Higher point symmetries of nuclei and their implications

Transcription

1 Higher point symmetries of nuclei and their implications Institute of Physics, Dep. Math. Phys.,UMCS, Lublin, Poland Saariselka 2009

2 COLLABORATION Jerzy Dudek, IPHC/IReS, Strasbourg, France Artur Dobrowolski, Institute of Physics Dep. Math. Phys.,UMCS, Lublin, Poland, Katarzyna Mazurek Niewodnicza«ski Institute of Nuclear Physics, Kraków, Poland Marek Mi±kiewicz, II LO im. Hetmana Jana Zamoyskiego, ul. Ogrodowa 16, PL Lublin, Poland

3 What is presented? 1 Symmetries

4 Spacetime versus intrinsic symmetries G lab (G int = G G) Assume that a nucleus is considered in the center of mass which is xed in the space. The remaining non-relativistic space symmetry is G lab = O(3). Every nuclear Hamiltonian has to be invariant in respect to the orthogonal group O(3). The intrinsic symmetries G int can be considered as direct product of the intrinsic space type group G and the intrinsic symmetries related to other properties of the system G (isospin, seniority, charges, number of particles, etc.).

5 Intrinsic groups G Jin-Quan Chen, Jialun Ping & Fan Wang: Group Representation Theory for Physicists, World Scientic, Def. For each element g of the group G, one can dene a corresponding operator g in the group linear space L G as: g S = Sg, for all S L G. The group formed by the collection of the operators g is called the intrinsic group of G. IMPORTANT PROPERTY: [G, G] = 0 The groups G and G are antyisomorphic.

6 Intrinsic variables Laboratory frame {q λµ }. Intrinsic frame {q λµ } N variables to N variables transformation? q (q, Ω), There are redundant variables. Three additional conditions required: F i (q, Ω) = 0, i = 1, 2, 3. The conditions dene physical meaning of Euler angles.

7 Single and Double Groups Example of the simplest tetrahedral group T: T e G ē G 3C C2 4C 3 4 C3 4C C 3 Time inv. Γ a Γ ω ω ω 2 ω 2 b Γ ω 2 ω 2 ω ω b Γ a Γ c Γ ω ω ω 2 ω 2 b Γ ω 2 ω 2 ω ω b where ω = exp(2π/3).

8 Selection rules for emg transitions i.r. of T d group labelled A 1, A 2, E, T 1 and T 2 are 1,1,2,3 and 3 dimensional, respectively. The electric dipole operator decompose into T 2 tensor of the T d group. The electric quadrupole operator decompose into the sum of E and T 2 tensors of the T d group. The electric octupole operator decompose into the sum of A 1, T 1 and T 2 tensors of the T d group.

9 Selection rules for emg transitions Electric transitions from triplets T d group. dipole T 1 A 2 + E + T 1 + T 2 dipole T 2 A 1 + E + T 1 + T 2 quadrupole T 1 A 2 + E + 2T 1 + 2T 2 quadrupole T 2 A 1 + E + 2T 1 + 2T 2 octupole T 1 ALL Representations octupole T 2 ALL Representations There are no usable independent of model selection rules.

10 1/3 The collective variables α λµ dened as: R(θ, φ) = R αλµ Y λµ(θ, φ) are irreducible tensors in respect to the group SO(3). The tensors Y λµ and α λµ can be transformed to the intrinsic frame. Then the equation of nuclear surface in the intrinsic frame is: R(θ, φ ) = R αλµ Y λµ(θ, φ ) The collective variables α λµ are spherical tensors in respect to G. λµ λµ

11 Harmonic oscillator in α The kinetic energy: T = 1 B λ α λµ α λµ 2 For xed λ: T = T vib + T rot + T coupl λµ T coupl complicated. T vib = B λ α λµ α λµ 2 T rot = 1 ω n ω n J nn 2 nn

12 The quadrupole Bohr Hamiltonian Bohr Hamiltonian in quadrupole coordinates (intrinsic frame). Notation: a µ = α 2µ Ĥ 2B = 1 2 [ 2 2 2B a0 2 J 2 n J n (a 0, a 2 ) (3a2 0 + ] 6a2 2 )2 2 8a2 2(3a2 + V (a 0, a 2 ) 0 2a2 2 )2 n a 2 2 The variables a 2 and a 0 are D 2h invariants. For a 2 0 the Hamiltonian Ĥ 2B is also D 2h invariant. For a 2 = 0 the Hamiltonian Ĥ 2B is axially symmetric.

13 Symmetry and the quadrupole Hamiltonian The general form of the quadrupole Hamiltonian: Ĥ 2 = Ĥ 2 (ˆπ 2µ, ˆα 2µ Ĵ k ) = = C n 2,...,n 2,m 2,...,m 2,k 1,k 2,k 3 n 2,...,n 2 m 2,...,m 2 k 1,k 2,k 3 ˆπ n 2 2, 2... ˆπ n 2 22ˆα m 2 2, 2... ˆα m 2ˆ 22 J k 1 k ˆ 2 k ˆ 3 1 J2 J3 Assume D 2h symmetry. The odd terms in ˆα 2,±1 = 0. The cross terms Ĵl Ĵk = 0 for l k

14 Symmetry and the quadrupole Hamiltonian The Hamiltonian can be constructed only from invariants. More restrictive requirement: the Hamiltonian can be constructed from invariant collective variables and invariants constructed from the angular momentum operators In the second case the Hamiltonian Ĥ 2 depends only on a 0, a 2, conjugated momenta and angular momentum operators which give the lowest term in the same form as it is in the Bohr Hamiltonian. The general Hamiltonian can contain terms ˆα 2,±1, which are not present in the Bohr Hamiltonian.

15 Tetrahedral Hamiltonian The only invariant under T d octupole variable is: ξ = i(α 32 α 3, 2 ) = Im α 32 C. Wexler, G. Dussel PR C 60(1999) Motion only in ξ, moment of inertia components: J lm = 0 for l m, J 1 = J 2 = J 3 Remark: In Wexler article J k < 0 (error).

16 Tetrahedral Hamiltonian A possible form of the tetrahedral octupole Hamiltonian: Ĥ T = Ĥ T (ˆπ β, ˆβ, ˆπ ξ, ˆξ, Ĵk) = = n,m,n,m k 1,k 2,k 3 C n,m,n,m,k 1,k 2,k 3 ˆπ n β ˆβ mˆπ n ξ ˆξ m (Ĵ1) k 1 (Ĵ2) k 2 (Ĵ3) k 3 = = Ĥ ξ (ˆπ β, ˆβ, ˆπ ξ, ˆξ) + Ĥ grot ( ˆβ, ˆξ, Ĵ 2, T 3 2 T 3 2,... ) + +Ĥ βξj (ˆπ β, ˆβ, ˆπ ξ, ˆξ, Ĵk), where T 3 µ = ((J J) 2 J) 3 µ.

17 Harm. Osc. type of tetrahedral Hamiltonian Classic (e.g., derived from h.o.) type of the tetrahedral Hamiltonian: Ĥ T = Ĥ ξ (ˆπ β, ˆβ, ˆπ ξ, ˆξ) + k (Ĵ)2 J k ( ˆβ, ξ) The hamiltonian has T d symmetry. The eigenfuctions belong to the scalar representation. There is no characteristic degeneracies (1,2,3) in its energy spectrum. The eigenfunctions ψ njmk (β, ξ, Ω) = u n (β, ξ)r J MK (Ω)

18 Simple tetrahedral Hamiltonian One needs to add to the Hamiltonian higher order terms in angular momentum operators. Ĥ T = Ĥ T + Ĥ grot( ˆβ, ˆξ, (Ĵ)2, T 3 2 T 3 2,... ), where T 3 µ = ((J J) 2 J) 3 µ. The hamiltonian has the full T d symmetry. There are characteristic degeneracies (1,2,3) in its energy spectrum. The eigenfunctions φ νjm (β, ξ, Ω) = nk cnk νjm u n(β, ξ)rmk J (Ω)

19 Symmetries, general collective Hamiltonian Ĥ = Ĥ(q 1, q 2,..., q s ) where q k (π, α, J l ) are invariants of the symetry group G

20 Collective electric transition operators Eisenberg, Greiner, Nuclear theory, The collective transition operator in the laboratory frame: Q coll λµ = ν D λ µν(ω) Qcoll λν Q λν coll = 3ZRλ { 0 ᾱ λν + λ + 2 4π 2 4π } (λ 1 0λ 2 0 λ0)(ᾱ λ1 ᾱ λ2 ) λ ν (2λ1 + 1)(2λ 2 + 1) λ 1 λ 2 where the transformation to the intrinsic frame is: 2λ + 1 α λµ = µ D (λ) µ µ (Ω)α(λ) µ

21 Pure octupole model collective Eλ transitions IF the Euler angles are chosen to x octupoles in the principal axes frame. For pure octupole T d collective model (ᾱ 3µ = 0 for µ ±2) the operators: Q coll 1µ Q coll 2µ = 0, because of ( ) = 0. = 0, because of ( ) = 0. The only non-zero moment is the octupole one.

22 Quadrupole + octupole model collective Eλ transitions The Euler angles are now chosen to x quadrupoles in the principal axes frame. There are no extra conditions for octupole variables.

23 Collective Eλ transitions, intrinsic frame The intrinsic part of the dipole transition operator: Q coll 10 = 3 { 3ZR π 7 ᾱ 22 ᾱ π 35ᾱ21ᾱ ᾱ 20 ᾱ } ᾱ 22 ᾱ ᾱ2 1ᾱ 7

24 Collective Eλ transitions, intrinsic frame Examples of intrinsic parts of transition operators: The quadrupole operator: Q coll 20 = 3ZR 0 2 { ᾱ 20 4π + 1 ( 10 5π 7 ᾱ20ᾱ ᾱ2 2ᾱ ᾱ30ᾱ 30 2ᾱ 3 1 ᾱ ᾱ3 3ᾱ 33 )} NO tetrahedral collective variable!!!!!!

25 The collective functions φ Jκ (ᾱ)= intrinsic function R JMν (Ω)= rotational function Ψ κjmν (ᾱ, Ω) = φ Jκ (ᾱ)r JMν (Ω) The reduced (in respect to M) matrix elements: Ψ κ J ν Q λ Ψ κjν = µ φ J κ Qλµ φ Jκ R J ν Dλ µ R Jν

26 Intrinsic collective functions q stands for quadrupole band, t for tetrahedral one. φ q ({η}, { ᾱ 2 }; ᾱ 2, ᾱ 3 ) = u 0 (η q 20 ; ᾱ 20 ᾱ 20 )u 0 (η q 22 ; ᾱ 22 ᾱ 22 ) u 0 (η q 30 ; ᾱ 30)u 0 (η q 31 ; ᾱ 31)u 0 (η q 32 ; ᾱ 32)u 0 (η q 33 ; ᾱ 33) u 0 (η q 31 ; ᾱ 31)u 0 (η q 32 ; ᾱ 32)u 0 (η q 33 ; ᾱ 33) φ t ({η}, { ᾱ 3 }; ᾱ 2, ᾱ 3 ) = u 0 (η t 20; ᾱ 20 )u 0 (η t 22; ᾱ 22 )u 0 (η t 30; ᾱ 30 ) u 0 (η t 31; ᾱ 31)u 1 (η t 32; ᾱ 32)u 0 (η t 33; ᾱ 33) u 0 (η t 31; ᾱ 31)u 0 (η t 32; ᾱ 32 ᾱ 32)u 0 (η t 33; ᾱ 33) Here η = mω/.

27 Single η approximation Assuming all η equal: φ q Q22 φ q = 0.24 ZR 2 0 [ ᾱ ᾱ 20 ᾱ 22 ] φ q Q20 φ q = 0.24 ZR 2 0 [ ᾱ η 2 ] φ t Q20 φ t = ZR η 2 ) ᾱ 22 φ q Q10 φ t = ZR 0 ( η exp η2 4 ( ᾱ ᾱ ᾱ 322 )

28 Intrinsic matrix elements γ = 0.6 Log tq20t^2 qq10t^ Η Figure: A=156, Z=64, β = 0.23, γ = 0.6, ξ = 0.12, Log(B(E 2, tt)/b(e 1, qt))

29 Intrinsic matrix elements γ = 0.6 Log tq20t^2 qq32t^ Η Figure: A=156, Z=64, β = 0.23, γ = 0.6, ξ = 0.12, Log(B(E 2, tt)/b(e 3, qt))

30 Intrinsic matrix elements γ = 0.6 Log pq2p^2 qq10p^ Η Figure: A=156, Z=64, β = 0.23, γ = 0.6, ξ = 0.12, Log(B(E 2, pp)/b(e 1, qp))

31 Conclusion Symmetries The transition probabilities: T (E; L = 1) = Eγ 3 B(L = 1) T (E; L = 2) = Eγ 5 B(L = 2) T (E; L = 3) = Eγ 7 B(L = 3) Weak E2 transitions within the tetrahedral band. Possible E1 transitions between tetrahedral and quadrupole band. Very weak E3 transitions between tetrahedral and quadrupole band.

32 SUMMARY Symmetries??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????

33 Intrinsic matrix elements γ = 0.6 (1/5) Log redqq2q^ Η Figure: A=156, Z=64, β = 0.23, γ = 0.6, ξ = 0.12, Log( Ψ qj=0 Q 2 Ψ qj=0 2 )

34 Intrinsic matrix elements γ = 0.6 (2/5) Log tq20t^ Η Figure: A=156, Z=64, β = 0.23, γ = 0.6, ξ = 0.12, Log( φ t Q20 φ t 2 )

35 Intrinsic matrix elements γ = 0.6 (3/5) Log qq10t^ Η Figure: A=156, Z=64, β = 0.23, γ = 0.6, ξ = 0.12, Log( φ q Q10 φ t 2 )

36 Intrinsic matrix elements γ = 0.6 (4/5) Log tq20t^2 qq10t^ Η Figure: A=156, Z=64, β = 0.23, γ = 0.6, ξ = 0.12, Log(B(E 2, tt)/b(e 1, qt))

37 Intrinsic matrix elements γ = 0.6 (5/5) Log qq32t^ Η Figure: A=156, Z=64, β = 0.23, γ = 0.6, ξ = 0.12, Log( φ q Q32 φ t 2 )

38 Summary Symmetries??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????