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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 )
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Struc ture and proper t ies o f meta ls and a l loys 15
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
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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 .
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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.
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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].
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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 .
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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
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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 .
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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
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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 ,
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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
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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 )
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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] .
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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
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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
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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.
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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
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Struc ture and proper t ies o f meta ls and a l loys 35
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
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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
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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
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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