Two-Nucleon Densities

Single-Nucleon Densities

This web page presents two-nucleon densities calculated for a variety of nuclei in the range A=2-12. Corresponding single-nucleon densities can be found here. These are from variational Monte Carlo calculations (VMC) using either the Argonne v18 two-nucleon and Urbana X three-nucleon potentials (AV18+UX) or one of the Norfolk Δ-full chiral effective field theory interactions: NV2+3-Ia, -Ia*, -Ib*, -IIa*, -IIb*.

Original density and momentum distribution results for AV18+UX were reported in:
Wiringa, et al., Phys. Rev. C 89, 024305 (2014).

The results with Norfolk chiral EFT interactions are reported in:
Piarulli, et al., Phys. Rev C 107, 014314 (2023).

Please cite the above papers when using results from these pages.

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);
Pastore, et al., Phys. Rev. C 90, 024321 (2014).

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.

An excellent overall review of quantum Monte Carlo methods for nuclei can be found at:
Carlson, et al., Rev. Mod. Phys. 87, 1067 (2015)

The largest nuclei are evaluated using the cluster VMC (CVMC) method.

The CVMC method is described in
Pieper, et al., Phys. Rev. C 46, 1741 (1992) for A=16 with AV1 4+UVII
Lonardoni, et al., Phys. Rev. C 96, 024326 (2017) for A=16,40 with AV18+UIX.

The Norfolk interactions are described and used in the following papers:
Piarulli, et al., Phys. Rev. C 94, 054007 (2016).
Piarulli, et al., Phys. Rev. Lett. 120, 052503 (2018).
Baroni, et al., Phys. Rev. C 98, 044003 (2018).
Schiavilla, et al., Phys. Rev. C 99, 034005 (2019).

Density distributions are provided for proton-proton, neutron-proton and neutron-neutron (where different) pairs. The densities are for the same wave functions used in generating the two-nucleon momentum distributions given here.

Following are figures and files that tabulate the two-nucleon densities to give an overall view of their shapes. The normalization is chosen such that:

Z(Z-1)/2 = ∫ d3r ρpp(r)
NZ = ∫ d3r ρnp(r)
N(N-1)/2 = ∫ d3r ρnn(r)

Where pp and nn densities are the same, as in T=0 nuclei, we give only one set.

2H(1+)
AV18
Figure
Table 		2H(1+)
NV2-Ia
Figure
Table 		2H(1+)
NV2-Ib
Figure
Table 		2H(1+)
NV2-IIa
Figure
Table 		2H(1+)
NV2-IIb
Figure
Table
3H(1/2+)
AV18+UX
Figure
Table 		3H(1/2+)
NV2+3-Ia
Figure
Table 		3H(1/2+)
NV2+3-Ia*
Figure
Table 		3H(1/2+)
NV2+3-Ib*
Figure
Table 		3H(1/2+)
NV2+3-IIa*
Figure
Table 		3H(1/2+)
NV2+3-IIb*
Figure
Table
3He(1/2+)
AV18+UX
Figure
Table 		3He(1/2+)
NV2+3-Ia
Figure
Table 		3He(1/2+)
NV2+3-Ia*
Figure
Table 		3He(1/2+)
NV2+3-Ib*
Figure
Table 		3He(1/2+)
NV2+3-IIa*
Figure
Table 		3He(1/2+)
NV2+3-IIb*
Figure
Table
4He(0+)
AV18
Figure
Table 		4He(0+)
AV18+UX
Figure
Table
 4He(0+)
NV2+3-Ia
Figure
Table 		4He(0+)
NV2+3-Ia*
Figure
Table 		4He(0+)
NV2+3-Ib*
Figure
Table 		4He(0+)
NV2+3-IIa*
Figure
Table 		4He(0+)
NV2+3-IIb*
Figure
Table
5H(1/2+)
AV18+UX
Figure
Table
6He(0+)
AV18+UX
Figure
Table 		6He(0+)
NV2+3-Ia
Figure
Table 		6He(0+)
NV2+3-Ia*
Figure
Table 		6He(0+)
NV2+3-Ib*
Figure
Table 		6He(0+)
NV2+3-IIa*
Figure
Table 		6He(0+)
NV2+3-IIb*
Figure
Table 		6Li*(0+)
AV18+UX
Figure
Table 		6Be(0+)
AV18+UX
Figure
Table
6Li(1+)
AV18+UX
Figure
Table 		6Li(1+)
NV2+3-Ia
Figure
Table 		6Li(1+)
NV2+3-Ia*
Figure
Table 		6Li(1+)
NV2+3-Ib*
Figure
Table 		6Li(1+)
NV2+3-IIa*
Figure
Table 		6Li(1+)
NV2+3-IIb*
Figure
Table
7He(3/2-)
AV18+UX
Figure
Table 		7Li*(3/2-)
AV18+UX
Figure
Table 		7Be*(3/2-)
AV18+UX
Figure
Table 		7B(3/2-)
AV18+UX
Figure
Table
7Li(3/2-)
AV18+UX
Figure
Table 		7Li(3/2-)
NV2+3-Ia
Figure
Table 		7Li(3/2-)
NV2+3-Ia*
Figure
Table 		7Li(3/2-)
NV2+3-Ib*
Figure
Table 		7Li(3/2-)
NV2+3-IIa*
Figure
Table 		7Li(3/2-)
NV2+3-IIb*
Figure
Table 		7Be(3/2-)
AV18+UX
Figure
Table 		7Be(3/2-)
NV2+3-IIb*
Figure
Table
8He(0+)
AV18+UX
Figure
Table 		8He(0+)
NV2+3-Ia
Figure
Table 		8He(0+)
NV2+3-Ia*
Figure
Table 		8He(0+)
NV2+3-Ib*
Figure
Table 		8He(0+)
NV2+3-IIa*
Figure
Table 		8He(0+)
NV2+3-IIb*
Figure
Table 		8C(0+)
AV18+UX
Figure
Table
8Li(2+)
AV18+UX
Figure
Table 		8Li(2+)
NV2+3-Ia
Figure
Table 		8Li(2+)
NV2+3-Ia*
Figure
Table 		8Li(2+)
NV2+3-Ib*
Figure
Table 		8Li(2+)
NV2+3-IIa*
Figure
Table 		8Li(2+)
NV2+3-IIb*
Figure
Table 		8B(2+)
AV18+UX
Figure
Table 		8B(2+)
NV2+3-IIb*
Figure
Table
8Be(0+)
AV18+UX
Figure
Table 		8Be(0+)
NV2+3-Ia
Figure
Table 		8Be(0+)
NV2+3-Ia*
Figure
Table 		8Be(0+)
NV2+3-IIb*
Figure
Table
9Li(3/2-)
AV18+UX
Figure
Table 		9Li(3/2-)
NV2+3-Ia
Figure
Table 		9Li(3/2-)
NV2+3-Ia*
Figure
Table 		9Li(3/2-)
NV2+3-IIb*
Figure
Table 		9C(3/2-)
AV18+UX
Figure
Table 		9C(3/2-)
NV2+3-Ia
Figure
Table 		9C(3/2-)
NV2+3-Ia*
Figure
Table 		9C(3/2-)
NV2+3-IIb*
Figure
Table
9Be(3/2-)
AV18+UX
Figure
Table 		9Be(3/2-)
NV2+3-Ia
Figure
Table 		9Be(3/2-)
NV2+3-Ia*
Figure
Table 		9Be(3/2-)
NV2+3-Ib*
Figure
Table 		9Be(3/2-)
NV2+3-IIa*
Figure
Table 		9Be(3/2-)
NV2+3-IIb*
Figure
Table
10He(0+)
AV18+UX
Figure
Table		 10Be(0+)
AV18+UX
Figure
Table 		10Be(0+)
NV2+3-Ia
Figure
Table 		10Be(0+)
NV2+3-Ia*
Figure
Table 		10Be(0+)
NV2+3-IIb*
Figure
Table 		10C(0+)
AV18+UX
Figure
Table 		10C(0+)
NV2+3-Ia
Figure
Table 		10C(0+)
NV2+3-Ia*
Figure
Table 		10C(0+)
NV2+3-IIb*
Figure
Table
10B(3+)
AV18+UX
Figure
Table 		10B(3+)
NV2+3-Ia
Figure
Table 		10B(3+)
NV2+3-Ia*
Figure
Table 		10B(3+)
NV2+3-IIb*
Figure
Table
11Li(3/2-)
AV18+UX
Figure
Table 		11B(3/2-)
AV18+UX
Figure
Table 		11B(3/2-)
NV2+3-Ia
Figure
Table 		11B(3/2-)
NV2+3-Ia*
Figure
Table 		11B(3/2-)
NV2+3-IIb*
Figure
Table 		11C(3/2-)
AV18+UX
Figure
Table 		11C(3/2-)
NV2+3-Ia
Figure
Table 		11C(3/2-)
NV2+3-Ia*
Figure
Table 		11C(3/2-)
NV2+3-IIb*
Figure
Table
12Be(0+)
AV18+UX
Figure
Table 		12Be(0+)
NV2+3-Ia*
Figure
Table 		12Be(0+)
NV2+3-IIb*
Figure
Table 		12B(1+)
AV18+UX
Figure
Table 		12B(1+)
NV2+3-Ia
Figure 1
Table 		12C(0+)
AV18+UX
Figure
Table 		12C(0+)
NV2+3-Ia
Figure
Table 		12C(0+)
NV2+3-Ia*
Figure
Table 		12C(0+)
NV2+3-IIb*
Figure
Table
16O(0+)
CVMC
AV18
Figure
Table 		16O(0+)
CVMC
AV18+UIX
Figure
Table
40Ca(0+)
CVMC
AV18
Figure
Table 		40Ca(0+)
CVMC
AV18+UIX
Figure
Table

Robert B. Wiringa
Last update December 4, 2024