Two-Nucleon Momentum Distributions

This web page presents two-nucleon momentum distributions calculated for various light nuclei in the range A=3-12 as discussed in Wiringa, Schiavilla, Pieper, and Carlson: Phys. Rev. C 89, 024305 (2014) (or arXiv:1309.3794). Corresponding single-nucleon momentum distributions can be found here. These are from variational Monte Carlo calculations (VMC) using the Argonne v18 two-nucleon and Urbana X three-nucleon potentials (AV18+UX). (Urbana X is intermediate between the Urbana IX and Illinois-7 models; it has the form of UIX supplemented with a two-pion S-wave piece, while the strengths of its terms are taken from the IL7 model. It does NOT have the three-pion-ring term of IL7.)

These VMC wave functions are the starting trial functions for a number of recent Green's function Monte Carlo (GFMC) calculations:
Brida, et al., Phys. Rev. C 84, 024319 (2011);
McCutchan, et al., Phys. Rev. C 86, 024315 (2012);
Pastore, et al., Phys. Rev. C 87, 035503 (2013);
Datar, et al., Phys. Rev. Lett. 111, 062502 (2013).

More details of the wave function construction can be found in
Wiringa, Phys. Rev. C 43, 1585 (1991) for A=3,4;
Pudliner, et al., Phys. Rev. C 56, 1720 (1997) for A=6,7;
Wiringa, et al., Phys. Rev. C 62, 014001 (2000) for A=8;
Pieper, et al., Phys. Rev. C 70, 044310 (2002) for A=9,10.

Some of the results are distributions for relative pair momenta q = (k1-k2)/2 projected into total pair spin S=0 or 1 and isospin T=0 or 1. Other results are generated as a function of both q and total pair momentum Q = (k1+k2) projected into pp, np, and nn pairs.. The momentum calculations are based on the method discussed in:
Schiavilla et al., Nucl. Phys. A449, 219 (1986);
with algorithmic improvements given in:
Schiavilla et al., Phys. Rev. Lett. 98, 132501 (2007).

ST Pair momentum distributions as functions of q

Following are files for various nuclei tabulating and illustrating the ST-projected pair momentum distributions ρST(q) as a function of the relative momentum q = (k1-k2)/2 between pairs. The normalization of these distributions is given by

NST = 1/(2π)3 ∫ d3q ρST(q)

where NST is the total number of pairs with total spin S=0 or 1 and T=0 or 1.

3He(1/2+)
Figure
Table
4He(0+)
Figure
Table
6Li(1+)
Figure
Table
7Li(3/2-)
Figure
Table
8Be(0+)
Figure
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9Be(3/2-)
Figure
Table
10B(3+)
Figure
Table
12C(0+)
PRELIMINARY
Figure
Table

NN Pair momentum distributions as functions of q

Alternatively, we can project NN pair momentum distributions ρNN(q), i.e., as pp, pn, and nn pairs, again as function of the relative momentum q = (k1-k2)/2 between pairs.
4He(0+)
Figure
Table
6He(0+)
Figure
Table
6Li(1+)
Figure
Table
8He(0+)
Figure
Table
8Be(0+)
Figure
Table
10B(3+)
Figure
Table
12C(0+)
PRELIMINARY
Figure
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NN Pair momentum distributions as functions of Q

NN pair momentum distributions can also be computed as a function of of the total center-of-mass pair momentum Q = (k1+k2).
3He(1/2+)
Figure
Table
4He(0+)
Figure
Table
6He(0+)
Figure
Table
6Li(1+)
Figure
Table
8He(0+)
Figure
Table
8Be(0+)
Figure
Table
12C(0+)
PRELIMINARY
Figure
Table

Pair momentum distribution functions of both q and Q

3He(1/2+)

Following are files for 3He with tabulations of the pn and pp momentum distributions along with figures to give an overall view of the distributions. Each file is for a given total Q = (k1+k2) and contains a tabulaton of ρpN(q,Q) as a function of relative q = (k1-k2)/2 The normalization is chosen such that:

ρpN(Q) = 1/(2π)3 ∫ d3q ρpN(q,Q)

and the integrals up to 5 fm-1 in q are given at the top of the file. The total normalization is:

NpN = 1/(2π)3 ∫ d3Q ρpN(Q)

where NpN is the total number of pN pairs in the nucleus.

Q = 0.00 fm-1
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Q = 0.25 fm-1
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Q = 0.50 fm-1
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Q = 0.75 fm-1
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Q = 1.00 fm-1
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Q = 1.25 fm-1
Figure
Table

4He(0+)

Following are files for 4He with tabulations of the pn and pp momentum distributions along with figures to give an overall view of the distributions. Normalizations as given above.

Q = 0.00 fm-1
Figure
Table
Q = 0.25 fm-1
Figure
Table
Q = 0.50 fm-1
Figure
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Q = 0.75 fm-1
Figure
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Q = 1.00 fm-1
Figure
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Q = 1.25 fm-1
Figure
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Q = 1.50 fm-1
Figure
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Q = 1.75 fm-1
Figure
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Q = 2.00 fm-1
Figure
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Q = 0.05 fm-1
Figure
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Q = 0.30 fm-1
Figure
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Q = 0.55 fm-1
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Q = 0.10 fm-1
Figure
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Q = 0.35 fm-1
Figure
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Q = 0.60 fm-1
Figure
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Q = 0.15 fm-1
Figure
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Q = 0.40 fm-1
Figure
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Q = 0.65 fm-1
Figure
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Q = 0.20 fm-1
Figure
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Q = 0.45 fm-1
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Q = 0.70 fm-1
Figure
Table

6Li(1+)

Following are files for 6Li with tabulations of the pn and pp momentum distributions along with figures to give an overall view of the distributions. Normalizations as given above.

Q = 0.00 fm-1
Figure
Table

7Li(3/2-)

Following are files for 7Li with tabulations of the pn and pp momentum distributions along with figures to give an overall view of the distributions. Normalizations as given above.

Q = 0.00 fm-1
Figure
Table

8Be(0+)

Following are files for 8Be with tabulations of the pn and pp momentum distributions along with figures to give an overall view of the distributions. Normalizations as given above.

Q = 0.00 fm-1
Figure
Table

10B(3+)

Following are files for 10B with tabulations of the pn and pp momentum distributions along with figures to give an overall view of the distributions. Normalizations as given above. (correction made 11/8/2014)

Q = 0.00 fm-1
Figure
Table

12C(0+)

Following are files for 12C with tabulations of the pn and pp momentum distributions along with figures to give an overall view of the distributions. Normalizations as given above. (entry added 11/8/2014)

PRELIMINARY
Q = 0.00 fm-1
Figure
Table

Robert B. Wiringa
Last update Sat Nov 8, 2014