Single-Nucleon Densities

Two-Nucleon Densities

This web page presents single-nucleon densities calculated for a variety of nuclei in the range A=2-12. Corresponding two-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.

The VMC wave functions used here 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 AV14+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 neutron spin-down, neutron spin-up, proton spin-down, and proton spin-up, for the M=J state. The densities are for the same wave functions used in generating the single-nucleon momentum distributions given here

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

ANS = ∫ d3r ρNS(r)

where NS denotes proton or neutron, spin up or down, and ANS is the total number (out of A) nucleons with the given nucleon-spin projection. Where proton and neutron density distributions are the same, as in T=0 nuclei, we give only one set, and similarly, if spin-up and spin-down projections are the same, as in 0+ states, we give totals only.

2H(1+)
AV18
Figure 1
Figure 2
Table 		2H(1+)
NV2-Ia
Figure 1
Figure 2
Table 		2H(1+)
NV2-Ib
Figure 1
Figure 2
Table 		2H(1+)
NV2-IIa
Figure 1
Figure 2
Table 		2H(1+)
NV2-IIb
Figure 1
Figure 2
Table
3H(1/2+)
AV18+UX
Figure 1
Figure 2
Table

3H(1/2+)
NV2+3-Ia
Figure 1
Figure 2
Table

3H(1/2+)
NV2+3-Ia*
Figure 1
Figure 2
Figure 3
Table 		3H(1/2+)
NV2+3-Ib*
Figure 1
Figure 2
Table

3H(1/2+)
NV2+3-IIa*
Figure 1
Figure 2
Table

3H(1/2+)
NV2+3-IIb*
Figure 1
Figure 2
Figure 3
Table
3He(1/2+)
AV18+UX
Figure 1
Figure 2
Table

3He(1/2+)
NV2+3-Ia
Figure 1
Figure 2
Table

3He(1/2+)
NV2+3-Ia*
Figure 1
Figure 2
Figure 3
Table 		3He(1/2+)
NV2+3-Ib*
Figure 1
Figure 2
Table

3He(1/2+)
NV2+3-IIa*
Figure 1
Figure 2
Table

3He(1/2+)
NV2+3-IIb*
Figure 1
Figure 2
Figure 3
Table
4He(0+)
AV18
Figure 1
Figure 2
Table 		4He(0+)
AV18+UX
Figure 1
Figure 2
Table 		4He(0+)
NV2+3-Ia
Figure 1
Figure 2
Table 		4He(0+)
NV2+3-Ia*
Figure 1
Figure 2
Table 		4He(0+)
NV2+3-Ib*
Figure 1
Figure 2
Table 		4He(0+)
NV2+3-IIa*
Figure 1
Figure 2
Table 		4He(0+)
NV2+3-IIb*
Figure 1
Figure 2
Table
5H(1/2+)
AV18+UX
Figure 1
Figure 2
Table
6He(0+)
AV18+UX
Figure 1
Figure 2
Table 		6He(0+)
NV2+3-Ia
Figure 1
Figure 2
Table 		6He(0+)
NV2+3-Ia*
Figure 1
Figure 2
Table 		6He(0+)
NV2+3-Ib*
Figure 1
Figure 2
Table 		6He(0+)
NV2+3-IIa*
Figure 1
Figure 2
Table 		6He(0+)
NV2+3-IIb*
Figure 1
Figure 2
Table 		6Li*(0+)
AV18+UX
Figure 1
Figure 2
Table 		6Be(0+)
AV18+UX
Figure 1
Figure 2
Table
6Li(1+)
AV18+UX
Figure 1
Figure 2
Table

6Li(1+)
NV2+3-Ia
Figure 1
Figure 2
Table

6Li(1+)
NV2+3-Ia*
Figure 1
Figure 2
Figure 3
Table 		6Li(1+)
NV2+3-Ib*
Figure 1
Figure 2
Table

6Li(1+)
NV2+3-IIa*
Figure 1
Figure 2
Table

6Li(1+)
NV2+3-IIb*
Figure 1
Figure 2
Figure 3
Table
7He(3/2-)
AV18+UX
Figure 1
Figure 2
Table 		7Li*(3/2-)
AV18+UX
Figure 1
Figure 2
Table 		7Be*(3/2-)
AV18+UX
Figure 1
Figure 2
Table 		7B(3/2-)
AV18+UX
Figure 1
Figure 2
Table
7Li(3/2-)
AV18+UX
Figure 1
Figure 2
Table

7Li(3/2-)
NV2+3-Ia
Figure 1
Figure 2
Table

7Li(3/2-)
NV2+3-Ia*
Figure 1
Figure 2
Figure 3
Table 		7Li(3/2-)
NV2+3-Ib*
Figure 1
Figure 2
Table

7Li(3/2-)
NV2+3-IIa*
Figure 1
Figure 2
Table

7Li(3/2-)
NV2+3-IIb*
Figure 1
Figure 2
Figure 3
Table 		7Be(3/2-)
AV18+UX
Figure 1
Figure 2
Table

7Be(3/2-)
NV2+3-Ia*
Figure 1
Figure 2
Figure 3
Table 		7Be(3/2-)
NV2+3-IIb*
Figure 1
Figure 2
Figure 3
Table
8He(0+)
AV18+UX
Figure 1
Figure 2
Table 		8He(0+)
NV2+3-Ia
Figure 1
Figure 2
Table 		8He(0+)
NV2+3-Ia*
Figure 1
Figure 2
Table 		8He(0+)
NV2+3-Ib*
Figure 1
Figure 2
Table 		8He(0+)
NV2+3-IIa*
Figure 1
Figure 2
Table 		8He(0+)
NV2+3-IIb*
Figure 1
Figure 2
Table 		8C(0+)
AV18+UX
Figure 1
Figure 2
Table
8Li(2+)
AV18+UX
Figure 1
Figure 2
Table

8Li(2+)
NV2+3-Ia
Figure 1
Figure 2
Table

8Li(2+)
NV2+3-Ia*
Figure 1
Figure 2
Figure 3
Table 		8Li(2+)
NV2+3-Ib*
Figure 1
Figure 2
Table

8Li(2+)
NV2+3-IIa*
Figure 1
Figure 2
Table

8Li(2+)
NV2+3-IIb*
Figure 1
Figure 2
Figure 3
Table 		8B(2+)
AV18+UX
Figure 1
Figure 2
Table

8B(2+)
NV2+3-Ia*
Figure 1
Figure 2
Figure 3
Table 		8B(2+)
NV2+3-IIb*
Figure 1
Figure 2
Figure 3
Table
8Be(0+)
AV18+UIX
Figure 1
Figure 2
Table 		8Be(0+)
AV18+UX
Figure 1
Figure 2
Table 		8Be(0+)
NV2+3-Ia
Figure 1
Figure 2
Table 		8Be(0+)
NV2+3-Ia*
Figure 1
Figure 2
Table 		8Be(0+)
NV2+3-IIb*
Figure 1
Figure 2
Table
9Li(3/2-)
AV18+UX
Figure 1
Figure 2
Table

9Li(3/2-)
NV2+3-Ia
Figure 1
Figure 2
Table

9Li(3/2-)
NV2+3-Ia*
Figure 1
Figure 2
Figure 3
Table 		9Li(3/2-)
NV2+3-IIb*
Figure 1
Figure 2
Figure 3
Table 		9C(3/2-)
AV18+UX
Figure 1
Figure 2
Table

9C(3/2-)
NV2+3-Ia
Figure 1
Figure 2
Table

9C(3/2-)
NV2+3-Ia*
Figure 1
Figure 2
Figure 3
Table 		9C(3/2-)
NV2+3-IIb*
Figure 1
Figure 2
Figure 3
Table
9Be(3/2-)
AV18+UX
Figure 1
Figure 2
Table

9Be(3/2-)
NV2+3-Ia
Figure 1
Figure 2
Table

9Be(3/2-)
NV2+3-Ia*
Figure 1
Figure 2
Figure 3
Table 		9Be(3/2-)
NV2+3-Ib*
Figure 1
Figure 2
Table

9Be(3/2-)
NV2+3-IIa*
Figure 1
Figure 2
Table

9Be(3/2-)
NV2+3-IIb*
Figure 1
Figure 2
Figure 3
Table 		9B(3/2-)
NV2+3-Ia*
Figure 1
Figure 2
Figure 3
Table 		9B(3/2-)
NV2+3-IIb*
Figure 1
Figure 2
Figure 3
Table
10He(0+)
AV18+UX
Figure 1
Figure 2
Table 		10Be(0+)
AV18+UX
Figure 1
Figure 2
Table 		10Be(0+)
NV2+3-Ia
Figure 1
Figure 2
Table 		10Be(0+)
NV2+3-Ia*
Figure 1
Figure 2
Table 		10Be(0+)
NV2+3-IIb*
Figure 1
Figure 2
Table 		10C(0+)
AV18+UX
Figure 1
Figure 2
Table 		10C(0+)
NV2+3-Ia
Figure 1
Figure 2
Table 		10C(0+)
NV2+3-Ia*
Figure 1
Figure 2
Table 		10C(0+)
NV2+3-IIb*
Figure 1
Figure 2
Table
10B(3+)
AV18+UX
Figure 1
Figure 2
Table

10B(3+)
NV2+3-Ia
Figure 1
Figure 2
Table

10B(3+)
NV2+3-Ia*
Figure 1
Figure 2
Figure 3
Table 		10B(3+)
NV2+3-IIb*
Figure 1
Figure 2
Figure 3
Table
11Li(3/2-)
AV18+UX
Figure 1
Figure 2
Table 		11B(3/2-)
AV18+UX
Figure 1
Figure 2
Table 		11B(3/2-)
NV2+3-Ia
Figure 1
Figure 2
Table 		11B(3/2-)
NV2+3-Ia*
Figure 1
Figure 2
Table 		11B(3/2-)
NV2+3-IIb*
Figure 1
Figure 2
Table 		11C(3/2-)
AV18+UX
Figure 1
Figure 2
Table 		11C(3/2-)
NV2+3-Ia
Figure 1
Figure 2
Table 		11C(3/2-)
NV2+3-Ia*
Figure 1
Figure 2
Table 		11C(3/2-)
NV2+3-IIb*
Figure 1
Figure 2
Table
12Be(0+)
AV18+UX
Figure 1
Figure 2
Table 		12Be(0+)
NV2+3-Ia*
Figure 1
Figure 2
Table 		12Be(0+)
NV2+3-IIb*
Figure 1
Figure 2
Table 		12B(1+)
AV18+UX
Figure 1
Figure 2
Table 		12B(1+)
NV2+3-Ia
Figure 1
Figure 2
Table 		12C(0+)
AV18+UIX
Figure 1
Figure 2
Table 		12C(0+)
AV18+UX
Figure 1
Figure 2
Table 		12C(0+)
NV2+3-Ia
Figure 1
Figure 2
Table 		12C(0+)
NV2+3-Ia*
Figure 1
Figure 2
Table 		12C(0+)
NV2+3-IIb*
Figure 1
Figure 2
Table
16O(0+)
CVMC
AV18
Figure 1
Figure 2
Table 		16O(0+)
CVMC
AV18+UIX
Figure 1
Figure 2
Table
40Ca(0+)
CVMC
AV18
Figure 1
Figure 2
Table 		40Ca(0+)
CVMC
AV18+UIX
Figure 1
Figure 2
Table

Robert B. Wiringa
Last update February 20, 2025