The Low Density Phases of the Electron Gas

For f i f t y y e a r s t h e phases o f t h e e l e c t r o n energy dominates t h e k i n e t i c ene rgy , so t h a t t h e g a s ( a l s o known a s j e l l i u m o r t h e Fermion one e l e c t r o n s must c r y s t a l l i z e . S ince these e a r l y component plasma) have been t h e o b j e c t o f c a l c u l a t i o n s , t h e r e have been numetous t h e o r e t i c a l i n t e r e s t , which began when Bloch investigation^^'^'^ of t h e r e l a t i v e s t a b i l i t i e s d i scove red t h a t t h e Sommerfeld model f o r of t hese t h r e e s t a t e s (paramagnet ic , d e s c r i b i n g e l e c t r o n s i n s imple me ta l s w i l l f e r romagne t i c , and c r y s t a l ) wi thout any consensus e x h i b i t a fer romagnet ic s t a t e i n t h e Hartree-Fock on when o r i f t h e fer romagnet ic t r a n s i t i o n would approximat ion f o r e l e c t r o n dens i t i e s occur and a t what d e n s i t y Wigner c r y s t a l l i z a t i o n corresponding t o cesium meta l and lower. I n t h i s would occur . The e n e r g i e s o f t he va r ious phzses approximat ion, Paul i e x c l u s i o n i s t he only a r e c l o s e t o each o t h e r over a wide ranye of mechanism f o r e l e c t r o n s t o c o r r e l a t e s o t h a t a d e n s i t i e s . To a c c u r a t e l y compute t he t r a n s i t i o n t r a n s i t i o n t o a magnetic s t a t e i s expected . A d e n s i t y one must be a b l e t o t r e a t c o r r e l a t i o n few y e a r s l a t e r , wigner2 pointed o u t , t h a t a s e f f e c t s e q u a l l y w e l l i n each phase. t h e e l e c t r o n i c d e n s i t y i s lowered, t h e p o t e n t i a l *This r e sea rch was suppor ted i n p a r t by the Nat ional Resource f o r Computation i n Chemistry under a g r a n t from t h e Nat ional Science Foundation (Grant No. CHE-7721305) and t h e Basic Energy Sciences D iv i s ion o f t h e U.S. Department o f Energy (Con t r ac t No. W=7405-ENG-48). Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980744 C7-296 JOURNAL DE PHYSIQUE Monte Car lo s imu la t ion i s such a method f o r c a l c u l a t i n g t h e p r o p e r t i e s o f many-body sys tems a c c u r a t e l y . Only r e c e n t l y have computers been s u f f i c i e n t l y powerful t o t a c k l e t h e s e quantum many-body problems. 'Ihe work o f Kalos , Levesque, and v e r l e t 7 on t h e ground s t a t e o f hard sphe re Bosons i s a landmark c a l c u l a t i o n . S i m i l a r s i m u l a t i o n s of Lennard-Jones ~ o s o n s ~ have agreed v e r y w e l l w i th exper imenta l d a t a on t h e low tempera ture p r o p e r t i e s o f Helium 4. Recent ly , we have gene ra l i zed t h e quantum Monte Car lo method s o a s t o t r e a t Fermion sys tems and have performed s i m u l a t i o n s o f t h e ground s t a t e o f t he e l e c t r o n gas. The d e t a i l s o f t h e method w i l l be g iven e lsewhere . Here we wish t o summarize t h e r e s u l t s t h a t concern t h e phases o f t h e e l e c t r o n gas a t low d e n s i t y . In t h e quantum Monte Car lo method, a t r i a l f u n c t i o n o f t h e S l a t e r Ja s t row form9 ( a product o f p a i r c o r r e l a t i o n f a c t o r s and a S l a t e r de terminant o f o r b i t a l s ) i s employed t o s p e c i f y t h e symmetry of t h e p a r t i c l e s , t o keep t h e popu la t ion of random walks i n t h e impor tant r eg ions o f phase space and t o reduce t h e v a r i a n c e o f t h e c a l c u l a t e d energy. The mean v a l u e of t h e c a l c u l a t e d energy w i l l be e x a c t l y t h e ground s t a t e energy. It i s i n t h i s s t a t i s t i c a l s e n s e t h a t t h e s imu la t ion i s e x a c t . As wi th most s t o c h a s t i c methods, t h e v a r i a n c e o f t h e energy w i l l be i n v e r s e l y p ropor t iona l t o t h e number o f s t e p s o f t h e random walk. For t h e f l u i d phases cons ide red , t h e o r b i t a l s used were p l ane waves wi th v e c t o r s l y i n g w i t h i n t h e Fermi sea . I n t h e paramagnetic f l u i d t h e s p a t i a l s t a t e s a r e doubly occupied; i n t h e fer romagnet ic phase they a r e s i n g l y occupied . Hence, t h e fer romagnet ic f l u i d i s ant i symmetr ic w i th r e s p e c t t o t h e i n t e r change o f a l l p a r t i c l e s ; t h e paramagnetic on ly wi th r e s p e c t t o l i k e sp ins . I n t h e Boson f l u i d , t h e S l a t e r de terminant i s n o t used. For t h e c r y s t a l l i n e phase , t h e o r b i t s a r e ~ a u s s i a n s , c e n t e r e d around BCC l a t t i c e s i t e s . Table I c o n t a i n s t h e energy of t h e fou r phases t h a t we have s t u d i e d a s a f u n c t i o n o f rs , t h e mean i n t e r p a r t i c l e spac ing i n u n i t s o f Bohr r a d i i . The energy d i f f e r e n c e between a Bose and Fermi c r y s t a l i s l e s s t han t h e accuracy o f our c a l c u l a t i o n . Hence, we cannot d i s c u s s t h e magnetic o r d e r i n g w i t h i n t h e c r y s t a l . I n F ig . 1 a r e p l o t t e d t h e e n e r g i e s r e l a t i v e t o t h e lowest 2 Boson energy t imes r . P l o t t e d i n t h i s manner one can see t h e sma l l d i f f e r e n c e s i n energy amongst t h e phases. The Boson sys tem c r y s t a l i z e s a t r = 160. 'Ihe Fermion sys tem undergoes two phase t r a n s i t i o n s ; p o l a r i z a t i o n a t r = 75 and Wigner c r y s t a l l i z a t i o n a t r = 100. The me l t i ng o f t h e Wigner c r y s t a l occu r s because , a s t h e d e n s i t y i n c r e a s e s , t h e z e r o p o i n t motion o f t h e e l e c t r o n s d i s r u p t s t h e l a t t i c e . A l t e r n a t i v e l y , t h e l o c a l i z a t i o n energy o f t h e e l e c t r o n s about t h e l a t t i c e s i t e s becomes l a r g e r t han t h e k i n e t i c energy o f t h e l i q u i d . Table I1 c o n t a i n s Lindemann' s r a t i o , t h e rms d e v i a t i o n o f an e l e c t r o n from t h e n e a r e s t l a t t i c e s i t e i n u n i t s o f t h e n e a r e s t ne ighbor d i s t a n c e . ' Ih is r a t i o e q u a l s 0.25 a t m e l t i n g f o r t h e Boson sys tem and 0.30 f o r t h e Fermion system. Th i s v a l u e a t me l t i ng i s abou t twice t h a t found c l a s s i c a l l y and ve ry s i m i l a r t o t h a t found8 i n s o l i d Helium 4 . The r eason why e l e c t r o n s want t o Table I II ' I I I I I 1 h e ground s t a t e energy o f the charged Fermi and Base systems. h e density parameter, r,, i s t h e Wrgner sphere r a d i u s i n u n l t s of Bohr r a d i i . me e n e r g i e s are Rydbergs and t h e d i g i t s i n p a r e n t h e s i s r e p r e s e n t t h e e r r o r bar i n t h e l a s t dec lmal p l a c e . l h e four phases a r e : paramagnet ic or unpolar ized Ferrni f l u l d (PMF); t h e ferromagnetic or p o l a r ~ z e d Fermi f l u i d ( M F ) , t h e Bose f l u r d (BF); and t h e Bose c r y s t a l w ~ t h a BCC l a t t i c e . 1 3 0 . 0