MICROSCOPIC MODEL FOR THE NUCLEAR INERTIA TENSOR

Transcription

1 MICROSCOPIC MODEL FOR THE NUCLEAR INERTIA TENSOR D. N. POENARU,, R. A. GHERGHESCU, W. GREINER IFIN-HH, PO Box MG-6, 775 Buchaest, Romaia Fafut Istitute fo Advaced Studies, J. W. Goethe Uivesity, Pf 9, D-654 Fafut am Mai, Gemay Received Decembe, 5 The wave fuctio of a spheoidal hamoic oscillato without spi-obit iteactio is expessed i tems of associated Laguee ad Hemite polyomials. The paiig gap ad Femi eegy ae foud by solvig the BCS system of two equatios. Aalytical elatioships fo the matix elemets of ietia ae obtaied fuctio of the mai quatum umbes ad potetial deivative. The esults give fo 44 Nd ad 5 Cf ae compaed with a hydodyamical model.. INTRODUCTION The potetial eegy suface [, ] i a multi-dimesioal hypespace of defomatio paametes β, β,, β gives the geealied foces actig o the ucleus. Ifomatio coceig how the system eacts to these foces is cotaied i a teso of ietial coefficiets {B ij }. Ulie the potetial eegy E = E( β ) which oly depeds o the uclea shape, the ietic eegy is detemied by the cotibutio of the shape chage dβ dβ i j E = Bij( ) β () dt dt ij, = whee B ij is the ietia (o the effective mass) teso. I a pheomeological appoach based o icompessible iotatioal flow, the value of a effective mass B i is usually close to the educed mass µ= ( AA/ A) M i the exit chael of the biay system. Hee M is the ucleo mass. Oe may use the Wee Wheele appoximatio []. The micoscopic (caig) model itoduced by Iglis [4] leads to much lage values of the ietia. By assumig the adiabatic appoximatio the shape vaiatios ae slowe tha the sigle-paticle motio. Accodig to the caig model, afte icludig the BCS paiig coelatios [5, 6], the ietia teso is give by [7, 8]: ν H/ β µ µ H/ β ν B = ( u υ + u υ ) + P () ij i j ( E E ) ν µ µ ν ν + νµ µ Rom. Jou. Phys., Vol. 5, Nos., P , Buchaest, 5 ij

2 88 D. N. Poeau, R. A. Gheghescu, W. Geie whee H is the sigle-paticle Hamiltoia allowig to detemie the eegy levels ad the wave fuctios ν ; u ν ad υ ν ae the BCS occupatio pobabilities, E ν is the quasipaticle eegy, ad P ij gives the cotibutio of the occupatio umbe vaiatio whe the defomatio is chaged. Whe the shape is detemied by oe defomatio paamete, ε: P d d d d ε = λ 5 ( ) ( ) ( ) ( ) λ 8 E + εν λ + εν λ d d d d ν ν ε ε ε ε dλ ν dh/ dε ν ( ε ) d ν λ ν dh/ dε ν dε dε Simila to the shell coectio eegy, the total ietia is the sum of cotibutios give by potos ad eutos B = B p + B. The deomiato is miimum fo the levels i the eighbouhood of the Femi eegy. A lage value of ietia is the esult of a lage desity of levels at the Femi suface. I the peset wo we coside a sigle-paticle model of a spheoidal hamoic oscillato without spi-obit iteactio fo which the caig appoach allows to obtai aalytical elatioships of the uclea ietia. The esult may be used to test complex compute codes oe should develop i a ealistic teatmet of the fissio dyamics based o the defomed two cete shell model [9, ].. NUCLEAR SHAPE PARAMETRIZATION The shape of a spheoid with semiaxes a, c (c is the semiaxis alog the symmety) expessed i uits of the spheical adius R = A / may be detemied by a sigle defomatio coodiate which ca be oe of the followig quatities: the semiaxes atio c/a; the ecceticity e= ( a/ c); the defomatio δ= 5(. c a)( / c + a), o the quadupola defomatio [] ε= ( c a) /( c+ a) which will be used i the followig, ad accodig to which the two oscillato fequecies ae expessed as: ε ε ( ) ( ) ω() ε =ω + ; ω () ε =ω () ad by taig ito accout the coditio of the volume cosevatio ωω = ( ω ) whee ω = MeV, oe has 4A / ε ( ) ω =ω ε + 7 / A paticulaly iteestig value is ε =. 6 of a supedefomed spheoid with the atio ca / =. (4)

3 Micoscopic model fo the uclea ietia teso 89. SPHEROIDAL HARMONIC OSCILLATOR The eigevalues [] i uits of ω ae give by: E = ω ( + ) + ω ( + / ) = ω [ N + / +ε( N/ )] (5) whee the quatum umbes ad ae oegative iteges. Thei summatio gives the mai quatum umbe N = +. I uits of ω oe has a liea vaiatio of the eegy levels i fuctio of defomatio ε. By icludig the vaiatio of ω with ε, oe obtais the aalytical elatioship ε i = Ei/ ω = [ N + / +ε( N )][ ( 7)] / / ε / + ε/ (6) i uits of ω. Due to the Pauli piciple, each eegy level ε i, with quatum umbes ad N, ca accomodate g= + ucleos. Oe has a umbe of ( N + )( N + ) ucleos i a completely occupied shell chaacteied by the mai quatum umbe N, ad the total umbe of states fo the lowest N + N shells is ( N + )( N + ) = ( N + )( N + )( N + ) /. Fo each value of N N = thee ae N + levels with =,,,, N. Whe the defomatio ε> iceases, a level with = deceases i eegy ad the oe with = N iceases. Fo some paticula values of the defomatio paamete thee is a cossig of seveal levels i the same poit leadig to a degeeacy followed by a empty gap. If o spi-obit couplig is cosideed fo the vaishig defomatio paamete, ε =, oe has the followig sequece of magic umbes:, 8,, 4, 7,, 68, 4, If ow ε=. 6 they become, 4,, 6, 8, 4, 6, 8,, 4, 8, The ow expeimetal values ca be obtaied oly with a spi-obit couplig icluded. The spi Σ cotibutes with poitive o egative values (up o dow) fo evey state with quatum umbes, =,,,... ad m=, hece i a system of cylidical coodiates ( ρ, ϕ, ) the wave fuctio [, ] ca be witte as Ψ= mσ =ψm ( ρφ ) ( ϕψ ) ( ) χσ ( ) (7) The eigefuctios ae give by m m η m( ) m m ( ) m N ( ) e L α α ψ ρ = η η = ψ η Φ ( ) m ϕ = e π imϕ (8) (9)

4 9 D. N. Poeau, R. A. Gheghescu, W. Geie 4 ξ () e () N () H α α ψ = ξ = ψ ξ () whee m L ae the associated (o geealied) Laguee polyomials ad H ae the Hemite polyomials. The ew dimesio-less vaiables η ad ξ ae defied by whee the quatities ρ η=, ξ= α α () 6 / ω 6 A M / ω α = A α = Mω ω ω ω () which deped o the ucleo mass, M, posses a dimesio of a legth. Thei umeical values, i fm, ca be estimated by owig that / M 4. 5 MeV fm ad ω = 4A / MeV. The omaliatio costats ( m! N ) ( ) =, N = ( ) + m! π! ae obtaied fom the othoomaliatio coditios () m( ) m ψ ρψ ( ) d ( ) ( )d ρρ ρ=δ, ψ ψ =δ (4) π Φm ( ϕ) Φm( ϕ)dϕ=δmm (5) Oe should tae ito accout that the factoial!=!=. We shall substitute the wave fuctios ψm ( η ) ad ψ ( ) ξ i the equatio () of the uclea ietia. 4. NUCLEAR INERTIA By igoig the spi-obit couplig the Hamiltoia of the hamoic spheoidal oscillato cotais the ietic eegy ad the potetial eegy tem, V: ω ( ) ( ) +εη+ εξ / V( η, ξ; ε ) = ω η + ω ξ = (6) [7 ε (9 + ε)]

5 5 Micoscopic model fo the uclea ietia teso 9 Now we ae maig some chages i the equatio (), by deotig the defomatio β = ε, by assumig [8, ] that the deivative H/ ε= V/ ε, ad that the tem P ij ca be eglected. o whee ω 4/ {[ ( 6) 9] [ ( ) 9] } d V = ε ε+ + η+ ε ε+ ξ dε [7 ε (9 + ε)] dv = f () f() d ε η+ ε ξ ω ε ( 6) 9 ( ) 9 f = εε+ + ; f = ε ε+ [7 ε (9 + ε)] [7 ε (9 + ε)] 4/ 4/ Fo a sigle defomatio paamete the ietia teso becomes a scala (7) (8) (9) ν V/ ε µ µ V/ ε ν B u υ u υ () ε = ( ) ( E E ) ν µ + µ ν νµ ν + µ whee the summatio is pefomed fo all states ν, µ tae ito cosideatio i the paiig iteactio. 4.. PAIRING INTERACTION We coside a set of doubly degeeate eegy levels { ε i } expessed i. Calculatios fo eutos ae simila with those fo potos, hece uits of ω fo the momet we shall coside oly potos. I the absece of a paiig field, the fist Z/ levels ae occupied, fom a total umbe of t levels available. Oly few levels below () ad above ( ) the Femi eegy ae cotibutig to the paiig coelatios. Usually =. If g s is the desity of states at Femi eegy obtaied fom the shell coectio calculatio g s = dz/ d ε, expessed i umbe of levels pe ω spacig, the level desity is half of this quatity: g = gs/. We ca choose as computig paamete, the cut-off eegy (i uits of ω ), Ω. Let us tae the itege pat of the followig expessio Ω g / = = () s Whe fom calculatio we get > Z/ we shall tae = Z/ ad similaly if > Z/ we coside = Z/. t t

6 9 D. N. Poeau, R. A. Gheghescu, W. Geie 6 The gap paamete = G u υ ad the Femi eegy with paiig coellatios λ (both i uits of ω ) ae obtaied as solutios of a oliea system of two BCS equatios = f = i ( ε λ ) + ε λ () f = G () ( ε λ ) + = whee = Z/ + ; = Z/ +. i f i The paiig iteactio G is calculated fom a cotiuous distibutio of levels λ+ω g ()d ε ε = G (4) λ Ω ( ε λ ) + whee λ is the Femi eegy deduced fom the shell coectio calculatios ad is the gap paamete, obtaied fom a fit to expeimetal data, usually tae as = / A ω. Both p ad decease with iceasig asymmety ( N Z) / A. Fom the above itegal we get g ( λ )l Ω G (5) Real positive solutios of BCS equatios ae allowed if G > (6) ε λ i.e. fo a paiig foce (G-paamete) lage eough at a give distibutio of levels. The system ca be solved umeically by Newto-Raphso method efiig a iitial guess λ = ( ε + ε )/( + ) + G( )/ s d d s s d s d G s d ( εd εs) 4 = / (7) whee ε s, s ae the eegy ad degeeacy of the last occupied level ad ε d, d ae the same quatities fo the ext level. Solutios aoud magic umbes, whe, have bee deived by Kuma et al. [4].

7 7 Micoscopic model fo the uclea ietia teso 9 As a cosequece of the paiig coelatio, the levels situated below the Femi eegy ae oly patially filled, while those above the Femi eegy ae patially empty; thee is a give pobability fo each level to be occupied by a quasipaticle ε λ υ = (8) ( ) ε λ + o a hole u = υ. Oly the levels i the ea viciity of the Femi eegy (i a age of the ode of aoud it) ae iflueced by the paiig coelatios. Fo this easo, it is sufficiet fo the value of the cut-off paamete to exceed a give limit Ω, the value i itself havig o sigificace. 4.. VARIATION WITH DEFORMATION The followig elatioship allows to calculate the effective mass i uits of / ( ω ) 9 f f f f B ( ) ε = u u ( E E ) ν µ + µ ν νµ ν + µ ω ν η+ ξ µ µ η+ ξ ν Matix elemets ae calculated by pefomig some itegals m f εη+ f εξ m =δ N N N N () () m m mm d m m m f ηη + e ηl ( ) ( ) d e ( ) ( ) η L η ξ ξ H H ξ ξ + m m m η ξ + f dηη e L ( η) L ( ) d e ( ) ( ) η ξξ H H ξ ξ υ υ (9) whee so that ξ N N d e ( ) ( ) ξ H H ξ ξ =δ () d e m ( ) m Nm m m N η ( ) L ηη L η η = δ () m f () εη+ f () εξ m =δ N N N N. m m mm m m d m f + e η L ( ) ( ) d e ( ) ( ) L f ξ δ ηη η η +δ H ξξ H ξ ξ

8 94 D. N. Poeau, R. A. Gheghescu, W. Geie 8 Next we ca use the elatioships [6]: ( ) d m + e η m m + m! ηη L ( ) L ( ) ( ) η η = δ + m +! () ξ d e H ( ) ( ) H ξξ ξ ξ = ( ) = π! δ ( )( ) + +δ δ 4 () which wee obtaied by usig the ecuece elatios ad othoomaliatio coditios fo Hemite polyomials ad associated Laguee polyomials. L( ) ( ) x = L + x L + ( x) ; + ( + ) L ( x) = [(+ + ) x] L ( x) ( + ) L ( x) + (4) H ( x) = xh ( x) H ( x) (5) with paticula values L ( x ) =, L ( x ) = + x, H( x ) =, H( x) = x. Evetually fom the sum of equatio (9) oe is left with a impotat diagoal cotibutio ad two odiagoal tems ω f 9 ( uνυν) B ε = δ ( ) ( ) 4 δ mm f + m + + f + / δ E ν= ν ω i 9 f ( u ν υµ + uµ υν) B ε = δ ( )( ) 4 δ mm + + δ + ( Eν + Eµ ) ν µ (6) (7) ω 9 f ( u ν υµ + uµ υν) B ε = δ ( ) 4 δ mm δ ( Eν + Eµ ) ν µ (8) whee i ad f have bee defied above. I ode to pefom the summatios of the odiagoal tems fo a state with a cetai ν (specified quatum umbes m ) oe has to coside oly the states with µ ν ad = ; m = m fo which = + o = espectively. Fially oe aives at the uclea ietia i uits of / MeV by addig the thee tems ad dividig by ω. Thee ae seveal hydodyamical fomulae [5] of the mass paametes. Fo a spheical liquid dop with a adius R =. 49A / fm oe has B i () = = MAR 5 5 =. 49 A / fm =. 485A / i MeV fm MeV Whe the spheoidal defomatio is switched o it becomes

9 9 Micoscopic model fo the uclea ietia teso 95 i 8 9 ε ε = + ε 4/ B () B i () [7 ε (9 + ε )] ( ε ) (9) Good esults fo the fissio halflives of actiides have bee obtaied by usig a ietia lage by about a ode of magitude. 5. RESULTS The mai esult of the peset study is epeseted by the equatios (6 8), which could be used to test complex compute codes developed fo ealistic sigle-paticle levels, fo which it is ot possible to obtai aalytical elatioships. Fig.. Top: compaiso of the effective mass (i uits of / MeV) calculated by usig the caig model fo the poto plus euto level schemes, oly fo eutos, as well as fo the iotatioal spheoidal ad spheical shapes. Bottom: shell coectios fo eutos ad potos, oly fo eutos, paiig coectios, ad shell plus paiig coectios. O the left had side 44 Nd, o the ight had side 5 Cf. Nuclea ietia of 44 Nd ad 5 Cf calculated with the hydodyamical equatios fo a spheical liquid dop ad spheoidal shapes ae illustated i Fig.. Oe ca see how B i () iceases whe the mass umbe of the ucleus

10 96 D. N. Poeau, R. A. Gheghescu, W. Geie Bi ε is iceased. The iotatioal value ( ε ) mootoously iceases with the spheoidal defomatio paamete ε. Due to the fact that i this sigle cete model the ucleus oly became loge without developig a ec ad eve aivig to a scissio cofiguatio whe the defomatio is iceased, the educed mass is ot eached as it should be i a two cete model []. As expected, the caig ietia calculated by usig the aalytical elatioships (6 8) of the spheoidal hamoic oscillato shows vey poouced fluctuatios which ae coelated to the shell coectios (calculated with the macoscopic-micoscopic method [7]) plotted at the bottom of Fig.. I coclusio, by usig the wave fuctios of the spheoidal hamoic oscillato (the simplest vaiat of the Nilsso model) without spi-obit couplig oe ca obtai aalytical elatioships fo the caig ietia. Cosequetly the esult may be coveietly used to test complex compute codes developed fo ealistic two cete shell models. This sigle cete oscillato is ot able to descibe fissio pocesses eachig the scissio cofiguatio o goud states with eced i o diamod shapes. Whe the defomatio paamete is iceased the ucleus became loge ad loge without developig a ec ad eachig the touchig poit cofiguatio. I this way it is ot possible to obtai at the limit a uclea ietia equal to the educed mass of the fial fagmets i a pocess of fissio, alpha decay o cluste adioactivity. The caig ietia is lage tha the hydodyamical oe fo a spheoidal shape which is highe tha that of a spheical ucleus. Acowledgmets. This wo was patly suppoted by the Cete of Excellece IDRANAP ude cotact ICA-CT--7 with Euopea Commissio, Bussels, by Deutsche Foschugsgemeischaft, Bo, by Gesellschaft fü Schweioefoschug (GSI), Damstadt, ad by the Miisty of Educatio ad Reseach, Buchaest. REFERENCES. D. N. Poeau ad W. Geie, i Nuclea Decay Modes, (Istitute of Physics Publishig, Bistol, 996) p D. N. Poeau ad I. H. Plosi, i ef. [] p. 4.. R. A. Gheghescu, D. N. Poeau ad W. Geie, Phys. Rev., C 5 (995) 66; Z. Phys. A 54 (996), D. R. Iglis, Phys. Rev., 96 (954), 59; 97 (955), 7; (956), J. Badee, L. Coope ad J. Schieffe, Phys. Rev, 8 (957), S. T. Belyaev, K. Da. Vides. Sels. Mat. Fys. Medd., (959), No.. 7. D. R. Bes, K. Da. Vides. Sels. Mat. Fys. Medd., (96), No.. 8. M. Bac, J. Damgaad, A. Jese, H. C. Pauli, V. M. Stutisy ad G. Y. Wog, Rev. Mod. Phys., 44 (97),. 9. R. A. Gheghescu, Phys. Rev., 67 (), 49.. R. A. Gheghescu, D. N. Poeau ad W. Geie, i Fissio ad Popeties of Neuto-Rich Nuclei, Poc. of the Iteat. Cof., Saibel Islad, Floida,. J. H. Hamilto, H. K. Cate ad A. V. Ramayya Eds. (Wold Scietific, Sigapoe, ) 77.

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