Skyrmions, Tetrahedra and Magic Numbers

Michael Atiyah’s interest in Skyrmions and their relationship to monopoles and instantons is recalled. Some approximate models of Skyrmions with large baryon numbers are then considered. Skyrmions having particularly strong binding are clusters of unit baryon number Skyrmions arranged as truncated tetrahedra. Their baryon numbers, $B = 4 \,, 16 \,, 40 \,, 80 \,, 140 \,, 224$, are the tetrahedral numbers multiplied by four, agreeing with the magic proton and neutron numbers $2 \,, 8 \,, 20 \,, 40 \,, 70 \,, 112$ occurring in the nuclear shell model in the absence of strong spin-orbit coupling.

[1]  N. Manton.  Evidence for tetrahedral structure of Calcium-40 , 2020, 2002.08744.

[2]  J. Rawlinson.  Coriolis terms in Skyrmion quantization , 2019, Nuclear Physics B.

[3]  N. Manton,et al.  Oxygen-16 spectrum from tetrahedral vibrations and their rotational excitations , 2019, International Journal of Modern Physics E.

[4]  C. Halcrow,et al.  Vibrational modes of Skyrmions , 2018, Physical Review D.

[5]  N. Manton,et al.  Rolling Skyrmions and the nuclear spin-orbit force , 2018, Nuclear Physics B.

[6]  N. Manton,et al.  Quantized Skyrmions from SU(4) weight diagrams , 2017, 1712.07786.

[7]  J. Rawlinson.  An alpha particle model for Carbon-12 , 2017, Nuclear Physics A.

[8]  M. Atiyah,et al.  Complex geometry of nuclei and atoms , 2016, Topology and Physics.

[9]  M. Atiyah.  Geometric Models of Helium , 2017, 1703.02532.

[10]  Yong-Liang Ma,et al.  Recent progress on dense nuclear matter in skyrmion approaches , 2016, 1612.06600.

[11]  D. Harland,et al.  A point particle model of lightly bound skyrmions , 2016, 1612.05481.

[12]  M. Atiyah,et al.  Anyons in geometric models of matter , 2016, 1611.04047.

[13]  N. Manton,et al.  A dynamical $\alpha$-cluster model of $^{16}$O , 2016, 1608.05048.

[14]  C. Halcrow.  Vibrational quantisation of the B = 7 Skyrmion , 2015, 1511.00682.

[15]  N. Manton,et al.  Scattering of nucleons in the classical Skyrme model , 2015, 1505.06843.

[16]  D. Harland,et al.  Skyrmions with low binding energies , 2015, 1501.05455.

[17]  M. Atiyah,et al.  Time evolution in a geometric model of a particle , 2014, 1412.5915.

[18]  N. Manton,et al.  A Skyrme model approach to the spin-orbit force , 2014, 1410.0880.

[19]  N. Manton,et al.  States of carbon-12 in the Skyrme model. , 2014, Physical review letters.

[20]  N. Manton,et al.  Quantization of $T_d$ - and $O_h$-symmetric Skyrmions , 2014, 1402.6994.

[21]  N. Manton,et al.  Gravitational instantons as models for charged particle systems , 2013, 1301.1624.

[22]  N. Manton,et al.  Skyrmions up to Baryon Number 108 , 2012, 1210.1712.

[23]  M. Atiyah,et al.  Geometric models of matter , 2011, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[24]  D. Freed.  Pions and generalized cohomology , 2006, hep-th/0607134.

[25]  R. Battye,et al.  Skyrmions and the α-particle model of nuclei , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[26]  N. Schunck,et al.  Nuclear tetrahedral symmetry: possibly present throughout the periodic table. , 2002, Physical review letters.

[27]  R. Battye,et al.  Skyrmions, fullerenes and rational maps , 2001, hep-th/0103026.

[28]  R. Battye,et al.  Solitonic fullerene structures in light atomic nuclei. , 2000, Physical review letters.

[29]  S. Jarvis.  A rational map of Euclidean monopoles via radial scattering , 2000 .

[30]  T. Gisiger,et al.  Recent mathematical developments in the Skyrme model , 1998, hep-th/9812148.

[31]  K. Yabana,et al.  TETRAHEDRAL AND TRIANGULAR DEFORMATIONS OF Z= N NUCLEI IN MASS REGION A 60-80 , 1997, nucl-th/9711012.

[32]  N. Manton,et al.  Rational maps, monopoles and skyrmions , 1997, hep-th/9705151.

[33]  N. Turok,et al.  Normal Modes of the {ital B}=4 Skyrme Soliton , 1997, hep-th/9704012.

[34]  N. Manton,et al.  Skyrme crystal from a twisted instanton on a four-torus , 1994, hep-th/9409182.

[35]  M. Atiyah,et al.  Geometry and kinematics of two Skyrmions , 1993 .

[36]  E. Braaten,et al.  Novel structure of static multisoliton solutions in the Skyrme model , 1990 .

[37]  Shtrikman,et al.  Skyrmion crystals and their symmetries. , 1989, Physical review. D, Particles and fields.

[38]  M. Atiyah,et al.  Skyrmions from instantons , 1989 .

[39]  J. Verbaarschot,et al.  DENSE SKYRMION SYSTEMS , 1989 .

[40]  S. Shtrikman,et al.  A new skyrmion crystal , 1988 .

[41]  N. Manton.  Is the $B=2$ Skyrmion Axially Symmetric? , 1987 .

[42]  Cook,et al.  Face-centered-cubic solid-phase theory of the nucleus. , 1987, Physical review. C, Nuclear physics.

[43]  E. Witten,et al.  Static Properties of Nucleons in the Skyrme Model , 1983 .

[44]  K. Lezuo.  Ground State Rotational Bands in 16O, 40Ca and 208Pb? , 1975 .

[45]  K. Wildermuth,et al.  Experimental evidence for cluster structures in light and medium weight nuclei , 1960 .

[46]  K. Bechert,et al.  NONLINEAR FIELD THEORY , 1956 .

[47]  Eugene P. Wigner,et al.  On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei , 1937 .