chapter 1 structure and properties of metals and alloys 1995 studies in surface science and...

8/11/2019 Chapter 1 Structure and Properties of Metals and Alloys 1995 Studies in Surface Science and Catalysis

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Ch a p te r 1

S T R U C T U R E A N D P R O P E R T I E S O F M E T A L S A N D A L L O Y S

1.1 A m icroscopic theory of so l ids

1 .1 .1 Th e q u a n tu m th e o r y o f p u r e me ta l s

I t i s imposs ib le to presen t in a s ing le chapte r an exac t theory of meta ls and a l loys ,

o r o f p h e n o me n a , s u c h a s th o s e f o r min g th e b a s i s o f e l e c t r o n - s p e c t r o s c o p ie s , th a t a r e u s e d

to s tudy and to es tab l ish the e lec t ron ic s t ruc ture of meta l l ic ca ta lys ts . However , i t i s f e l t

th a t a b o o k o n c a ta ly s i s b y a l lo y s s h o u ld a t l e a s t in t r o d u c e s o me o f th e imp o r ta n t t e r ms

(band s t ruc ture , dens i ty of s ta tes , photoemiss ion f rom the va lence band , e tc . ) on bas is o f

s o me v e r y s imp le th e o r e t i c a l c o n s id e r a t io n s ; i t i s n o t o u r a mb i t io n to a c h ie v e mo r e th a n

that.

A l l mo d e r n b o o k s o n u n d e r g r a d u a te p h y s ic a l c h e mis t r y [ 1 - 3 ] o f f e r a n in t r o d u c t io n

to q u a n tu m me c h a n ic s , wh ic h i s th e b a s i s o f th e c h e mic a l b o n d th e o r y . Th e r e a d e r i s th u s

expec ted to be famil ia r wi th te rms such as the wave or s ta te func t ion (e .g . Z or ~ t ) , the

Ha mi l to n ia n o r to ta l e n e r g y o p e r a to r I t / a n d th e Sc h r 6 d in g e r e q u a t io n :

(/-~ - E) z = 0 (1)

where E is the s teady s ta te to ta l energy of the sys tem, the s ta te o f which is descr ibed by

func t ion Z . The to ta l energ y opera to r 121 can be sp l i t in to two par ts , the k ine t ic energ y

opera tor T and the po ten t ia l energy opera tor V. Opera tor T is a di f f erent ia l o p e r a to r a n d

thus equ a t ion 1 is a d i f fe ren t ia l equa t ion o f second order . The tex t book s [1-3] o f fe r a lso

a n in t r o d u c t io n to th e f o r m o f f u n c t io n s Z f o r h y d r o g e n a to m, f o r th e h y d r o g e n - l ik e a to ms

( l i th ium, sod ium, po tass ium, e tc . ) and for func t ions (orb i ta ls ) o f some o ther a toms . With

me ta l s a n d a l lo y s we a r e in te r e ste d in th e f o r m o f th e

sol id

o r

crys tal orbi ta l s ~ .

Le t u s

summar ize some of the i r bas ic fea tures [4] .

We sha l l mos t ly be in te res ted in meta ls o r a l loys in a c rys ta l l ine form. Such bodies

d i s t in g u i s h th e ms e lv e s b y h a v in g a p e r io d ic p o te n t i a l V , s o i f we c o n s id e r a l in e a r o n e -

d imens iona l sys tem of an in f in i te length and wi th a per iod ic i ty , i .e . la t t ice cons tan t , a , the

e lec t ron dens i ty p , which is p ropor t iona l to the probabi l i ty o f f ind ing an e lec t ron in a un i t

vo lu m e, i .e . g t*g t wh ere g t* is the com plex conju ga ted fo rm of ~ , wi l l be the sam e a t a l l

p laces d i f fe r ing by a ; the re fore we s ta te tha t


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St ruc tu re and p rope r t i e s o f m e ta l s and a l l oys 9

W e subs t i t u t e t he c rys t a l o rb i t a l by a l i nea r com bina t ion o f a tom ic o rb i t a l s I~ n

n ind i ca t ing the i r p l ace

w i t h

Vrr = L.C.A.O. = n Cn *n (7)

in t he Schr6d inge r equa t ion 1 and r ead i t a s :

2 n C n ( f l - E) On -" 0 (8)

To conve r t equa t ion 8 , w h ich i s a d i f f e r en t i a l equa t ion , i n to an a lgebra i c r e l a t i on be tw een

num er i ca l va lues , w e m ul t i p ly i t by

0*e and

in tegra te ov er the space of N a to ms. In th is

w ay , by w r i t i ng t7t i n an ex t ended fo rm ( the use fu lness o f it w i l l be seen im m edia t e ly ) , w e

ob ta in t he r e l a t i on

]~n Cn { I **e (T "1- Vcrystal Uat- Uat) 0Pnd~ -

g I

**e *n

d ' l : } = 0

(9)

T + Uat i s H ~ the Ha mil to nian of a f ree a to m o f the e le men t o f the chain and d 'c the

elem ent of the space . Sub st i tu t ing th is exp ress ion for H ~ and ca l l ing Vcryst- Uat "- A V , w e

ob ta in

]~n Cn {I 0*e

( H~

AV ) *n d'l~- g I 0*e On

d'c} = 0

(10)

W e n o w a s s u m e t h a t t h e o v e r l a p b e t w e e n

*n

and **e i s zero wh en n i s not equa l to e (n

and e deno te d i f f e ren t pos i t i ons o f a tom s) , bu t i s un i ty w hen e equa l s n . Thus

I 0*e *n d'c = o and I **n *n d'c = 1 (11 )

Fur the r , i n ou r approx im a t ion w e keep on ly t he fo l low ing t e rm s o f equa t ion 10

-- Eat (e=n) =

I I ~ e H ~ Cn d1: (1 2)

= 0 (ee:n)

= 13 (e=n_+l)

I *e m v *n =

0 (e=n) (13)

= 0 (e>n_+l)


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10 chapte r 1

The second te rm (e=n) is ze ro because of the def in i t ion of the ' ze ro ' po ten t ia l energy . I f

we choose n , somewhere in the cha in , we a re then le f t wi th a genera l equa t ion for Cn:

(14)

s ince a l l o ther te rms vanish in our approximat ion def ined by equa t ions 11 to 13 . Then

a n + 1 + a n _ 1

E = a + [3 (15)

Cn

Th e B lo c h th e o r e m c a n b e f u l f i l l e d b y t a k in g

C,, = exp ( ik x) (16)

s ince

~kx,, ikx -x

0 = ~ e g O , ,= e . ~ e r e x g O ( x -x ,, ) (17)

Th e t e r m r e p r e s e n te d b y th e s u mma t io n h a s th e p r o p e r t i e s o f th e f u n c t io n U( x ) . By

subs t i tu t ing the express ion for Cn in equa t ion 15 the energy E is

E ( k ) = a

+ [3 (e -~ + e +U'a)=a +213 co ska (18)

Severa l f ea tures of th is equa t ion a re very in te res t ing . There a re N d if fe ren t va lues of k ,

thus there a re N d if fe ren t c rys ta l o rb i ta ls ~k and N d if fe ren t energy leve ls . The leve ls E(k)

form, for a la rge Ns , a quas i -cont inuous band of energ ies be tween

Emax

and

Emin,

a n d f r o m

equa t ion 18 the band wid th is

Emax-Em~n=4p

(19)

In o ther words , the band wid th is p ropor t iona l to the over lap or hopping in tegra l B . The

(n-1)d orb i ta ls (n be ing the pr inc ipa l quantum number of va lence e lec t rons ) over lap less

than do the ns orb i ta ls . In the rough approximat ion which we have cons idered , the bands

are separa ted and the (n- l ) d -band is nar row and the (n) s -band is b road . The d-band has

then 5N leve ls , bu t the s -band on ly N leve ls . The dens i ty of s ta tes be tween E and E+dE is

as a consequence h igher in the d-band than in the s -band . These a re the main p ieces of

informat ion which one needs in order to unders tand chapte r s 2 and 3 .


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Struc ture and proper t ies o f meta ls and a l loys 11

E ( k )

l

- l ' t 0

O

k !

O

> N ( E )

> N ( E )

f i g u r e 1

l ef t: E n e r g y a s a f u n c t i o n o f t h e w a v e - v e c t o r k, f o r a h y p o th e t i c a l o n e - d i m e n s i o n a l

c r y s t a l ( a c h a i n o f a t o m s ) w i t h a la t t ic e c o n s t a n t a .

r i g h t: D e n s i t y o f s t a t e s c u r v e , c o r r e s p o n d i n g t o E ( k ) s h o w n i n t h e le f t p a r t .

The func t ion cos (k a ) i s de f ined over the in te rva l k f rom - r t /a to +r t /a and in f igure

1 6 has a nega t ive va lue , an d tx is taken as the a rb i t ra ry ze ro . W here the s lope of the

func t ion is s teep , the re a re on ly a few s ta tes ( i .e . few k ' s ) in a g iven range of energy , bu t

wh e r e th e s lo p e i s lo w, a s in th e n e ig h b o u r h o o d o f g /a , th e r e a r e ma n y . I n o th e r wo r d s ,

the d ens i ty of s ta tes N(E ) is a func t ion wh ich increases wi th (dE/dk) -1, as i s seen in f igure

2 . We s h a l l me e t th e t e r m

d e n s i t y o f s t a t e s

a t many p laces in th is book .

In the f ree e lec t ron approximat ion and for an one d imens iona l so l id cha in , ~k is

equa l to A e ikx and the S chr6d inger equa t ion is used w i th V equa l to ze ro . I t r eads

d2 ~ k 8 n 2m

+ ~ E q ~ k = 0

d ~ h

(20)

Su b s t i tu t io n o f ~ k b y th e f u n c t io n o f f r e e e l e c t r o n s in th e Sc h r 6 d in g e r e q u a t io n p r o d u c e s


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12 chap te r 1

k Z h 2

E - (21)

8~;2m

w hich can be com pared w i th t he N ew ton ian r e l a t i on w h ich says t ha t E equa l s pZ /2m , w i th

p be ing the m om entu m . Indeed , fo r f ree e l ec t rons t he m o m e ntu m p equa l s h k o r i n o the r

w ords , k i s a m om en tum in un i t s o f h .

E l k )

- n k 0 k n

G t3

a 2 - 2r~ 1

a 2

a 2 + 2 1 3 2

a l - 2 ~ 1

a l + 2

[31

f i g u r e 2

E k ) , d i s p e r s i o n f u n c t i o n s , f o r a

h y p o t h e t i c a l o n e d i m e n s i o n a l

c r y s t a l a t o m i c c h a in ) w i th tw o

o r b i ta l s , c o r r e s p o n d i n g t o t h e

e n e r g i e s z 1 a n d z2 o n e a c h

a t o m .

j 3 1 , j 3 2 < O ; I ~ 1 1 < I j% l

The in t e rva l

- n / a

to

n / a

fo rm s a k - space in to w h ich a l l poss ib l e non-equ iva l en t k ' s

a re p l aced ; i t i s ca l l ed t he f i r s t Br i l l ou in zone . W e have seen th rough our d i scuss ion o f t he

Bloch theorem tha t k equa l s 2ng /N a , o r i n o the r w ords k i s r e l a t ed to t he r ec ip roca l l a t t i ce

con stan t , a -1. Th e interv al

- n / a

to

n / a

i s t hus r ec ip roca l w i th r e spec t t o r ea l space . Because

p equa l s hk , t h i s i n t e rva l i s a l so a m om entum space , i n to w h ich a l l poss ib l e s t a t e s , each

c h a r a c t e r i z e d b y i t s e n e r g y a n d k - n u m b e r o r m o m e n t u m , a r e p l a c e d .

In h ighe r approx im a t ions t han tha t co r r e spond ing to t he f r ee e l ec t ron m ode l , t he

m om entum i s no t equa l t o hk . H ow ever , w i th r ega rd to va r ious fo rces F , hk s t i l l behaves

l ike a m om entum , s ince F i s a lw ays equa l t o hk ' . The re fo re , k can be ca l l ed a pseudo-m o-

m e n tum , i n un it s o f h . Th i s i s an im por t an t po in t fo r unde r s t an d ing the an g le - r e so lve d

va lence -band pho toem iss ion , w h ich i s d i scussed in chap te r s 2 and 3 .

Le t u s now m ake the s t ep f rom cha ins o f a tom s to tw o-d im ens iona l f l a t a r r ays .

N ow , fo r a squa re l a t t i ce


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Struc ture and proper t ies o f meta ls and a l loys

13

~ ,

= P C r , ~ ~ , . , ~ (22 )

r , s

wi th

O r , s

= e X p

i k r X r + i k s Y s ) (23)

and the energy is

E k ) = a +

213(cosk~a + coskya )

(24)

and it forms a plane in the E(kx, ky) space (see f igure 3, under a) on the lef t)

The f i r s t Br i l lou in zone shown under b) in f ig .3 is now two d imens iona l , a s i s a lso

the whole k- space . I t i s o f ten usefu l to show the func t ion E(k) in a more s imple way .

The n E(k) is ca lcu la ted for se lec ted va lues of k , fo r exam ple a long certa in l ines in the

Br i l lou in zone , such as , f rom the po in t k (0 ,0) to the po in t k0 t /a ; 0 ) , e tc . , a s i s shown in

f igure 3 le f t under c ) . The dens i ty of s ta tes cor responding to the energ ies be tween ~ + 413

and o t - 4g is shown in the lower r igh t comer [4] .

In th ree d im ens ions , wi th E (k x, k y , k z ) the p ic tures a re s l igh t ly more complica ted

but no t conceptua l ly very d i f fe ren t .

The mathemat ica l theory of g roups teaches us tha t in each space hav ing t rans la t i -

o n a l s y mme t r y a n d in wh ic h Bo r n - Ka r ma n c o n d i t io n s h o ld , th e r e a r e a lwa y s 1 4 d i f f e r e n t

Brava is la t t ices . Both the rea l and the rec iproca l o r k space a re spaces wi th a la t t ice , i .e .

t r ans la t iona l symmetry , and th is means tha t each la t t ice type in one space mus t have a

counte rpar t in the o ther ( i .e . r ec iproca l) space . For example , i t fo l lows tha t bcc ( rea l space)

transla tes into fcc (reciproc al) and fcc ( real) translates into bcc ( reciprocal) , e tc . Fur t her ,

i t f o l lo ws th a t th e v e c to r k a n d th e p s e u d o - mo me n tu m p h a v e in a c r y s ta l o f r e c ta n g u la r

form the same d irec t ions in the rea l and in the rec iproca l space . This enables us to

in d ic a te th e r e a l mo v e me n t o f e l e c t r o n s b y mo v e me n ts o f k - s t a t e s in th e r e c ip r o c a l s p a c e .

This i s aga in an impor tan t s ta tement for the descr ip t ion and unders tanding of the angle -

reso lved e lec t ron photo emiss io n . I f e lec t rons proceed in the k- space der ived , fo r exam ple ,

f rom a cubic c rys ta l in a ce r ta in d i rec t ion , they do the same in the rea l c rys ta l too , and a t

the sur face they cont inue to pass in to vacuum without r e f rac t ion because the kx ,ky-compo-

nents a re prese rved . F igure 4 shows a Br i l lou in zone of an fcc rea l - space la t t ice .

The func t ion E(k) , ca l led of ten the d ispers ion law, is usua l ly theore t ica l ly ca lcu la -

ted for ce r ta in se lec ted va lues of k , fo r example , fo r k ' s a long wel l chosen l ines in te rcon-

nec t ing im por tan t po in ts on the Br i l lou in zones . These po in ts a re denoted by le t te rs F , X,

K, W e tc . ( see a lso chapte r s 2 and 3) . The typ ica l fo rm of such E(k) - sec t ions a re shown

in f igure 5a by resu l ts fo r copper . When the k-poin ts cor responding to the h ighes t energy

leve ls s t i l l occupied by e lec t rons a re in te rconnec ted by a p lane , the so ca l led Fermi sur face


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14 chap ter 1

is c rea ted ( f igure 5b) .

(a)

k ,

t ~ / a

k~

a

y

(b)

bl[ a

: ~ / a

~/a w

kl

Energy

(c)

= 4~

o{

~ + 4 ~

(0,0)

n,a, O) n/a,n /a)

(0,0)

X M F

NIE)

f i g u r e 3 .

A h y p o t h e t i c a l t w o d i m e n s i o n a l c r y s t a l .

T h r e e r e p r e s e n t a t i o n s o f E k ) f o r t h e s b a n d .

a ) E n e r g y s u r f a c e f o r o n e q u a r t e r o f t h e B r i l l o u i n z o n e .

b ) C o n s t a n t - e n e r g y c o n t o u r s, i l lu s t r a ti n g t h e s y m m e t r y o f t h e z o n e .

c ) E n e r g y p l o t t e d o v e r a t r ia n g u l a r p a t h o f k va lu e s, s h o w i n g m i n i m u m a n d m a x i m u m

energ ies , an d dens i ty o f s ta tes .

f o r s y m b o l s F , X a n d M s e e f i g u r e 4 )


9/66


k

z

1

t d l

y

f igure 4 .

Bri l lou in zone in a reciprocal space wi th b .c .c .

latt ice, c orre spon ding to fc .c . lat t ice in the

real space.

f igure 5 .

Ban d s t ruc tu re a ) and

Ferm i sur face b ) f o r copper .

In a) the Cu 3d bands are

label led; the dashed curve

shows the 4 s band pred ic t ed

wi thou t any m ix ing wi th

the d band [4]

E n e r g y [

F X W

C 3 '

,,,,~

I

{ b )


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16 cha pter 1

Th e o c c u p a t io n o f E ( k ) - l e v e l s a t t e mp e r a tu r e s a b o v e a b s o lu te z e r o i s g o v e r n e d b y

the Fermi-Dirac d is t r ibu t ion func t ion [1-4] .

E - E v ) ]

(25)

f ( E ) = [ 1 + e x p ( - 1

k T

Equat ion 25 s ta tes tha t as the tempera ture approaches ze ro , a l l leve ls be low EF, the Fermi

energy , bec om e o ccupied ( i.e . f(E) tends to un i ty ) and a ll leve ls above E v beco me vacant .

Thus for meta ls hav ing a pseudo-cont inu ous band , E F is the h ighes t o ccupied leve l a t the

abso lu te ze ro . B y us ing eq ua t ion 25 in s ta t is tica l the rmo dyna m ics , one can d er ive tha t E F

is the to ta l f r ee energy of the e lec t rons , pe r e lec t ron , i .e . i t i s the e lec t rons ' chemica l

po ten t ia l . A t equi l ib r iu m there is on ly a sing le va lue of EF for the w hole sys tem .

The Fermi energy is a to ta l energy , i .e . i t inc ludes a lso the e lec t ros ta t ic po ten t ia l

energy , such as tha t due to the contac t po ten t ia l be tween meta ls , and o ther s imila r te rms .

I t i s the re fore a lmos t a lways necessary to take EF as the ze ro re fe rence leve l , because i t i s

a lway s d i f f icu l t and usua l ly impo ss ib le to es tab l ish the exac t pos i t ion of EF with regard to

th e v a c u u m le v e l Eva c. The work func t ion of the meta l , O, i s on ly approximate ly (0-2 eV)

equal to Evac-EF.

I f one connec ts by a cont inuous sur face in k- space a l l k ' s cor responding to EF, the

so-ca l led Fermi sur face a r ises . For f ree e lec t rons , as can be seen f rom equa t ion 21 , th is

sur face is a sphere (EF - k2) . For o ther , h igher approx imat ion s th is sphere is deforme d; an

e x a mp le o f a Fe r mi s u r f a c e wh ic h h a s b e e n e s ta b l i s h e d e x p e r ime n ta l ly a s we l l a s b y

theore t ica l ca lcu la t ions is shown in f igure 5b .

The geometr ic form of the Fermi sur faces is a l ready known for mos t meta ls [7] .

The main techniques to es tab l ish the form of the Fermi sur face a re those assoc ia ted wi th

the so-ca l led de Haas and van Alphen e f fec t and the sk in e f fec t [7] .

An impor tan t f ea ture of a meta l o r an a l loy is the dens i ty of s ta tes a t the Fermi

leve l N(Ev) ; th is va lue can be de te rmined by measurements o f the magne t ic suscept ib i l i ty

and the hea t capac i ty a t low tempera ture [7] .

In the d iscuss ion on the e lec t ron ic s t ruc ture of meta ls and a l loys and i t s r e la t ion to

e lec t ron spec troscopies , the mos t impor tan t quant i ty is mos t p robably the dens i ty of s ta tes ,

N(E) . This i s because a s imple re la t ion ex is ts be tween the d is t r ibu t ion of the photoemit ted

e lec t rons I (E) , and the in tegra l dens i ty of s ta tes taken over a l l ang les N(E) : to a good

a p p r o x ima t io n

I E ) = c o n s t . M ~ i . N E ) in in a rN E ) ~ 2

(26)

wh e r e

Mf, i

i s

~ffinal H'

~ i n i t i a l

dx and H' i s the per turba t ion caus ing the t r ans i t ion f rom the

init ia l into the f inal s ta te . The density of s ta tes for the f ree electron in the f inal s ta te is

propor t iona l to ~ /E, so tha t fo r h igh energ ies i t changes c om para t i ve ly l i t t le over the


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Struc ture and proper t ies o f meta ls and a l loys

17

energy range of in te res t . This means tha t the d is t r ibu t ion I (E) measured a t the de tec tor i s

on ly a s l igh t ly deformed dens i ty of s ta tes for the sys tem before ion isa t ion , N ( E ) i n i t i a v

I n th e a p p r o x ima t io n o f ' n e a r ly f r e e e l e c t r o n s ' , t h e d e n s i ty o f s t a t e s f u n c t io n

resembles tha t shown in f igure 6 . This s imple form a l ready re f lec ts the main fea tures

observed exper imenta l ly , and there fore in theore t ica l d iscuss ions the N(E) func t ion is o f ten

schem at i ca l l y

pic tured as in f igure 6 ( see chapte r s 2 and 3) .

F ig u r e 5b s h o ws th e Fe r mi s u r f a c e o f c o p p e r . I f th i s we r e a me ta l wh ic h c o u ld b e

e x a c t ly d e s c r ib e d b y a f r e e e l e c t r o n mo d e l , th e Fe r mi s u r f a c e wo u ld b e p e r f e c t ly s p h e r ic a l .

The "necks" s t ick ing ou t towards the (111) faces a re caused by the per iod ic c rys ta l

po ten t ia l V.

f i gure 6 .

D en s i ty o f s ta t e s i n a band

(Ema~ Emin) or a model o f

near l y f r e e e l ec trons

N I E )

i

I .

s

i \

/ I

\

I

I

I

I , / E

I

, , J

E m i n E m a x

- E

m a x

E

Th e v o lu me o f th e s p a c e e n c lo s e d b y th e Fe r mi s u r f a c e d e p e n d s o n th e to ta l

number of e lec t rons n in the sys tem; for f ree e lec t rons , the sur face a rea is p ropor t iona l to

n ~'3. K n o w in g th a t, we s h a ll n o w ma k e a h y p o th e t i c a l e x p e r ime n t : we r e p la c e s o me c o p p e r

a to ms b y a to ms wi th mo r e th a n o n e v a le n c e e le c t r o n , f o r e x a mp le , b y z in c o r a lu min u m.

This increases the vo lu m e und er the E F sur face and s ince the sph ere can not in gen era l case

c o n t in u e to g r o w in to th e h ig h e r Br i l lo u in z o n e , b e c a u s e o f a g a p in e n e r g y o n th e

Br i l lou in zone face , the sphere - l ike form wil l p robably be deformed to f i l l up the s ta tes

near to and jus t unde r the Br i l lou in zon e faces. H ow ever , i t is a lso poss ib le tha t i f the

a l loy could have another c rys ta l lographic s t ruc ture than tha t o f copper , and as a conse-

q u e n c e to h a v e a n o th e r f o r m o f Br i l lo u in z o n e , th e a d d i t io n a l e l e c t r o n s c o u ld b e b e t t e r

a c c o mmo d a te d a t lo we r e n e r g ie s , f o r e x a mp le , in a mo r e s p h e r e - l ik e b o d y . Th u s , th e

a v e r a g e n u mb e r o f e l e c t r o n s p e r a to m in s u c h c a s e s wi l l d i c t a t e th e c r y s ta l lo g r a p h ic

s t ruc ture of the a l loy . Hume-Rothery has formula ted severa l very usefu l ru les re la t ing the

m os t s tab le s t ruc ture to the average num ber of e lec t rons [8], and a l though so m e de ta i ls o f

h is theory a re no t longer va l id , the bas ic idea is p robably sound . There a re a lso some

p a p e r s wh ic h t r y to r e l a t e th e Hu me - Ro th e r y ' s s t r u c tu r a l c h a n g e s in a l lo y s to th e c h a n g e s


12/66

18 cha pter 1

in the catalytic activity [9] .

Three approximat ions in the descr ip t ion of the behaviour of e lec t rons in a per iod ic

poten t ia l have so fa r been ment ioned: ( i ) a mode l o f f ree e lec t rons ; ( i i ) a mode l o f near ly

f ree e lec t rons ; ( i i i ) a mode l o f e lec t rons t igh t ly bound to the a toms ( t igh t b ind ing

approximat ion , wi th L .C.A.O. used as a t r ia l func t ion) .

These approximat ions a re usefu l to e luc ida te the te rms of which the theory and an

e x p e r ime n ta l i s t ma k e u s e a n d to id e n t i f y th e p h e n o me n a ty p ic a l f o r s y s te ms wi th a

per iod ic po ten t ia l , bu t a l l th ree approximat ions ment ioned a re unsu i tab le for quant i ta t ive

pred ic t ions . Approximat ions i and i i exaggera te the de- loca l iza t ion of e lec t rons , whi le

approximat ion i i i cons iders the e lec t rons as too s t rongly bound and too much loca l ized on

the ind iv idua l a toms . S ince a l l th ree a re one-e lec t ron approximat ions and take the e lec t ron-

e lec t ron in te rac t ions (Coulombic and exchange in te rac t ions ) impl ic i t ly in the average

poten t ia l , they do no t t r ea t th is par t icu la r aspec t p roper ly . The h igher approximat ions t ry to

improve on th is s i tua t ion . Of many ways of do ing i t tha t a re descr ibed in the l i te ra ture , we

s h a l l me n t io n o n ly th e f o l lo win g o n e s [ 1 0 - 1 6 ] : ( 1 ) Au g me n te d P la n e Wa v e ( APW) a n d

r e la te d th e o r ie s , [ 1 0 ,1 ] a n d th e Ko r r in g a - Ko h n - Ro s to k e r ( KKR) a p p r o x ima t io n s [ 1 2 ] ,

which bo th a t tempt to improve the cons t ruc t ion of the wave func t ion ; (2 ) e lec t ron dens i ty

method (Kohn, Sham, Lang [13 ,14] ) which expl ic i t ly t r ea ts the e lec t ron-e lec t ron in te rac t i -

ons . Jus t a f ew remarks about these theor ies fo l low.

One poss ib i l i ty for improving the cons t ruc ted wave func t ion is to cu t the c rys ta l in

a space where the e lec t rons behave as essen t ia l ly f ree and in a space where they behave as

b e in g b o u n d t ig h t ly to th e n u c le i : t h i s i s wh a t th e APW ( Au g me n te d P la n e Wa v e ) th e o r y

does . The Schr6dinger equa t ion is then so lved ins ide a spher ica l po ten t ia l wa l l o f a r ad ius

R. The e lec t ros ta t ic po ten t ia l o f the nuc leus is hypothe t ica l ly conta ined in th is sphere ,

be ing zero ou ts ide . The so lu t ion for the sphere resembles tha t fo r f ree a toms , be ing a

l inear combina t ion of p roduc ts o f the rad ia l func t ions and spher ica l ha rmonics . The

coef f ic ien ts o f the l inear combina t ion a re then chosen in such a way tha t the so lu t ions

match smooth ly , on the sur face of the sphere , the p lane waves which descr ibe the

beh avi ou r of e lectro ns outs id e th e s phere [ 10, 11 ] .

Another technique for cons t ruc t ing a wave func t ion or c rys ta l o rb i ta l which would

descr ibe proper ly the de loca l ized charac te r o f the e lec t rons in the meta l i s the theory

s u g g e s te d b y Ko r r in g a a n d b y Ko h n a n d Ro s to k e r ( KKR) [ 1 2 ] . I n th e KKR th e o r y th e

a tomic spheres a re aga in cons idered . We can then imagine tha t a t the sur face of a ce r ta in

a tom ic sphere there is a so lu t ion w hich w e sha l l ca ll the ou tgo ing func t ion ~out , and the

same holds for a l l o ther a tomic spheres . As a consequence a t the sur face of our f i r s t

c h o s e n s p h e r e a c o mb in a t io n

C I ) i n

of a l l o ther waves ex is ts . The two func t ions

C I ) i n

and Oout

a re pu t equa l on the sur face of the a tomic sphere , and they a re mutua l ly re la ted as

sca t te red and inc ident waves , wi th sca t te r ing depending on the po ten t ia l ins ide the a tomic

spheres .


13/66

Structure and proper t ies of metals and al loys 19

Both the APW and the KKR techniques descr ibe the poten t ia l ins ide the spheres as

an ar t i f ic ia l po ten t ia l , which has no components ou ts ide the sphere; i t i s cal led muff in- t in

poten t ia l . Both techniques have also been appl ied to a l loys (see below).

Another successfu l approach to the problems of the descr ip t ion of the so l id s ta te

has been suggested [13-16] . The au thors of these papers have shown that the system of

many elect rons can be to tal ly descr ibed by the elect ron densi ty n(r ) , and have in t roduced a

funct ion E(n(r) ) , by the fo l lowing equat ion [13 ,14]

E ( n ( r ) ) = T [n ( r ) ] -

(27)

N n ( r ) e 2 n ( r ) n ( r / / )

E Z e 2 f I r_ R M I d r + - - f ir _-- ~i

M---1

2

+ E , o ._ , o + E .x c h ( n ( r ) )

In th is equat ion T s tands for the k inet ic energy of the non- in teract ing elect rons wi th

densi ty n(r ) , the second term is for nuclei -elect ron in teract ions wi th the nuclei a t the

pos it ions RM in the lat t ice, the third term is the mutual Co ulom bic interac tion of electrons ,

Eion_io n

i s for Coulombic repuls ion of nuclei and the las t term is the exchange energy . The

func t ion E(n ( r ) ) has a min imum when n ( r ) co r responds to t he g round s t a t e dens i ty and the

minimal energy i s then taken as the ground s ta te energy of the system. The pract ical

approximat ion i s to wri te down the equat ion for one elect ron funct ions wi th an effect ive

poten t ia l and wi th the exchange term wri t ten in the so cal led local -densi ty approximat ion .

The s imp les t fo rm o f t he who le theo ry i s fo rmu la t ed fo r a model wi th an un i fo rm

cont inuous posi t ive background, wi th elect rons as d iscrete charges on i t . This i s the so

ca l l ed Je l l i um model . Many p rob lems o f chemiso rp t ion and p romoter e f fec t s have been

successfu l ly a t tacked by th is theory and many importan t conclusions der ived [15,16] . To

our knowledge i t has no t been used for a l loys , for which i t i s no t wel l su i ted .

1 .1 .2 Paul ing ' s theory of pure metals

This theory was formulated [17,18] a t a t ime when the chemical bonding was

usual ly descr ibed in terms of e lect ron pai rs and resonance s t ructures , wi th the real

s t ruc tu re somewhere in be tween them. Phys i c i s t s never r esponded to Pau l ing ' s i dea ' s wi th

much en thusiasm, but a l l h is ideas , including the theory of metals , are ex t remely popular

among chemis t s . Tha t i s t he main r eason why they a re p resen ted and ana lyzed be low. The

other reason i s to demonst rate that in real i ty i t i s hard ly possib le to avoid a more d i f f icu l t

theory [4-16] by accept ing one such as that o f Paul ing . I t i s no t possib le to use semi-

empir ical approaches based on vague reasoning and yet be ab le to make r e l i a b l e predict i-

ons.


14/66

20 chapte r 1

Paul ing ana lyzed the c rys ta l lographic s t ruc tures and d is tances be tween a toms for

var ious meta l l ic e lements . In order to be ab le to compare s t ruc tures wi th var ious coord ina-

t ion numbers , CN (CN is 8 for bcc , 12 for f cc ) and wi th var ious numbers of e lec t rons

ava i lab le for bonding ( tha t i s the va lency , v ) . Pau l ing in t roduced the so ca l led s ing le bond

rad ius , R(1) for a l l e lements , de f ined as

R(n) = R(1) - 0 .60 0 In n (28)

where R is in A and n is the bond order . For metals the la tter is

n = ( v / f .N . ) ( 2 9 )

The ana ly t ica l fo rm of equa t ion 28 and the va lue of 0 .6 ,h , fo r the pre logar i thmic cons tan t

were der ived f rom R(1) , R(2) and R(3) of e thane , e thene and e thyne . With molecu les used

for th is ca l ib ra t ion the order n was thus a lways grea te r than one , bu t fo r meta ls i t i s

a lways less than one . However , Pau l ing assumed the same equa t ion to ho ld for bo th cases ,

the cons tan t be ing on ly s l igh t ly ad jus ted , f rom 0 .7A for ca rbon-carbon bonds to 0 .6 /k for

a l l o ther bonds . The sys tem of s ing le bond- rad i i i s shown in f igure 7 [ 17].

Z . 5

2 . 0

1.5

1.0

0 . 5

0 . 0

T h e f ir s t 1 0 n g p e r io d

. o

c o ~

- \ - " -%

S o ~

~ o

T, -,o _ _o=o= g~

. o - ~ - ~ . o

V " ~ , - O - o _ o _ o _ O

C.,r J F e I N iC U I I A S s ; ~ r

- M n C o Z n G e

T h e s e c o n d lo n g p e r i o d

o

z ' \ o . _ - o ' - o = _ ~ o " ~ I ~ " . . .

N b , ~ 1 7 6 A q J J 5 O j n

I T r ! Rh I Cd Sn ie I

M o ~

P d

R u

o S i n g l e - b o n d m e t a l l i c

r a d i i

o O c t a h e d r a l r a d i i

A T e t r a h e d r a l r a d i i

t t I 1 1 1 t

18 2 0 3 0 3 6 4 0 5 0 5 4

A K r X e

f igure 7

Single bond radii as calculated by Pau ling fo r the indicated metals and semiconducting

elements [17].


15/66

Struc ture and proper t ies o f me ta ls and a l loys 21

In the same f igure the te t rahedra l r ad i i a re a lso p lo t ted ; they a re rea l fo r s ,p -

e lements and f ic t i t ious for the t r ans i t ion meta ls . The s t ra igh t l ine of the f i r s t pe r iod is

descr ibed by : R(1) = Rl(SP 3) = 1 .825 - 0 .043 z wh ere z i s the num ber o f e lec t rons

outs ide the a rgon she l l . Trans i t ion meta ls show a contrac t ion in R(1) and accord ing to

Paul ing [17,18] th is i s due to the par t ic ipa t ion of d -orb i ta ls in the meta l l ic bond . He

th e r e f o r e in t r o d u c e d th e c o n c e p t o f d - c h a r a c te r 8 ( in %) , a q u a n t i ty e x p r e s s in g e x a c t ly h o w

mu c h o f th e b o n d in g i s d u e to th e d - e le c t r o n s . Wi th th i s 8 , h e wr o te th e e mp i r i c a l

equa t ion for s ing le bond rad ius R(1) :

R( 1 ) = R l ( 5 , z ) = 1 .82 5- 0 .0 4 3 z - ( 1 .6 0 0 - 0 .1 0 0 z ) 8

(30)

Th e f o r m o f e q u a t io n 3 0 wa s c h o s e n to d e s c r ib e th e r e s u l t s a n d to f i t t h e c u r v e s o f R( 1 )

v s z s h o wn a b o v e o n ly f o r th e f i r s t r o w o f t r a n s i t io n me ta l s . To u n d e r s ta n d h o w Pa u l in g

obta in ed the po in ts necessary to der ive the abso lu te va lues of the cons tan ts in equa t ion 30

we mu s t lo o k to h i s t r e a tme n t o f th e e l e c t r o n ic s t r u c tu r e o f th e ma g n e t i c e l e me n ts i r o n ,

coba l t and n icke l .

Paul ing specula ted tha t each meta l has th ree types of o rb i ta l : ( i ) a tomic orb i ta ls

in to which unpa ired as wel l as pa i red e lec t rons can be p laced ; ( i i ) va lence orb i ta ls in to

w hich e lec t rons wh ich form the meta l l ic bonds a re p laced ; ( i ii ) me ta l l ic o rb i ta ls wh ich a re

u n o c c u p ie d a n d wh ic h me d ia te "u n h in d e r e d r e s o n a n c e " . To b e a b le to e x p la in th e u s e o f

f r a c t io n a l n u mb e r s o f e l e c t r o n s wh e n d e s c r ib in g th e b o n d in g , Pa u l in g a s s u me d th a t a me ta l

c a n h a v e s e v e r a l ima g in a r y e x t r e me s t r u c tu r e s , wh ic h a r e mix e d in c e r t a in p r o p o r t io n s to

g ive the rea l s t ruc ture . The rea l s t ruc ture is tha t which resu l ts in the exper imenta l ly - found

magne t ic moments . This i s i l lus t ra ted by tab le 1 [17] . I t i s assumed tha t n icke l has two

s t r u c tu r e s wh ic h a r e mix e d in th e p r o p o r t io n s 3 0 % ma g n e t i c n ic k e l a n d 70 % n o n -

ma g n e t i c , in wh ic h a l l e l e c t r o n s a r e p a i r e d . Th i s mix tu r e l e a d s to th e e x p e r ime n ta l ly f o u n d

ma g n e t i c mo me n t p e r a to m o f 0 .6 Bo h r ma g n e to n s , a n d in a s imi la r wa y th e s t r u c tu r e s o f

c o b a l t a n d i r o n a r e mix e d to p r o d u c e th e e x p e r ime n ta l ly d e te r min e d v a lu e o f th e ma g n e t i c

m o m e n t p e r a t o m .

By c o n s t r u c t in g s u c h h y p o th e t i c a l s t r u c tu r e s a n d mix in g th e m in th e in d ic a te d wa y ,

Paul ing a lso ca lcu la ted the 8% charac te r , ( the las t co lumn of tab le 1) . He ca lcu la ted va lues

for the d-charac te r fo r i ron , coba l t and n icke l and wi th them he c rea ted equa t ion 30 for

R( 1 ) . Th i s e q u a t io n h a s to f i t a l l p o in t s wh ic h h a v e b e e n c a lc u la te d f r o m R( n ) ' s o f

in d iv id u a l me ta l s . Th e n , h e u s e d th e R( 1 ) v a lu e s to c a lc u la te 8 f o r n o n - ma g n e t i c me ta l s

a n d p r o d u c e d th e t a b le o f v a le n c ie s a n d % d - b a n d c h a r a c te r ( s e e t a b le 2 ) , v a lu e s o f wh ic h

s o o n b e c a me v e r y p o p u la r a mo n g s t c h e mis t s . Th e r e h a v e a l s o b e e n a t t e mp ts [ 2 1 ] to a p p ly

the 8 va lues to expla in resu l ts on a l loys and even on su lph ides .


16/66

22 chapter 1

table 1

Percentage d-character of cobal t , n ickel and copper (Paul ing theory)

(brackets ind icate bonding orb i ta ls)

M eta l

Co(B)

Co (B)

Ni(A)

Ni(B)

Cu (A)

Cu (B)

Outer electrons

3d

4s I 4p

~ T T ~

T, T I / ~

T$ T T ' ~ - ~ ~ . 9

T ~ T ~ i - - - o

~ ].

T~ T~T~ I 1 - ~

P~eso-

nance

rat io

35

65

30

70

25

75

Percen tag e d - ch arac te r

35~oo X z~ + 6~ o o X 3/~ = 39.5 %

30~ 00 X 2/~ -Jl- 70~ 00 X 3/~ _-400 -/0

25/~00 X 3/~ _Jr- 75/~00 X 2/~ -- 3 5 .7 %

table 2

Percentage d-character (d%) and valency (v) of e lements in the f i rs t ser ies of t ransi t ion

metals

V d%

Sc 3 20

Ti 4 27

V 5 35

Cr 6.3 39

Mn 6.4 40.1

Fe 5 .78 39.7

Co 6 39.5

Ni 6 40.0

Cu 5.5 36

How ever , the quest ion i s whe ther the popular ity of the 5 values i s jus t i fied . T hey

have been der ived f rom hypothet ical e lect ronic s t ructures using empir ical equat ions for


17/66

Structure and proper t ies of metals and al loys

23

R(1) ' s . I t i s doubtfu l whether equat ion 28 f rom which the argument s tar t s and which holds

for C-C bonds and the bond order n greater than one, can be appl ied to metal -metal bonds

and n less than one. Hume-Rothery [20] co l lected some resu l t s which cont rad icted

Pau l ing ' s s ta tements on th is po int . How ever , even i f equat ion 28 w ere of general appl ica-

bil i ty (as some modern authors assume [21]) a very mildly cri t ical reader would st i l l f ind

many quest ionable s teps in the procedure lead ing to the tab le of valencies and 8 values .

1 .1 .3 The Eng el -B rew er theory of metals and al loys

This theory has a number of features that are s imi lar to the ideas of Paul ing :

d i rected valencies , an importan t ro le of hybr id izat ion of orb i ta ls on atoms const i tu t ing the

metal , widely changing valencies and the omnipresent e lect ron pai rs .

Brewer i l lus t rates h is theory wi th the example of tungsten [26] , The conf igurat ion

of tungsten in a free atom ground state is d4s 2. However, the two s-electrons form,

accord ing to Brewer , a c losed shel l , which i s non-b inding and which in the so l id s ta te

causes repuls ion of o ther tungsten atoms. However , the conf igurat ion dSs i s on ly 33 ,5

kJ/mol (8 kcal /mol) above the ground s ta te , th is d i f ference being cal led promot ion energy

of the d 5 s conf iguration , and the d4sp conf igurat ion i s 230kJ/mol (55 kcal /mo l) a bove the

ground s ta te . Upon forming the metal , the energy of the das 2 conf igurat ion i s supposed to

be lowered by 569kJ/mol (136 kcal /mol) , the dSs conf igurat ion by 890kJ/mol (211

kcal /mol) and dnsp conf igurat ion by 569k J/mol (136 kcal /mol) . We shall now ex am ine the

procedure by which the numerical values are ob tained .

Fol lowing Hume-Rothery , Engel [24] associated crystal lographic s t ructures wi th

numbers of valence electrons in certain orbitals, i .e. with certain electronic configurat ions.

Having in mind the elements : sodium (bcc) , magnesium (hcp) and aluminum (fcc) , wi th

one, two and three valence elect rons respect ively , he suggested that the t ransi t ion elements

with the con figurat ion dn-ls should have a bc c structure, w ith dn-2sp they should have the

hexagonal close-packed structure and with dn-3sp2 the fcc-s t ructure whe re n i s numbe r o f

valence elect rons . Of course, some smal l deviat ions in n ( for example al loys) are to lerated .

Vice versa , knowing the crystal lographic s t ructure one can determine the number and

distribution of the valence electrons over the orbitals. The authors of the theory [24-27]

assumed further that the contribution per s or p electron is given by the interpolat ion l ine,

which connects the poin ts for metals having no b inding by d-elect rons , and serves as a

calibrat ion (see figure 8).

The contribution to the binding strength by d-electrons is calculated in the

fo l lowing way. The promot ion energy i s subt racted f rom the subl imat ion energy: the

former i s f ixed by the crystal lographic s t ructure of the metal in quest ion . The s t ructure

determines , namely , how many elect rons should be in the s and p orb i ta ls .


18/66

24 chapter 1

60

f i g u r e 8

*~ 50

B r e w e r - E n g e l t h e o r y o f m e t a l s -3

B o n d i n g e n e r g y k c a l / m o l e e l e c tr o n ) c 4 0

o f t h e i n d i ca t ed e l ec t ro n s 4 s , p o r -~

3 d , r e sp . ) a s a f u n c t i o n o f t h e p o s i t i o n ~ 30

i n th e p e r i o d i c t a bl e. E l e m e n t s o f th e

f i r s t l o n g p e r i o d a r e s h o w n . -a 2 0

u

E ,p th e u p p er cu rve ) i s e s t i m a t ed b y I O

in ter~ex t rapola t ion .

E a ca l c u l a t e d a s d e s c r i b e d i n t h e t e xt . o

C o S c T i V C r M n I re C o N i C u Z n

I I I l i 1 1 I I

XX

X

3d

F

o = -

/ 1 . l l 1 1 1 t [

0 I 2 3 4 5 4 3 2 I 0

N o . o f u np a i r e d e l e c t r o n s p e r a t o m

The to tal cont r ibu t ion by s , p bonding i s then subt racted , values being taken f rom graphs

such as that in f igure 8 , and the res t o f the b inding energy i s d iv ided by the number of

unpai red d-elect rons . For example, hcp cobal t i s expected to have the conf igurat ion dTsp .

From the sum of a l l d -orb i ta ls , two should be occupied by pai rs of e lect rons and three by

unpa i red e l ec t rons . The max imum poss ib l e number o f unpa i red e l ec t rons i s cons idered as

the ground s ta te conf igurat ion . As can be seen f rom f igure 8 , whi le the cont r ibu t ion to the

binding energy by s ,p orb i ta ls increases monotonical ly wi th atomic number , the cont r ibu t i -

on by unpai red d-elect rons decreases . By ci rcu lar argument , the au thors [24-27] ra t ional ize

the crystal lographic pat terns in the per iod ic tab le of e lements , us ing values such as those

shown in f igure 8 . Somet imes the ass ignment of the most s tab le conf igurat ion appears to

be easy , as wi th molybdenum and tungsten , bu t in o ther cases var ious conf igurat ions lead

to very s imi lar energ ies and thus to uncer tain t ies , such as i s the case wi th y t t r ium and

zi rconium.

The Engel -Brewer theory has a lso been appl ied to problems of the s tab i l i ty and

crystal lographic s t ructure of a l loys , in par t icu lar to s t ructures of some in termetal l ic

compounds. Such compounds are formed when a metal on the lef t -hand s ide of the

per iodic tab le ( i . e . a metal wi th almost empty d-orb i ta ls) i s combined wi th a metal on the

r igh t -hand s ide, where elements have several d -orb i ta ls wi th pai red d-elect rons . Brewer

stated [26] that " the use of empty orb i ta ls of hafn ium and tan talum by the non-bonding

( i .e . pai red) e lect rons of osmium or p lat inum could opt imize the use of avai lab le orb i ta ls

and elect rons , and approach the opt imal b inding achieved by tungsten" . Using the example

of Hfl r 3 B rew er i l lus t rated how di f f icu l t i t i s to ma ke qu ant i ta tive pred ict ions of heats of

al loy (compound) format ion , that i s , to go beyond qual i ta t ive pred ict ions . Never theless , the

number of cases of b inary and ternary al loys where the pred ict ions are sat i sfy ing i s


19/66


20/66

26 chapter 1

where P and Q are cons tants , f (c) i s a symmetr ica l funct ion of the molar ra t ios , and for an

al loy A B form ing a sol id so lut ion i t is XA(1-XA), XA being the m ole fract ion of co m po ne nt

A. A~)* is the d ifference in the v alues of ~* for the tw o elem ents, ~* bein g to a f i rst

appro xima t ion the w ork funct ion ~; Anws i s the d i f ference in the va lues of e lec t ron

dens i t i es in the Wigner-Sei tz ce l l s , a l l cor responding to A and B, respect ive ly . Miedema

showed tha t a more se l f -cons i s tent sys tem of enthalpies of format ion, in be t ter agreement

wi th va lues known f rom exper iment , can be obta ined i f one uses the adjus ted ~* values

tabula ted by the author . The di f ference between ~ and ~* i s smal l for p la t inum (5 .55 vs

5 .65 V) , but somewhat l a rge for some other e lements ( for Zr , 3 .15 vs 4 .05 V) . Miedema

sug ges ted calcu lat ing the densi t ies nws by using

(B/Vm)v~,

where B i s t he bu l k modul us o f

com press ibi l i ty and V m the m olar volum e.

The idea behind equat ion 31 i s tha t e lec t rons are t ransfer red f rom atoms of a meta l

of lower e lec t ronegat iv i ty to a toms of a meta l of h igher e lec t ronegat iv i ty . According to

Miedema [32] , the charge t ransfer red per a tom mT~ can be calculated by

AZ A = 1.2 (1-XA) A~* (32)

This m eans tha t in Hf l r 3 about 0 .7 of an e lec t ron per hafnium atom i s t ransfer red f ro m

hafnium to i r id ium. The Engel -Brewer theory, which a l so expla ins the high s tabi l i ty of th i s

compound (see 1 .1 .3) , assumes an opposi te e lec t ron t ransfer . The exper imenta l resul t s ,

e .g . core level shi f t s , on var ious compounds of th i s type indica te tha t most l ike ly there i s

no e lec t ron t ransfer a t a l l ( see chapter 3) , but format ion of s t rong par t ia l ly- local ized bonds

takes p lace be tween unl ike e lements ( see chapter 2) .

The prac t ica l success of th i s theory i s indisputable . I t i s a lmost imposs ible to check

the s tabi l i ty exper imenta l ly and to make some predic t ions concerning phase diagrams of

a l l a l loys of potent ia l in teres t for mater ia l sc iences . Miedema 's theory, however , of fers a

cer ta in tool for ma king rough but useful predic t ions , where exper im enta l resul t s a re

lacking.

The theore t ica l background of the theory i s however weak. I t i s too s t rongly

as soc i a t ed w i t h t he a s sumed cha rge t r ans f e r be t ween t he componen t s o f a l l oys , and moreo-

ver , whi le the work funct ion ~ i s indeed a measure of the e lec t ronegat iv i ty of meta l

sur faces , a subs tant ia l cont r ibut ion to ~ i s made by the sur face dipole , which i s not present

in the bulk a t the Wigner-Sei tz ce l l boundar ies , where the charge t ransfer should take

place.

1.1.5

The quan t um t heory o f a l l oys

Quan t um mechan i ca l ca l cu l a t i ons on sma l l o rgan i c mol ecu l e s can ach i eve a ve ry

high accuracy, which i s imposs ible to achieve wi th la rge sys tems of in terac t ing par t ic les ,


21/66

Structure and proper t ies of meta l s and a l loys

27

such as sol id crystals . Yet the fact that the potent ial in the sol id can be taken as periodic,

and Born-Karman condi t ions can be assumed to be ful l f i l l ed ( see 1 .1 .1) , a l lows us to be

somewhat prec i se when t rea t ing the proper t ies of l a rge s ingle crys ta l s of meta l l i c e lements

(see for example a compar i son of ca lcula ted band s t ruc tures wi th those der ived f rom

e l ec t ron pho t oemi s s i on i n chap t e r 3 ) . However , when a r andom a l l oy i s f o rmed wi t h

elemen ts A and B, the m ole f rac t ions be ing x A and xB, the per iodic i ty of the p otent ia l i s

abol i shed and the degree of sophis t ica t ion which i s needed for a descr ip t ion of the same

accuracy i s cons iderably enhanced.

Fau l kne r summar i zed t he ea r l y deve l opment o f t he quan t um t heory o f a l l oys i n a

paper [33] which we shal l fo l low.

The potent ia l in the a l loy A-B a t a point r can be wr i t t en as a sum of cont r ibut ions

from different lat t ice si tes

(Rn):

V( r) : n Wn ( r - Rn) (33)

V n

i s V A or VB according to the a to m on s i te n , but the fac t tha t in rand om al loys V i s no

longer per iodic i s a problem in the descr ip t ion of a l loys . There are severa l ways of coping

wi th th i s d i f f icul ty , but we shal l ment ion only three of them.

(1) In the R i g i d B an d T heory (R B T) one neg lec t s t he d i f fe r ence be t ween A and B and

assumes tha t the only consequence of subs t i tu t ing A for B i s tha t the common band i s

occu pied to a h igh er o r lowe r degree , v iz . E F i s shif ted , jus t by adding e lec t rons to or

ext rac t ing them f rom the pool of e lec t rons under the Fermi sur face [34,35] . A consequence

of th i s model i s tha t charge i s f ree ly t ransfer red f rom one component to another , for

example , f rom copper to n ickel . The RBT was for a long t ime the bas i s of ear ly theor ies

of catalysis by al loys [9,19,36], but the total fai lure of this theory, and of ideas behind i t ,

to expla in the photoemiss ion resul t s ( see chapters 2 and 3) s topped i t s appl ica t ion af ter

about 1968 , when the papers by Spicer appeared [37] .

(2) The next l evel of approximat ion i s a model of a v i r tua l c rys ta l wi th an average

potent ial VAV [38,39] on each lat t ice point :

WAy = XA VA(r) +

XBVB(r) (34)

The di f ference V Av(r) - Vo(r) , wh ere Vo(r ) is the ideal pe riodic poten t ial , can be t reated as

a per turbat ion and i t l eads to smal l devia t ions f rom the Eo(k ) funct ion for the per iodic

potent ia l . I t has been shown tha t th i s i s a l so a ra ther poor approximat ion.

(3) Higher approximat ions s tem f rom the theory of mul t ip le sca t ter ing phenomena. This i s

appropr ia te , because the crys ta l orbi ta l ~ i s , in the context of these theor ies , cons t ructed in

such a way tha t the de local iza t ion of e lec t rons outs ide the a tomic spheres i s formal ly

desc r i bed by wave func t i ons whi ch l ook l i ke a combi na t i on o f " i ncomi ng" waves w i t h


22/66

28 chapter 1

waves " sca t tered" by sur rounding a toms. Atoms A and B are in th i s way cons idered as

unl ike sca t terers conver t ing the incoming funct ion in to di f ferent sca t tered funct ions , by the

opera t ion of potent ia l s V A and VB. The opera tor w hich re lates the incom ing and sca t tered

waves is t , there being different values tA and tB for each type of atoms. In early at tempts

an averaged sca t ter ing opera tor was used.

t a v = XA tA + x B tB (3 5)

but the resul t s were even worse than wi th the approximat ion of the vi r tua l c rys ta l . The

break through came when Soven [40,41] sugges ted us ing the fol lowing pic ture . A vi r tua l

crys ta l i s cons t ructed which has an in i t i a l ly undetermined coherent potent ia l W(r) on each

si te. The scat ter ing is caused by local deviat ions from this potent ial , so that the scat ter ing

operators are:

tA = (V A- W ) + (V A- W ) CJ tA

tB = ( V B - W ) + ( V 8 - W ) C J tB ( 3 6 )

In equat ion 36,

fur ther be low.

s tands for the so-ca l led Green opera tor , which wi l l be br ief ly d i scussed

The reader i s a l ready fami l iar wi th the Schr6dinger equat ion"

[ -h2/2m) V 2 + V ( r ) ] ~ = E ~

(37)

which in the opera tor form reads as"

12I~ = E ~ or ( E - 121) ~ = 0 (38)

The Green opera tor i s def ined by an analogous opera tor equat ion:

( E - 121) G = 1 (39 )

and i t i s very useful in descr ib ing sca t ter ing phenomena or o ther quantum mechanica l

problems, such as the cons t ruct ion of wave funct ions for crys ta l s , which have a s imi lar

s t ruc ture . For example , in the formal i sm and the language of the mul t ip le sca t ter ing, the

solut ion of equat ion 37 is wri t ten as [33]:

V - - r - I ( ~ Jo ]~ n tn ~ ] / ~ n ( 4 0 )


23/66


24/66

30 chap t e r 1

5 --1

7 7 % C u , 2 3 % N , |

o ? . _

, / h x \

-

zf , /

I

- \ \ \ - ,~ , _

i / / , " \ _

o

/

i . . . . . . r - . . . . . " l

- 0 . 7 - 0 . 6 - 0 . 5 - 0 . 4 - 0 . 3 - 0 . 2 - 0 . I 0

E N E R GY B E L O W E f ( r y d b e r g s )

4 O

3 5

~

3O E

o

2 5 ~

o

2 0 ~

i---

~5 ~

LL

I 0 o

> -

5 ~

Z

0

f i gure 9

Dens i t y o f s t a tes f o r a Cu-Ni

al loy and the projected densi t ies

of s tates fo r the Cu an d Ni s ites .

Not ice: the UPS experimental

resul t s are at lowest energies

de formed by t he ar t e fac t s o f t he

experimental techniques but the

region round E i s correct ly pro-

bed [44b1.

f i gure 10

C o m p a r i s o n o f X P S v a l e n c e- b a n d

s p e c tr u m f o r a n e v a p o r a t e d C u - N i

al loy sample wi th a densi ty of

s ta tes f o r C u o . 6 N i o . 4 ca lcu la t ed

in CPA. f rom re f 42)

I0 ! ! Q i I .~

/

/

1

0.5

0 1 9

8 6 4 2 0 -2

B I N D IN G E N E R G Y ( e V )

The C oheren t Po t en t i a l Approx i ma t i on (C PA) has been worked ou t i n g r ea t de t a i l

[43,44] and fur ther modi f ied and extended to ordered a l loys . I t goes much beyond the

scope of th i s book to d i scuss these developments , but we refer the in teres ted reader to

some se l ec t ed pape r s [45 ] . An ex t ended compar i son o f expe r i men t a l and t heore t i ca l UPS

and XPS intens i t i es has been presented [46] .


25/66

Structure and proper t ies of metals and al loys 31

Theo ret ical calcu lat ions using the CPA [47 ] and exper imental X PS resu l t s [48]

have been compared and found to agree for the Pd-Hx system (foreign atom is p laced

in ters t i t ia l ly ) ; the exper imental ly found hydrogen- induced s ta tes cen tered at 5 .4 eV below

the Fermi level s t rongly suppor t the idea that hydrogen i s p resent as a toms and not as

pro tons which have in jected thei r e lect rons in to the d-holes of pal lad ium. In ters t i t ia l a l loys

behave probably in a s imi lar way.

In relat ion to the var ious semi-empir ical theor ies of a l loys (see sect ions 1 .1 .3 and

1.1 .4) we note that the al loys of s-metals have been also analyzed by CPA theory [49] .

For these al loys the CPA theory pred icts the ex is tence of var ious local ized s ta tes in the

whole range of energ ies of valence elect rons .

There are a lso some o ther sophis t icated methods which ei ther general ly or for some

special cases are s t i l l bet ter su i ted for exact calcu lat ions than the or ig inal form of the CPA

theo ry [50,51 ] .

Al loying i s known to cause some red is t r ibu t ion of e lect rons on the ind iv idual

atoms of a l loy components . For example, pal lad ium in s i lver has a narrower d-band than

in pure pal lad ium. In consequence, f rom a cer ta in d i lu t ion up (about 60% si lver) , the

whole narrowed band of s ta tes local ized around pal lad ium atoms fal l s below the Fermi

level . This band therefore becomes fu l ly occupied at the expense of the s-band . The

narrowing of the d-band resu l t s f rom the d iminished over lap of the pal lad ium orb i ta ls (see

sect ion 1 .1 .1 for the relat ion betwee n ove r lap and band wid th) , and f rom the suppress ing

of the d-d elect ron repuls ion by d i lu t ion . These and s imi lar ef fects should also ex is t in

some o ther a l loys and one has a lways to consider the possib i l i ty of a change in e lect ron

conf igurat ion caused by al loy ing . More in t r igu ing i s the quest ion to what ex ten t charge

t ransfer between al loy components occurs . In chapters 2 and 3 we wi l l d iscuss some

resu l t s deal ing wi th th is po in t , bu t f i rs t we consider a t tempts to deal theoret ical ly wi th th is

p rob lem.

Kfib ler e t a l . [52] used the Augmented Spher ical Wave method and the local

funct ional densi ty theory for sel f -consis ten t calcu lat ions; they calcu lated the to tal and

par t ia l densi ty of s ta tes , that i s , expressed per component and per orb i ta l o f a g iven

symmetry , and f rom that they determined the occupat ion of s , p and d orb i ta ls of

ind iv idual components . They concluded , for example, that in Zr3Pd the pal lad ium atoms

receive 0 .6 s ,p e lect rons per a tom from the zi rconium atoms which each lose 0 .2 s ,p e lec-

t rons per a tom. The in t ra-atomic t ransfer of s to d e lect rons on pal lad ium is calcu lated to

be near ly zero . A closer inspect ion of calcu lat ions made by the same technique [52]

reveals that the conf igurat ions for pure metals do not agree wi th the exper imental resu l t s

(e .g . coppe r has 9 .5 d-elect rons , ins tead of 10) , so that the pred icted cha rge t ransfers in

al loys might be doubted . Pred ict ions for core level sh i f t s are a lso presented [52] .

Ear ly s tud ies of the M6ssbauer ef fect on al loys of t ransi t ion metals revealed

somewhat large i somer sh i f t s which were at f i rs t explained so lely by charge t ransfer


26/66

32 chapter 1

between the a l loy components ( see chapter 3) . However , l a ter thorough theore t ica l work

revealed tha t the or ig inal s t ra ight forward explanat ion needed ser ious cor rec t ions [53] .

When an a tom A wi th a spat ia l ly extended p- or d-orbi ta l i s squeezed in a l a t t i ce of

another e lement B wi thout tha t par t i cular fea ture , e lec t rons on the far reaching p-orbi ta l s

s imula te in the space around B a toms a charge t ransfer . An analys i s shows tha t indeed the

p- l ike charge increases on B , but tha t th i s i s mainly due to the ta i l ing of the p-orbi ta l s

f rom A into the B a tomic spheres and not to an increase of p-e lec t rons on the on s i t e

orbi ta l s of B , or to o ther bonding ef fec t s . At tempts were made to subt rac t the ta i l ing ef fec t

or pseudo-charge t ransfer f rom the to ta l charge t ransfer , the la t t e r be ing ca lcula ted by

integra t ing the dens i ty of e lec t rons wi thin the Wigner-Sei tz or a tomic spheres . Resul t s of

these very del ica te ca lcula t ions are shown in f igure 11.

o l ' ' ' ' ' ' ' I . . . . ' ' t < 1

Tai l Only _ / M od Mu lhken i

o . o . . . . . . . . . . . . I . . . . . . . . . . . . .

A -

u Compounds o

Pt Compou nds [ ]

I r Compou nds , ,

I t . P t ,and Au

0.2 Si tes r i , .~. . ,

o, \ \

O0

j

-0,1 +

Hf Ta W Re Os I r Pt HI Ta W Re Os I r Pt

-01

-02

f i g u re 1 1

R es id u a l ch a rg e t ra n s fer a t I r ,

P t a n d A u a n d t h e o t h e r a t o m i c

s i te s in co m p o u n d s o f th e C sC I

s t ru c tu re co m p o u n d s a f t e r th e

e f f ec ts o f ta i lin g w ere su b t ra c -

ted . Th is invo lves taking the

to ta l ch a rg e t ra n s fer a n d su b -

s tract ing o f the ta i l ing charge .

T h e r ig h t h a n d co lu m n sh o w s

the resu l ts fo r the res idua l char-

g e w h en th e m o d i f i ed Mu l l i ken

sch em e i s u sed to e s t im a te th e

ta il ing . The le f t han d co lum n

sh o w s th e resu l t s w h en o ver la p

con tr ibu t ions to the ta i l ing are

n eg lec ted a n d i s sh o w n to p ro v i -

d e so m e sen se o f h o w sen s i t i ve

the resu l t s a re to the t rea tm en t o f th e o ver la p . T h e o p en sym b o l s w e re o b ta in e d u s in g

e l e m e n t a l v o lu m e s , h o w e v e r , w h e n h a f n i um c o m p o u n d s h a v e v o l u m e s a f e w p e r c e n t

sm a l l er th a n th e su m s o f th e e l em en ta l vo lu m es a n d th e so l id sym b o l s in d ica te th e

co n seq u en ce s i f t h is vo lu m e co n t ra c t io n i s a ss ig n ed to the h a fn iu m s it e. ~ ro m re f 5 3


27/66

Structure and proper t ies of metals and al loys

33

For cataly t ical ly in teres t ing al loys , for example p lat inum-gold , the charge t ransfer

to p lat inum is +0 .2 elect rons when correct ion for ta i l ing i s no t made, bu t wi th th is

correct ion i t i s -0 .1 . The d i f ference in calcu lated charge t ransfer values , which i s on ly the

resu l t o f the use of d i f feren t methods of calcu lat ions can be as large as 0 .5 e lect rons .

Conclusions are therefore very caut iously drawn [53] ; too much should not be read in to

our f igure 11 and perhaps of greates t in teres t i s that once charge ta i l ing i s accounted for ,

the remain ing changes in e lect ron counts are consis ten t wi th a p icture where elect ronegat i -

v i t ies increase as one t raverses the 5d row from hafn ium to the elements to i t s r igh t , wi th

gold however being less e lect ronegat ive than p lat inum and perhaps i r id ium as wel l . Whi le

consis ten t wi th some not ions th is i s inconsis ten t wi th many not ions concern ing the

chemist ry of go ld" . The d i f f icu l ty in assess ing theoret ical ly the ex ten t of the charge

t ransfer i s c lear ly and s imply demonst rated by th is : one has to calcu late charge t ransfer of

the order of 0 .1 e lect rons per a tom wi th up to 10 elect rons being involved , no t knowing

exact ly over what space one has to count the elect rons belonging to each component .

In chapter 3 , several e lect ron densi ty contour maps are presented ( f igures 13 ,21 ,22)

which also touch the problem of the charge t ransfer . Actual ly they do not show much that

one would cal l charge t ransfer . The reader wi l l a l so f ind in chapter 3 o ther v iews and

prel iminary conclusions on the problem of charge t ransfer as der ived f rom the to tal i ty of

al l results presented by that chapter.

F inal ly , we ment ion the elect ronic theory of order ing and segregat ion in cataly t ical -

ly less in teres t ing al loys such as these formed between s imple metals , for example, a lkal i

and nob le meta l s [54 ]. A n impor t an t s t ep is m ade by u s ing microscop ic quan tum mechan i -

cal calcu lat ions to pred ict macroscopic thermodynamic behaviour . Values for charge

t ransfer between al loy components are der ived , for example, for the 1 :1 al loys: Li -Cs, 0 .25

elect ron/atom from Cs to Li ; Na-Cr , 0 .27 elect ron/at f rom Cs to Na; Cu-Au, 0 .11 elect ron

per / a t f rom Cu to Au ; Ag-Au , 0 .13 e l ec t ron /a t f rom Ag to Au .

We have seen above how di f f icu l t i t i s to pred ict , i f on ly in a qual i ta t ive way, the

main features of a l loys; use of the coherent po ten t ia l approximat ion (CPA) theory was the

f i rs t real b reakht rough. To pred ict the proper t ies quant i ta t ively , e .g . the amount of charge

t ransfer , i s again a task an order of magni tude more d i f f icu l t . However , cataly t ic chemists

want to know the composi t ion and the elect ronic s t ructure of a l loy surfaces as wel l as

thei r p roper t ies in chemisorp t ion and catalysis . I t i s an ex t remely demanding task , bu t

some progress has a l ready been made in th is d i rect ion [55-57] . For example a pred ict ion

has been made concern ing the b inding energy on a surface of an n ickel -copper a l loy [56] .

Atomic adsorp t ion on the hol low square of four a toms i s considered and the conclusion i s

reached that the b inding energy on th is c luster i s so much inf luenced by the average

composi t ion of the al loy that i t d rops by a factor of two when going f rom pure n ickel to

very d i lu te n ickel -copper a l loys . However an effect o f th is s ize seems to be at var iance

wi th the exper imental resu l t s ment ioned in o ther par ts of th is book.


28/66

34 chapte r 1

The problem of adsorp t ion on a l loys has been a lso approached theore t ica l ly by

other au thors , fo r example , in the book by van Santen [59] , whose t r ea tment i s s impler

than tha t in [56] a l though i t goes fur ther in apply ing the theory . Van Santen conc ludes tha t

there is an e lec t ron ic s t ruc ture e f fec t on the b ind ing energy of hydrogen a toms on an a l loy

c lus te r where the average number of va lence e lec t rons in the a l loy is a var iab le .

The n icke l-copper sys tem is a f avour i te sub jec t fo r ca lcu la t ions , s ince i t concerns

somewhat l igh t e lements , and a lso many of the exper imenta l r esu l ts r e la te to i t . Cas te l lan i

[60] s tud ied i t by Extended Hticke l Theory and conc luded tha t the re should be a charge

trans fe r and an e f fec t o f copper on adsorp t ion proper t ies o f n icke l . P lac ing of n icke l in to a

copper matr ix causes accord ing [60] a decrease of about 30 kJ /mol in the hea t o f

a d s o r p t io n o f c a r b o n mo n o x id e , wh e n c o mp a r e d wi th n ic k e l in a n ic k e l ma t r ix . An e v e n

mo re pro nou nced charge t r ans fe r f rom cop per to n icke l was found in [61 ] .

S imp le LCAO th e o r ie s a r e g o o d e n o u g h to e x p lo r e n e w p h e n o me n a [ 59 - 6 3 ] s u c h

as adsorp t ion on mul t icomponent sys tems , and to in t roduce the te rms necessary for the

descr ip t ion of the exper imenta l r esu l ts , bu t they a re no t very re l iab le for making quant i ta t i -

v e p r e d ic t io n s . Ho we v e r , s o me p io n e e r in g wo r k in h ig h e r a p p r o x ima t io n h a s a l r e a d y b e e n

car r ied ou t too . Musca t [64] has s tud ied hydrogen adsorp t ion on the (111) face of copper ,

in the sur face of which a n icke l a tom impur i ty is p laced . Binding energy was f i r s t

ca lcu la ted for a c lus te r us ing 19 muf f in- t in po ten t ia ls o f e i ther pure copper or wi th one

nicke l a tom ins tead of one copper a tom ( see f igure 12) .

f i g u r e 1 2

M o d e l c l u s t e r s

HCu 9 or HCu 8Ni

u s e d i n

c a l c u l a t io n s b y M u s c a t . C i r c l e s - u p m o s t l a y e r,

t r i a n g l e s - t h e l a ye r u n d er i t , sq u a re - a n a t o m

i n t h e n ex t l o w er l a ye r . C ro ss i n d i ca t e s t h e

p o s i t i o n o f H . P l a c i n g N i i n p o s i t i o n 1 c h a n g e s

t he o n e e l e c t r o n e n e r g y o f t h e s y s t e m b y 0 . 4 0

a .u . i n o t h e r w o r d s t h e e n s e m b l e N iC u 2 o f

n e a r e s t a t o m s b e h a v e s v e r y d i ff e r e n t ly f r o m t h e

e n s e m b l e C u 3 ). H o w e v e r p l a c i n g o f N i in p o s i-

t i o n 2 , 3 o r 4 ca u se s a n e f f e c t 4 -5 t i m es sm a l -

l e r a n d w i t h N i i n 5 a n d 6 t h e e f f e c t i s z e ro

t h i s i s a n eg l i g i b l e i n t e ra c t i o n t h ro u g h t h e

meta l ) .

I n th e f o l lo win g s t e p , th e e n e r g y c h a n g e wa s c a lc u la te d d u e to th e c lu s te r b e in g e mb e d d e d

in to a n e f f e c t iv e me d iu m c o n s i s t in g o f a h o mo g e n e o u s e le c t r o n g a s . Th e r e s u l t s we r e q u i t e

in te res t ing : the e f fec t o f in t roduc ing a n icke l a tom in a copper matr ix is on ly impor tan t

when a n icke l a tom is in the pos i t ion 1 ( see f igure 12) . In pos i t ion 2 i t i s f ive t imes


29/66


smal le r and in pos i t ion 3 i t i s smal l and of the oppos i te s ign ; in pos i t ion 5 i t has no

inf luence a t a l l . Obvious ly , ha rd ly any of the e f fec t o f n icke l i s th rough- the- la t t ice , v iz .

th rough the co l lec t ive meta l p roper t ies ; i t i s c lea r ly a shor t r ange , chemica l bond e f fec t .

We can ex trapola te th is conc lus ion and say tha t on the (111) sur face of copper -n icke l

a l loys one can expec t four types of t r i - a tomic c lus te r each showing a d is t inc t ly d i f fe ren t

b ind ing energy towa rds hydrogen : Ni 3, Ni2Cu, NiCu2, Cu3 , the las t show ing very we ak

binding; we m ay then t ry to expla in chemisorp t ion and ca ta ly t ic r esu l ts by th is mo de l , in

wh ich the en sem ble s ize (Ni 3, Ni2 . .. ) p lays the m os t impo r tan t ro le . W e sha l l tu rn to th is

po in t in chapte r 9 and sha l l see tha t th is approach can be success fu l .

The resu l ts fo r the ca lcu la ted charge t r ans fe r and l igand e f fec ts show an in te res t ing

and c lea r p ic ture : the s impler the theory is , the more pronounced is the charge t r ans fe r .

The reader wi l l f ind fur ther d iscuss ion on th is po in t in chapte r 3 and e lsewhere in th is

book .

1.2

S o m e r e s u l t s o f th e t h e o r y o f c h e m i s o r p t i o n o n m e t a l s a n d a l l o y s

1.2 .1 Genera l f ea tures of chemisorp t ion - a qua l i ta t ive p ic ture based on quantum chemica l

ca lcu la t ions

1 2 1 1 Chemisorption of atoms

Th e r e a d e r w i l l b e f a mi l i a r w i th th e q u a n tu m me c h a n ic a l th e o r y o f b o n d in g in a

mo le c u le A- B . Th e s imp le s t c a s e i s th a t o f two a to ms e a c h h a v in g o n e v a le n c e e le c t r o n in

one a tomic orb i ta l ( see [1-4] o r , more advanced tex t [65] ) . The main te rms and resu l ts o f

such a theory a re summar ized in f igure 13 and the in format ion conta ins the f i r s t ingred ien t

needed to bu i ld up a qua l i ta t ive p ic ture of a chemisorp t ion theory . A so l id is ac tua l ly a

g ian t molecu le . Due to the mutua l in te rac t ion of a l l a toms in the so l id , the molecu la r

energy leve ls fo rm a whole band of leve ls ( see 1 .1 .1 and f igure 14 be low) . In ana logy

wi th d ia to mic mo le c u le s , th e lo we r p a r t o f th e b a n d i s c a l l e d b o n d in g a n d th e u p p e r o n e

ant ibonding . W e have presen ted a s imple theory of band form at ion in sec t ion 1 .1 .1 f rom

wh ich we kno w tha t the band w id th is p ropor t iona l to the matr ix e lem ent 13 (1 .1 .1 ,

equation 19) .

In pr inc ip le , bo th the meta l a toms , on le f t -hand s ide of f igure 15 , and the adsorbed

a toms or molecu les a t h igh sur face coverages can form a band of energy leve ls [66] .

However , le t us s ta r t wi th the s imples t case . A s ing le a tom, wi th one e lec t ron in a s ing le

va lence orb i ta l on a s ing le energy leve l , in te rac ts wi th a so l id , the e lec t rons of which

occupy a band of energ ies ( f ig .15 , le f t -hand s ide) The in te rac t ion of an a tom with a so l id

can be descr ibed by one of the two l imi t ing cases :

( i) the in te rac t ion is we ak and leads to a b roadenin g of leve l A in to a v i r tua l band


30/66

3 6 c h a p t e r 1

( i i )

i n s i d e t h e m e t a l b a n d ;

t h e i n t e r a c t i o n i s s t r o n g a n d c a n b e d e s c r i b e d a s t h e f o r m a t i o n o f a p s e u d o m o l e c u l e

f r o m a t o m A a n d o n e o r s e v e r a l s u r f a c e a t o m s o f th e s u r fa c e ; b y i n t e r a c t i o n w i t h

t h e so l i d , th e e n e r g y l e v e l s o f t he p s e u d o m o l e c u l e s a r e b r o a d e n e d a g a i n i n t o a

n a r r o w b a n d .

B e f o r e b o n d f o r m a t i o n

B e f o r e b o nd fo r m a t i o n

. E

b b ~

m e t a l a t o m

~ B "

a m p l i t u d e s

~a b "

/

/

/

/

/

\

\

a b

I

E A B

/ ~ a b a n t i b o n d i n g M . O . \ \

/ I \

/

/ I /

/

I /

A E /

/

I /

, z

/

I /

\ \ I E b B

~ B ' b o n d i n g M .O .

a f t e r

ond

f o r m a t i o n '

b a

b a

E a , ~ a

a d s o r b a t e a t o m

f i g u r e 1 3

L e f t a n d r i g h t : e n e r g y l e v e ls o f a t o m s b a n d a , b e f o r e a m o l e c u l e A B h a s b e e n f o r m e d .

I n th e m i d d l e , e n e r g y l e v e ls b o n d i n g - B , a n t i b o n d i n g a b ) c o r r e s p o n d i n g t o t h e m o l e c u l e

A B a r e f o r m e d b y t h e i n t e r a c ti o n s o f e l e c tr o n s o n t he l e v e l b a n d a . I n t h e c r u d e s t

a p p r o x i m a t i o n :

2 8 9

A E = [4 V 2 , b ,a + ( E b - E a ) } ,

with Vo,a = ~dO n ~ dr,

E a = ] , / 2 ( E b + E a ) + { V 2 b , a + 1 /4 ( E b - E o ) 2 } 8 9


31/66

St r u c tu r e a n d p r o p e r t i e s o f me ta l s a n d a l lo y s

3 7

a) b )

c ) - - - - ~N ( E )

~

N = 8

N oo N oo

f i g u r e 1 4

One d im ens iona l cha in o f a toms , w i t h i nd i ca t ed numb ers o f a toms N

a) b ) - energy bands

c) - dens i ty o f s ta tes curve , correspo nding to b)

l

Th e th e o r y f o r b o th l imi t in g c a s e s wa s d e v e lo p e d in th e 6 0 ' s , a n d in s o me e a r l i e r

p io n e e r in g p a p e r s [ 67 ] . W h a t f o l lo ws i s b a s e d o n s o m e o f th e o r ig in a l p a p er s [ 6 8 -71 ] a n d

s o me r e v ie ws [ 1 5 ,1 6 ,59 ] . Le t u s s t a r t th e d i s c u s s io n wi th th e we a k - b o n d l imi t .

f i g u r e 1 5

Wea k bond l im i t in t he f orm at i on o f a bond be tw een a t om A s i ng le orb it a l, s i ng le

e l ec tron) an d a t oms o f me ta ls . Me ta l e l ec t rons occupy t he energy ban d up t o F erm i

Energy . By in teract ion, l eve l A b roaden s and becom es par t ly occupied.

The in te rac t ion of an e lec t ron in

a n E A

leve l ( f igure 15 , r igh t) wi th e lec t rons in the

band ( f igure 15 , le f t ) leads to two e f fec ts on the

E n

level:

( i) the E A leve l i s sh i f ted on the energy sca le , and


32/66

38 chap te r 1

( ii ) by the in t e rac t ion w i th a tom s , the e l ec t rons o f w h ich fo rm the band , i t i s b roade -

ned.

The b roa den ed l eve l E A can be fu l ly o r pa r t i a ll y occu p ied by e l ec t rons o r be com ple t e ly

em p ty , i n w h ich case i t r ep resen t s an A + ion. W hen the ene rgy band fo r t he m e ta l i s

na r row , t he b roaden ing o f t he adso rp t ion l eve l i s l e s s p ronounced than w hen the band i s

broad.

Le t u s now m ake a s t ep t o t he s t rong-bond l im i t and cons ide r t he so -ca l l ed

pseud om olecu le s . The s t rong pseu dom olecu la r i n t e rac t ion sp l it s t he E A l eve l i n to tw o

broadened l eve l s : a bond ing and an an t ibond ing one , w i th a gap in be tw een . Th i s i s

s im i l a r t o t he s i t ua t ion w i th t he m olecu le A B in f i gu re 13 . The an t ibond ing band can be

ei ther abo ve or be lo w the Fe rmi level E F, or i t can b e spl i t by EF in to an o ccu pie d and

unoccupied par t . This las t case i s shown in f igure 16.

d 1

S

before

interact ion

- - E F ~

E A

E

-E A-

N(E)

metal ,

on t ibond ing

EA- metal ,

bonding

f i g u r e 1 6

S t r o n g c h e m i s o r p t i o n p s e u d o m o l e c u l e s ) l i m it . C h e m i s o r p t i o n o f a t o m A o n a t r a n s i t io n

m e t a l w i t h s a n d d b a n d s . L e f t - b e f o r e i n t e r a c t io n , r i g h t - a f t e r i n t e r a c t i o n .

W e sha l l now inves t iga t e w he the r and how the s im ple t heo ry can exp la in t he t r ends

in t he chem iso rp t ion bond s t r eng th fo r adso rbed a tom s w he n m e ta l s o f t he pe r iod ic t ab l e

a r e c o m p a r e d .

Theore t i ca l ana lys i s [15 ,16 , 5 9 , 68 -7 1] show s tha t w e can m ake the fo l low ing

s im ple s t a t em en t s .

( i) The pos i t i on o f t he E A l eve l s (bands ) and the i r s epa ra t ion f rom each o the r a r e m a in ly

due to the in terac t ion of

chapter 1 structure and properties of metals and alloys 1995 studies in surface science and...

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