The collective model from a Cartan-Weyl perspective

Transcription

1 The collective model from a Cartan-Weyl perspective Stijn De Baerdemacker Veerle Hellemans Kris Heyde Subatomic and radiation physics Universiteit Gent, Belgium INT workshop on new approaches in nuclear many-body theory October 9, 7, Seattle WA, USA stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT / 6

2 The geometrical collective model The basics of the geometrical collective model The physics motivation The collective variables in the Cartan-Weyl scheme An algebraic description within the Cartan-Weyl scheme 3 Test application in quantum shape phase transitions conclusions & outlook stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT / 6

3 The geometrical collective model The basics of the geometrical collective model The physics motivation The collective variables in the Cartan-Weyl scheme An algebraic description within the Cartan-Weyl scheme 3 Test application in quantum shape phase transitions conclusions & outlook stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT 3 / 6

4 What s so geometrical about the geometrical model? Macroscopically, the atomic nucleus can be compared to a charged liquid drop. Deviations from the sphere are developed in multipole orders. Up to nd order Radius R(θ, φ) = R [ + α Y (θ, φ)] α µ :: collective quadrupole a coordinates a L = tensor stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT / 6

5 A gallery of shapes Radius R(θ, φ) = R [ + α Y (θ, φ)] Rotation to the intrinsic system α = β cos γ α = α = β/ sin γ α = α = β is a measure for the deformation, γ for the triaxiality stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT 5 / 6

6 The nuclear chart around Z=8 shell closure Collective excitation modes are very important in the low-energy spectra Renewed interest due to the quantum shape phase transitions stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT 6 / 6

7 All kinds of shapes neighbouring isotope chains Os :: triaxial nuclei Pt :: the γ-softie Pb :: 3 coexisting families Hartree-Fock mean field calculation A minimum can be found at γ A. Ansari, Phys. Rev. C 38 (988) 953. Calculations in the framework of analytically solvable potentials L. Fortunato et. al., Phys. Rev. C7 (6) 3. stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT 7 / 6

8 All kinds of shapes neighbouring isotope chains Os :: triaxial nuclei Pt :: the γ-softie Pb :: 3 coexisting families Potential Energy Surface calculation Very flat γ dependence R. Bengtsson et al., Phys. Lett. B 83 (987) (). stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT 7 / 6

9 All kinds of shapes neighbouring isotope chains Os :: triaxial nuclei Pt :: the γ-softie Pb :: 3 coexisting families Interacting Boson Model calculation Extension to coexisting configurations points towards spherical-oblate-prolate structure ( ) 368 ( ) 333 ( ) 65 (8 ) 6 (6 ) 738 ( ) 337 ( ) 95 V. Hellemans, private communication (7) stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT 7 / 6

10 input/output of the geometrical model The Bohr Hamiltonian Ĥ = ˆT + V (α) kinetical egergy describes the stiffness of the surface ˆT = B π π + B 3 [πα] π +... V α describes small deformations γ? Use a Taylor expansion Collective potential.8.6. β Os V (α) = c (α α) + c 3 ([αα] α) c (α α) + c 5 ([αα] α)(α α) +... stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT 8 / 6

11 The geometrical collective model The basics of the geometrical collective model The physics motivation The collective variables in the Cartan-Weyl scheme An algebraic description within the Cartan-Weyl scheme 3 Test application in quantum shape phase transitions conclusions & outlook stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT 9 / 6

12 The rules of the game Commutation relations fix the structure [π µ, α ν ] = i δ µν, [α µ, α ν ] =, [π µ, π ν ] = The algebraic structure of the geometrical model is contained in the following recoupling formula (α α)(π π ) = (α π )(α π ) + 3i (α π ) ([απ ] () [απ ] () + [απ ] (3) [απ ] (3) ) It comprises the generators of the direct product group Ẑ = α α Ẑ = π π Ẑ 3 = α π SU(, ) O(5) { i L M = [απ ] () M i O M = [απ ] (3) M stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT / 6

13 Why SU(, ) O(5)? The Hamiltonian Ĥ = B π π + B 3 π [απ] + c (α α) + c 3 ([αα] α) + c (α α) + c 5 ([αα] α)(α α) + c 6 (α α) 3 + d 6 ([αα] α) +... SU(, ) basic block α α = β The radial dependence Basis is known O(5) basic block [αα] α = 7 β3 cos 3γ The angular dependence Cartan-Weyl basis stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT / 6

14 The Cartan-Weyl basis of O(5) Cartan s theorem Every semi simple algebra of dimension n and rank r can be rotated to a natural basis for which [H i, H j ] =, [H i, E α ] = α i E α, {i, j} r [E α, E β ] = N α+β E α+β [E α, E α ] = α i H i {α, β} n r Rotation {L m, O m } {X i, Y j, T µν } Group reduction is clear O(5) O() = SU() SU() }{{}}{{}}{{}}{{} v X (X,M X ) (X,M Y ) Y_ T/, / O 3 X + T/,/ 3 L A natural Cartan basis emerges vx (M X, M Y ) T /, / Y + T /,/ X_ stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT / 6

15 Action of the O(5) generators on the natural basis (i) Y_ T/, / X_ 3 O X + T/,/ 3 T /, / Y + T /,/ L {X ±, X } and {Y ±, Y } span standard SU() algebras According Racah: T µ,ν acts as a bispinor of character / in the SU() SU() space [X, T µν ] = µt [X ±, T µν ] = µν The action of T µ,ν on a basis state vx (M X, M Y ) is thus T ( µ)( ± µ + )T µν vxm X M Y = a + v, X +, M X + µ, M Y + ν + a v, X, M X + µ, M Y + ν Applying the Wigner Eckart theorem twice ( ) ( a ± = ( ) k X ± X X ± X M X µ µ M X M Y ν ν M Y µ±ν ) vx ± T vx stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT 3 / 6

16 Action of the O(5) generators on the natural basis (ii) For every v and X :: two unknown matrix elements two conditions needed C [O(5)] = (X + Y [TT ] () ) [T µν, T µ ν ] = cx m X + c Y m Y m = {µν, µ ν } norm selection rules (X =... v ) intermediate state method renders double reduced matrix elements Example: Action of T /,/ T vxm X M Y = (X +MX +)(X +M Y +)(v X )(v+x +3) (X +)(X +) (X MX )(X M Y )(v X +)(v+x +) (X )(X +) vx + M X + M Y + vx M X + M Y + stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT / 6

17 Tensor character of the α variables basis block of the potential β 3 cos(3γ) [αα] () α α α Need for matrix elements α L vx (M X, M Y ) α µ v X (M X, M Y ) α {α } and {α, α, α, α } have bispinor and Wigner Eckart α α, {α, α, α, α } α vxm X M Y αµν v λλ X M X M Y ( ) ( = ( ) k X λ X X λ X M X µ M X M Y ν M Y α character respectively µν ) vx α λ v X stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT 5 / 6

18 Closed expressions for the α matrix elements What is used [T µν, α ( )µ ν µ ν ] = δ µµ δ νν α [T µν, α ] = [α λλ µ,ν, α λ λ µ ν ] = α µ,ν α α = β = Z Seniority selection rules :: α is a v = tensor Bispinor { } can be expressed in terms of biscalar {} matrix elements Closed expressions for the matrix elements result What is obtained: matrix elements of α vxm X M Y α v +, XM X M Y = β vxm X M Y α v, XM X M Y = β (v X +)(v+x +3) (v+3)(v+5) (v X )(v+x +) (v+)(v+3) stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT 6 / 6

19 Projection to the physical basis Experimental spectra have good angular momentum quantum number L The Hamiltonian is a scalar with respect to the angular momentum algebra O(3) v= O [L L, C G ] [L, C G ] = 7 8 L Only the angular momentum projection is a good quantum number in the Cartan basis A rotation brings the natural- to the physical basis stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT 7 / 6

20 All ingredients are ready α Matrix elements in O(5) basis are derived Inclusion of SU(, ) basis is straightforward in a similar fashion Diagonalising = choosing a basis Harmonic oscillator = choosing ω H = B π π + ξv vib + ( ξ)v γ-ind Margetan & Williams Phys. Rev. C5 (98) n first mat shape HO ξ Computer code is now under continuous development to diagonalise general collective potentials. Present status: upto β stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT 8 / 6

21 The geometrical collective model The basics of the geometrical collective model The physics motivation The collective variables in the Cartan-Weyl scheme An algebraic description within the Cartan-Weyl scheme 3 Test application in quantum shape phase transitions conclusions & outlook stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT 9 / 6

22 Test application in quantum shape phase transitions Quantum shape phase transitions cover a large part of the model Hilbert space Ideal testground for the method Upto β, 3 meaningful limits result 3 limits vibrational limit γ-independent rotor axial deformed rotor V γ β V 6 5 γ β V 6 5 γ β stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT / 6

23 3 limits 3 limits vibrational limit γ-independent rotor axial deformed rotor 8 Trivial limit Large degeneracies B(E ) addition rule V 6 5 γ β.. E [MeV] stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT / 6

24 3 limits 3 limits V vibrational limit γ-independent rotor axial deformed rotor 5 γ β E [MeV] Remaining seniority symmetry β-excitation band stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT / 6

25 3 limits 3 limits Highly pronounced bands Rotation-Vibration model V vibrational limit γ-independent rotor axial deformed rotor 5 γ β E [MeV] E [MeV] stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT / 6

26 Along the transition path: energy spectrum g band nd K= K= K= st ξ ξ ξ vibrator γ-independent rotor axial rotor vibrator stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT / 6

27 Along the transition path: quadrupole moments Quadrupole moments. ground. ground.. Q = ˆQ µ = 3ZR π α µ.. α µ is seniority v = tensor Q for vib. γ-ind. rotor transition Other observables (B(E), ρ(e)) ξ ξ γ-ind. rotor axial rotor vibrator stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT 3 / 6

28 The geometrical collective model The basics of the geometrical collective model The physics motivation The collective variables in the Cartan-Weyl scheme An algebraic description within the Cartan-Weyl scheme 3 Test application in quantum shape phase transitions conclusions & outlook stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT / 6

29 Conclusions & outlook Collectivity accounts for a lot of the physics around Z = 8 shell closure All necessary matrix elements of a more general type of potential can be calculated in a Cartan-Weyl scheme First test applications in the framework of quantum shape phase transitions renders reliable results Further terms need to be included to study more general collective structures (e.g. triaxiality, shape coexistence), needed for the collectivity around the Z = 8 closed shell Possible extension to higher rank algebras (O(7) octupole degrees of freedom and beyond)... stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT 5 / 6

30 Thank you for you attention! stijn de baerdemacker (ghent university) The collective model from a Cartan-Weyl perspective INT 6 / 6