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 (unless otherwise noted) 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.)

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Results for some of the Norfolk NV2+3 potentials are now being added:
Piarulli, et al., Phys. Rev. C 94, 054007 (2016).
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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);
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.

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., arXiv:1705.04337 for A=16,40 with AV18+UIX.

The results are generated as distributions 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:

where NS denotes proton or neutron, spin up or down, and A_NS 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+) Figure 1 Figure 2 Table	3He(1/2+) Figure 1 Figure 2 Table
	4He(0+) Figure 1 Figure 2 Table	4He(0+) AV18 Figure 1 Figure 2 Table
5H(1/2+) Figure 1 Figure 2 Table
	6He(0+) Figure 1 Figure 2 Table	6Li*(0+) Figure 1 Figure 2 Table	6Be(0+) Figure 1 Figure 2 Table
		6Li(1+) Figure 1 Figure 2 Table
	7He(3/2-) Figure 1 Figure 2 Table	7Li*(3/2-) Figure 1 Figure 2 Table	7Be*(3/2-) Figure 1 Figure 2 Table	7B(3/2-) Figure 1 Figure 2 Table
		7Li(3/2-) Figure 1 Figure 2 Table	7Be(3/2-) Figure 1 Figure 2 Table
	8He(0+) Figure 1 Figure 2 Table	8Li(2+) Figure 1 Figure 2 Table	8Be(0+) Figure 1 Figure 2 Table	8Be(0+) AV18+UIX Figure 1 Figure 2 Table	8B(2+) Figure 1 Figure 2 Table	8C(0+) Figure 1 Figure 2 Table
		9Li(3/2-) Figure 1 Figure 2 Table	9Be(3/2-) Figure 1 Figure 2 Table		9C(3/2-) Figure 1 Figure 2 Table
	10He(0+) Figure 1 Figure 2 Table		10Be(0+) Figure 1 Figure 2 Table	10B(3+) Figure 1 Figure 2 Table	10C(0+) Figure 1 Figure 2 Table
		11Li(3/2-) Figure 1 Figure 2 Table		11B(3/2-) Figure 1 Figure 2 Table	11C(3/2-) Figure 1 Figure 2 Table
			12Be(0+) Figure 1 Figure 2 Table	12B(1+) Figure 1 Figure 2 Table	12C(0+) Figure 1 Figure 2 Table	12C(0+) AV18+UIX Figure 1 Figure 2 Table
						16O(0+) CVMC AV18+UIX Figure 1 Figure 2 Table	16O(0+) CVMC AV18 Figure 1 Figure 2 Table
							40Ca(0+) CVMC AV18+UIX Figure 1 Figure 2 Table	40Ca(0+) CVMC AV18 Figure 1 Figure 2 Table

Robert B. Wiringa
Last update June 16, 2022