Skyrmions, Alpha particles and Oxygen-16
Nicholas S. Manton
DAMTP, University of Cambridge
[email protected]
ECT* Trento, September 2016
Skyrme Model
I
SU(2) Skyrme field
U(x) = σ(x) 12 + iπ(x) · τ
with σ 2 + π · π = 1. Boundary condition U(x) → 12 as
|x| → ∞.
I
Current (gradient of U) is Ri = (∂i U)U −1 .
I
Baryon number B is the topological charge, the degree
(winding) of U as map from R3 to SU(2),
Z
1
εijk Tr(Ri Rj Rk ) d 3 x .
B=−
24π 2
I
Static Skyrme Energy E (Skyrme units) is
Z 1
1
2
Tr([Ri , Rj ][Ri , Rj ]) + mπ Tr(12 − U) d 3 x .
− Tr(Ri Ri ) −
2
16
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
π̂(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.
Constant energy density surfaces of B=1 to B=8 Skyrmions
(with mπ = 0) [R. Battye and P. Sutcliffe]
B=1 Skyrmion (two different orientations)
B=2 Skyrmion [D. Feist]
B=3 Skyrmion
B=4 Skyrmion – Alpha particle
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.
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 = 21 . Excited states (Delta-resonances) have J = I = 23 .
I
B = 2: 2 H (Deuteron), with J = 1 and I = 0.
I
B = 3: 3 H and 3 He, with J =
I
B = 4: 4 He (Alpha particle), with J = I = 0.
1
2
and I = 21 .
B=6 Skyrmion (two different orientations)
I
The B = 6 Skyrmion has D4d symmetry. Its two rotational
generators give Finkelstein-Rubinstein constraints
π
ei 2 L3 eiπK3 |Ψi = |Ψi
eiπL1 eiπK1 |Ψi = −|Ψi .
I
(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.)
Allowed states have Isospin 0 (6 Li), with spin/parity
J P = 1+ , 3+ , 4− , 5+ , 5− , · · · ,
and Isospin 1 (6 He ,6 Li ,6 Be), with spin/parity
J P = 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.
B=8 Skyrmion (mπ = 1)
I
Beryllium-8’s clear rotational band with states of spin/parity
J P = 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 J P = 0− , 2− states in addition to the
known J P = 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 J P = 0+
ground states.
28.1 MeV J=0 + , I=2
27.8 MeV J=0 +, I=2
Helium−8
27.5 MeV J=0 + , I=2
27.0 MeV J=0 + , I=2
26.3 MeV J=0 +, I=2
Carbon−8
19.3 MeV J=3 + , I=1
17.0 MeV
J=2 + ,
I=1
19.1 MeV J=3 +, I=1
18.6 MeV J=3 +, I=1
16.6 MeV J=2 +, I=1
16.4 MeV J=2 + , I=1
Lithium−8
Boron−8
11.4 MeV J=4 +, I=0
3.0 MeV
J=2 +, I=0
J=0 +, I=0
Beryllium−8
Energy level diagram for nuclei with baryon number 8
Skyrmions of Large Baryon Number
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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.
B=24 Skyrmion
B=32 Skyrmion
B=31 Skyrmion (B=32 with corner cut off)
B=108 Skyrmion [P.H.C. Lau]
B = 12 Skyrmions and Carbon-12
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There are two Skyrmions of very similar energies – and
both apparently stable. One is a triangle of B = 4 cubes
with D3h symmetry, the other a linear chain with D4h
symmetry.
I
P.H.C. Lau and NSM have quantised their rotational motion
as rigid bodies. The relevant inertia coefficients V11 = V22
and V33 have been calculated for each Skyrmion.
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Allowed states for the triangular Skyrmion have spin/parity
J P = 0+ , 2+ , 3− , 4− , 4+ , 5− , 6+ , 6− , 6+ in rotational bands
with K = 0, 3, 6. Their energies are
1
1
1
J(J + 1) +
−
K2 .
E(J, K ) =
2V11
2V33 2V11
These match well the experimental rotational band of the
ground state.
The Hoyle State of Carbon-12
I
Our model suggests that the 0+ Hoyle state corresponds
to a linear chain of alpha particles.
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The J P = 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.
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The Skyrme model makes predictions for several spin 6
states.
B=12 Skyrmion with D3h symmetry
B=12 Skyrmion with D4h symmetry
45
40
35
30
E (MeV)
25
20
15
Hoyle Band
10
5
Ground-State Band
0
0
5
10
15
20
25
30
35
40
45
J(J+1)
Rotational energy spectrum of the two B=12 Skyrmions,
compared with data
States of Oxygen-16
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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.
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The configurations all have D2 symmetry so miss the 1−
vibrational states.
Scattering channel of four alpha particles
Energy level diagram for 16 O states. Solid ≡ Skyrme model.
Circle/Triangle ≡ +/− parity. Hollow ≡ Experiment.
Summary
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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 for 7 Li/7 Be and 16 O.