Nuclear Physics (9 th lecture)

Transcription

1 Nucea Physics (9 th ectue) Cotet The ucea she mode (cotd.) The successes ad faiues of the ucea she mode o Nucea spi ad paity o Nucea magetic momets o Nucea quadupoe momets Nucea modes #4: The ucea coective mode Raiwate appoximatio Cosequeces of ucea defomatio o Nisso-schemes The ucea she mode (cotd.) We assumed spheica ceta potetia m,, Y,,, m,79 = (d) 8 p = (s) 6,99 = (4f) 4 4 7, = (p) 6 4,,,...,,,... m st case: the ifiite we potetia M k, V E, kr (+) Sum p = (s) 4,49 = (p) 6 8 8? V Magic umbes:, 8,, 8,, 8, 6 - adia quatum umbe - obita quatum umbe m - magetic quatum umbe k,, The fist magic umbes ae okay, but the est? V, if R,, if R The ucea she mode (cotd.) Descates spheica p (+)(+) (+) s 6 p 6 s d p 6 4f 4 4 s 4d 4 g g 7 The hamoic osciato ca descibe oy the fist magic umbes. Somethig moe shoud be icuded! Magic umbes:, 8,, 8,, 8, 6 V V m d case: the hamoic osciato potetia The ucea she mode (cotd.) d case: the Saxo-Woods potetia V R Aaytica soutio of the Schödige equatio is ot possibe. Numeica soutios show that the hamoic osciato eves sepaate accodig to the quatum umbe. The magic umbes caot be expaied! Somethig ese is eeded! V Magic umbes:, 8,, 8,, 8, 6 e R a Saxo-Woods is somewhee i betwee 4

2 The ucea she mode (cotd.) E.P. Wige, M. Goeppet-Maye, H. Jese (Nobe-pize 96) Thee shoud be a spi-obit iteactio tem! V m V LS,, E,, LS The tota agua mometum: J L S J L S LS fom whee: LS J L S The eigevaue of the LS opeato ss Sice s ad, if the eigevaue of the LS opeato, if Degeeatio gets spit! If the V LS () potetia is attactive, the = +/ state gets owe eegy ad the = / state gets highe eegy. The ucea she mode m Remembe:,, Y,,,,...,,,... m these ae the atua quatum umbes,, m I ucea physics the otatio of the states is NOT accodig to the mai quatum umbe ike i atomic physics (sice the mai quatum umbe depeds o the potetia), but o the adia quatum umbe Exampe: d + = =/ 6 + Eegy Magic umbes The ucea she mode Excited states: Simpe case: eve-eve coe + oe patice (p o ) outside the coe. The poto ad euto potetias ae sighty diffeet (Couomb potetia). Sige-patice excitatio eegies easoaby we descibed. Moe compicated case: odd-odd uceus two patices (,p) outside the coe. The esidua iteactio betwee the upaied euto ad poto shoud aso be take ito cosideatio. 7 8

3 The ucea she mode Nucea spi ad paity (goud state): Eve-eve ucei: I =, p = + (because of the paiig) Eve-odd ucei: I = (the agua mometum of the upaied) p = (-) Odd-odd ucei: the coupig betwee the two upaied uceo shoud aso be take ito accout. /(-) 9 The ucea she mode (cotd.) Nucea magetic momets: Remembe: I geea: μ g J m N g is the gyomagetic facto Dipoe magetic momets ca be oigiated fom two souces: evovig chage (e.g. potos o > obits) μ gmn Nuceos have itisic magetic momets: μs gssmn m s (p) =,79846 m N m s () =,948 m N, whee m N is the ucea mageto. Pobem: Sice i geea μ μ μ g g s g s s e mn M p, g g s, theefoe m ad ae ot paae! Oy the poectio of m o ca be measued! μ s meff g gs mn s ad the diectio: μeff g mn g gs mn 8 ev T Nucea magetic momets (cotd.) s Fom the pevious: g g gs The usua tick: fom s we get s ad s s ss ss Usig these g g gs Sice we get g g gs g ad meff g gs g mn poto euto g g s,86,86 Fou cases: upaied poto = +/ = / upaied euto = +/ = / Nucea magetic momets (cotd.) Schmidt ies upaied euto upaied poto The Schmidt-ies oigiatig fom the she mode ae the imitig vaues. The she mode is a exteme appoximatio.

4 The ucea she mode (cotd.) Nucea quadupoe momets The quadupoe momet of the upaied poto: Qp z R A Fo a poto hoe : Q p Qp Expeimeta fact: Qexp Q p (sometimes eve a facto of )! Nucide Z N Chaacte Q exp 7 O K 9 7 Lu Bi euto = / poto = / mid she = 7/ + poto = 9/ [mb] Q Q SM [mb] Ratio p z Y,.6 8,64 mb The ucea she mode (cotd.) Nucea quadupoe momets Quadupoe measuemets fo odd-a ucei. The hoizota axis is eithe euto umbe o poto umbe, whicheve is odd Fo a she mode, those ucei with cosed shes shoud be spheicay symmetic ad have o quadupoe momet Betwee magic umbes the quadupoe momets ca be eay high the she mode ca ot expai that. 4 ) Idicatios Nucea modes #4: The ucea coective mode We have see that the she mode caot expai some obsevatios: Nucei betwee the cosed shes have quadupoe momets eve i the goud state. How ca the goud state ucea quadupoe momets be much age tha pedicted by the she mode? Why ae the magetic momets diffeig fom the Schmidties?..etc. ) Thee wee some simpificatio i the sige-patice she mode: The ucea Hamitoia: Hˆ A i ~ i V mi sige-patice mea ucea opeatos potetia Assumptio was: the shape of the mea ucea potetia is ceta depeds oy o the absoute vaue of i : V( i ) (tue?) simia shape as the ucea desity (sice the uceos ceate it). Is that ecessaiy spheica? Possibe soutio: Keep the assumptio that the shape is simia to the shape of the ucea desity (sice the potetia oigiates fom the uceos) Aow the ucea desity to be defomed potetia gets aso defomed Keep the ucea voume costat (icompessibe fuid) i 6 4

5 How to detemie the defomatio? Iteative appoach: Note: is a vecto, it depeds o the diectio as we!,, Hatee-Fock (Bogoiubov) method Nucea desity () Y() Nucea potetia V() Detemie Y by miimizig the eegy <Y H Y> It coveges oy sowy (if covegig at a) fo may patices. Sow, difficut ad cumbesome cacuatios (atisymmetizig, paiig, etc.) 7 Raiwate appoximatio Maiage of the iquid dop mode ad the she mode. Cocept: uceus = coe + vaece uceo(s) Coe () Eegy miimisatio defomabe iquid dop Nuceo defoms the coe she mode Leo James Raiwate (97-986) Nobe-pize 97 Nucea potetia V() Nuceo eegy vaiatio by petubatio 8 Raiwate appoximatio (cotd.) Simpest case: Oy oe uceo outside the coe The coe ca be defomed to eipsoid Voume coseved: 4 p 4 R p A B A R Fo sma defomatios: B R The potetia is oigiay spheica, but it gets aso defomed: V V, whee cos cos dv The petubed potetia: V V V cos d (to the fist ode) z 9 y x Raiwate appoximatio (cotd.) The tota eegy of the uceus E E C E N coe vaece uceo a) The upetubed coe is spheica: EC E whee EC S is the igidity of the coe. dv V cos d C S... b) The petubed eegy of the vaece uceo: den EN EN... EN EN d * E N V d (petubatio theoy) R Hee is the upetubed wave fuctio: m,, Y, dv Theefoe E N R d Y cos d d Note: sice is desity, its dimesio is m. The dimesio of R is m.

6 V dv d dv R d Y cos E N d dv d d theefoe V R R d V RR R The mass quadupoe momet of the uceo: Q N * cos d R d Y cos d dv d R d QN QN We get R R R EN V R R R U whee U V QN The eegy of the uceus: E EC EN EC S EN U QN Q E S U This has a miimum, if U N S Note that the sig of the defomatio is the same as the sig of Q N : The uceo attacts the coe, ad defoms it. The tota quadupoe momet of the uceus: Q Q C Q If we kow the, the eectic quadupoe momet of the coe ca be cacuated: Q C cos d whee is the poto-desity. 4p Z 4 Usig Z A B we get Q C cos d ZR 4pA B ZR U Fiay we have: Q 4 QN S Q>>Q N is possibe fo sma S ( soft coe, easiy defomabe) Few exampes: 9 Qmeas 8Bi6 douby magic coe (igid) + poto: QN 8 Rb Q,b 87 Rb Q,4 b (magic umbe of eutos) S Q,6 b S Q,4 b N Cosequeces of coe defomatio (ad potetia defomatio) ) Nisso eve schemes The degeeated eves spit with defomatio magic umbes may chage! ) This gives a bette desciptio fo the ucea spis ad paity 6