Structure of 3He

Using electron scattering data, the diffraction pattern off 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{3}$$\end{document}He shows it to be an equilateral triangle possessing dihedral D3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{3}$$\end{document} point group symmetry (PGS). Previous work showed that 4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{4}$$\end{document}He is a 3-base pyramid with C3v\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{3v}$$\end{document} PGS. 6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{6}$$\end{document}Li is predicted to have C2v\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2v}$$\end{document} PGS. As nuclear A→\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \rightarrow $$\end{document} large, atomic nuclei enter into the ‘protein folding problem’ with many possible groundstate PGS competing for lowest energy.


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, three 2 P 1 2 , one 4 P 1 2 and three 4 D 1 2 states that have specific wavefunctions chosen for their analytical tractability and physical plausiblity. Even so, the author had to exclude a region of configuration space in order to establish a 'hole' . The author calls this 'a three-nucleon repulsive core' . Technically, this paper has physics errors because it has cross-terms between the different irreducible representations of S 3 . The final paper reviewed here is the calculation 6 of the 3 He form factor in the mesonexchange model. In the authors' words: 'The charge form factors show a striking disagreement with experiment: the theoretical momentum transfer at the first minimum is too high and the height of the second maximum is too low.' In addition, the famous 'hole' in the charge distribution for r = 0 is not reproduced. Of course, the meson-exchange model has other issues beyond the A = 3 system, but they will not be discussed. This concludes the short literature review. Here it is shown that 3 He is an equilateral triangle.
We calculate the charged form factor, F ch , which in one-photon exchange, is The nuclear charge density ρ(r) for 3 He need not be spherically symmetric. The charge density for point nucleons is ( τ 3i is the z-component isospin operator for nucleon numbered i) Since the proton itself has charge density ρ p (r) , then ρ(r) is the convolution and now where F pt (q 2 ) is the charge form factor using point nucleons and F q (q 2 ) is the Fourier transform of ρ p which is the familiar dipole form factor 7 (1 + q 2 (.054842fm 2 )) −2 . Experimentalists normalize the charge form factor by F ch =F ch /Z (Z = nuclear charge) so the normalized F ch (0) = 1 ; we will call the normalized charge form factor, 'the charged form factor' .
(1) F ch (q 2 ) = ρ(r) sin qr qr d 3 r www.nature.com/scientificreports/ We now construct � pt (r 1 , s 1 , t 1 , . . .) in which we indicate the position, spin and isospin variables. The S = 1/2, T = 1/2 supermultiplet has symmetry group S 3 irreducible representations (IR), due to the Pauli Principle. Furthermore, as explained below, Quantum Chromodynamics (QCD) requires that groundstate nuclei have PGS. The vertices of the lattice in the center-of-mass system are (c is the base)

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, the triangle is equilateral. The orientation of the lattice plane is immaterial, because the pointlike charge density depends only on inner products a i · a j (due to the fact that the electron beam is not coherent). Using group theory 8  In Eq. (6), the C i are constants. The reason for the decomposition is that the QCD Hamiltonian has effective nucleon s i ·s j spin terms which mix the IR of S 3 . For the basic spatial wavefunction, we take the zero phonon harmonic oscillator (H.O.). The harmonic oscillator parameter is the zero point energy (which is related to the Uncertainty Principle) of the nucleon in the atomic nucleus. We define φ(r 1 , r 2 , r 3 ) to be where C = 1 α 3/2 ( 2 π ) 9/4 and α is the H.O. length parameter. In Eq. (7), the '123' only reference the spatial variables r 1 , r 2 , r 3 . For simplicity of notation, hereafter, we put a 1 ≡ a, a 2 ≡ b, a 3 ≡ c . The spin-isospin wavefunction ST(123) is where χ(2) ↓ is nucleon 2 spin down, and τ (3) ↑ is nucleon 3 isospin up (a proton). We now construct the individual wavefunctions of Eq. (6) by introducing the Young Tableau of Fig. 1 and the idempotent operators A , S which respectively are the antisymmetrizer and symmetrizer. 3 He has positive parity, with P the parity operator. In Table 1, we give the PGS parameters for 3 He and 4 He. Basically, the extra nucleon in 4 He sits on a withdrawn equilateral triangle base of 3 He. This is understandable because of the existence of a 4-body force in 4 He, discussed below. The radial (scalar) point nucleon nuclear density is which is displayed in Fig. 3. We see 3 He has a 'hole' at the center. Finally, we calculate the root-mean-squared R He mass radius of 3 He, which is This is done by assembling a master table of expectation values, such as φ(132 The mass radius is the physical extent of the wavefunction (size of nucleus).
The atomic nucleus is the solution of the N-quark low energy semi-relativistic Hamiltonian. For N=3, reference 9 solved for the complete J π N, family using the 2-body charmonium potential 10 . (Recently, four-quark matter has been found, and it is anticipated it may have a PGS shape 11 . This further substantiates the Charmonium potential.) Kiefer was able to predict the known N, J π states and all the known photon decay amplitudes for transitions to the nucleon groundstate. Reference 12,13 solved the N = 6 quark problem and showed that the physical deuteron was due to quark-exchange Feynman diagrams. Reference 14 showed that the quark-exchange forces give rise to effective nucleon-nucleon potentials. Reference 15 showed that QCD has 2-, 3-, 4-body quark exchange forces. Finally, reference 16 was able to concatenate the N-quark Hamiltonian into a nuclear code. This showed that the atomic nucleus groundstate has PGS, while excitations are coherent (keeping the nuclear bonds intact: rotations and vibrations) and incoherent (breaking the nuclear bonds). The saturation of atomic forces is due to the fact that nucleons have only three quarks to exchange: the four-body quark exchange force, due to the (10) www.nature.com/scientificreports/ gluon 4-body interaction, is the strongest binding mechanism for the atomic nucleus, reference 16 . The nuclear code can be expanded to predict groundstate spins, by noting that the 2-body, 3-body bonds are spin-dependent. For example, the binding energy of the A = 6 nucleus is E(6) = E 4 + 2E 3 + 4E 2 , where E i are the binding energies of the i-body bonds, so the E 2 spins cancel, leaving 2E 3 -spins ( J π = 1 + ). Similarly, E(7) = E 4 + 3E 3 + 4E 2 , with the E 2 spins canceling leaving 3 E 3 spins ( J π = 3 2 − ). The key quantity allowing the nuclear interactions to occur is the overlap of nucleon wavefunctions in the atomic nucleus. There are two radii for the nucleon: the electromagnetic and the mass. For the nucleon in the S = T = 1/2 state, the former radius-squared is r 2 Q−N which is a negative value for the neutron. Physically speaking, the electromagnetic radius measures the internal charge distribution while the mass radius r M measures the physical size. In realty, the neutron mass radius and the proton mass radius are nearly identical in value because the gluons in Quantum Chromodynamics do not couple to electric charge. The mass radius of the nucleon is 9 ∼ 1.38−1.40 fm, showing that the atomic nucleus has overlaping nucleon wavefunctions, allowing QCD color interactions to occur between colorless hadrons. An important experiment that can be conducted is high-energy elastic scattering off 6 Li, which is predicted to have C 2v PGS, reference 16 . However, as the nuclear A → large, it becomes very difficult to ascertain the geometry of the groundstate wavefunction, the 'protein folding problem' . For large A nuclei, one must consider that the Jahn-Teller effect 17 may appear, changing the assumed PGS.