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Sample records for 9j-symbols

  1. Semiclassical analysis of the Wigner 9j symbol with small and large angular momenta

    SciTech Connect

    Yu Liang; Littlejohn, Robert G.

    2011-05-15

    We derive an asymptotic formula for the Wigner 9j symbol, in the limit of one small and eight large angular momenta, using a gauge-invariant factorization for the asymptotic solution of a set of coupled wave equations. Our factorization eliminates the geometric phases completely, using gauge-invariant noncanonical coordinates, parallel transports of spinors, and quantum rotation matrices. Our derivation generalizes to higher 3nj symbols. We display without proof some asymptotic formulas for the 12j symbol and the 15j symbol in the Appendices. This work contributes an asymptotic formula of the Wigner 9j symbol to the quantum theory of angular momentum and serves as an example of a general method for deriving asymptotic formulas for 3nj symbols.

  2. Maximum J Pairing and Asymptotic Behavior of the 3j and 9j Coefficients

    NASA Astrophysics Data System (ADS)

    Hertz-Kintish, Daniel; Zamick, Larry; Kleszyk, Brian

    2014-09-01

    We investigate the large j behavior of certain 3 j and 9 j symbols, where j is the total angular momentum of one particle in a given shell. Our motivation is the problem of maximum J pairing in nuclei, along with the more familiar J = 0 pairing. Maximum J pairing leads to an increase in J = 2 coupling of two protons and two neutrons relative to J = 0 . We find that a coupling unitary 9 j symbol (U 9 j) is very weak as j increases, leading to wavefunctions which are to an excellent approximation single U 9 j coefficients. Our study of the large j behavior of coupling unitary 9 j symbols is through the consideration of the case when the total angular momentum I is equal to Imax - 2 n and Imax ≡ 4 j - 2 , where n = 0 , 1 , 2 , ... . We here derive asymptotic approximations of coupling 3 j symbols and find that the 3 j ~j - 3 / 4 in the high j limit. One major analytical tool we used is the Stirling Approximation. Through analytical, numerical, and graphical methods, we show the power law behavior of the coupling unitary 9 j symbols in the n / j << 1 limit, i.e. U 9 j ~j-n . Power-law behavior is evident if there is a linear dependence of ln | U 9 j | vs. ln j . We also present some examples of percent errors in our approximations. We investigate the large j behavior of certain 3 j and 9 j symbols, where j is the total angular momentum of one particle in a given shell. Our motivation is the problem of maximum J pairing in nuclei, along with the more familiar J = 0 pairing. Maximum J pairing leads to an increase in J = 2 coupling of two protons and two neutrons relative to J = 0 . We find that a coupling unitary 9 j symbol (U 9 j) is very weak as j increases, leading to wavefunctions which are to an excellent approximation single U 9 j coefficients. Our study of the large j behavior of coupling unitary 9 j symbols is through the consideration of the case when the total angular momentum I is equal to Imax - 2 n and Imax ≡ 4 j - 2 , where n = 0 , 1 , 2 , ... . We here

  3. Exact computation and large angular momentum asymptotics of 3nj symbols: Semiclassical disentangling of spin networks.

    PubMed

    Anderson, Roger W; Aquilanti, Vincenzo; da Silva Ferreira, Cristiane

    2008-10-28

    Spin networks, namely, the 3nj symbols of quantum angular momentum theory and their generalizations to groups other than SU(2) and to quantum groups, permeate many areas of pure and applied science. The issues of their computation and characterization for large values of their entries are a challenge for diverse fields, such as spectroscopy and quantum chemistry, molecular and condensed matter physics, quantum computing, and the geometry of space time. Here we record progress both in their efficient calculation and in the study of the large j asymptotics. For the 9j symbol, a prototypical entangled network, we present and extensively check numerically formulas that illustrate the passage to the semiclassical limit, manifesting both the occurrence of disentangling and the discrete-continuum transition.

  4. Exact computation and large angular momentum asymptotics of 3nj symbols: Semiclassical disentangling of spin networks

    SciTech Connect

    Anderson, Roger W.; Aquilanti, Vincenzo; Silva Ferreira, Cristiane da

    2008-10-28

    Spin networks, namely, the 3nj symbols of quantum angular momentum theory and their generalizations to groups other than SU(2) and to quantum groups, permeate many areas of pure and applied science. The issues of their computation and characterization for large values of their entries are a challenge for diverse fields, such as spectroscopy and quantum chemistry, molecular and condensed matter physics, quantum computing, and the geometry of space time. Here we record progress both in their efficient calculation and in the study of the large j asymptotics. For the 9j symbol, a prototypical entangled network, we present and extensively check numerically formulas that illustrate the passage to the semiclassical limit, manifesting both the occurrence of disentangling and the discrete-continuum transition.

  5. High-nuclearity mixed-valence clusters and mixed-valence chains: general approach to the calculation of the energy levels and bulk magnetic properties.

    PubMed

    Clemente-Juan, J M; Borrás-Almenar, J J; Coronado, E; Palii, A V; Tsukerblat, B S

    2009-05-18

    A general approach to the problem of electron delocalization in the high-nuclearity mixed-valence (MV) clusters containing an arbitrary number of localized spins and itinerant electrons is developed. Along with the double exchange, we consider the isotropic magnetic exchange between the localized electrons as well as the Coulomb intercenter repulsion. As distinguished from the previous approaches dealing with the MV systems in which itinerant electrons are delocalized over all constituent metal sites, here, we consider a more common case of systems exhibiting partial delocalization and containing several delocalized domains. Taking full advantage of the powerful angular momentum technique, we were able to derive closed form analytical expressions for the matrix elements of the full Hamiltonian. These expressions provide an efficient tool for treating complex mixed-valence systems, because they contain only products of 6j-symbols (that appear while treating the delocalized parts) and 9j-symbols (exchange interactions in localized parts) and do not contain high-order recoupling coefficients and 3j-symbols that essentially constrained all previous theories of mixed valency. The approach developed here is accompanied by an efficient computational procedure that allows us to calculate the bulk thermodynamic properties (magnetic susceptibility, magnetization, and magnetic specific heat) of high-nuclearity MV clusters. Finally, this approach has been used to discuss the magnetic properties of the octanuclear MV cluster [Fe(8)(mu(4)-O)(4)(4-Cl-pz)(12)Cl(4)](-) and the diphthalocyanine chains [YPc(2)].CH(2)Cl(2) and [ScPc(2)].CH(2)Cl(2) composed of MV dimers interacting through the magnetic exchange and Coulomb repulsion.

  6. A Theorem for Two Nucleon Transfer

    NASA Astrophysics Data System (ADS)

    Zamick, Larry; Mekjian, Aram

    2004-05-01

    We use the short notation for a unitary 9j symbol U9j(Ja,Jb)=<(jj)Ja(jj)Ja|(jj)Jb(jj)Jb>I=0 The wave fcn of a state of 44Ti with ang momentum I can be written as sum D(Jp,Jn) [(jj)Jp (jj)Jn]I. For the I=0 ground stae Jp=Jn. We found a new relationship SumJp U9j(Jp,Jx) D(Jp,Jp)= 1/2 D(Jx,Jx) for T=0 and =-D(Jx,Jx) for T=2. We could explain this by regarding U9j for even Jp,Jx as a square matrix hamiltonian, which, when diagonalized has eigenvalues of 1/2(triply degenerate) and -1(singly degenerate) corresponding to T=0 and T=2 respectively.*This theorem is useful,in the context of 2 nucleon transfer, for counting the number of pairs of particles in 44Ti with even Jx.The expressions simplifies to 3|D(Jx,Jx|^2,thus eliminating a complex 9jsymbol A deeper understanding of this result arises if we consider the strange interplay of angular momentum and isospin. Consider the interaction 1/4-t(1).t(2),which is unity for T=0 states and zero for T=1. For n nucleons with isospin T the eigenvalues are n^2/8+n/4-T(T+1)/2 But if we evaluate this with the usual Racah algebra then we note that in the single j shell the interaction can also be written as <(jj)Ia V (jj)Ia>= (1-(-1)^Ia)/2 i.e. the interaction acts only in odd J states since they have isospin T=0.In 44Ti the matrix element of the hamiltonian is [2+2U9j(Jp,Jx)].Connecting this with the isospin expression gives us the eigenvalues above for U9j. * L.Zamick, E. Moya de Guerra,P.Sarriguren,A.Raduta and A. Escuderos, preprint.