VMC Spectroscopic Overlaps

Part I: Pickup Reactions

This web page presents recent spectroscopic overlaps calculated for a variety of nuclei in the range A=3-12. Corresponding one-nucleon densities 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) or the Norfolk Δ-full chiral effective field theory interactions NV2+3-Ia, -Ia*, and -IIb*.

A number of these results have served as input to papers on specific reactions, including:
Mecca, et al., Phys. Lett. B 798, 134989 (2019);
Crespo, et al., Phys. Lett. B 803, 135355 (2020);
Cravo, et al., Phys. Lett. B 859, 139087 (2024).

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.

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 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).

Following are figures and tabulations of the single-nucleon r-space amplitudes A(r) (in fm^-3/2) and the integrated spectroscopic factors with the normalization:

S.F. = Integral ( A²(r) r² dr ) .

Momentum-space amplitudes are available upon request.

In the following, states are designated by ^AZ(J^π;T) except the T is omitted for states of the lowest isospin available to that nucleus. Second excited states of the same quantum numbers are denoted by ^AZ(J^π₂;T).

Robert B. Wiringa
Last update November 12, 2024

VMC Spectroscopic Overlaps

Part I: Pickup Reactions

Part II: Stripping Reactions

Older VMC Spectroscopic Overlaps

GFMC Spectroscopic Overlaps

³H(1/2+)-> AV18+UX ²H(1+)+n Figure ²H(1+)+n Table	³H(1/2+)-> NV2+3-Ia ²H(1+)+n Figure ²H(1+)+n Table	³He(1/2+)-> AV18+UX ²H(1+)+p Figure ²H(1+)+p Table	³He(1/2+)-> NV2+3-Ia ²H(1+)+p Figure ²H(1+)+p Table
	⁴He(0+)-> AV18+UX ³H(1/2+)+p Figure ³H(1/2+)+p Table ³He(1/2+)+n Figure ³He(1/2+)+n Table	⁴He(0+)-> NV2+3-Ia ³H(1/2+)+p Figure ³H(1/2+)+p Table ³He(1/2+)+n Figure ³He(1/2+)+n Table	⁴He(0+)-> NV2+3-Ia* ³H(1/2+)+p Figure ³H(1/2+)+p Table ³He(1/2+)+n Figure ³He(1/2+)+n Table	⁴He(0+)-> NV2+3-IIb* ³H(1/2+)+p Figure ³H(1/2+)+p Table ³He(1/2+)+n Figure ³He(1/2+)+n Table
	⁶He(0+)-> AV18+UX ⁵H+p Summary ⁵H(1/2+)+p Figure ⁵H(1/2+)+p Table ⁵H(5/2+)+p Figure ⁵H(5/2+)+p Table ⁵H(3/2+)+p Figure ⁵H(3/2+)+p Table ⁵He+n Summary ⁵He(3/2-)+n Figure ⁵He(3/2-)+n Table ⁵He(1/2-)+n Figure ⁵He(1/2-)+n Table	⁶Li(1+)-> AV18+UX ⁵He+p Summary ⁵He(3/2-)+p Figure ⁵He(3/2-)+p Table ⁵He(1/2-)+p Figure ⁵He(1/2-)+p Table
	⁷He(3/2-) -> AV18+UX ⁶He(0+)+n Figure ⁶He(0+)+n Table ⁶He(2+)+n Figure ⁶He(2+)+n Table	⁷Li(3/2-;3/2) -> AV18+UX ⁶He(0+)+p Figure ⁶He(0+)+p Table ⁶Li(0+;1)+n Figure ⁶Li(0+;1)+n Table	⁷Be(3/2-;3/2) -> AV18+UX ⁶Li(0+;1)+p Figure ⁶Li(0+;1)+p Table ⁶Be(0+)+n Figure ⁶Be(0+)+n Table	⁷B(3/2-) -> AV18+UX ⁶Be(0+)+p Figure ⁶Be(0+)+p Table ⁶Be(2+)+p Figure ⁶Be(2+)+p Table
		⁷Li(3/2-) -> AV18+UX ⁶He+p Summary ⁶He(0+)+p Figure ⁶He(0+)+p Table ⁶He(2+)+p Figure ⁶He(2+)+p Table ⁶Li+n Summary ⁶Li(1+)+n Figure ⁶Li(1+)+n Table ⁶Li(3+)+n Figure ⁶Li(3+)+n Table ⁶Li(0+;1)+n Figure ⁶Li(0+;1)+n Table	⁷Li(3/2-) -> NV2+3-Ia ⁶He+p Summary ⁶He(0+)+p Figure ⁶He(0+)+p Table ⁶He(2+)+p Figure ⁶He(2+)+p Table ⁶Li+n Summary ⁶Li(1+)+n Figure ⁶Li(1+)+n Table ⁶Li(3+)+n Figure ⁶Li(3+)+n Table ⁶Li(0+;1)+n Figure ⁶Li(0+;1)+n Table	⁷Li(3/2-) -> NV2+3-Ia* ⁶He+p Summary ⁶He(0+)+p Figure ⁶He(0+)+p Table ⁶He(2+)+p Figure ⁶He(2+)+p Table ⁶Li+n Summary ⁶Li(1+)+n Figure ⁶Li(1+)+n Table ⁶Li(3+)+n Figure ⁶Li(3+)+n Table ⁶Li(0+;1)+n Figure ⁶Li(0+;1)+n Table	⁷Li(3/2-) -> NV2+3-IIb* ⁶He+p Summary ⁶He(0+)+p Figure ⁶He(0+)+p Table ⁶He(2+)+p Figure ⁶He(2+)+p Table ⁶Li+n Summary ⁶Li(1+)+n Figure ⁶Li(1+)+n Table ⁶Li(3+)+n Figure ⁶Li(3+)+n Table ⁶Li(0+;1)+n Figure ⁶Li(0+;1)+n Table	⁷Be(3/2-) -> AV18+UX ⁶Be+n Summary ⁶Be(0+)+n Figure ⁶Be(0+)+n Table ⁶Be(2+)+n Figure ⁶Be(2+)+n Table ⁶Li+p Summary ⁶Li(1+)+p Figure ⁶Li(1+)+p Table ⁶Li(3+)+p Figure ⁶Li(3+)+p Table ⁶Li(0+;1)+p Figure ⁶Li(0+;1)+p Table	⁷Be(3/2-) -> NV2+3-Ia ⁶Be+n Summary ⁶Be(0+)+n Figure ⁶Be(0+)+n Table ⁶Be(2+)+n Figure ⁶Be(2+)+n Table ⁶Li+p Summary ⁶Li(1+)+p Figure ⁶Li(1+)+p Table ⁶Li(3+)+p Figure ⁶Li(3+)+p Table ⁶Li(0+;1)+p Figure ⁶Li(0+;1)+p Table
	⁸He(0+)-> AV18+UX ⁷He+n Summary ⁷He(3/2-)+n Figure ⁷He(3/2-)+n Table ⁷He(1/2-)+n Figure ⁷He(1/2-)+n Table	⁸Li(2+)-> AV18+UX ⁷He+p Summary ⁷He(3/2-)+p Figure ⁷He(3/2-)+p Table ⁷He(5/2-)+p Figure ⁷He(5/2-)+p Table	⁸Li(2+)-> NV2+3-Ia ⁷He+p Summary ⁷He(3/2-)+p Figure ⁷He(3/2-)+p Table ⁷He(5/2-)+p Figure ⁷He(5/2-)+p Table	⁸Li(2+)-> NV2+3-Ia* ⁷He+p Summary ⁷He(3/2-)+p Figure ⁷He(3/2-)+p Table ⁷He(5/2-)+p Figure ⁷He(5/2-)+p Table	⁸Li(2+)-> NV2+3-IIb* ⁷He+p Summary ⁷He(3/2-)+p Figure ⁷He(3/2-)+p Table ⁷He(5/2-)+p Figure ⁷He(5/2-)+p Table
		⁸Li(2+)-> AV18+UX ⁷Li+n Summary ⁷Li(3/2-)+n Figure ⁷Li(3/2-)+n Table ⁷Li(1/2-)+n Figure ⁷Li(1/2-)+n Table ⁷Li(7/2-)+n Figure ⁷Li(7/2-)+n Table ⁷Li(5/2-)+n Figure ⁷Li(5/2-)+n Table ⁷Li(5/2^-₂)+n Figure ⁷Li(5/2^-₂)+n Table ⁷Li(7/2^-₂)+n Figure ⁷Li(7/2^-₂)+n Table	⁸Li(2+)-> NV2+3-Ia ⁷Li+n Summary ⁷Li(3/2-)+n Figure ⁷Li(3/2-)+n Table ⁷Li(1/2-)+n Figure ⁷Li(1/2-)+n Table ⁷Li(7/2-)+n Figure ⁷Li(7/2-)+n Table ⁷Li(5/2-)+n Figure ⁷Li(5/2-)+n Table ⁷Li(5/2^-₂)+n Figure ⁷Li(5/2^-₂)+n Table ⁷Li(7/2^-₂)+n Figure ⁷Li(7/2^-₂)+n Table	⁸Li(2+)-> NV2+3-Ia* ⁷Li+n Summary ⁷Li(3/2-)+n Figure ⁷Li(3/2-)+n Table ⁷Li(1/2-)+n Figure ⁷Li(1/2-)+n Table ⁷Li(7/2-)+n Figure ⁷Li(7/2-)+n Table ⁷Li(5/2-)+n Figure ⁷Li(5/2-)+n Table ⁷Li(5/2^-₂)+n Figure ⁷Li(5/2^-₂)+n Table ⁷Li(7/2^-₂)+n Figure ⁷Li(7/2^-₂)+n Table	⁸Li(2+)-> NV2+3-IIb* ⁷Li+n Summary ⁷Li(3/2-)+n Figure ⁷Li(3/2-)+n Table ⁷Li(1/2-)+n Figure ⁷Li(1/2-)+n Table ⁷Li(7/2-)+n Figure ⁷Li(7/2-)+n Table ⁷Li(5/2-)+n Figure ⁷Li(5/2-)+n Table ⁷Li(5/2^-₂)+n Figure ⁷Li(5/2^-₂)+n Table ⁷Li(7/2^-₂)+n Figure ⁷Li(7/2^-₂)+n Table	⁸B(2+)-> AV18+UX ⁷Be+p Summary ⁷Be(3/2-)+p Figure ⁷Be(3/2-)+p Table ⁷Be(1/2-)+p Figure ⁷Be(1/2-)+p Table ⁷Be(7/2-)+p Figure ⁷Be(7/2-)+p Table ⁷Be(5/2-)+p Figure ⁷Be(5/2-)+p Table ⁷Be(5/2^-₂)+p Figure ⁷Be(5/2^-₂)+p Table ⁷Be(7/2^-₂)+p Figure ⁷Be(7/2^-₂)+p Table	⁸B(2+)-> NV2+3-Ia ⁷Be+p Summary ⁷Be(3/2-)+p Figure ⁷Be(3/2-)+p Table ⁷Be(1/2-)+p Figure ⁷Be(1/2-)+p Table ⁷Be(7/2-)+p Figure ⁷Be(7/2-)+p Table ⁷Be(5/2-)+p Figure ⁷Be(5/2-)+p Table ⁷Be(5/2^-₂)+p Figure ⁷Be(5/2^-₂)+p Table ⁷Be(7/2^-₂)+p Figure ⁷Be(7/2^-₂)+p Table
		⁹Li(3/2-) -> AV18+UX ⁸He+p Summary ⁸He(0+)+p Figure ⁸He(0+)+p Table ⁸He(2+)+p Figure ⁸He(2+)+p Table ⁸Li+n Summary ⁸Li(2+)+n Figure ⁸Li(2+)+n Table ⁸Li(1+)+n Figure ⁸Li(1+)+n Table ⁸Li(3+)+n Figure ⁸Li(3+)+n Table	⁹Li(3/2-) -> NV2+3-Ia ⁸He+p Summary ⁸He(0+)+p Figure ⁸He(0+)+p Table ⁸He(2+)+p Figure ⁸He(2+)+p Table ⁸Li+n Summary ⁸Li(2+)+n Figure ⁸Li(2+)+n Table ⁸Li(1+)+n Figure ⁸Li(1+)+n Table ⁸Li(3+)+n Figure ⁸Li(3+)+n Table	⁹Li(3/2-) -> NV2+3-Ia* ⁸He+p Summary ⁸He(0+)+p Figure ⁸He(0+)+p Table ⁸He(2+)+p Figure ⁸He(2+)+p Table ⁸Li+n Summary ⁸Li(2+)+n Figure ⁸Li(2+)+n Table ⁸Li(1+)+n Figure ⁸Li(1+)+n Table ⁸Li(3+)+n Figure ⁸Li(3+)+n Table		⁹C(3/2-) -> AV18+UX ⁸C+n Summary ⁸C(0+)+n Figure ⁸C(0+)+n Table ⁸C(2+)+n Figure ⁸C(2+)+n Table ⁸B+p Summary ⁸B(2+)+p Figure ⁸B(2+)+p Table ⁸B(1+)+p Figure ⁸B(1+)+p Table ⁸B(3+)+p Figure ⁸B(3+)+p Table *⁸B(0+;2)+p Figure ⁸B(0+;2)+p* Table	⁹C(3/2-) -> NV2+3-Ia ⁸C+n Summary ⁸C(0+)+n Figure ⁸C(0+)+n Table ⁸C(2+)+n Figure ⁸C(2+)+n Table ⁸B+p Summary ⁸B(2+)+p Figure ⁸B(2+)+p Table ⁸B(1+)+p Figure ⁸B(1+)+p Table ⁸B(3+)+p Figure ⁸B(3+)+p Table *⁸B(0+;2)+p Figure ⁸B(0+;2)+p* Table
			⁹Be(3/2-) -> AV18+UX ⁸Li+p Summary ⁸Li(2+)+p Figure ⁸Li(2+)+p Table ⁸Li(1+)+p Figure ⁸Li(1+)+p Table ⁸Li(3+)+p Figure ⁸Li(3+)+p Table ⁸Be+n Summary ⁸Be(0+)+n Figure ⁸Be(0+)+n Table ⁸Be(2+)+n Figure ⁸Be(2+)+n Table ⁸Be(2⁺₂)+n Figure ⁸Be(2⁺₂)+n Table	⁹Be(3/2-) -> NV2+3-Ia ⁸Li+p Summary ⁸Li(2+)+p Figure ⁸Li(2+)+p Table ⁸Li(1+)+p Figure ⁸Li(1+)+p Table ⁸Li(3+)+p Figure ⁸Li(3+)+p Table ⁸Be+n Summary ⁸Be(0+)+n Figure ⁸Be(0+)+n Table ⁸Be(2+)+n Figure ⁸Be(2+)+n Table ⁸Be(2⁺₂)+n Figure ⁸Be(2⁺₂)+n Table	⁹Be(3/2-) -> NV2+3-Ia* ⁸Li+p Summary ⁸Li(2+)+p Figure ⁸Li(2+)+p Table ⁸Li(1+)+p Figure ⁸Li(1+)+p Table ⁸Li(3+)+p Figure ⁸Li(3+)+p Table ⁸Be+n Summary ⁸Be(0+)+n Figure ⁸Be(0+)+n Table ⁸Be(2+)+n Figure ⁸Be(2+)+n Table ⁸Be(2⁺₂)+n Figure ⁸Be(2⁺₂)+n Table	⁹Be(3/2-) -> NV2+3-IIb* ⁸Li+p Summary ⁸Li(2+)+p Figure ⁸Li(2+)+p Table ⁸Li(1+)+p Figure ⁸Li(1+)+p Table ⁸Li(3+)+p Figure ⁸Li(3+)+p Table ⁸Be+n Summary ⁸Be(0+)+n Figure ⁸Be(0+)+n Table ⁸Be(2+)+n Figure ⁸Be(2+)+n Table ⁸Be(2⁺₂)+n Figure ⁸Be(2⁺₂)+n Table
			¹⁰Be(0+) -> AV18+UX ⁹Li+p Summary ⁹Li(3/2-)+p Figure ⁹Li(3/2-)+p Table ⁹Li(1/2-)+p Figure ⁹Li(1/2-)+p Table ⁹Be+n Summary ⁹Be(3/2-)+n Figure ⁹Be(3/2-)+n Table ⁹Be(1/2-)+n Figure ⁹Be(1/2-)+n Table	¹⁰Be(0+) -> NV2+3-Ia ⁹Li+p Summary ⁹Li(3/2-)+p Figure ⁹Li(3/2-)+p Table ⁹Li(1/2-)+p Figure ⁹Li(1/2-)+p Table ⁹Be+n Summary ⁹Be(3/2-)+n Figure ⁹Be(3/2-)+n Table ⁹Be(1/2-)+n Figure ⁹Be(1/2-)+n Table	¹⁰B(3+) -> AV18+UX ⁹Be+p Summary ⁹Be(3/2-)+p Figure ⁹Be(3/2-)+p Table ⁹Be(5/2-)+p Figure ⁹Be(5/2-)+p Table	¹⁰B(3+) -> NV2+3-Ia ⁹Be+p Summary ⁹Be(3/2-)+p Figure ⁹Be(3/2-)+p Table ⁹Be(5/2-)+p Figure ⁹Be(5/2-)+p Table	¹⁰C(0+) -> AV18+UX ⁹C+n Summary ⁹C(3/2-)+n Figure ⁹C(3/2-)+n Table ⁹C(1/2-)+n Figure ⁹C(1/2-)+n Table	¹⁰C(0+) -> NV2+3-Ia ⁹C+n Summary ⁹C(3/2-)+n Figure ⁹C(3/2-)+n Table ⁹C(1/2-)+n Figure ⁹C(1/2-)+n Table
				¹¹B(3/2-) -> AV18+UX ¹⁰Be+p Summary ¹⁰Be(0+)+p Figure ¹⁰Be(0+)+p Table ¹⁰Be(2+)+p Figure ¹⁰Be(2+)+p Table ¹⁰Be(2⁺₂)+p Figure ¹⁰Be(2⁺₂)+p Table ¹⁰B+n Summary ¹⁰B(3+)+n Figure ¹⁰B(3+)+n Table ¹⁰B(1+)+n Figure ¹⁰B(1+)+n Table *¹⁰B(0+;1)+n Figure ¹⁰B(0+;1)+n* Table ¹⁰B(1⁺₂)+n Figure ¹⁰B(1⁺₂)+n Table	¹¹B(3/2-) -> NV2+3-Ia ¹⁰Be+p Summary ¹⁰Be(0+)+p Figure ¹⁰Be(0+)+p Table ¹⁰Be(2+)+p Figure ¹⁰Be(2+)+p Table ¹⁰Be(2⁺₂)+p Figure ¹⁰Be(2⁺₂)+p Table ¹⁰B+n Summary ¹⁰B(3+)+n Figure ¹⁰B(3+)+n Table ¹⁰B(1+)+n Figure ¹⁰B(1+)+n Table *¹⁰B(0+;1)+n Figure ¹⁰B(0+;1)+n* Table ¹⁰B(1⁺₂)+n Figure ¹⁰B(1⁺₂)+n Table	¹¹C(3/2-) -> AV18+UX ¹⁰C+n Summary ¹⁰C(0+)+n Figure ¹⁰C(0+)+n Table ¹⁰C(2+)+n Figure ¹⁰C(2+)+n Table ¹⁰C(2+₂)+n Figure ¹⁰C(2+₂)+n Table ¹⁰B+p Summary ¹⁰B(3+)+p Figure ¹⁰B(3+)+p Table ¹⁰B(1+)+p Figure ¹⁰B(1+)+p Table *¹⁰B(0+;1)+p Figure ¹⁰B(0+;1)+p* Table ¹⁰B(1⁺₂)+p Figure ¹⁰B(1⁺₂)+p Table	¹¹C(3/2-) -> NV2+3-Ia ¹⁰C+n Summary ¹⁰C(0+)+n Figure ¹⁰C(0+)+n Table ¹⁰C(2+)+n Figure ¹⁰C(2+)+n Table ¹⁰C(2+₂)+n Figure ¹⁰C(2+₂)+n Table ¹⁰B+p Summary ¹⁰B(3+)+p Figure ¹⁰B(3+)+p Table ¹⁰B(1+)+p Figure ¹⁰B(1+)+p Table *¹⁰B(0+;1)+p Figure ¹⁰B(0+;1)+p* Table ¹⁰B(1⁺₂)+p Figure ¹⁰B(1⁺₂)+p Table
			¹²Be(0+) -> AV18+UX ¹¹Be+n Summary ¹¹Be(1/2-)+n Figure ¹¹Be(1/2-)+n Table ¹¹Be(3/2-)+n Figure ¹¹Be(3/2-)+n Table ¹¹Be(3/2-₂)+n Figure ¹¹Be(3/2-₂)+n Table	¹²B(1+) -> AV18+UX ¹¹Be+p Summary ¹¹Be(1/2-)+p Figure ¹¹Be(1/2-)+p Table ¹¹Be(3/2-)+p Figure ¹¹Be(3/2-)+p Table ¹¹Be(5/2-)+p Figure ¹¹Be(5/2-)+p Table ¹¹B+n Summary ¹¹B(3/2-)+n Figure ¹¹B(3/2-)+n Table ¹¹B(1/2-)+n Figure ¹¹B(1/2-)+n Table ¹¹B(3/2-₂)+n Figure ¹¹B(3/2-₂)+n Table	¹²C(0+) -> AV18+UX ¹¹B+p Summary ¹¹B(3/2-)+p Figure ¹¹B(3/2-)+p Table ¹¹B(1/2-)+p Figure ¹¹B(1/2-)+p Table ¹¹B(3/2-₂)+p Figure ¹¹B(3/2-₂)+p Table ¹¹C+n Summary ¹¹C(3/2-)+n Figure ¹¹C(3/2-)+n Table ¹¹C(1/2-)+n Figure ¹¹C(1/2-)+n Table ¹¹C(3/2-₂)+n Figure ¹¹C(3/2-₂)+n Table	¹²C(0+) -> NV2+3-Ia ¹¹B+p Summary ¹¹B(3/2-)+p Figure ¹¹B(3/2-)+p Table ¹¹B(1/2-)+p Figure ¹¹B(1/2-)+p Table ¹¹B(3/2-)₂+p Figure ¹¹B(3/2-₂)+p Table	¹²C(0+) -> NV2+3-Ia* ¹¹B+p Summary ¹¹B(3/2-)+p Figure ¹¹B(3/2-)+p Table ¹¹B(1/2-)+p Figure ¹¹B(1/2-)+p Table ¹¹B(3/2-)₂+p Figure ¹¹B(3/2-₂)+p Table	¹²C(0+) -> NV2+3-IIb* ¹¹B+p Summary ¹¹B(3/2-)+p Figure ¹¹B(3/2-)+p Table ¹¹B(1/2-)+p Figure ¹¹B(1/2-)+p Table ¹¹B(3/2-)₂+p Figure ¹¹B(3/2-₂)+p Table