Principios para diseño de interfaz de usuario

F i g u r e 4 .3 E x perim ent - free ato m difference profiles fo r th e th ree principle directional C o m p to n profiles o f alum in ium m easured on b o th s p e c tr o m e te r system s. G ood agree­ m ent is necessarily observed between ex perim ent a n d free ato m a t high values of m om entum .

am e riciu m system ).

T h e v ariation o f d ete cto r efficiency w ith energy for th e ,#*Au system w as corrected fo r b y th e M onte C arlo sim ulatio n of th e s ca tte rin g and a b s o r p tio n processes in th e ger­ m a n iu m d etector cry sta l described in section 3.3.1. No c o rre c tio n for efficiency is neces­ s a r y fo r th e low energy sy stem since th e d ete cto r is v irtu a lly 100% efficient o v er the C o m p to n peak region (see figure 3.2).

4 .5 . R e s u lts a n d D is c u s s io n

4 . 5 . 1 . T h e B a n d S t r u c t u r e C a l c u l a t i o n s

T h e experim ental resu lts have been com pared with four c a lc u la tio n s as in d ic a te d in s e c tio n 4.2. The A P W m ethod used by K ubo e t al (1976) to c a lc u la te C o m p to n profiles p ro v id e s a good descrip tio n o f th e b and s tru c tu re and th e d e n s ity o f s ta te s and ac cu ra tely re p ro d u ce s th e F erm i surface. T h e calculatio n o f m om entum d e n s ity was ca rrie d o u t over 273 reciprocal lattice v ectors w ith a cubic m esh of 505 k-p o in ts in th e irredu cible volum e eq u iv a le n t to 1/48 o f th e Brillouin zone. It does n o t, however, in c lu d e term s d esc rib in g the effects of electron exchange and correlatio n in th e lattice p o te n tia l.

T h e self-consistent field ex a ct exchange H artree-F ock ca lc u la tio n m ade by C a u s a e t al (1981) uses a LC A O m ethod to o b tain th e electronic w a v e fu n c tio n w ith each atom ic o r b i t a l consisting o f a linear com bination o f four G aussian b a s is fu n ction s fittin g each S la te r- ty p e o rb ital. T h e exclusion of electron-electron sh ield in g effects by th is m ethod t e n d s to overestim ate th e individual one electro n p o ten tials a lth o u g h th is is cancelled to s o m e ex te n t by th e absence o f a correlatio n te rm in the la ttic e p o te n tia l. As a consequence o f t h is cancellation th is calcu lation is found to reproduce a c c u ra te ly th e F erm i su rface .

T h e calculation m ade by T aw il (1975) generates th e ele c tro n ic charge d is trib u tio n fro m a set of C arte sia n -G a u ssia n prim itive w avefunctions o b ta in e d from a self-con sisten t m odified tig h t binding model. T h e value o f th e Ferm i energy d edu ced from th is model, h o w ever, produced a d ifferent connectivity w ith th e F erm i s u rfa c e a t th e W sy m m e try

- 5 5 -

poin ts o f th e W igner-Seitz cell to th a t observed experim entally (see C rac k n ell and W ong, 1973) an d consequently was n o t used to determ ine th e d irectional profiles. Instead T aw il chose a v a lu e of Er to reproduce th e required connectivity which in cre ase d th e num ber of b and ele ctro n s to 3.1. W hen a com pensatory rescaling factor is in tro d u c e d (i.e. 0.97), how­ ever, th e co nsequential d isto rtio n o f the C om p ton lineshape near t h e F erm i surface is so significant t h a t the 111 direction al profile c a n n o t be used for m ea n in g fu l com parison w ith ex p e rim en t. A lthough the 110 and 100 directions for this calculation a r e presented here it m ust be em ph asised th a t th e ir reliability is also suspect.

T h e fo u rth theory cu rren tly available is th a t of Sacchetti (1984, p riv ate com m unica­ tion) w hich em ploys th e charg e density of M oruzzi, W illiam s an d J a n a k (1978) to calcu­ late th e C om p to n profile w ithin the slowly varying den sity a p p ro x im a tio n (i.e. a m odification o f th e local den sity approxim ation) and includes a c o rre la tio n term in the one-electro n p oten tial. T here is no indication how well th is th e o ry describes th e F erm i surface.

T h eo ries which ca lcu late th e m om entum w avefunctions by F o u rie r tran sform in g the solu tio n s to th e K ohn-Sham eq u a tio n s (i.e. tho se based on th e local d en sity ap prox im atio n such a s t h a t of Sacchetti) do n o t yield the tru e m om entum d e n s ity because correlation effects a r e n o t properly tre a te d (see C h ap ter 6). They m ay be c o rre c te d by adding th e L a m -P la tz m a n term (eq. 2.33) to th e C om pton profile as o utlined in section 2.7. T h e in p u t for «J|j(r )|(<0 in eq. 2.33, w hich represents th e C om pton profile o f the hom ogeneous in te ra c tin g electron gas w ith th e (local) charge density, p(r), is ca lc u la te d from an an aly tic expression for th e 3D m om entum density, given in section 6.3.1.1. p(r) is tak en from th e ta b u la te d den sity functional m uffin-tin charge densities by M oruzzi e t al (1978). T h is cal­ cu la tio n is relatively sim ple for th e regions of co n stan t charge d e n s ity occupied by th e co n d u c tio n electrons b u t becom es m ore com plicated in th e core re g io n . T h e effect o f th e correc tio n is to sm ear o u t th e C om pton profile obtain ed from t h e K ohn-Sham “ pseudo- w a v efu n c tio n s” i.e. it prom otes electrons to higher m om entum s ta te s as described in c h a p te r 6.

F igure 4.4 shows the L am -P latzm an c o rre c tio n for alum inium along w ith the in tera ctin g - free electron gas difference profile, J ‘nt from which th e form er was derived. B oth sh ift electron density from low to high m om entum . C om parison o f th e tw o lineshapes shown in th is figure in dicates t h a t alth o u g h both have a sim ilar shape, the L am -P latz m a n correction has a significantly larger effect on the to ta l profile. Its co n trib u ­ tion to th e to ta l energy of th e system m ay be d ed u c ed from the second m om ent of th e profile th ro u g h th e ap p lication of the virial th e o re m (see section 2.2). It a p p a ren tly increases th e electronic kinetic energy by ap p ro x im a te ly 30 eV per ato m (com pared to a cohesive en ergy o f 7.28 eV for alum inium ). W h ilest th is value is o f th e co rrec t o rd e r of m ag n itu d e little significance can be placed on it b ecause th e calcu latio n is very p rone to sm all e rro rs in the L am -P latzm an correction a t h igh m om entum . Indeed, a s h ift of only 0.023 % J (0 ) in th e baseline o f Aj££*|(q) changes t h e k in e tic energy by 30 eV p er atom .

4 .5 .2 . D iff e re n c e P ro file s

B oth se ts of exp erim ental directional differen ce profiles are show n in figure 4.5, to g e th e r w ith th e v aria tio n predicted by th e A P W ca lculation. It is im m ediately obvious t h a t b o th ex p e rim en t and th eo ry d isplay a very sm a ll anisotrop y, th e differences am o u n t­ ing to ~ ± 1 /4 % J(0) a t low m om en ta. T h e am e riciu m sp ectro m eter is m uch m ore suc­ cessful in m easu rin g these effects because of th e s u p e rio r statistic al accuracy o f its d a t a (<r ~ s 1 /1 0 % J(0 ) c.f. r 3 /1 0 % J(0 ) for th e ,#*Au system ). S ystem atic erro rs w hich m ay plague th e in dividual profiles are isotropic and th e re fo re of no consequence in th e direc­ tional difference. O n th e o th e r h and th e an is o tro p y is com parable to th e sta n d a rd devia­ tion o f th e higher energy d a ta where the o scillation s a re n o t so well established. T h e m ag­ n itu d e o f th ese oscillations im plies a relatively s p h eric al F erm i su rface such a s th a t illus­ tra te d by figure 4.6, which is calculated for a free ele ctro n gas, an d is ch arac teristic o f a good free electro n m etal. It can be seen from th is figure, which assum es th ree electrons per p rim ativ e cell, t h a t the free electron F erm i sp h ere o f alum inium ex te nd s to th e th ird Bril­ louin zone alth o u g h only p a r t o f th is s tru c tu re is sh ow n (see M ackintosh, 1963). T h e iso-

111-100 (198A

)

Pz (a.u.)

F i g u r e 4 .5 E xperim ental anisotropies for d a ta acquired on th e am e riciu m and gold spec­ tro m eter sy stem s. T he solid line shows th e correspon d in g A P W theory difference profiles w hich have been convoluted w ith a G au ssian o f 0 .5 7 a.u. and 0.4 0 a.u. FW HM fo r com parison w ith m easurem ents m ade w ith th e 24,Am and ‘®*Au isotopes respectively. T h e full o rd in a te scales displayed are a p p ro x im a te ly J(0) for th e gold system d a ta and ' / c J(0) for th e m ore sensitive a m e ric iu m system d ata .

F ig u r e 4 .8 Reduced zone represent a l ion of th e free electron F erm i sph ere o f w ith 3 valence electron s per p rim itiv e cell. T h e second Brillouin zone is w ith electrons. T h e com plex s tru c tu ré of th e th ird zone is only show n in

a lu m in iu m a lm o s t full p a r t (a fte r

tro p ic n a tu re o f th e m om entum d is trib u tio n o f a free electron gas is in d icate d by th e sphericity of th e co n to urs in th e second an d th ird zones respectively.

T he sm all scale of th e ex p erim ental anisotropy is in good agreem en t w ith th e results of the 2-D ACAR experim ent of M ader et al (1976) and can be used to d ifferentiate betw een th e available theories. R eferring to figure 4.7, th e T aw il th eo ry clearly ov eresti­ m ates th e m ag n itu d e o f th e m easured difference for th e 110 - 100 an iso tro p y by roughly a fa cto r of fo ur w hereas th e slowly v arying density calculatio n of S acc h etti p redicts no significant profile difference w hich is m uch closer to th e experim en tal re su lt. T he self- con sisten t ex a ct exchange H artree-F ock calculation on average ov erestim a tes the m agni­ tu d e of th e profile an isotropies by roughly a fa cto r o f tw o , alth o u g h th ey a re of the sam e frequency an d occur in ap p roxim ately in th e sam e position as th ose d ete cted experim en­ tally . F ro m figures 4.5 an d 4.7 it is a p p a re n t th a t th e m ost successful th eo ry in predictin g th e observed profile anisotropies o f alum inium is th e A PW calculatio n o f K ubo e t al, hence th e inclusion of th is calculation alongside ex p e rim en t in figure 4.5. N o t only is the scale of th e a n iso tro p y predicted correctly b u t, as th e m ore precise low energy d a ta shows, the oscillations in th e d a ta are followed closely by t h e A P W theory.

In an a tte m p t to in te rp re t th e observed profile anisotrop ies a ca lcu latio n based on a sim ple W igner-S eitz m odel was perform ed (see section 2.5.3) in w hich free electron spheres were ce n tred on th e [111] an d [200] reciprocal la ttic e sites each w eighted u niform ly w ith th e high o rd e r m om en tum coefficients, I Ao (0) I * (i.e. 0.0097 and 0.0030 respectively) ta k e n from M ader e t al (1976). T h e d irec tiona l profiles w ere then o b tain ed by in te g ra tin g o ver successive p lanes in th e direction perpen dicular to th e ap p ro p riate la ttic e vector. T h is essen tially involved sum m ing th e squares of th e ch o rd s defined by th e in terse ctio n of th e plane o f in te g ra tio n w ith th e free electron spheres a t 0.01 a.u. in terv als. T h e m ag nitud e o f the difference profiles predicted by th is m etho d below th e F erm i m om entu m is com p arab le to ex p e rim en t. Above pr, how ever, th e period w ith w hich th ey oscillate d eviates from t h a t observed d u e to th e neglection o f th e th ird n e a re s t neighbour coefficients. C onsequently th is ca lcu latio n is to o a p p ro x im a te to describe th e anisotropic b ehavior ab ove th e F erm i