Construction of spherical harmonics and Clebsch–Gordan coefficients

Caprio, M. A.; Rowe, D.J.; Welsh, Trevor A.

doi:10.1016/j.cpc.2008.12.039

Cited by 26 publications

(36 citation statements)

References 33 publications

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“…Several advantages result from this choice of dynamical subgroup chain: (i) with the now available SO(5) Clebsch-Gordan coefficients [26,27], and explicit expressions for SO(5) reduced matrix elements, matrix elements of BM operators can be calculated analytically; (ii) by appropriate choices of SU(1,1) modified oscillator representations, the β basis wave functions range from those of the U(5) ⊃ SO(5) harmonic vibrational model to those of the rigid-beta wave function of the SO(5)-invariant Wilets-Jean model; and (iii) with these SU(1,1) representations, collective model calculations converge an order of magnitude more rapidly for deformed nuclei than in U(5) ⊃ SO(5) bases. Thus, the ACM combines the advantages of the BM and the IBM and makes collective model calculations a simple routine procedure [4,[28][29][30].…”

Section: Acm Model Calculationsmentioning

confidence: 99%

Algebraic Collective Model and Nuclear Structure Applications

Thiamová¹,

Abolghasem²

2014

Acta Phys. Pol. B

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We present results that can be obtained with an algebraic version of Bohr's collective model with the aim to identify the limitations of the model as much as it successes. A special focus is placed on the analysis of triaxial nuclei. The first, simple application of the algebraic collective model to 188,192 Os is performed with the aim to identify, among the observed J = 0 + states, the best candidates for a β vibration. A conclusion of this analysis is that beta bandheads in these nuclei are to be expected at a much higher energy than the energy of the observed 0 2 states. This finding is in agreement with the vibration-rotation model predictions and the predictions of the symplectic model. A decisive role played by β and γ fluctuations for a correct description of the amplitudes of the quasi-γ-band staggerings is shown. In particular, it is revealed that β fluctuations act differently in the axially symmetric and the triaxial regime.

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Section: Acm Model Calculationsmentioning

confidence: 99%

Algebraic Collective Model and Nuclear Structure Applications

Thiamová¹,

Abolghasem²

2014

Acta Phys. Pol. B

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“…In the Maple [17] code presented here, the use of files of highly accurate precomputed SO(5) ⊃ SO(3) Clebsch-Gordan coefficients together with an exact analytic expression for the SO(5)-reduced matrix elements of SO(5) spherical harmonics enable these computationally intensive methods to be avoided entirely. These files of SO(5) ⊃ SO(3) Clebsch-Gordan coefficients were computed [18,19] using the algorithm developed in [20]. This algorithm, which also calculates SO(5) spherical harmonics, was based on the methods of [1] for calculating model SO (5) wave functions.…”

Section: Introductionmentioning

confidence: 99%

A computer code for calculations in the algebraic collective model of the atomic nucleus

Welsh

Rowe

2016

Computer Physics Communications

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A Maple code is presented for algebraic collective model (ACM) calculations. The ACM is an algebraic version of the Bohr model of the atomic nucleus, in which all required matrix elements are derived by exploiting the model's SU(1,1)×SO(5) dynamical group. This paper reviews the mathematical formulation of the ACM, and serves as a manual for the code.The code enables a wide range of model Hamiltonians to be analysed. This range includes essentially all Hamiltonians that are rational functions of the model's quadrupole momentsqM and are at most quadratic in the corresponding conjugate momentaπN (−2 ≤ M, N ≤ 2). The code makes use of expressions for matrix elements derived elsewhere and newly derived matrix elements of the operators [π ⊗q ⊗π]0 and [π ⊗π]LM . The code is made efficient by use of an analytical expression for the needed SO(5)-reduced matrix elements, and use of SO(5) ⊃ SO(3) Clebsch-Gordan coefficients obtained from precomputed data files provided with the code.The parameter a in the basis {|(a, λ) ν } for L 2 (R + , β 4 dβ) is a useful scale parameter that implicitly defines the U(1) ⊂ SU(1, 1) subgroup (see Section III). The group SO(5) in the chain (5) is the group of linear transformations of the five quadrupole moments {q M } that leave β 2 invariant, and SO(3) is the rotational subgroup that transforms the quadrupole moments as a basis for the 5-dimensional L = 2 irrep. An extra 'missing label' α in the range 1 ≤ α ≤ d vL is needed to distinguish the multiplicity, d vL , of SO(3) irreps of the same angular momentum L in an SO(5) irrep of seniority v (seniority is the SO(5) analogue of angular momentum). This multiplicity is given [18,24] by

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“…Thus, the relatively small differences between Figs. 3(a) and 3(b) can be attributed to SO (5) centrifugal coupling effects between the rotational and vibrational degrees of freedom that are included in Fig. 3(a) but not in 3(b).…”

Section: The Bohr Collective Modelmentioning

confidence: 99%

“…we can generate the complete set of SO(5) spherical harmonics and, from them, derive all required SO(5) Clebsch-Gordan coefficients [9,5]. We also obtain [5] the SO(5)-reduced matrix elements…”

Section: The Bohr Collective Modelmentioning

confidence: 99%

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The fundamental role of symmetry in nuclear models

Rowe

2013

AIP Conference Proceedings

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Abstract. The purpose of these lectures is to illustrate how symmetry and pattern recognition play essential roles in the progression from experimental observation to an understanding of nuclear phenomena in terms of interacting neutrons and protons. We do not discuss weak interactions nor relativistic and sub-nucleon degrees of freedom. The explicit use of symmetry and the power of algebraic methods, in combination with analytical and geometrical methods are illustrated by their use in deriving a shell-model description of nuclear rotational dynamics and the structure of deformed nuclei.

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Construction of spherical harmonics and Clebsch–Gordan coefficients

Cited by 26 publications

References 33 publications

Algebraic Collective Model and Nuclear Structure Applications

Algebraic Collective Model and Nuclear Structure Applications

A computer code for calculations in the algebraic collective model of the atomic nucleus

The fundamental role of symmetry in nuclear models

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