Bulletin of the American Physical Society
2005 APS April Meeting
Saturday–Tuesday, April 16–19, 2005; Tampa, FL
Session Z13: Nuclear Theory II |
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Sponsoring Units: DNP Chair: Ruprecht Machleidt, University of Idaho Room: Marriott Tampa Waterside Room 12 |
Tuesday, April 19, 2005 3:30PM - 3:42PM |
Z13.00001: R-Matrix Method Using the Polar Form of the Schroedinger Operator Charles Weatherford Any linear operator ($\hat A$) can be decomposed into a product of an Hermitian times a Unitary operator [P-O L\"owdin, ${\bf Linear\ Algebra\ for\ QuantumTheory}$,Wiley 1998, New York], as per ${\hat A} \equiv {\hat H}_1{\hat U_1} ={\hat U_2}{\hat H_2}$ where ${\hat H}_1 = ( {\hat A}{\hat A}^{\dagger})^{1/2}$, ${\hat U}_1 = {\hat H}_1^{-1} {\hat A}$, ${\hat H}_2 = ( {\hat A}^{\dagger}{\hat A})^{1/2}$, ${\hat U}_2 = {\hat A}{\hat H}_2^{-1} $. Such decompositions constitute what is called the polar form of the operator. A version of the R-matrix scattering theory will be presented employing the polar form of the time-independent Schroedinger operator (SO). The SO is not Hermitian in the continuum within the finite R-matrix sphere. An approximate inverse of the SO is constructed by diagonalizing its positive definite Hermitian component. An approximate expression for the scattered wave is obtained by projecting onto these eigenstates. In the process, a new type of minimum principal obtains such that minimization of the positive definite eigenvalues produces the most rapidly convergent series for the operator inverse and therefore, the solution. This algorithm is described and applied to several model problems which constitute a proof of principle. [Preview Abstract] |
Tuesday, April 19, 2005 3:42PM - 3:54PM |
Z13.00002: DWBA (d,N) Calculations Including Dirac Phenomenological Potentials and an Exact Treatment of Finite-range Effects Eric Hawk, James McNeil An algorithm for the inclusion of both Dirac phenomenological potentials and an exact treatment of finite-range effects within the DWBA is presented. The numerical implementation of this algorithm is used to calculate low-energy deuteron stripping cross sections, analyzing powers, and polarizations. These calculations are compared with experimental data where available. The impact of using several commonly employed nuclear potentials (Reid soft-core, Bonn, Argonne v18) for the internal deuteron wave function is also examined. [Preview Abstract] |
Tuesday, April 19, 2005 3:54PM - 4:06PM |
Z13.00003: The number of $J=0$ pairs in $^{44,46,48}$Ti Larry Zamick, Alberto Escuderos, Aram Mekjian In the single $j$-shell, the configuration of an even--even Ti isotope consists of 2 protons and $n$ neutrons. The $I=0$ wave function can be written as $$\Psi=\sum_{Jv} D(J,Jv) [(j^2)^J_\pi (j^n)^J_\nu ]^{I=0},$$ where $v$ is the seniority quantum number. There are several states with isospin $T_{\rm min}=|(N-Z)/2|$, but only one with $T_{\rm max}=T_{\rm min}+2$. By demanding that the $T_{\rm max}$ wave function be orthogonal to the $T_{\rm min}$ ones, we obtain the following relation involving a one-particle cfp: $$ D(00)=\frac{n}{2j+1} \sum_J D(J,Jv)(j^{n-1}(jv=1)j|}j^nJ) \sqrt{2J+1} $$ This leads to the following simple expressions for the number of $J=0$ $np$ pairs in these Ti isotopes: \begin{itemize} \item For $T=T_{\rm min}$, \ \ \ \# of pairs $(J_{12}=0)=2|D(00)|^2/n$ \item For $T=T_{\rm max}$, \ \ \# of pairs $(J_{12}=0)=2n|D(00)|^2=\frac{2n(2j+1-n)}{(2j+1)(n+1)}$ \end{itemize} For $^{44}$Ti we have also the result for {\em even} $J_{12}$ $$ \#\ {\rm of}\ nn\ {\rm pairs}\ =\ \#\ {\rm of}\ pp\ {\rm pairs}\ =\ \#\ {\rm of}\ np\ {\rm pairs}\ =\ |D(J_{12},J_{12})|^2 $$ [Preview Abstract] |
Tuesday, April 19, 2005 4:06PM - 4:18PM |
Z13.00004: Alternate derivation of the Ginocchio-Haxton relation $[(2j-3)/6]$ Alberto Escuderos, Larry Zamick We want the number of states with total angular momentum $J=j$ for 3 identical particles (e.g. neutrons) in a $j$ shell. We form states $M_1>M_2>M_3$ with total $M=M_1+M_2+M_3$. Consider first all states with $M=j+1$. Next form states by lowering $M_3$ by one. All such states exist because the lowest value of $M_3$ is $(j+1)-j-(j-1)=-j+2$. So far we have the total number of states with $J > j$ and $M=j$. The additional states with $M=j$ are the states with $J=j$. These additional states have the structure $M_1,M_2,M_2-1$ because if we try to raise $M_3$ we get a state not allowed by the Pauli principle, namely $M_1,M_2,M_2$. The possible values of $M_1,M_2$ are respectively $j-2n$ and $1/2+n$, where $n=0,1,2\cdots$. The total number of $J=j$ states is $N=\bar{n}+1$ (with $\bar{n}=n_{\rm max}$), while $\bar{n}$ itself is the number of seniority 3 states. The condition $M_1>M_2$ leads to $\bar{n}<(2j-1)/6$ or $N<(2j+5)/6$. This is our main result. It is easy to show that this is the same as the G-H relation\footnote{J.N.~Ginocchio and W.C.~Haxton, {\em Symmetries in Science VI}, ed. by B.~Gruber and M~Ramek, Plenum, New York (1993)} (see also Talmi's 1993 book) $\bar{n}=[(2j-3)/6]$, where $[]$ means the largest integer. Since $2j$ is an odd integer, $(2j-1)/6$ is either $I, I-1/3$ or $I-2/3$, where $I$ is an integer. If the value is $I$, then $\bar{n}=[(2j-3)/6]=[I-1/3]=I-1$. It is easy to show agreement in the other 2 cases as well. The number of $J=j$ states for the 3-particle system is equal to the number of $J=0$ states for a 4-particle system. [Preview Abstract] |
Tuesday, April 19, 2005 4:18PM - 4:30PM |
Z13.00005: Bound nucleons have unique masses that govern elemental properties Eugene Pamfiloff It is know that measured binding energies associated with elements require equivalent energy to break the nuclear bond of a nucleus. Based upon the proposals contained in a recent published work [1] and with support from experimental high-energy data, it can be shown that a portion of listed binding energies are attributed to bound nucleons having a unique mass for every element. The figures show, relative to the hydrogen proton, that of the: a) 1.112 MeV binding energy per nucleon for $^{2}$H, 44{\%} or 0.486 MeV represents a change in mass for the proton and neutron; b) of 5.629 MeV binding energy per nucleon for $^{7}$Li, 87{\%} or 4.890 MeV represents a change of mass for each nucleon; c) likewise, $^{56}$Fe has 8.811 MeV binding energy per nucleon and of this 92{\%} or 8.119 MeV represents a change in mass for each nucleon; and $^{232}$Th has 7.639 MeV binding energy per nucleon and of this, 90{\%} or 6.848 MeV represents a change in mass for each nucleon. This demonstrates that the nucleons of each element have unique masses. It can be shown that if three protons are removed from $_{82}$Pb the result is not $_{79}$Au. We conclude and predict that in addition to the Z number, elemental properties are determined by the unique proton and neutron masses for each element. [1] \href{mailto:megforce@physast.uga.edu}{megforce@physast.uga.edu} ``The Order of the Forces'' [Preview Abstract] |
Tuesday, April 19, 2005 4:30PM - 4:42PM |
Z13.00006: Chaos as a Bridge between Classical Determinism and Quantum Probability Wm. C. McHarris Chaos provides the fundamental determinism so dear to Einstein, yet for all practical purposes it must be treated statistically, as proposed by the Copenhagen school. Thus, both Einstein and Bohr could have been correct in their debates. In a series of papers I have demonstrated that a number of the so-called imponderables or paradoxes generated in the Copenhagen interpretation of quantum mechanics have parallel explanations in the realm of nonlinear dynamics and chaos theory [i.a., J. Opt. B: Quantum and Semiclass. Opt. {\bf 5}, S442 (2003)]. These include exponential decay laws, interpretations of Bell-type inequalities, spontaneous symmetry breaking, and even diffraction. I give a brief overview of these, concentrating on the interpretation of the CHSH inequality (an experimentally friendly Bell-type inequality), demonstrating that here one is comparing correlated versus uncorrelated statistics more than quantum versus classical mechanics--- nonlinear classical dynamical systems have been shown to have sufficient long-range correlations, as codified by the entropy of nonextensive thermodynamics, to raise the upper bound imposed by Bell-type inequalities into the range of quantum mechanics. As a result, many of the experiments ruling out ``local reality'' are perhaps moot. [Preview Abstract] |
Tuesday, April 19, 2005 4:42PM - 4:54PM |
Z13.00007: Variable Nuclear Barrier Heights as Irregular Potential Waves Due to Various Nuclear Motions Stewart Brekke The nuclear potential barrier height is an irregular wave due to random and periodic motion nuclear motions such as vibration, rotation and orbiting. Due to the vibrations and other nuclear motions, the potential well is vibrating irreglarly also. Assume the nuclear motion is a three dimensional oscillator were r = \{(AcosX)$^2 +$ (AcosY$^2 +$ AcosZ)$^2$)\}$^{1/2}$. For cos =0, r = 0 min, cos =RMScos, r=1.22A average, cos=1, r= 1.707A max. Therefore, using V = kq(1)q(2)/r the barrier height ranges from V=infinitely high, = 0.816q(1)q(2)/A on average, to a low of 0.577q(1)q(2)/A where A= average amplitude of nuclear vibration, q(1) is the nuclear charge, q(2) = charge of incoming or outgoing particle. Nuclear motion makes the gravitational and magnetic fields irregular wave also. [Preview Abstract] |
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Z13.00008: Variable Cross Sections Due to Nuclear Vibrations Stewart Brekke Due to random nuclear vibrations the cross sectional area for inoming particles to a nucleus is a variable. If b = AcosY is the impact parameter in one dimension, then the cross section $\sigma$=$\pi$(A cosY)$^{2}$ where A=amplitude of vibration. Therefore, $\sigma$=$\pi$(A)$^{2}$ maximum, $\sigma$=$\pi$(0.707A)$^{2}$ average rms, and $\sigma$=0 minimum values for the variable nuclear cross sections per nucleus. [Preview Abstract] |
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Z13.00009: Classical Model of Rutherford-Santilli Neutron Robert Driscoll Model n(RS), isomer of n, Bohr's real ground state of H$^{+}$: proton p and electron e, separated 0.81 Fermi, in circular orbits about c.m., with parallel magnetic dipole moments (MDMs) normal to orbital plane. Binding force: Coulomb less magnetic. Each momentum: mass x velocity less charge x (vector potential A at particle). Assuming unmutated p, and n data: orbital e velocity v is 0; slight mutation: v/c $<<<$ 1. (Ref. 1.) Mutated e: mass, 2.5 x (mass of free e); spin, 0.038$\hbar$/2; g, 0.52; MDM, 3.6 x E(-26) S.I. Stability requires external A, 0[0.01 S.I.], found in atomic nuclei. The n(RS) \textbf{$\to $} n by spin flip of e; e captured by positive constituent of p; gamma photon emitted. (Ref. 2.) \newline \newline 1. R. M. Santilli, \textit{Hadronic Journal 13}, 513 (1990); \textit{Chinese Journal Sys. Eng. {\&} Elec. 6}, No. 4, 177 (1995) \newline 2. R. B. Driscoll, \textit{Hadronic Journal 27}, No. 6 (2004) [Preview Abstract] |
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