Two-Nucleon Momentum Distributions

Single-Nucleon Momentum Distributions

This web page presents two-nucleon momentum distributions calculated for various light nuclei in the range A=3-12. Corresponding single-nucleon momentum distributions 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 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).

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 two-nucleon densities corresponding to these wave functions are given here.

ST Pair momentum distributions as functions of q

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

N_ST = 1/(2π)³ ∫ d³q ρ_ST(q)

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

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+UX
Figure
Table
Table (v18) 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

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

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

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
⁸Be(0+)
AV18+UX
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

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

¹²C(0+)
AV18+UX
Figure
Table

NN Pair momentum distributions as functions of q

Alternatively, we can project NN pair momentum distributions ρ_NN(q), i.e., as pp, pn, and nn pairs, again as function of the relative momentum q = (k₁-k₂)/2 between pairs.

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+UX Figure Table Table (v18)	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
	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(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
		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
	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
		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
			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
			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
			¹⁰Be(0+) AV18+UX 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
					¹²C(0+) AV18+UX Figure Table	¹²C(0+) NV2+3-Ia* Figure Table

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).

³H(1/2+) AV18+UX Figure Table	³He(1/2+) AV18+UX Figure Table
	⁴He(0+) AV18+UX Figure Table Table (v18)
	⁶He(0+) AV18+UX Figure Table	⁶Li(1+) AV18+UX Figure Table
	⁸He(0+) AV18+UX Figure Table		⁸Be(0+) AV18+UX Figure Table
			¹⁰Be(0+) AV18+UX Figure Table	¹⁰B(3+) AV18+UX Figure Table
					¹²C(0+) AV18+UX Figure Table

Pair momentum distribution functions of both q and Q

³He(1/2+)

Following are figures and tabulations of the pn and pp ρ(q,Q) momentum distributions.

Figures

The first figures show the total ρ_NN(q,Q) over q = 0-5 fm-1 for values of Q from 0 up to 5 fm-1:
pp pairs are shown in red and np pairs in blue.
The next figures show contributions coming from pairs with relative separation less than (r<) or greater than (r>) 2.5 fm.
Additional figures show the same data separately for pp or np with solid circles for r< pairs and open circles for r> pairs
and a solid line for the sum of all pairs.
Curves are normalized to integrate to the total number of np (2) or pp (1) pairs.

³He(1/2+)
AV18+UX
(q,Q=0-0.5)
(q,Q=0-1)
(q,Q=0-5)
(q,Q,r<2.5)
(q,Q,r>2.5)
np(q,Q,r<,r>)
pp(q,Q,r<,r>)
³He(1/2+)
NV2+3-Ia*
(q,Q=0-0.5)
(q,Q=0-1)
(q,Q=0-5)
(q,Q,r<2.5)
(q,Q,r>2.5)
np(q,Q,r<,r>)
pp(q,Q,r<,r>)
³He(1/2+)
NV2+3-IIb*
(q,Q=0-0.5)
(q,Q=0-1)
(q,Q=0-5)
(q,Q,r<2.5)
(q,Q,r>2.5)
np(q,Q,r<,r>)
pp(q,Q,r<,r>)

Tables

Following are files for the total momentum distributions.
In each file there are successive tables of ρ_NN(q = 0-6 fm-1,Q) for fixed total Q from 0 to 5 fm-1.
Before each Q table, the integral over q for that Q is computed.
At the end of the file the corresponding integral over Q is given.
Over the range of (q,Q) computed, > 2.01 out of 2 np pairs are counted and > 1.00 out of 1 pp pair.
There are also files with tabulations for r< and r>.
For r < 2.5 fm there are ≈ 1.18 np pairs and ≈ 0.50 pp pairs.
For r > 2.5 fm there are ≈ 0.83 np pairs and ≈ 0.50 pp pairs.

³He(1/2+)
AV18+UX
Table (all)
Table (r<)
Table (r>)
³He(1/2+)
NV2+3-Ia*
Table (all)
Table (r<)
Table (r>)
³He(1/2+)
NV2+3-IIb*
Table (all)
Table (r<)
Table (r>)

⁴He(0+)

Following are figures and tabulations of the pn and pp ρ(q,Q) momentum distributions.

Figures

⁴He(0+)
AV18+UX
(q,Q=0-0.5)
(q,Q=0-1)
(q,Q=0-5)
(q,Q,r<2.5)
(q,Q,r>2.5)
np(q,Q,r<,r>)
pp(q,Q,r<,r>)
⁴He(0+)
NV2+3-Ia*
(q,Q=0-0.5)
(q,Q=0-1)
(q,Q=0-5)
(q,Q,r<2.5)
(q,Q,r>2.5)
np(q,Q,r<,r>)
pp(q,Q,r<,r>)
⁴He(0+)
NV2+3-IIb*
(q,Q=0-0.5)
(q,Q=0-1)
(q,Q=0-5)
(q,Q,r<2.5)
(q,Q,r>2.5)
np(q,Q,r<,r>)
pp(q,Q,r<,r>)

Tables

Following are files for the total momentum distributions.
In each file there are successive tables of ρ_NN(q = 0-6 fm-1,Q) for fixed total Q from 0 to 5 fm-1.
Before each Q table, the integral over q for that Q is computed.
At the end of the file the corresponding integral over Q is given.
Over the range of (q,Q) computed, > 3.98 out of 4 np pairs are counted and > 0.99 out of 1 pp pair.
There are also files with tabulations for r< and r>.
For r < 2.5 fm there are ≈ 2.80 np pairs and ≈ 0.68 pp pairs.
For r > 2.5 fm there are ≈ 1.18 np pairs and ≈ 0.32 pp pairs.

⁴He(0+)
AV18+UX
Table (all)
Table (r<)
Table (r>)
⁴He(0+)
NV2+3-Ia*
Table (all)
Table (r<)
Table (r>)
⁴He(0+)
NV2+3-IIb*
Table (all)
Table (r<)
Table (r>)

⁶Li(1+)

Following are figures and tabulations of the pn and pp ρ(q,Q) momentum distributions.

Figures

The first figures show the total ρ_NN(q,Q) over q = 0-5 fm-1 for values of Q from 0 up to 5 fm-1:
pp pairs are shown in red, nn pairs in green, and np pairs in blue.
The next figures show contributions coming from pairs with relative separation less than (r<) or greater than (r>) 2.5 fm.
Additional figures show the same data separately for pp or np with solid circles for r< pairs and open circles for r> pairs
and a solid line for the sum of all pairs.
Curves are normalized to integrate to the total number of np (9) or pp (3) pairs.

⁶Li(1+)
AV18+UX
(q,Q=0-0.5)
(q,Q=0-1)
(q,Q=0-5)
(q,Q,r<2.5)
(q,Q,r>2.5)
np(q,Q,r<,r>)
pp(q,Q,r<,r>)
⁶Li(1+)
NV2+3-Ia*
(q,Q=0-0.5)
(q,Q=0-1)
(q,Q=0-5)
(q,Q,r<2.5)
(q,Q,r>2.5)
np(q,Q,r<,r>)
pp(q,Q,r<,r>)
⁶Li(1+)
NV2+3-IIb*
(q,Q=0-0.5)
(q,Q=0-1)
(q,Q=0-5)
(q,Q,r<2.5)
(q,Q,r>2.5)
np(q,Q,r<,r>)
pp(q,Q,r<,r>)

Tables

Following are files for the total momentum distributions.
In each file there are successive tables of ρ_NN(q = 0-6 fm-1,Q) for fixed total Q from 0 to 5 fm-1.
Before each Q table, the integral over q for that Q is computed.
At the end of the file the corresponding integral over Q is given.
Over the range of (q,Q) computed, > 8.96 out of 9 np pairs are counted and > 2.98 out of 3 pp pairs.
For AV18+UX there are also files with tabulations for &rlt; and &rgt;.
For r < 2.5 fm there are ≈ 3.89 np pairs and ≈ 0.92 pp pairs.
For r > 2.5 fm there are ≈ 5.07 np pairs and ≈ 2.06 pp pairs.

⁶Li(1+)
AV18+UX
Table (all)
Table (r<)
Table (r>)
⁶Li(1+)
NV2+3-Ia*
Table (all)
Table (r<)
Table (r>)
⁶Li(1+)
NV2+3-IIb*
Table (all)
Table (r<)
Table (r>)

¹²C(0+)

Following are figures and tabulations of the pn and pp ρ(q,Q) momentum distributions.

Figures

The first figures show the total ρ_NN(q,Q) over q = 0-5 fm-1 for values of Q from 0 up to 5 fm-1:
pp pairs are shown in red, nn pairs in green, and np pairs in blue.
For the AV18+UX case there are additional sets of figures:
These show contributions coming from pairs with relative separation less than (r<) or greater than (r>) 2.5 fm.
Additional figures show the same data separately for pp or np with solid circles for r< pairs and open circles for r> pairs
and a solid line for the sum of all pairs.
Curves are normalized to integrate to the total number of np (36) or pp (15) pairs.

¹²Be(0+)
AV18+UX
(q,Q=0-1)
(q,Q=0-2)
(q,Q=0-4)

¹²Be(0+)
NV2+3-Ia*
(q,Q=0-1)
(q,Q=0-2)
(q,Q=0-4)

¹²C(0+)
AV18+UX
(q,Q=0-0.5)
(q,Q=0-1)
(q,Q=0-5)
(q,Q,r<2.5)
(q,Q,r>2.5)
np(q,Q,r<,r>)
pp(q,Q,r<,r>)
¹²C(0+)
NV2+3-Ia*
(q,Q=0-1)
(q,Q=0-2)
(q,Q=0-4)

Tables

Following are files for the total momentum distributions.
In each file there are successive tables of ρ_NN(q = 0-6 fm-1,Q) for fixed total Q from 0 to 5 fm-1.
(For NV2+3-Ia* q = 0-5 fm-1, Q = 0-4 fm-1).
Before each Q table, the integral over q for that Q is tabulated.
At the end of the file the corresponding integral over Q is given.
Over the range of (q,Q) computed, 35.6 out of 36 np pairs are counted and 14.8 out of 15 pp pairs.
For AV18+UX there are also files with tabulations for r< and r>. For r < 2.5 fm there are 12.7 np pairs and 4.2 pp pairs.
For r > 2.5 fm there are 22.9 np pairs and 10.6 pp pairs.

¹²Be(0+)
AV18+UX
Table (all)

¹²Be(0+)
NV2+3-Ia*
Table (all)

¹²C(0+)
AV18+UX
Table (all)
Table (r<)
Table (r>)
¹²C(0+)
NV2+3-Ia*
Table (all)

Robert B. Wiringa
Last update October 9, 2023

Two-Nucleon Momentum Distributions

Single-Nucleon Momentum Distributions

ST Pair momentum distributions as functions of q

NN Pair momentum distributions as functions of q

NN Pair momentum distributions as functions of Q

Pair momentum distribution functions of both q and Q

3He(1/2+)

Figures

Tables

4He(0+)

Figures

Tables

6Li(1+)

Figures

Tables

12C(0+)

Figures

Tables

³He(1/2+)

⁴He(0+)

⁶Li(1+)

¹²C(0+)