Spherical shell model

Nilsso Model Spherical Shell Model Deformed Shell Model Aisotropic Harmoic Oscillator Nilsso Model o Nilsso Hamiltoia o Choice of Basis o Matrix Elemets ad Diagoaliatio o Examples. Nilsso diagrams

Spherical shell model Nuclear properties described i terms of ucleos cosidered as idepedet particles movig i a average potetial create by all ucleos. Experimetal evidece for shell effects: Existece of magic umbers:,8,,8,5,8,6 Large sigle particle separatio eergies Nuclei are strogly boud at shell closures Derivatio of the average field from microscopic two-body forces (selfcosistet Hartree-Fock method). Assume the existece of such a potetial ad costruct it pheomeologically Characteristics of the potetial: ( ) V V r = r r= r r< R > ( ), > V r r R

Spherical potetials Ifiite square well Harmoic oscillator Woods-Saxo potetial ( ) for V r = V r R = for r > R V r M r ( ) = ω ( ) V r V = ( ) exp r R / a Eige-fuctios ψ ( ) ( Ω) j kr Y m ψ R ( r) Y ( Ω) m r / ( ) = ( ) R r r e L r umerically E(, ) = ξ MR ξ : root of j = ( ξ) Eige-eergies (, ) = ω ( /) = ω ( N /) E itermediate

Spherical potetials Woods-Saxo potetial ( ) V r V = ( ) exp r R / a Harmoic oscillator V r M r ( ) = ω Ifiite square well ( ) for V r = V r R = for r > R

Spherical potetials H.O. W.S. Square

Spherical potetials & spi-orbit V ( r) = Mω r C s D ( ) s = j s ( ) ω ( ) ( ) E D Cε, = / SO ε ε SO SO = for j = = for j = Δ = ε SO

Deformed shell model Spherical potetial well valid for closed shells Far from closed shells: deformed sigle particle potetial Experimetal evidece: Existece of rotatioal bads: I(I) spectra Large quadrupole momets ad quadrupole trasitio probabilities Sigle particle structure AisotropicHarmoicOscillator Geeralied Woods-Saxo V ( r θϕ) LS r,, = V exp a V = λ V r p s ( (, θ, ϕ) ) R( θϕ, ) ( θϕ, )

Aisotropic Harmoic Oscillator Ellipsoidal distributio: Aisotropic Harmoic Oscillator as average field m H = Δ x y m ( ω ) x ωy ω Frequecies are proportioal to the iverse of the ellipsoid axes For axially symmetric shapes, we itroduce the parameter δ ω ω δ = ω ω = ωx = ωy = ω δ 4 ω = ω δ i ω = ω R a i From volume coservatio ωxωω y = ( ) ω /6 4 6,.95 ω( δ) = ω δ δ δ β 7

Aisotropic Harmoic Oscillator Itroducig dimesioless coordiates through the oscillator legth we get b( δ ) = r' r/ b mω = ( δ ) ( δ) ( δ) H mω m x y 4 ω( δ) ω( δ ) 4 = Δ x y δ ( x y ) δ ( δ) = Δ ω ( δ) δ ( ) ω ( δ) δ m mω mω ω 4 π r δ ω Δ 5 = = H H δ Axial symmetry: cylidrical basis ( δ) r Y ( Ω) { N,, ρ, m, ms} ε ω ω ω i= x, y, N = = m x y (, ρ, m ) = ii = ( ρ m ) N = ω N δ ρ

Aisotropic Harmoic Oscillator Eige-states characteried by φ ψ ψ mm Ωπ Ω= ± = [ N m ] m π ( ) ( ) m R, σ = ( ρ ) ( ) χm ( σ) m ( ρ ) L ( ρ ) ρ ( ) H ( ) π ρ s ρ s ψ m ψ ρ e im ϕ N

Aisotropic Harmoic Oscillator Eergy level structure: N= ε ω ω ω i= x, y, (, ρ, m ) = i = ( ρ m ) N = ω N δ 9 ( m ρ ) = ( ) N = ε ω ω δ N = ρ m m l ρ Ω deg 5/, 7/ 4 /, / eergy = = = = /, 5/ / /, / deformatio /

The Nilsso model: Hamiltoia Axially symmetric harmoic oscillator potetial spi-orbit term l term H H C s D = N = ω δ Δ β ( ) r r Y κ ω s μ N C = κω D = κμω N ( ) = N N

The Nilsso model: Hamiltoia H F F E = H κω F δ = κ ηu s μ = N = N ω ( δ) κω f /6 4 6 4 π δ δ r Y s μ 7 5 N N,Z<5 5<Z<8 8<N<6 8<Z 6<N κ.8.67.67.577.65 μ.6.4.65.5

s The Nilsso model: Basis odiagoal i basis ad { N,, ρ, m, ms} For large deformatios s, ca be eglected: Asymptotic quatum umbers { N,, ρ, m, ms} : [ Nm ] Ωπ For small deformatios δ-terms ca be eglected: Spherical basis Nilsso used basis { N m m } Diagoal terms { N,, j, Ω},,, s H N,, m, ms = N ω N,, m, m N,, m, m = N,, m, m s ( ) s s

The Nilsso model:matrix elemets Matrix elemets ' m' m' s s mm s = ' m = m', m' ± m = m' ±, m' s s s m m = m' m' s s, m ±, s, m, ± = m ± m, m, ± s, m, ± =± m ( )( ) ' 5 ' m' Y m = i m ' m' ' 4π ' m = m' m = m' = ', ' ± N = N' ± s s

The Nilsso model: Matrix elemets Radial matrix elemets ( )! ( /) r N = e L r b b Γ b r b / ( / ) ( ' )!( )! ' ( ) μ ν ( ' ' /) ( /) Γ ( p σ )!( ' ) ( ) ( ) ( σ ν ) N r N = b Γ Γ ' '!! σ σ σ! σ! σ μ '!! p = ( ' ) μ = p ' / ν = p / N = ( ) / N, N ± admixtures ( /) ( /) = = N r N N ( ) N r N = / ( )( ) N r N = / ( )( ) N r N = / / ( )( ) N r N =

The Nilsso model α Nilsso states: i C α ; α { } { Nm m } = Ω π i α s N Ω Nmm s Ωπ N N N N = Ω = / = Ω = / / / Ω= / = Ω = 5/ / / 5/ Ω= / / / Ω= / = Ω = 7/ 5/ 5/ 5/ 7/ Ω= 5 / Ω= / 5/ 5/ / / / Ω= / / / / /

The Nilsso model: N= Diagoaliatio i blocks Ω,π (with or without ΔN= admixtures) H = H κω F = H κω ηu s μ { } δ 4 6 η = δ δ κ 7 /6 U = 4 π r 5 Y N = Ω= / =, = Nm m s N = μ= s = Y = ηu s μ =

The Nilsso model: N= N = Ω= / =, = Nm m s N = Ω= / =, = Nm m s, - N = μ= s = 4 π 4 π η ry = η r Y = η 5 5 r = 5 5 Y = = 4π 4π 5 ηu s μ = η N = μ= s = 4 π 4 π η ry = η r Y = η 5 5 r = 5 5 Y = = 4π 4π 5 ηu s μ = η ηu s μ - = η ηu s μ - = - η η± η η λ = η 9 λ η

The Nilsso model: N= { }{ } N = Ω= 5/ =, = =, = Nmms ηu s μ = η N = Ω= / Nmms, - ηu s μ = η ηu s μ - = η ηu s μ = - η η N = Ω= / Nmms,, - ηu s μ = η ηu s μ = ηu s μ = η ηu s μ = η ηu s μ - = 6 ηu s μ - = - η η 6 η

The Nilsso model: N= N = Ω= 7 / ( ημ) N = Ω= 5/ μ 6 η μ N = Ω= / 4 ημ η 5 5 ημ 5 μ N = Ω=/ 4 6 ημ η 5 5 6 ημ 5 4 η μ η 5 5 η μ 5

The Nilsso model: ΔN= Ω=/ with ΔN= admixtures 44 4 4 44 4 η 6 η η 6 η η η 5 7 5 7η 6 η η η 7 7 44 η μ η 5 4 4 η 6μ 5 5η 6 4 44 7 4 η μ 7 η 4 6

A Z Nucleus N K π Al Na F 7 4 5/ / 9 9 / Si 7 4 5/ 5 Mg 5/ O 9 8 / Ne 9 9 / 9 4 5 / 7 Be Li 4 /

Spherical levels split ito (j)/ levels Levels (Ωπ) are twofold degeerate Asymptotic q-umbers ot coserved for small deformatios but useful to classify levels For positive deformatios (PROLATE SHAPES), levels with lower Ω are shifted dowwards For egative deformatios (OBLATE SHAPES), levels with lower Ω are shifted upwards

N m = ρ Nm = Ω = = N =/ m / N = Ω =/ = m = / = m = / N = Ω =/ = m = / N = Ω =/ = m = / N = Ω =/ = m = = m = / = m = / / = m = / N = Ω =5/ = m = 5/ N = Ω =/ = m = / = m = / = m = / = m = / N = Ω =/ = m = / N = Ω=5/ = m = / = m = / = m = 5/ = m = 5/ N = Ω =7/ = m = 7 / Ω π

Laboratory frame ( ) Wave fuctios i the uified model Ψ φ r Φ( θ ) I Ψ = 6π { } I* ( ) ( ) ( ) I J I* IKM D ( ) ( ) ( ) MK θk φk r DM K θk φk r k Itrisic wave fuctios (Nilsso) Rotatio matrices Z lab. M I R J Z itr. Ω K

-j abγ a b c abαβ c γ = ( ) c α β γ