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Sample records for 9j-symbols

  1. Couplings and recouplings of four angular momenta: Alternative 9j symbols and spin addition diagrams.

    PubMed

    Santos, Robenilson F; Bitencourt, Ana Carla P; Ragni, Mirco; Prudente, Frederico V; Coletti, Cecilia; Marzuoli, Annalisa; Aquilanti, Vincenzo

    2017-04-01

    The Wigner 9j symbols of the first kind-also known as Fano X-coefficients-serve to connect different addition schemes of four angular momenta, widely known examples being the LS and the jj couplings in atomic, molecular, and nuclear spectroscopies. Here, we also consider alternative sequences of binary couplings of four angular momenta, which are dealt through the 9j symbols of the second kind, and are explicitly given by the pentagonal (or Biedenharn-Elliott) identity. These coefficients are essential ingredients in the quantum-mechanical treatments of rotational and polarization phenomena in reaction dynamics and photoinduced processes. We also emphasize the combinatorial structure underlying the extended construction of a previously introduced truncated icosahedral "abacus", and provide extensions useful for algebraical manipulations, semiclassical interpretations, and computational applications, including all the 120 addition schemes.

  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. Degeneracies when T=0 two body interaction matrix elements are set equal to zero: Talmi's method of calculating coefficients of fractional parentage to states forbidden by the Pauli principle

    NASA Astrophysics Data System (ADS)

    Robinson, Shadow J. Q.; Zamick, Larry

    2001-11-01

    In a previous work [S.J.Q. Robinson and Larry Zamick, Phys. Rev. C 63, 064416 (2001)] we studied the effects of setting all two body T=0 matrix elements to zero in shell model calculations for 43Ti (43Sc) and 44Ti. The results for 44Ti were surprisingly good despite the severity of this approximation. In single-j shell calculations (fn7/2) degeneracies arose between the T=12 I=(12)-1 and (132)-1 states in 43Sc as well as the T=12 I=(132)-2, (172)-1, and (192)-1 in 43Sc. For 44Ti the T=0 states 3+2, 7+2, 9+1, and 10+1 are degenerate as are the 10+2 and 12+1 states. The degeneracies can be explained by certain 6j symbols and 9j symbols either vanishing or being equal as indeed they are. Previously we used Regge symmetries of 6j symbols to explain the vanishing 6j and 9j symbols. In this work a simpler, more physical method is used. This is Talmi's method of calculating coefficients of fractional parentage (cfp) for identical particles to states which are forbidden by the Pauli principle. This is done for both the one particle cfp to handle 6j symbols and the two particle cfp for the 9j symbols. From this we learn that the common thread for the angular momenta I for which the above degeneracies occur is that these angular momenta cannot exist in the calcium isotopes in the f7/2 shell. There are no T=32 f37/2 states with angular momenta 12, 132, 172, and 192. In the same vein there are no T=2 f47/2 states with angular momenta 3, 7, 9, 10, or 12. For these angular momenta, all the states can be classified by the dual quantum numbers (Jπ,Jν).

  4. Asymptotic Analysis of Spin Networks with Applications to Quantum Gravity

    NASA Astrophysics Data System (ADS)

    Haggard, Hal Mayi

    comparison of this spectrum with that found in loop gravity shows excellent agreement. This provides a simplified derivation of the quantization of space that strengthens earlier proposals along these lines. The second application is an asymptotic formula for the 9j-symbol including its amplitude, phase, and all of the phase adjustments. The 9j-symbol is a more complex spin network than has been treated at this level of detail before and arises as part of the vertex amplitude in spin foams, the loop gravity analog of the path integral approach to quantum gravity. More broadly this quantitative result provides further motivation for developing the asymptotics of higher 3nj-symbols; in the long term these asymptotics, which are accurate even for small quantum numbers, may furnish an effective computational tool for bridging loop gravity predictions to testable experiments.