8. Two Ion Interactions

Transcription

1 8. Two on ntetons The moels of mgnet oe hve een se on mltonns of the fom on J J zw. Wht s the physl ogn of ths two on ntetons 8. Dpol nteton The ogn of lge moleul fels nnot e the wek mgnet pol nteton CD gµ B 3R R δ Exese: lulte the oe of mgntue of the pol nteton fo moments of µ B septe y Å R 5 R J J

2 8. Exhnge nteton Ogn: - Eletostt nteton etween Eletons - ul nple ntsymmet wve funtons ex Exese: how y onseng the nteton of two eletons, n sttes n tht the exhnge ntegl s ex g e πε Chpte 8 Two on ntetons ex J J

3 , πε,, e E Unpetue soluton fo eletons: Unpetue polem... mgne t to e solve -ssume eletons to e n ffeent otl sttes wth nely the sme enegy e.g. vl f the two nule e well septe, f they e lose then the one eleton polem goun stte wll not e egenete n we wll get snglet goun stte wth ntsymmet spn wve funton pt... s n the e tom - notton fo the spn pt of the wve funton -... lte Detemnnt,, Two eletons wth / -fol egeney: Chpte 8 Two on ntetons 3 3

4 Chpte 8 Two on ntetons etuton oulom : we hve to gonlze oulom efne... { } { } oulom oulom oulom oulom πε e

5 Chpte 8 Two on ntetons 5 oulom ow mgne the polem of eleton n stte > n eleton n stte > wthout ntsymmetzng the wve funton: t hs lso -fol egenete goun stte: φ φ φ φ Φ Φ oulom

6 Chpte 8 Two on ntetons 6,, z y x ul - pnopetos: x x z z y y x x

7 Chpte 8 Two on ntetons 7 oulom Φ Φ πε e... nste of ntsymmetze wve funtons we n use et pouts of sngle eleton sttes n onse n exhnge ntetons etween the spns., olem: the vlue of epens on the hoe of... Tkng the set wll gve ompletely ffeent n!!! eng,rev. Mo. hys , Remey: hosen suh tht they e stongly lolze oun tom

8 Chpte 8 Two on ntetons 8 Exhnge nteton n hypothetl hge Bosons,, γ γ γ γ γ γ γ ymmetze tte Bosons: etuton oulom : we hve to gonlze oulom... { } { } oulom oulom oulom oulom 9-fol egeney: pn sttes efne

9 Chpte 8 Two on ntetons 9 [ ] oulom,, z y x ± ± ± ± m m m m m m m m y x z m Bqut nteton fo etls see next pge

10 Chpte 8 Two on ntetons ome etls on pn Mtes: z y x

11 ... we onse two toms eh hvng eletons n unflle shells,.e. ong to uns ule. ow oes the exhnge nteton look lke Does t ontn qut tems eletons wth /,,3, πε,,,3, E e Unpetue polem... mgne t to e solve Unpetue soluton fo eletons: -ssume eletons to e n ffeent otl sttes enegy - notton fo the spn pt of the wve funton,,,, wth nely the sme Chpte 8 Two on ntetons

12 Chpte 8 Two on ntetons lte Detemnnt Defntons: 6-fol egeney: CD BD BC AD AC AB e e AB πε πε etuton oulom : gonlze oulom oulom

13 oulom

14 ... Rewtng ths esult s 6x6 mtx...e.g. t follows oulom 3....e. o qut nteton! Chpte 8 Two on ntetons 3 3 n e lge euse they e lote on the sme tom n stlze n tplet: onseng ths petuton fst n gonlzng the 6x6 mtx see pevous pge, we fn 9 fol egenete goun stte see next pge

15 Chpte 8 Two on ntetons 5 9 fol egenete goun stte fo n exhnge only: These 9 sttes oespon extly to the 9 sttes of the two ntetng osons polem susse efoe. oweve, n ontst to the Boson exhnge nteton swthng on the emnng smlle exhnge ntetons of the eleton polem yels the followng mtx fo the enegy see next pge, O BQUADRATC TERACTO:

16 Chpte 8 Two on ntetons 6 oulom oulom... pn opetos fom olem see ove

17 eletons wth /,..., πε,,..., < E e out tte π χ χ... χ k k k Antsymmetzng Opeto: A - lte Detemnnt: D A π emutton Opeto! Chpte 8 Two on ntetons 7! emuttons

18 Chpte 8 Two on ntetons 8 < oulom oulom e D D π π πε π π!!!, A A... only ontutons fom o... π π π π πε oulom e <!! k k k k k χ χ χ χ χ π χ exhnges wve funtons χ exhnges spnos, [Chek ths y pplyng: ]... χ χ

19 e D oulom D π [ ] π πε < π π π [ ]π < < wth the spno pt of the stte veto efne s π χ χ... χ e n k k k k πε e k k k k πε wth Ths we ntoue to sum ove ll, wthout the estton < Chpte 8 Two on ntetons 9 n se some of the eletons nvolve e on the sme on, one hs to onse fst ths oelton n susequently the smlle exhnge etween ffeent ons. ml to the - eleton polem ths les to fo poof see eng,rev. Mo. hys Fo ons wth otl moment L>: JL Lg J J... g J -J ex g ex ex < ex J J

20 opetes of the exhnge ntegl ex Fo eletons entee t the sme tom these hve to e n ffeent otl sttes wth the sme enegy,.e. n n unflle shell: usully postve stlzng the tplet stte onsstent wth un s st ule leng to n ntsymmet sptl wve funton Fo eletons on ffeent toms et exhnge : postv/negtve feomgnetsm/ntfeomgnetsm The Bethe-lte uve. Chpte 8 Two on ntetons

21 Exhnge nvolvng moe thn eletons: 8.3 net Exhnge upeexhnge n on ystls Cystl tutue of MnO eest negho ps of Mn Mngnese ons e Connete v O - Oxygen toms Chpte 8 Two on ntetons

22 Chpte 8 Two on ntetons

23 8. RY nteton n Metls RY... Ruemn, ttel, suy, Yosh Ogn: spn polston of the onuton eletons y lolze eletons... net exhnge mehnsm fo fee eletons t shows oslltoy ehvo RY os RY k F 3 see lso hpte kf Chpte 8 Two on ntetons 3

24 8.5 Othe Exhnge Mehnsms Doule Exhnge n vlene flututng ompouns The e g eleton on Mn 3 ste n hop to neghong Mn ste only f thee s vny of the sme spn hoppng oes not flp spn. The hoppng s fvoule n tems of knet enegy n theefoe the feomgnet, metll hgh onutvty stte s of lowe enegy. Chpte 8 Two on ntetons

25 Ansotop Exhnge nteton ml to supeexhnge, howeve the metng tom s not ffeent tom suh s oxygen, ut the exte spn ot stte of mgnet on of the sme kn. DM g ε γ Chpte 8 Two on ntetons 5 D DM...Dzyoloshnsky-Moy The exhnge veto D vnshes f thee s n nveson symmety t the mpont etween tom n. The effet s to nt spns y smll ngle. Exese: ese the ffeene etween nsotop exhnge ntetons, nsotopy of the exhnge, ystl fel nsotopy. γ