e i r t r a n s l a t i o n a l p r o p e r t i e s p r e c l u d e f o r m a t i o n of a s y l n m e t r i c a l s t r u c t u r e (There

is t o o

m u c h lost space!) However, if o n e m o r e a t o m is a d d e d to t h e p y r a m i d , w e then

have

an

octahedron

which

is

space-filling

with

translational

properties. Going f u r t h e r , c o m b i n a t i o n s of seven a t o m s are a s y m m e t r i c a l , b u t e i g h t a t o m s f o r m a c u b e w h i c h c a n be p r o p a g a t e d to infinity. Note turning the

t o p layer of four a t o m s (see

1.3.1.)

t h a t by

by 45 ~ , we have

h e x a g o n a l u n i t w h i c h is r e l a t e d to t h e h e x a g o n a l u n i t c o m p o s e d a t o m s , two t r i a n g l e s atop of e a c h o t h e r .

a

of 6

By t a k i n g P i n g - P o n g balls a n d

gluing t h e m t o g e t h e r to f o r m t h e p r o p a g a t i o n u n i t s as s h o w n in 1 . 3 . 1 . , o n e c a n get a b e t t e r p i c t u r e of t h e s e u n i t s . A l t h o u g h we c a n c o n t i n u e w i t h m o r e

a t o m s p e r p r o p a g a t i o n unit, it is

easy to s h o w t h a t all of t h o s e are r e l a t e d to t h e four basic p r o p a g a t i o n u n i t s f o u n d in t h e solid state (and d e p i c t e d in 1.3.1.), to wit:

1.3.2.-

Tetrahe dron Hexagon

(4) (6 or 8)

Octahedron

(6 )

Cube

(8)

We t h u s c o n c l u d e t h a t s t r u c t u r e s of solids are b a s e d , in g e n e r a l , u p o n four bas/c p r o p a g a t i o n units, w h i c h are s t a c k e d in a s y n m l e t r i c a l a n d s p a c e filling form to n e a r infinity. Variation of s t r u c t u r e in solids d e p e n d s u p o n w h e t h e r t h e o t h e r a t o m s f o r m i n g the s t r u c t u r e are larger or s m a l l e r t h a n

lO

Introduction to the solid state

the basic p r o p a g a t i o n u n i t s c o m p o s i n g the s t r u c t u r e . In m a n y cases they are smaller a n d will fit into the i n t e r s t i c e of the p r o p a g a t i o n unit. A good e x a m p l e is the p h o s p h a t e t e t r a h e d r o n , P 0 4 . The p4+ a t o m is small e n o u g h to fit into the c e n t e r (interstice) of the t e t r a h e d r o n formed by the four oxygen atoms. If we c o m b i n e it with E r 3+ (which is slightly smaller t h a n PO4), we obtain a tetragonal s t r u c t u r e , a sort of elongated cube of h i g h s y m m e t r y . But if we c o m b i n e it with La 3+ , w h i c h is larger t h a n P 0 4 3 - , a m o n o c l i n i c s t r u c t u r e with low s y m m e t r y results. Now, you m a y t h i n k t h a t this d i s c u s s i o n is mainly c o n j e c t u r e since t h e case in n a t u r e m a y differ considerably. Actually, it is easy to ascertain t h a t m o s t of the e l e m e n t s (which are h o m o g e n e o u s , i.e.- having the all of the s a m e type of atoms) form s t r u c t u r e s t h a t are either cubic or hexagonal. This is s h o w n in the following diagram: 1.3.3.-

[Structure of the Metallic Elements as Related to the Periodic TableI D

Body Centered Cubic

~

Face C e n t e r e d Oabic

Hexagonal Close-Packed '.'.','.'."

~X;~X',

~i::~::i::i"::~i::~::i!iiiiii~::~::!::"i::~::i!i~' ..... ::i~i-~

. . . . . . . . . . . . . . . . . . .

/"

. . . . . .

~

," 9 9 iT . . . . . . . . . . . . . . . . . . . .

:....:.~::: :.:r::..:::: ~ ] Z ' : ' : ' : ' : ' : [ ' : ' : ' : ' : ' ] : ' : ' : ' : ' : ' : l ~

[~'-

~

!Uii~ Ga ,x~ As !i!~ Br ~ ''""""

|xxxxx

.r

:xxxzx

e ~

:xx.xxx

9....-."----.,...-...~cc.:-,~:-:.:,.~:~-:.-~a~,~j~ ~ ~ HOi ~ , ~ .'.'.'.'.'.

~'.'.'.'."

ii~ii!i~

,',',',',',

I.'.'.'.'.'..'.p..:..'.'-..'.:i-.'.'...-..

,-. ',,' .., " , " , , " - , , , ',,",, ',,' ~ ~

~'Rf, HaUnh ..,

t

q

k

q

. 9. . .. . . -. . . . . . 9. .. . . .. . . .. . . .. . - . . . . . - .. . . . .. .. . .. . ... . -. . . .9 .. . -.. - .

xx K

. ...... .

~

~x'~i~iil~:~::| ~r :lr~ ! e mt~.~i~,~l~l~l,~,l~.ol~111~,~! .............................. . ......... . ....... , , ~.~l 41

"-'-'-'-'-

- 9 -.-..

xxx

"-'-%'-"

', ::::::::::::::::::::::: Np li! !]Aml cm jBIc ]

! E' I'mlM"l

.:

x

Irr I

Actinides (8+) It s h o u l d be clear t h a t the m e t a l e l e m e n t s have either a f a c e - c e n t e r e d ,

1.3 Propagation models and the close-packed solid body-centered

or

hexagonal

close-packed

structure.

II These

three

s t r u c t u r a l u n i t s are s h o w n in t h e following d i a g r a m : 1.3.4.-

Body-Centered Cubic

Face-Centered Cubic

Hexagonal Close-Packed

A b e t t e r p e r s p e c t i v e is given in t h e 1 . 4 . 1 4 d i a g r a m , g i v e n below. T h e r e i n , w e s h o w t h e s e v e n t y p e s of c l o s e - p a c k e d s t r u c t u r e s , i.e.- Bravais l a t t i c e s , t h a t a p p e a r in t h e solid s t a t e . Of t h o s e e l e m e n t s w h o s e s t r u c t u r e s a r e n o t i n d i c a t e d in t h e above d i a g r a m , t h e following t a b l e s h o w s t h e i r s t r u c t u r e : T a b l e 1-1 Crystal S t r u c t u r e s of S o m e E l e m e n t s G a s e s at R o o m T e m p

Liciuids a t R o o m T e m p .

O t h e r Solid M e t a l s

H, He = h e x a g o n a l i

i

i

Ne, Ar, Kr, Xe, R n

Ga = o r t h o r h o m b i c

Rhombohedral

(Solid = fcc)

Hg = r h o m b o h e d r a l

B, As, Sb, Bi Orthorhombic

F = cubic

I, S, N p

CI, Br = o r t h o r h o m b i c

P, Po - M o n o c l i n i e

N = hexagonal

In, S n - T e t r a g o n a l

O = cubic _

i

i

Co, Ni, Cu, e t c .

fcc= f a c e - c e n t e r e d cubic

i

You will n o t e t h a t e v e n t h o u g h s o m e e l e m e n t s a r e g a s e s or l i q u i d s at r o o m temperature, determined.

the

structure

that

they

S o m e of t h e t e m p e r a t u r e s

form required

in

the

solid

to c o n d e n s e

has

been

them into

t h e solid p h a s e are close to a b s o l u t e z e r o . But, if t h e a t o m s a r e n o t all of t h e s a m e k i n d , i.e.- h e t e r o g e n e o u s , w e h a v e

12

Introduction to the solid state

a different p r o b l e m . P a r t of t h e p r o b l e m lies in t h e fact t h a t 6 . 0 2 3 x 1 0 2 3 m o l e c u l e s c o m p r i s e one m o l e a n d we s y m m e t r i c a l m a n n e r to f o r m a c l o s e - p a c k e d represents

100.

I grams

of p o w d e r .

We

find

m u s t s t a c k t h e s e in a solid. For CaCO3, t h i s in

general

that

solid

s t r u c t u r e s a r e b a s e d on t h e l a r g e s t a t o m p r e s e n t , as well as h o w it s t a c k s together

in

space

c o m p o u n d s , t h i s is t h e ~ e n

(its

valence)

~ ,

filling-form.

For

most

inorganic

e.g.- oxides, silicates, p h o s p h a t e s ,

sulfates, b o r a t e s , t u n g s t a t e s , v a n a d a t e s , etc. T h e

few e x c e p t i o n s

involve

c h a l c o g e n i d e s , h a l i d e s , h y d r i d e s , etc., b u t even in t h o s e c o m p o u n d s , t h e s t r u c t u r e is b a s e d u p o n a g g r e g a t i o n of t h e l a r g e s t a t o m , i.e.- the sulfur a t o m in ZnS. Zinc sulfide exhibits two s t r u c t u r e s ,

sphalerite-

a cubic

a r r a n g e m e n t of t h e sulfide a t o m s , a n d w u r t z i t e - a h e x a g o n a l a r r a n g e m e n t of t h e

sulfide

atoms.

Divalent zinc

atoms

have e s s e n t i a l l y

the

same

c o o r d i n a t i o n in b o t h s t r u c t u r e s . T h i s b r i n g s u s to a d i s c u s s i o n of t h e s t r u c t u r e of solids a n d h o w s t r u c t u r e is d e t e r m i n e d . 1.4.- THE S T R U C T U R E OF SOLIDS We h a v e i n d i c a t e d t h a t solids c a n have several s y m m e t r i e s . Let u s n o w examine

the

structure

of solids

in

more

detail.

In

1895,

R6ntgen

e x p e r i m e n t a l l y d i s c o v e r e d "x-rays" a n d p r o d u c e d t h e first p i c t u r e of t h e b o n e s of t h e h u m a n h a n d . This w a s followed by w o r k b y y o n Laue in 1 9 1 2 w h o s h o w e d t h a t solid c r y s t a l s c o u l d a c t as diffraction g r a t i n g s to f o r m s y m m e t r i c a l p a t t e r n s of "dots" on a p h o t o g r a p h i c film w h o s e a r r a n g e m e n t depended

u p o n h o w t h e a t o m s w e r e a r r a n g e d in t h e solid. It w a s s o o n

r e a l i z e d t h a t t h e a t o m s f o r m e d "planes" w i t h i n t h e solid. In 1913,

Sir

William H e n r y Bragg a n d his son, William L a w r e n c e Bragg, analyzed t h e manner

in w h i c h s u c h x - r a y s w e r e r e f l e c t e d by p l a n e s of a t o m s in t h e

solid. T h e y certain

showed

angles,

that these

reflections

would

a n d t h a t t h e values w o u l d d e p e n d

be m o s t upon

intense

the

at

distance

b e t w e e n t h e p l a n e s of a t o m s in the c r y s t a l a n d u p o n t h e w a v e l e n g t h of t h e x-ray. T h i s r e s u l t e d in t h e Bragg e q u a t i o n : 1.4.1.-

n k = 2 d sin 0

13

1.4 The structure of solids

w h e r e d, t h e d i s t a n c e , is in a n g s t r o m s (,~ = I 0 -8 cm) b e t w e e n p l a n e s a n d 0 is t h e angle in d e g r e e s of t h e reflection. In Bragg's x - r a y d i f f r a c t i o n e q u a t i o n , i.e.- t h e a n g l e , 0 , is actually t h e angle b e t w e e n a given p l a n e of a t o m s in t h e s t r u c t u r e a n d t h e p a t h of t h e x - r a y b e a m . T h e unit, "d", is defined as t h e d i s t a n c e b e t w e e n p l a n e s of t h e lattice a n d k is t h e wavelength

of t h e

radiation.

It w a s

Georges

Friedel

who

in

1913

d e t e r m i n e d t h a t t h e i n t e n s i t i e s of r e f l e c t i o n s of x - r a y s f r o m t h e s e p l a n e s in t h e solid c o u l d be u s e d to d e t e r m i n e t h e s y m m e t r y of t h e solid. T h u s , by c o n v e n t i o n , we usually define p l a n e s , n o t p o i n t s , in t h e l a t t i c e . We c a n define t h e s t r u c t u r e of a n y given solid in t e r m s of its l a t t i c e p o i n t s . W h a t t h i s m e a n s is t h a t if we s u b s t i t u t e a p o i n t for e a c h a t o m (ion) c o m p o s i n g the

structure,

we find t h a t t h e s e

points

constitute

a lattice,

i.e.-

an

internal configuration, having certain s y m m e t r i e s . A lattice is n o t a s t r u c t u r e p e r se. A l a t t i c e is d e f i n e d as a s e t o f t h r e e -

d i m e n s i o n a l p o i n t s . T h e s e p o i n t s m a y , or m a y not, b e totally o c c u p i e d by t h e a t o m s c o m p o s i n g t h e s t r u c t u r e . C o n s i d e r a cubic s t r u c t u r e s u c h as t h a t s h o w n in t h e following d i a g r a m : 1.4.2.ann r

II

m

l

I

x

14

Introduction to the solid state

Here

we

have

dimensional

a

set

cubic

of atoms

pattern.

(ions.)

The

arranged

lattice

in

directions

a

simple

are

three-

defined,

by

convention, a s x , y & z. Again, b y c o n v e n t i o n , "x" is d e f i n e d as t h e r i g h t h a n d d i r e c t i o n f r o m t h e origin, "y" is in t h e v e r t i c a l d i r e c t i o n a n d "z" is at a n a n g l e to t h e p l a n e of x & y (In o u r c u b e , t h e a n g l e is 9 0 ~ . Note t h a t there

a r e e i g h t (8) c u b e s in o u r e x a m p l e . T h e u a l t - c e l l is t h e s m a l l e s t

cube. The

unit-cell

directions

are defined

as t h e

"lattice-unit-vectors".

T h a t is, t h e x, y, & z d i r e c t i o n s of t h e u n i t cell a r e v e c t o r s h a v i n g d i r e c t i o n s c o r r e s p o n d i n g to x @ ~ ; y @ ~ ; z @ c w i t h l e n g t h s o f e a c h u n i t - v e c t o r b e i n g e q u a l to 1.0. (Our n o t a t i o n for a v e c t o r h e n c e f o r t h is a l e t t e r w h i c h is " o u t l i n e d " , i.e.- t h e ~ u n i t - c e l l t r a n s l a t i o n v e c t o r ) . ~, the translation vector, 1.4.3.-

~

=

where n i, n 2 , and n a

is t h e n :

n l~

+n 2b

+n a

a r e i n t e r c e p t s of t h e u n i t - v e c t o r s , ~ ,

b , ~ , on

t h e x , y , z - d i r e c t i o n s in t h e lattice, r e s p e c t i v e l y . T h e u n i t c e l l v o l u m e is then1.4.4.-

V

=

{ ~. 9 ~

x c }

(This is a " d o t - c r o s s " v e c t o r p r o d u c t ) .

T h u s , it is e a s y to s e e t h a t in B r a g g ' s x - r a y d i f f r a c t i o n e q u a t i o n , t h e a n g l e , 0 , is a c t u a l l y t h e a n g l e b e t w e e n a g i v e n p l a ~ e of a t o m s in t h e s t r u c t u r e a n d t h e _tBth of t h e x - r a y b e a m . T h e u n i t , "d", is d e f i n e d a s t h e d i s t a n c e b e t w e e n p l a n e s o f t h e l a t t i c e a n d k is t h e w a v e l e n g t h of t h e r a d i a t i o n . As w e s a i d , w e u s u a l l y d e f i n e p l a n e s , n o t p o i n t s , in t h e lattice. T h e r e a s o n that

we

do

this

is t h a t

the

waves

of e l e c t r o m a g n e t i c

radiation

are

c o n s t r u c t i v e l y d i f f r a c t e d b y p l a n e s of a t o m s in t h e s o l i d r a t h e r t h a n p o i n t s in t h e l a t t i c e .

One system

that has come

"MILLER INDICES" w h i c h is r e p r e s e n t e d

into

general

u s e is t h a t of

by: { h , k , I }.

In t h i s s y s t e m , w e u s e a , b, & c (not u n i t - c e l l v e c t o r s ) to r e p r e s e n t

the

l e n g t h s o f i n t e r c e p t s w h i c h d e f i n e t h e p l a n e s w i t h i n t h e u n i t cell. M i l l e r Indices

are the reciprocals

of t h e i n t e r c e p t s ,

p l a n e o n t h e x , y , z - d i r e c t i o n s in t h e l a t t i c e .

a,

b & c, of t h e c h o s e n

::~

15

1.4 The structure of solids

To i l l u s t r a t e t h i s c o n c e p t , e x a m i n e t h e s y m m e t r y e l e m e n t s of o u r cube, s h o w n as follows: 1.4.5.i

ISymmetry Elements, of a Cube I

Allllllllil A.

L/

~~~V {olo}

{ 100}

{ool}

B.

{I 10)

{01 I)

{I 1 I I

i

/I 84

C.

V 12oo1

, ,V 1002}

1020}

y Tetrad Axes

Triad Axes

Diad Axes

16

Introduction to the solid state

Note t h a t in all cases, we have the i n t e r c e p t s of t h e a, b & c specified as Miller

Indices.

This

s i m p l y m e a n s t h a t given

these

indices,

we

can

d e t e r m i n e w h e r e t h e a t o m s lie w i t h i n the u n i t cell of t h e lattice. This is i n d i c a t e d in t h e following: 1.4.6.-

a, b, c (I12, 0 , 0 }

MILLER INDICES {200}

(0 , I I 2 , 0 } ( 0, 0, 1 / 2 )

(020} (002)

Note also t h e (100),

(110)

a n d (111) p l a n e s are illustrated. Planes are

i m p o r t a n t in solids b e c a u s e , as we will see, t h e y are u s e d to locate a t o m p o s i t i o n s w i t h i n t h e lattice s t r u c t u r e . The TETRAD, TRIAD , AND DIAD AXES ARE AI~O SHOWN IN P a r t D. T h e s e are r o t a t l o a a l s T m m e t r y

axes.

T h a t is, t h e t r i a d axis m u s t be r o t a t e d 3 - t i m e s in o r d e r to b r i n g a given c o r n e r b a c k to its original p o s i t i o n . The final f a c t o r to c o n s i d e r is t h a t of the ax~le b e t w e e n t h e x , y , a n d z d i r e c t i o n s in t h e lattice. In o u r e x a m p l e s so far, angles w e r e 9 0 ~ in all d i r e c t i o n s . If t h e angles are not 9 0 ~, t h e n we have a d d i t i o n a l lattices to define. For a given unit-ceU defined by the axes a, b a n d c, t h e c o r r e s p o n d i n g angles are defined as: a , ~ , 7 , w h e r e a is t h e angle in t h e x - d i r e c t i o n , etc. In 1921, E w a l d d e v e l o p e d a m e t h o d of calculating the s u m s of diffraction intensities

from different p l a n e s in the lattice by c o n s i d e r i n g

called

"Reciprocal

the

Lattice". The

reciprocal

lattice

w h a t is

is o b t a i n e d by

d r a w i n g p e r p e n d i c u l a r s to e a c h p l a n e in the lattice, so t h a t t h e axes of the reciprocal

lattice are p e r p e n d i c u l a r

to t h o s e of the crystal lattice.

This h a s t h e r e s u l t t h a t the p l a n e s of the r e c i p r o c a l lattice are at r i g h t angles (90 ~ to the real p l a n e s in the unit-cell. E w a l d u s e d a s p h e r e to represent

h o w the x - r a y s i n t e r a c t w i t h any given lattice p l a n e in t h r e e

d i m e n s i o n a l space. He e m p l o y e d w h a t is n o w called the E w a l d S p h e r e to s h o w h o w r e c i p r o c a l s p a c e could be utilized to r e p r e s e n t diffraction of xrays by lattice p l a n e s . E w a l d originally r ~ r o t e

the Bragg e q u a t i o n as:

17

1.4 The structure of solids 1.4.7.-

sin

q

=

n l/

=

! / d {hkl}

2d {hkl}

2li

Using this equation, Ewald applied it to the case of the diffraction s p h e r e w h i c h we s h o w in the foUowing diagram: 1.4.8.-

[The Ewald Sphere ]

1 d hkl

In this case, the x-ray b e a m e n t e r s the s p h e r e e n t e r s from the left and e n c o u n t e r s a lattice plane, L. It is t h e n diffracted by the angle 20 to t h e point on the sphere, P, w h e r e it is r e g i s t e r e d as a diffraction point on t h e reciprocal lattice. The d i s t a n c e b e t w e e n p l a n e s in the reciprocal lattice is given as 1/dhkl w h i c h is readily o b t a i n e d from the diagram. It is for t h e s e r e a s o n s , we c a n u s e the Miller Indices to indicate p l a n e s in the real lattice, b a s e d u p o n the reciprocal lattice. The reciprocal lattice is useful in defining some of the e l e c t r o n i c p r o p e r t i e s of solids. T h a t is, w h e n we have a s e m i - c o n d u c t o r (or even a c o n d u c t o r like a metal), we find t h a t the e l e c t r o n s are confined in a band, defined by t h e reciprocal lattice. This has i m p o r t a n t effects u p o n t h e conductivity of any solid a n d is k n o w n as the "band theory" of solids. It t u r n s out t h a t the reciprocal lattice is also the site of the BriUouin zones, i.e.- the "allowed" electron e n e r g y b a n d s in the solid. How t h i s o r i g i n a t e s is explained as follows.

18

Introduction to the solid state

The free e l e c t r o n r e s i d e s in a q u a n t i z e d e n e r g y well, defined by k (in w a v e - n u m b e r s ) . T h i s r e s u l t c a n be derived f r o m t h e S c h r 6 d i n g e r wavee q u a t i o n . However, in the p r e s e n c e of a periodic a r r a y of e l e c t r o m a g n e t i c p o t e n t i a l s arising from t h e a t o m s c o n f i n e d in a crystalline lattice, t h e e n e r g i e s of t h e e l e c t r o n s from all of t h e a t o m s are severely limited in orbit a n d are r e s t r i c t e d to specific allowed e n e r g y b a n d s . This p o t e n t i a l o r i g i n a t e s f r o m a t t r a c t i o n a n d r e p u l s i o n of t h e e l e c t r o n c l o u d s from t h e p e r i o d i c a r r a y of a t o m s in the s t r u c t u r e . S o l u t i o n s to this p r o b l e m w e r e m a d e by Bloch in

1930 w h o s h o w e d

t h e y h a d t h e form

(for a one-

d i m e n s i o n a l lattice): =

1.4.9.-

e ~ u(x) - one d i m e n s i o n a l

tIJk (a) = where

e ~a uk(x) - t h r e e d i m e n s i o n a l

k is t h e wave n u m b e r of t h e allowed b a n d as m o d i f i e d

by t h e

lattice, a m a y be x, y or z, a n d th (x) is a p e r i o d i c a l f u n c t i o n w i t h the s a m e p e r i o d i c i t y as t h e potential. One r e p r e s e n t a t i o n is s h o w n in t h e following d i a g r a m , given as 1.4. I0. on the n e x t page. We have s h o w n t h e least c o m p l i c a t e d one w h i c h

t u r n s o u t to be t h e

s i m p l e cubic lattice. S u c h b a n d s are called "Brilluoin" zones a n d , have

said,

are t h e

allowed

energy

bands

of e l e c t r o n s

in

as w e

any given

c r y s t a l l i n e lattice. A n u m b e r of m e t a l s a n d s i m p l e c o m p o u n d s have b e e n s t u d i e d a n d t h e i r Brilluoin s t r u c t u r e s d e t e r m i n e d . However, w h e n o n e gives a r e p r e s e n t a t i o n of t h e e n e r g y b a n d s in a solid, a ' q 3 a n d - m o d e l " is usually p r e s e n t e d . The following diagram, p r e s e n t e d as 1.4.11. on a s u c c e e d i n g page, s h o w s t h r e e b a n d m o d e l s as u s e d to d e p i c t e n e r g y s t a t e s of a n y given solid. T h e y i n c l u d e i n s u l a t o r s , s e m i - c o n d u c t o r s a n d m e t a l s (conductive).

This is. In the solid, e l e c t r o n s r e s i d e in t h e

v a l e n c e b a n d , (as defined by the BriUoin zones in the r e c i p r o c a l lattice) b u t c a n be e x c i t e d into the c o n d u c t i o n b a n d b y a b s o r p t i o n of energy. T h e e n e r g y gap of various solids d e p e n d s

upon the

nature

of t h e

atoms

c o m p r i s i n g t h e solid. S e m i - c o n d u c t o r s have a r a t h e r n a r r o w energy gap ( f o r b i d d e n zone) w h e r e a s t h a t of i n s u l a t o r s is wide (metals have little o r no gap). T h a t is, t h e allowed b a n d s in m e t a l s overlap w i t h the v a l e n c e

19

1.4 The structure of solids

1.4.10.- T h e Allowed E n e r g y B a n d s (Brillouin Zones) in a Crystal

Three Dimensional Lattice (d= lattice constant )

.i. i ==

.......................

I

I

i.

9

I

I

..... " ....... i ................... " ......... " ....

-|'--

. ....................

,

I

I

-"

!

i

i

i

I

9

I

I

':I'i~, =I~

T

--

_~ 9

.........i

--~

T

"--6-

Three Dimensional Lattice in Reciprocal Space (d= lattice constant) Band

2nd Allowed

j

Forbidden

Band

E I

-n

Id

+ n

Id

-u

id

+~Id

b a n d . Note t h a t e n e r g y levels of t h e a t o m s " A " i n 1.4.1 I. are s h o w n in t h e v a l e n c e b a n d . T h e s e will vary d e p e n d i n g u p o n t h e n a t u r e a t o m s p r e s e n t . E l e c t r o n s c a n be e x c i t e d into t h e s e low-lying s t a t e s , d e p e n d i n g u p o n t h e t e m p e r a t u r e of t h e m a t e r i a l . T h u s , w e find t h a t a total of t h r e e (3) f a c t o r s are n e e d e d to define a g i v e n lattice a n d its s t r u c t u r e . This is s h o w n as follows: 1.4.12.-

I

-

unit-cell a x e s , intercepts and angles

II - r o t a t i o n a l s y m m e t r y III-

localized s p a c e g r o u p s y m m e t r y

Introduction to the solid state

20

1.4.1 I.- E n e r g y B a n d Models U s e d to Depict E n e r g y S t a t e s of S o l i d s

Insulator ,,,,,

Semi-Conductor ,

Ill I

Conduction Band

Conduction Band

Forbidden Energy Gap

ForbiddenEnergy Gap Valence B and

A

A

A

A

A

A

A

A

A

A

A

A

Metal Conduction Band

Forbidden Energy Gap Valence Band

A It is t h e s e t h r e e

A

f a c t o r s (see

A

A

1.4.12.)

A

A

w h i c h give rise to t h e d i f f e r e n t

s y m m e t r i e s of solids: F a c t o r I gives rise to t h e 14 Bravais l a t t i c e s F a c t o r II g e n e r a t e s t h e 32 p o i n t - g r o u p s F a c t o r III c r e a t e s th e 2 32 s p a c e - g r o u p s In

these

three

factors,

each

contributes

to

the

total

number

of

s y m m e t r i c a l l a t t i c e s t h a t c a n a p p e a r in t h e solid state. T h i s will b e c o m e e v i d e n t in t h e following d i s c u s s i o n : In F a c t o r I (which we h a v e a l r e a d y c o n s i d e r e d ) , if a , b , c a n d r ~ , 7, t h e n w e have a d i f f e r e n t lattice t h a n t h a t d e f i n e d by a = b ~ c ; a = ~ ~ y T h e n u m b e r of c o m b i n a t i o n s t h a t we c a n m a k e f r o m t h e s e

3-1engths a n d 3 - a n g l e s is s e v e n (7) a n d

t h e s e define t h e 7 u n i q u e lattice s t r u c t u r e s , called BRAVAIS LATTICES. T h e s e h a v e b e e n given n a m e s , as s h o w n in 1.4.13. on t h e n e x t page. E a c h of t h e s e s e v e n l a t t i c e s m a y have s u b l a t t i c e s , t h e total b e i n g 14. If w e a r r a n g e t h e c r y s t a l s y s t e m s in t e r m s of s y m m e t r y , t h e c u b e h a s t h e

21

1.4 The structure of solids

1.4.13.-

CUBIC

TETRAGONAL

HEXAGONAL

ORTHORHOMBIC

TRIGONAL

MONOCLINIC

TRICLINIC h i g h e s t s y m m e t r y a n d t h e triclinic lattice, t h e l o w e s t s y m m e t r y . We c a n t h e r e f o r e a r r a n g e t h e s e v e n (7)

s y s t e m s into a h i e r a r c h y as s h o w n in t h e

following d i a g r a m : 1.4.14.-

Hierarchy of Crystal Systems HEXAGONAL TETRAGONAL TRIGONAL 0RTHORHOMBIC

IMON0CLINICl

ITR!CLINIr T h e h i g h e s t s y m m e t r y lattice is at t h e top, w h i l e t h e l o w e s t is at t h e b o t t o m . T h e s e 14 Bravais l a t t i c e s are s h o w n in 1.4.15. given o n t h e n e x t page. If w e n o w a p p l y rotational s y m m e t r y {Factor II) to t h e 14 Bravais l a t t i c e s , w e o b t a i n t h e 32 P o i n t - G r o u p s w i t h t h e factor of s y m m e t r y i m p o s e d u p o n t h e m . T h e s y m m e t r y e l e m e n t s t h a t have b e e n u s e d are s h o w n as follows: 1.4.16.-

Rotation axes Plane symmetry < horizontal

vertical

Inversion symmetry (mirror) T a b l e I-2, lists t h e s e o n t h e following page w i t h c o r r e s p o n d i n g s y m b o l s , and the relation between axes and angles associated with each structure. T h e s e are t h e 14 Bravais L a t t i c e s w h i c h are u n i q u e in t h e m s e l v e s .

22

Introduction to the solid state

1.4.15. THE 14 BRAVAIS SPACE LATTICES CUBIC LATTICES

Cubic-P

Cubic-I

I

ORTHORHOMBIC LATTICES

Cubic-F

Orthorhombic-P

TETRAGONAL LATTICES

0 r t h o r h o m b i c-C

Tetragonal-P

IVIONOCLINI(

Tetragonal-I Orthorhombi c-F

TRIGONAL-R

Orthorhombic-I t

Monoclinic P

HEXAGONAL - P

TRICLINIC - P Note t h a t only c u b i c , t e t r a g o n a l a n d o r t h o r h o m b i c l a t t i c e s h a v e 90 ~ a n g l e s in all l a t t i c e s d i r e c t i o n s . M o n o c l i n i c a n d h e x a g o n a l l a t t i c e s h a v e two 9 0 ~ d i r e c t i o n s w h i l e t r i g o n a l a n d triclinic l a t t i c e s h a v e none . T a b l e 1.2. lists t h e a n g l e s of all of t h e

14 Bravais lattices, w i t h lattice t y p e s , i.e.- body-

c e n t e r e d , etc., s y m b o l s a n d a n g l e s w i t h i n t h e l a t t i c e s d e f i n e d . in t r i g o n a l - a n d h e x a g o n a l - P s y n m l e t r i e s , u = 120 ~

Note t h a t

1.4 The structure of solids

23

TABLE 1 - 2 T h e F o u r t e e n (14) Bravais L a t t i c e s in T h r e e D i m e n s i o n s RESTRICTIONS LATTICES SYSTEM

ON UNIT CELL

IN S Y S T E M

CUBIC

3

L A T r I C E ~YMBOLS

AXE~ 0 R A N G L E S

P (primitive)

a=

I (body-

a = ~ =

centered)

b = c 7

=

90 ~

F(face - c e n t e r e d ) HEXAGONAL

1

a=b~c

(a=~= TETRAGONAL

TRIGONAL

2

P,I

1

a=b r

MONOCLINIC

2

~

P,C

a~b,c

I ,F

a=13=?=90

P, C

as

In 1 9 6 5 , W e i n r e i c h point-group

P

1

developed

symmetries.

These

120 o)

~ c

~

br

a=y=90 TRICLINIC

=

a=b=c a=~= ? <120 ~ 9 0 o > y < 120 ~

R

(Rhombohedral)

ORTHORHOMBIC 4

9 0 ~ ;~,

~

aab~c

a s e t of figures w h i c h i U u s t r a t e t h e 3 2 are

shown

in

1.4.16.,

given

on

the

following p a g e , a l o n g w i t h a p p r o p r i a t e Schoenflies s y m b o l s . T h e s e a r e a type of s h o r t - h a n d u s e d to i n d i c a t e t h e n u m b e r a n d t y p e of s y m m e t r y e l e m e n t s p r e s e n t in a given p o i n t - g r o u p . I n t h i s n o m e n c l a t u r e , t h e e l e m e n t s are d e f i n e d as s h o w n in 1.4.17. on a following p a g e .

24

Introduction to the solid state

1.4.16.

The 32 Point Group Symmetries

i

I Hexag~

m 7riclini~l

N-1

....

N]

~

___I 0rthorhombic1~

Ic2~[

There

N

~

is o n e o t h e r factor c o n t r i b u t i n g to t h e overall s y m m e t r i e s of t h e

lattice s t r u c t u r e . T h i s f a c t o r is t h a t of t h e local s y n ~ n e t r y of t h e a t o m i c

25

1.5 Determination of structure of compounds C = r o t a t i o n axis only

1.4.17.-

D = dihedral (rotation plus dihedral rotation axes) I = inversion symmetry T = tetrahedral symmetry 0 = octahedral symmetry g r o u p s w h i c h actually f o r m t h e s t r u c t u r e . E x a m p l e s a re t h e " s o l i d - s t a t e b u i l d i n g b l o c k s" given above, i.e.- t h e t e t r a h e d r o n - like g r o u p , P O 4 3 - , a n d t h e o c t a h e d r o n - like NbO6-. It is e a s y to see t h a t ff a s t r u c t u r e composed

of s u c h b u i l d i n g b l o c k s , t h e y will i m p o s e

s y n m ~ e t r y on t h e lattice, in a d d i t i o n to t h e o t h e r

is

a local s t r u c t u r a l

symmetries

already

p r e s e n t . T h e r e s u l t is t h a t F a c t o r III of 1.4.11. i m p o s e s f u r t h e r s y m m e t r y r e s t r i c t i o n s o n t h e 32 p o i n t g r o u p s a n d w e o b t a i n a total of 231 g r o u p s ~ We do n o t i n t e n d

to delve f u r t h e r

into

this aspect

~p~ace

of l a t t i c e

c o n t r i b u t i o n s to c r y s t a l s t r u c t u r e . It is sufficient to k n o w t h a t t h e y e x i s t . Having now covered

the

essential

parts

of lattice

structure,

we

will

e l u c i d a t e h o w a c t u a l s t r u c t u r e is d e t e r m i n e d for a given solid. 1.5. - DETERMINATION OF S T R U C T U R E OF C O M PO U N D S We w a n t to r e v i e w h o w o n e goes a b o u t actually d e t e r m i n i n g t h e s t r u c t u r e of a given solid. T h e r e are two f a c t o r s w e n e e d to c o n s i d e r : are

required

in

actually d e t e r m i n i n g

a structure?;

2)

1) w h a t s t e p s what

kind

of

i n f o r m a t i o n do w e o b t a i n ? A c r y s t a l is a p e r i o d i c a r r a y of a t o m s in w h i c h t h e i n t e r a t o m i c

distances

a n d i n t e r p l a n a r s p a c i n g are of t h e s a m e o r d e r of m a g n i t u d e as t h e w a v e l e n g t h s of t h e r e a d i l y available x - r a y s , e.g.- t h e K a r a d i a t i o n s : Mo = 0 . 7 1 1 A - Cu = 1 . 5 4 1 8 A 9 a n d Cr = 2 . 2 9 1 A. A c r y s t a l t h e r e f o r e a c t s as a t h r e e - d i m e n s i o n a l diffraction g r a t i n g for x-rays, a n d t h r e e e q u a t i o n s ( t h e Laue

equations)

must

be

satisfied

if

there

is

to

be

constructive

i n t e r f e r e n c e of m o n o c h r o m a t i c x - r a y s . T h e Laue e q u a t i o n s are:

1.5.1.-

a(~-~o)-hk;

b(~-~o)-kk;

c ( 7 - 7 o ) = Ik

26

Introduction to the solid state

w h e r e a, b a n d c are the r e p e a t d i s t a n c e s of the lattice, a a n d ao (etc.), are the direction cosines for the diffracted a n d i n c i d e n t respectively, a n d h, k, a n d 1 are integers defining the o r d e r

beams of t h e

p a r t i c u l a r diffracted beam. W. L. Bragg s h o w e d t h a t the above e q u a t i o n s are equivalent to the c o n d i t i o n for reflection of the x-rays by the plane w i t h indices h/el, namely: 1.5.2.-

nk

= 2d

sin0

O is the angle b e t w e e n the i n c i d e n t (or reflected) b e a m a n d the plane, w h i c h we define u s i n g hkl values. Here, d is the p e r p e n d i c u l a r d i s t a n c e b e t w e e n successive planes. In s t r u c t u r e and formulation d e t e r m i n a t i o n s of a given c o m p o u n d , we p r o c e e d as follows. First of all, u s i n g x-ray diffraction e q u i p m e n t available, one obtains a series of diffraction lines. One t h e n calculates the value of d, the distance b e t w e e n a d j a c e n t planes in the crystal lattice by u s i n g the Bragg Equation. The i n t e n s i t i e s are r e a d from the diffraction chart, scaled to the m o s t i n t e n s e line. If we do n o t k n o w w h a t the n a t u r e of the m a t e r i a l is, t h e n we use the o b t a i n e d p a t t e r n to d e t e r m i n e c o m p o s i t i o n . Using the set of diffraction lines, we pick the t h r e e m o s t i n t e n s e lines in t h e pattel'n. One example is s h o w n in 1.5.3. for ~ 1 1 0 1 8 . We wish to identify these lines from the p a t t e r n a n d t h e n to identify the c o m p o u n d . 1.5.3.- Diffraction P a t t e r n of a L a n t h a n u m Aluminate

[A Typical X-ray Diffraction Pattern Obtained From a Diffr~tometer I

0

I0

20

30

40 S0 2 8 Degrees

60

70

80

90 I00

By referring to the "POWDER DIFFRACTION FILE", p u b l i s h e d by t h e

27

1.5 Determination of structure of compounds AMERICAN , ~ X ~ ~

FOR T E ~ T I N O AND M A T E R I A I ~ 1916 R a c e St.,

Philadelphia, Penna. 19103, we c a n look u p t h e m o s t p r o b a b l e c o m p o s i t i o n . T h i s t u r n s o u t to be ~ 1 1 0 1 8 . Usually, we will k n o w h o w the material was made and the c o m p o n e n t s

u s e d to m a k e it. If not, w e

c a n analyze for c o n s t i t u e n t s . In t h i s case, w e w o u l d find La a n d AI, a n d w o u l d s u r m i s e t h a t we h a v e a n oxidic c o m p o u n d . We find t h a t t h e r e a r e two c o m p o u n d s possible, v i z . - ~ 0 3 a n d L a ~ l 1 0 1 8 . However, t h e 3 m o s t i n t e n s e l i n e s of I.aA103 are: 1.5.4.-

d = 2 . 6 6 A(100), 3 . 8 0 A(80) a n d 2 . 1 9 .h.(80).

T h i s does n o t fit o u r p a t t e r n . We do find t h a t t h e p a t t e r n for LaAII I O 1 8 is i d e n t i c a l to t h e x - r a y p a t t e r n we o b t a i n e d . U s i n g 1.5.3., d - v a l u e s c a n t h e n b e c a l c u l a t e d f r o m t h e 20 values u s i n g t h e B r a g g e q u a t i o n , for t h e h e x a g o n a l c o m p o s i t i o n , ~ 1 1 0 1 8 . T h e {h,k,l} values c a n b e c a l c u l a t e d f r o m s p e c i a l f o r m u l a s d e v e l o p e d for t h i s p u r p o s e . T h e s e are given in T a b l e 1-3, p r e s e n t e d on t h e n e x t page. T h e s e values a r e o b t a i n e d b y t r y i n g c e r t a i n values in t h e h e x a g o n a l f o r m u l a , a n d s e e i n g if t h e r e s u l t s c o n f o r m . W h a t t h i s m e a n s is t h a t we s u b s t i t u t e M i l l e r I n d i c e s into t h e f o r m u l a a n d see if t h e c a l c u l a t e d value m a t c h e s I / d 2. O n c e t h i s is d o n e , we h a v e c h a r a c t e r i z e d

o u r m a t e r i a l . B e c a u s e of t h e

p h y s i c a l g e o m e t r y of t h e x - r a y d i f f r a c t i o n g o n i o m e t e r device), one o b t a i n s values of 20 d i r e c t l y .

(angle-measuring

Note t h a t t h e e q u a t i o n s in T a b l e 1-3 for t h e h i g h s y r m n e t r y l a t t i c e s a r e r a t h e r s i m p l e w h i l e t h o s e for low s y m m e t r y l a t t i c e s are c o m p l i c a t e d . S i n c e we k n o w t h a t t h e c o m p o u n d , I.aAlllOl8 is h e x a g o n a l , we u s e t h e equation: 1.5.5.-

l/d 2=4/3

[( h 2 + k 2 + 1 2 ) /

a 2]

If we do this, we o b t a i n t h e d a t a in Table 1-4 also s h o w n o n t h e n e x t page. T h i s allows u s to d e t e r m i n e t h e u n i t cell l e n g t h s for o u r c o m p o u n d as: Hexagonal:

a o = 5.56. A - b o = 2 2 . 0 4 A

28

Introduction to the solid state

T a b l e I- 3 P l a n e S p a c i n g s for V a r i o u s L a t t i c e G e o m e t r i e s CUBIC 1/d 2 = h 2 + k 2 + 12/a 2

1/d 2=4/3

1/ d 2 = h

HEXAGONAL [(h 2 + k 2 + 1 2 ) /

TETRAGON/EL 2 + k 2 / a 2 + 12 / c 2

ORTHORHOMBIC 1 / d 2 = ( h 2 / a 2 ) + ( k 2 f b 2 ) + ( 1 2 / c 2 ) + 12/c 2

a 21

RHOMBOHEDRAL

lid 2 =

L_h_2 + k 2 + ! 2) s i n 2 a + 2 [ h k + kl + h l ) i c o s 2 a - cos a ) a2 ( 1 - 3 c o s 2 a + 2 c o s 3 a )

1/d 2={

MONOCLINIC 1 / s i n 2 b}{ h 2 /a 2 + ( k 2 s i n 2 b) /b 2 + 1 2

= v o l u m e of u n i t cell . S l l = b 2 c 2

sin 2 b 9 $33 $23

20 8.02 16.1 18.86 20.12 22.07 24.23 24.55 32.19 32.50

/ac}

TRICLn~C h 2 + $ 2 2 k 2 + $ 3 3 12 + 2 S 1 2 h k + 2 $ 2 3 kl + 2 S 1 3 hl

lid 2 =l/V2{ Sll where: V

/c 2-2hlcosb

= a2 b 2 sin 2

= a2 b c ( c o s

b cosg

g"

-cos

S12

=

abc 2

sin 2

a

9 $22

=

(cos a c o s b - c o s

a ) - $ 1 3 = ab 2 c ( c o s g

TABLE 1-4 C o n v e r s i o n of 20 V a l u e s of t h e Diffraction P a t t e r n to I / I0 d {hid} 2 q I / I0 16 11.02 002 36.18 74 27 6 4.81 004 39.39 29 32 4.71 001 40.94 66 28 4.41 012 42.79 46 I0 4.03 013 45.61 15 21 :~.67 001 53.36 11 3.63 014 58.57 36 44 2.78 110 60.07 60 15 2.76 008 67.35 48 18 14

a2 c 2 g)

-

cosa-cosb)

{hid} V a l u e s {hkl} d 114 2.48 2.29 023 2.20 0010 2.11 025 2.01 026 1.72 029 127 1.58 I. 5 4 0211 1.39 220 1.32 0214 1.04 2214

1.5 Determination of structure of compounds Additionally, we c a n list t h e

x-ray parameters

and

29 convert

them

to

s t r u c t u r a l factors as s h o w n . T h e s e values are a v e r a g e d over all of t h e r e f l e c t i o n s u s e d for calculation. Note t h a t t h i s p a t t e r n h a s several p l a n e s w h e r e t h e "d" value is m o r e t h a n t e n . Let u s c o n s i d e r o n e o t h e r e x a m p l e . S u p p o s e we o b t a i n e d t h e following s e t of 20 v a l u e s a n d i n t e n s i t i e s for a c o m p o u n d : TABLE ! - 5 DIFFRACTION LINES AND I N T E N S I T I E S OBTAINED Intensity

2__0 .in d e g r e e s v

96

29.09

100

33.70

56

48.40

50

57.48

14

60.28

18

78.38

10 14

90.55 80.78

17 19

97.80 118.26

11

121.00

7

144.34

9

148.49

T h e first t h r e e values are t h e s t r o n g e s t diffraction lines. After c a l c u l a t i n g "d" values a n d looking u p t h e set of s t r o n g lines w h i c h c o r r e s p o n d to o u r set, we find t h a t t h e p r o b a b l e c o m p o u n d is CdO 2 , or c a d m i u m p e r o x i d e . T h i s c o m p o u n d t u r n s o u t to b e cubic in s t r u c t u r e , w i t h ao

= 5.313 A.

When we

planes,

calculate

the

{h,k,l}

values

of t h e

diffracting

the

s t r o n g e s t line is f o u n d to be {200}. We c a n t h e n m a k e t h e d e t e r m i n a t i o n t h a t s i n c e Cd 2+ is a s t r o n g l y diffracting a t o m (it h a s h i g h a t o m i c w e i g h t , w h i c h is o n e w a y of s t a t i n g t h a t it h a s m a n y e l e c t r o n shells, i.e.- I s 2 2 s 2 2 p 6 3s 2 3 p 6 3 d 1 0 4 s 2 4 p 6 4 d 1 0 , t h e s t r u c t u r e is p r o b a b l y f a c e - c e n t e r e d

30

Introduction to the solid state

cubic. I n d e e d , t h i s t u r n s o u t to be t h e case. In t h e u n i t cell, Cd a t o m s a r e in

the

special

positions

of

:

{0,0,0},

{II2,112,1/2};

0,112.1/2};

{I 12,1 / 2,0}. T h e r e are four m o l e c u l e s p e r u n i t cell. We c o u l d c o n t i n u e f u r t h e r a n d c a l c u l a t e i n t e n s i t i e s , Ic , u s i n g a t o m i c s c a t t e r i n g f a c t o r s a l r e a d y p r e s e n t in prior l i t e r a t u r e . We w o u l d scale c a l c u l a t e d i n t e n s i t i e s to o u r o b s e r v e d i n t e n s i t i e s by S I c = S I o . We t h e n c a l c u l a t e a reliability factor, called R, from R = S (Io - Ic ) I S Io. A low value i n d i c a t e s t h a t o u r s e l e c t i o n of lattice p a r a m e t e r w a s c o r r e c t .

If not, we c h o o s e a slightly

different value a n d a p p l y it. The details of t h e p r o c e d u r e for d e t e r m i n i n g e x a c t s t r u c t u r e a n d a t o m i c p o s i t i o n s in t h e lattice are well k n o w n , b u t a r e b e y o n d t h e s c o p e of

t h i s C h a p t e r . However, for y o u r own edification, I

r e q u e s t t h a t you i n d e x t h e lines (calculate t h e {hkl} values) given above, s i m i l a r to t h a t p r e s e n t e d in Table 1-4. S u m m a r i z i n g to t h i s point, we have s h o w n t h a t only c e r t a i n p r o p a g a t i o n u n i t s c a n b e s t a c k e d to infinity to form c l o ~ - p a c k e d

solids. We have also

s h o w n h o w t h e u n i t s fit t o g e t h e r

solids w i t h s p e c i f i c

to f o r m specific

s y m m e t r i e s . T h e n a t u r e of t h e s t r u c t u r e of solids h a s also b e e n r e v i e w e d in s o m e

detail.

Now let u s look at t h e

solid f r o m t h e

standpoint

of

stacking and stacking defects. 1.6. - THE D E F E C T SOLID We have s h o w n t h a t b y s t a c k i n g a t o m s or p r o p a g a t i o n u n i t s t o g e t h e r , solid w i t h

certain

symmetry

aspects

results.

If we

have

done

a

this

p r o p e r l y , a p e r f e c t solid s h o u l d have r e s u l t e d w i t h no defects in it. Yet, d e f e c t s are r e l a t e d to t h e e n t r o p y of t h e solid, a n d a p e r f e c t solid w o u l d violate t h e entro/~

second

law of t h e r m o d y n a m i c s .

This

law s t a t e s t h a t z e r o

is only p o s s i b l e at absolute zero t e m p e r a t u r e . T h u s , m o s t of t h e

solids t h a t we e n c o u n t e r a r e d e f e c t - s o l i d s . It is n a t u r a l to a s k c o n c e r n i n g t h e n a t u r e of t h e s e defects. It s h o u l d be obvious t h a t m o s t of t h e m will be s t a c k i n g d e f e c t s or faults. N a t u r e (and i n d e e d w e

in o u r p r o p a g a t i o n u n i t

e x a m p l e as well) finds it i m p o s s i b l e to s t a c k a t o m s (molecules) in p e r f e c t o r d e r to infinity. Moreover,

even if we could o b t a i n a p e r f e c t

w o u l d likely b e n o n - r e a c t i v e a n d w o u l d be ~ n ~ j u / a r ~

solid, it

stable. The r e a s o n

1.6 The defect solid for t h i s is t h a t it is t h e

defects

31

in solids w h i c h

give t h e m

special

properties. C o n s i d e r t h e s u r f a c e of a solid. In t h e i n t e r i o r , we see a c e r t a i n s y m m e t r y which depends

upon the

structure

of t h e

solid. As we a p p r o a c h

the

s u r f a c e f r o m t h e i n t e r i o r , t h e s y m m e t r y b e g i n s to c h a n g e . At t h e v e r y surface, t h e s u r f a c e a t o m s see only half t h e s y m m e t r y t h a t t h e i n t e r i o r a t o m s do (and haft of t h e b o n d i n g as well). R e a c t i o n s b e t w e e n solids t a k e place at t h e surface. If t h e r e w e r e s o m e w a y to c o m p l e t e t h e s y m m e t r y of t h e s u r f a c e a t o m s , t h e n t h e y too w o u l d likely be n e a r l y n o n - r e a c t i v e . In a t h r e e - d i m e n s i o n a l

solid, we c a n conceive of t h r e e

m a j o r t y p e s of

defects, one-, two- a n d t h r e e - d i m e n s i o n a l in n a t u r e . T h e s e are c a l l e d : point, line (edge)

and volume

(plane),

respectively.

Point

defects

are

c h a n g e s at a t o m i s t i c levels, while line a n d v o l u m e d e f e c t s are c h a n g e s in s t a c k i n g of g r o u p s of a t o m s (molecules). An e a s y w a y to visualize p o i n t d e f e c t s is s h o w n in t h e following: 1.6.1.-

BUBBLE RAFT SHOWING EFFECTS OF VACANCY AND "IMPURITY" ON HEXAGONAL CLOSE-PACKING

A b u b b l e raft is m a d e by c r e a t i n g b u b b l e s in a s o a p solution w h i c h float to t h e s u r f a c e of t h e liquid (in t h i s case, w a t e r ) to c r e a t e a raft. T h e t r i c k is

32

Introduction to the solid state

to get t h e size of t h e b u b b l e s all t h e s a m e . If not, o n e d o e s n o t get a way to c o m p a r e t h e effects of close p a c k i n g u s i n g b u b b l e s to s i m u l a t e a t o m p o s i t i o n s in a lattice s t r u c t u r e . T h i s p r o b l e m w a s w o r k e d

on u n t i l t h e

investigator was successful. T h e first t h i n g o n e n o t i c e s is t h a t p a c k i n g defects h a v e a significant effect o n t h e close p a c k i n g of a b u b b l e raft. Two t y p e s of d e f e c t s c a n be s e e n . One is a "vacancy", t h a t is, t h e b u b b l e s u p p o s e d to b e t h e r e , is m i s s i n g . T h e o t h e r is a n " i m p u r i t y " , h e r e as a l a r g e r bubble. Note its effect on t h e d e g r e e of o r d e r i n g . Again, lattice c o m p e n s a t i o n is t h e n o r m a n d t h e c l o s e p a c k i n g is c o m p e n s a t e d b y a n a d j u s t m e n t in t h e lattice. Note t h a t m o s t of t h e b u b b l e s are all t h e s a m e a n d are h e x a g o n a l l y - c l o s e p a c k e d . You c a n i m a g i n e t h e effect o n s t a c k i n g in t h e lattice of a v e r y i m p u r e solid w h e r e t h e c o m p o u n d is o n l y 9 5 % p u r e . In t h e following is a n o t h e r view of d e f e c t s : 1.6.2.-

In t h i s d i a g r a m , we see t h r e e t y p e s of p o i n t defects. In a d d i t i o n to t h e v a c a n c y , we also see two t y p e s of s u b s t i t u t i o n a l defects. Both are d i r e c t s u b s t i t u t i o n s in t h e

"lattice",

or a r r a n g e m e n t

of t h e

a t o m s . One is a

s m a l l e r a t o m , while t h e o t h e r is l a r g e r t h a n t h e a t o m s c o m p r i s i n g

the

lattice. Note t h e difference, d u e to size of t h e i m p u r i t y , u p o n t h e o r d e r i n g

1.6 The defect solid

33

in the hexagonally c l o s e - p a c k e d lattice. In addition, we have also i n c l u d e d the "interstitial" atom, t h a t is, one t h a t is able to i n s i n u a t e itself into t h e i n t e r s t i c e s of the lattice. This c o m p l e t e s the list of possible point d e f e c t s in a m o n o - a t o m i c lattice. In the next Chapter, we will s h o w o t h e r p o i n t defects possible w h e n b o t h cation a n d anion a t o m - s p e c i e s are p r e s e n t in the lattice. In the following diagram, we see the effect of an e d g e dislocation u p o n hexagonal close p a c k i n g : 1.6.3.-

LINE DEFECTS DUE TO PACKING (1) AND DEFECTS DUE T 0 INHOMOGENEITIE S (2).

iql,O

, 4, ,ulu,,1P~r A e,'~'oToToT*~nlu~O~O I O

,e,'~:,e,

At "1" in 1.6.3., the h e x a g o n a l p a c k i n g h a s c h a n g e d to t h a t of cubic closepacking, c a u s i n g a line defect. At "2", the line defect is c a u s e d by lack of o r d e r i n g along the plane. This m a y also be r e g a r d e d as a series of p o i n t defects affecting ordering. What has h a p p e n e d is t h a t m a n y d e f e c t vacancies have c o n g r e g a t e d a n d have c a u s e d a b o u n d a r y b e t w e e n t h e "grains". Within the g r a i n - b o u n d a r y , the o r d e r i n g is u n i f o r m b u t differs from the next grain. It is actually a line i m p e r f e c t i o n (2-dimensional), o r an edge i m p e r f e c t i o n . A b e t t e r p e r s p e c t i v e of an edge dislocation is s h o w n in 1.6.4., p r e s e n t e d on the n e x t page.

Introduction to the solid state

34

1.6.4

-

A Line Defect[

4b-

Note t h a t in its s i m p l e s t form, an edge dislocation is an omission of a line of a t o m s c o m p o s i n g the lattice. A n o t h e r perspective is p r e s e n t e d in t h e following, s h o w n as 1.6.5. on the next page. In this diagram, the area within the lattice a r o u n d the line defect is u n d e r b o t h c o m p r e s s i o n a n d t e n s i o n due to the difference in a t o m - d e n s i t y as one p a s s e s t h r o u g h it in a direction p e r p e n d i c u l a r to the line defect. Thus, t h e r e is excess e n e r g y in the lattice due to the c o m p r e s s i v e - t e n s i l e forces p r e s e n t . This has the effect of causing the edge dislocations to p r o p a g a t e t h r o u g h the solid by a h o p p i n g m o t i o n until they reach t h e surface of the solid. T h e r e have b e e n several cases w h e r e it has b e e n possible to directly observe line i m p e r f e c t i o n s by suitable p r e p a r a t i o n and m i c r o s c o p i c e x a m i n a t i o n of the surface in reflected light. One example is the MgO crystal. MgO is a cubic crystal and it is possible to etch it along the {100} direction (this is the direction along the x-axis). What is o b s e r v e d is a series of surface lines of specific length. The r e a s o n t h a t

35

1.6 The defect solid 1.6.5.A n E d g e D i s l o c a t i o n in t h e Solid-' Note t h a t t h e L a t t i c e is U n d e r C o m p r e s s i o n and T e n s i o n on E i t h e r S i d e of t h e Line D e f e c t mmmmimm mmimili milllli

i|Pqimmi mBilnnn m m i m m i m i i m m i m m i m m i

m i i m m a mmme.w n

Another Perspective of Line D e f e c t s

mmmamrm

immrlmm, i m i m m i

Lmt.lmmmm immmmml

t h e s e defect lines s h o w up is t h a t they are m o r e easily e t c h e d by acid along the direction of the edge dislocation. The volume defect is s o m e w h a t m o r e difficult to visualize. In t h e following figure, given as 1.6.6. on the next page, a "screw dislocation" is shown, so-called b e c a u s e of its t o p o g r a p h y . This is an actual r e p r e s e n t a t i o n of a paraffin single crystal. The t o p - v i e w s h o w s the edges of the d i a m o n d - s h a p e d crystal w h i c h are s h a d e d for b e t t e r clarity. A side-view is also shown. At the b o t t o m of 1.6.6. is s h o w n h o w the dislocation line (line defect) b e c o m e s a screw-dislocation. If t h e g r o w t h p a t t e r n is spiral, there is an offset 'I)". The length of the spiral is, of c o u r s e r e l a t e d to "r", the r a d i u s of growth, a n d to Or, the c h a n g e in offset r a d i u s of growth. Referring b a c k to 1.6.4., one sees how the line defect is related to the spiral g r o w t h p a t t e r n , since the line d e f e c t p r o p a g a t e s to the surface w h e r e the actual crystal g r o w t h t a k e s place. T h u s , the volume defect is one w h i c h is g e n e r a t e d from the interior to the surface of the crystal.

36

Introduction to the solid state

1.6.6.-

Topology of a S c r e w Dislocation

TOP VIEW

G r o w t h Patt e r n of a Single Crystal of ParalTin - C35H 72

SIDE VIEW | |

I

I

I

I

I

I

illl II

I

L

I

I

i

I

|

I

I

I

I

I

b ::::::::::::::::::::

d

How a Dislocation Line Becomes a Screw Dislocation

j I I |

DISLOCATION LINE

It is t h e i n t r i n s i c defects t h a t have t h e m o s t i n t e r e s t for us, since t h e y affect t h e c h e m i c a l p r o p e r t i e s of t h e solid while e x t r i n s i c

defects have

little effect. E x t r i n s i c defects are t h e p r o p e r s t u d y for t h o s e i n t e r e s t e d in the

mechanics

of

solids,

particularly

metals.

A

broad

variety

of

1.6 The defect solid

37

c o m m e r c i a l p r o d u c t s are b a s e d u p o n c o n t r o l l e d - p o i n t - d e f e c t s . T h e s e include t r a n s i s t o r s , i n t e g r a t e d circuits, p h o t o s e n s o r s , color-television, f l u o r e s c e n t l a m p s , j u s t to n a m e a few. None of t h e s e would be p o s s i b l e w i t h o u t point defects. It is for this r e a s o n t h a t we devote the n e x t c h a p t e r to the point defect. SUGGESTED READING 1. G. W e i n r e i c h - " S o l i d s - E l e m e n t a r y T h e o r y for Advanced S t u d e n t s " - , J. Wiley & Sons, Inc., New York ( 1 9 6 5 ) . 2. A.J. D e k k e r - " S o l i d State Physics" - , P r e n t i c e - H ~ l ,

Inc., E n g l e w o o d

Cliffs, NJ (1958). (See Chaps. 1,2 &3 in p a r t i c u l a r ) . 3. J.M. Honig, " I m p e r f e c t i o n s in Crystal", J. Cherru Ed., 34, 224 ( 1 9 5 7 ) . 4. J.A. McMillan, " S t e r e o g r a p h i c P r o j e c t i o n s of t h e Colored Crystallographic Point Groups", Am. J. Phys., 35, 1049 ( 1 9 6 7 ) .