Skyrmions, Alpha particles and Oxygen-16

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Skyrmions, Alpha particles and Oxygen-16

Nicholas S. Manton

DAMTP, University of Cambridge N.S.Manton@damtp.cam.ac.uk

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Skyrme Model

I SU(2) Skyrme field

U(x) = σ(x) 12+iπ(x) · τ

with σ2+ π · π =1. Boundary condition U(x) → 12as

|x| → ∞.

I Current (gradient of U) is Ri = (∂iU)U−1.

I Baryon number B is the topological charge, the degree (winding) of U as map from R3_{to SU(2),}

B = − 1 24π2

Z

εijkTr(RiRjRk)d3x .

I Static Skyrme Energy E (Skyrme units) is Z −1 2Tr(RiRi) − 1 16Tr([Ri,Rj][Ri,Rj]) +m 2 πTr(12− U) d3x .

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Skyrmions

I A Skyrmion is a localised solution of minimal energy for given topological charge B. It is interpreted as an

unquantised nucleus with baryon (atomic mass) number B.

I A Skyrmion is a spatially interesting pion condensate, with its orientation free to vary.

I The Runge colour sphere records the normalised pion field ˆ

π(x). The colours are superposed on a constant energy

density surface.

I The orientational coordinates in space and isospace need to be quantised, using rigid body quantisation. Vibrational modes should also be considered.

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Constant energy density surfaces of B=1 to B=8 Skyrmions (with mπ =0) [R. Battye and P. Sutcliffe]

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Skyrmions from Rational Maps

I These Skyrmion solutions are found numerically, but the Rational Map Approximation gives rather accurate

analytical formulae for the fields [C. Houghton, NSM and P. Sutcliffe].

I The Rational Map Approximation separates the radial from the angular structure and clarifies the shapes, symmetries and approximate energies of Skyrmions.

I The classical binding energy of Skyrmions is too high, compared to nuclear binding energies, but variants of the Skyrme model can give better results.

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Quantised Skyrmions

I Skyrmions are quantised as rigid bodies and acquire spin and isospin. Skyrme field topology and Skyrmion

symmetries constrain the allowed states [Finkelstein-Rubinstein].

I Lowest-energy quantum states for small B were found by [Adkins, Nappi and Witten; Braaten and Carson; Walhout]:

I B = 1: Proton and neutron, with spin J = 1₂ and isospin I = 1₂. Excited states (Delta-resonances) have J = I = 3₂.

I B = 2: 2H (Deuteron), with J = 1 and I = 0.

I B = 3: 3H and 3He, with J = 1₂ and I = 1₂.

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I The B = 6 Skyrmion has D4d symmetry. Its two rotational

generators give Finkelstein-Rubinstein constraints eiπ2L3_eiπK3_{|Ψi = |Ψi}

eiπL1_eiπK1_{|Ψi = −|Ψi .}

(Note: Li, Ki are spin and isospin operators w.r.t.

body-fixed axes. One gets full spin and isospin multiplets w.r.t. space-fixed axes.)

I Allowed states have Isospin 0 (6Li), with spin/parity JP =1+,3+,4−,5+,5−, · · · ,

and Isospin 1 (6He ,6Li ,6Be), with spin/parity JP =0+,2+,2−, · · · .

I One sees the alpha + deuteron, or three-deuteron structure in the Lithium-6 states. The spectrum is qualitatively right, but the rigid-body approach does not give accurate energies.

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I Beryllium-8’s clear rotational band with states of spin/parity JP =0+_,₂+_,₄+_{is reproduced by the quantised Skyrmion.}

The physical ground state is a marginally unbound “molecule” of alpha particles. The energy level spacing is close to J(J + 1).

I For isospin 1 (Lithium-8 and Boron-8) the Skyrme model predicts low-energy JP ₌₀−_,₂− _{states in addition to the}

known JP =2+,3+states. These have not been seen, but may be hard to produce and observe.

I The isospin 2 quintet (Helium-8 etc.) has correct JP =0+ ground states.

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16.4 MeV Boron−8 J=2 , I=1+ 3.0 MeV 11.4 MeV 19.1 MeV 16.6 MeV 19.3 MeV _{18.6 MeV} 27.5 MeV 27.0 MeV 27.8 MeV 28.1 MeV 26.3 MeV 17.0 MeV Helium−8 Lithium−8 Beryllium−8 Carbon−8 J=0 , I=2+ J=0 , I=2+ J=0 , I=2+ J=0 , I=2+ J=0 , I=2+ J=3 , I=1+ _{J=3 , I=1}+ J=3 , I=1+ J=2 , I=1+ J=2 , I=1+ J=4 , I=0+ J=2 , I=0+ J=0 , I=0+

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Skyrmions of Large Baryon Number

I Beyond baryon number B = 8 one needs new methods to understand Skyrmions, and to construct configurations to relax numerically. Skyrmions are more compact when mπ ' 1 (its physical value).

I Gluing together B = 4 cubes with colours touching on faces works for B = 12, 16, 24, 32 [Feist].

I One can also cut chunks from the Skyrme crystal, a cubic array of half-Skyrmions, with exceptionally low energy per Skyrmion [Castillejo et al., Kugler and Shtrikman]. B = 32 and B = 108 illustrate this. One can cut single Skyrmions off the corners of these chunks, to obtain, e.g. B = 31 and B = 100.

I The significant Coulomb energy for large B is so far ignored.

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B = 12

Skyrmions and Carbon-12

I There are two Skyrmions of very similar energies – and both apparently stable. One is a triangle of B = 4 cubes with D3hsymmetry, the other a linear chain with D4h

symmetry.

I P.H.C. Lau and NSMhave quantised their rotational motion as rigid bodies. The relevant inertia coefficients V11=V22

and V33 have been calculated for each Skyrmion.

I Allowed states for the triangular Skyrmion have spin/parity JP =0+,2+,3−,4−,4+,5−,6+,6−,6+in rotational bands with K = 0, 3, 6. Their energies are

E (J, K ) = 1 2V11 J(J + 1) + 1 2V33 − 1 2V11 K2. These match well the experimental rotational band of the ground state.

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The Hoyle State of Carbon-12

I Our model suggests that the 0+ _{Hoyle state corresponds}

to a linear chain of alpha particles.

I The JP ₌₀+_,₂+_,₄+_{rotational band of Hoyle state}

excitations [M. Freer et al.] has much smaller slope than the ground state band.

I The ratio of slopes, and the ratio of the root mean square radii of the Carbon-12 ground state and Hoyle state, are well fitted by the Skyrmions.

I The Skyrme model makes predictions for several spin 6 states.

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0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 J(J+1) E ( M eV ) Hoyle Band Ground-State Band

Rotational energy spectrum of the two B=12 Skyrmions, compared with data

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States of Oxygen-16

I Four B = 4 cubes can be arranged as a tetrahedron or flat square. Rigid-body quantisation gives several rotational bands, but misses states associated with vibrations.

I [C. Halcrow, C. King and NSM] have considered a 2-parameter family of configurations of four cubes, and have quantised the shape parameters together with rotations.

I The configurations all have D2symmetry so miss the 1−

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Energy level diagram for16O states. Solid ≡ Skyrme model. Circle/Triangle ≡ +/− parity. Hollow ≡ Experiment.

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Summary

I The fundamental ingredient of the Skyrmion model of nuclei is the chiral field U. The protons and neutrons emerge from the quantisation of spin and isospin of the topological Skyrmions.

I Toroidal and cubic structures, with B = 2 and B = 4

respectively, are present in all Skyrmions, and easily seen. They contribute deuteron and alpha particle constituents of larger nuclei, if the overall isospin is zero. Their role in larger neutron-rich nuclei is not yet known.

I If Skyrmions are treated as rigid bodies, the binding

energies are too high, and charge densities have too much spatial variation. Recent work on Skyrmion quantisation allows for vibrations, and relative motion of sub-clusters. This gives improved models for7Li/7Be and16O.