Nucleon and nucleon-pair momentum distributions in $A\le 12$ nuclei

Background: Momentum distributions of individual nucleons and nucleon pairs reflect features of the short-range structure of nuclei and provide useful insights into various reactions on nuclei, such as $(e,e^{\prime }p)$ and $(e,e^{\prime }pp/pn)$ electrodisintegration processes or neutrino-nucleus interaction experiments.

Purpose: To provide the nuclear physics community with the results (available online) of a systematic study of single-nucleon momentum distributions and nucleon-pair and nucleon-cluster momentum distributions for $A\le 12$ nuclei.

Method: The realistic Argonne $v_{18}$ two-nucleon and Urbana X three-nucleon potentials are used to generate accurate variational Monte Carlo wave functions for the $A\le 12$ nuclei; quantum Monte Carlo methods are used to calculate the momentum distributions.

Results: Single-nucleon distributions are given, broken down into proton and neutron components and spin-up and spin-down components where appropriate. Nucleon-pair momentum distributions are given either in pair spin and isospin ST projection or for $pp$, $pn$, and $nn$ pairs. Nucleon-cluster momentum distributions include $dp$ in ${}^3$He, $tp$ and $dd$ in ${}^4$He, $\alpha d$ in ${}^6$Li, $\alpha t$ in ${}^7$Li, and $\alpha \alpha$ in ${}^8$Be.

Conclusions: The momentum distributions exhibit common characteristic shapes, with tensor correlations (or lack thereof) playing a dominant role in the 1.5–3 fm${}^{-1}$ range, while spin-isospin correlations dominate at higher momenta.

Figure 1

The total proton momentum distribution in the deuteron is shown by the red solid line; the contributions from $S$-wave and $D$-wave components are shown separately by blue and magenta dashed lines.

Figure 2

The spin-isospin densities for polarized ${}^3$He in the $M_J=+\frac{1}{2}$ state are shown by solid red (blue) lines for spin-up protons (neutrons) and by dashed magenta (cyan) lines for spin-down protons (neutrons).

Figure 3

The proton momentum distribution in ${}^4$He is shown for the AV18 + UX Hamiltonian by the red solid line, and for AV18 alone by the blue long-dash line; simplified calculations for AV18 using 1) only a central Jastrow correlation (J) is shown by the cyan short-dash line, and 2) including spin-isospin (but no tensor) correlations (4) is indicated by the magenta dash line.

Figure 4

The proton momentum distribution in ${}^4$He is shown by the red circles; the $tp$ cluster distribution is shown by the blue triangles and the $dd$ cluster distribution is shown by the magenta squares.

Figure 5

The proton (neutron) momentum distribution in ${}^6$He is shown by the red circles (blue squares); the difference of neutron and proton densities is shown by the magenta triangles with error bars.

Figure 6

The nucleon momentum distribution ${\rho }_N$=${\rho }_p+{\rho }_n$ in ${}^6$Li is shown by the red circles; the $\alpha d$ cluster distribution for the ground state is shown by blue point-up triangles and for the first excited state by magenta point-down triangles.

Figure 7

The proton (neutron) momentum distributions in ${}^7$Li are shown by the red circles (blue squares); the $\alpha t$ cluster distribution for the ground (second excited) state is shown by magenta point-down (cyan point-up) triangles.

Figure 8

The proton momentum distribution in ${}^8$Be is shown by the red circles; the $\alpha \alpha$ cluster distribution in the $0^{+}$ ground state is shown by the blue squares and in the $2^{+}$ first excited state by the magenta triangles.

Figure 9

The proton (neutron) momentum distribution in ${}^{10}$Be is shown by the red circles (blue squares); the difference of neutron and proton densities is shown by the magenta triangles with error bars.

Figure 10

The proton momentum distributions in all $T=0$ nuclei from $A=2$–12.

Figure 11

The nucleon-pair momentum distributions in ${}^4$He as a function of relative momentum $q$ is shown by the solid black line, while the projection into pairs of total spin-isospin $ST$ is given by the various symbols.

Figure 12

The nucleon-pair momentum distributions in ${}^9$Be as a function of relative momentum $q$ is shown by the solid black line, while the projection into pairs of total spin-isospin $ST$ is given by the various symbols.

Figure 13

The nucleon-pair momentum distributions in ${}^8$He as a function of relative momentum $q$ is shown by the solid black line, while the projection into $nn$, $np$, and $pp$ pairs is shown by the blue squares, magenta triangles, and red circles, respectively.

Figure 14

The nucleon-pair momentum distributions in ${}^8$He as a function of total pair momentum $Q$ is shown by the solid black line, while the projection into $nn$, $np$, and $pp$ pairs is shown by the blue squares, magenta triangles, and red circles, respectively.

Figure 15

The nucleon-pair momentum distributions in ${}^8$Be for back-to-back $Q=0$ pairs as a function of relative momentum $q$ is shown by red circles for $pp$ pairs and blue squares for $np$ pairs.

Figure 16

The proton-proton momentum distributions in ${}^4$He averaged over the directions of $\mathbf{q}$ and $\mathbf{Q}$ as a function of $q$ for several fixed values of $Q$ from 0 to 1.25 fm${}^{-1}$.

Figure 17

The proton-neutron momentum distributions in ${}^4$He averaged over the directions of $\mathbf{q}$ and $\mathbf{Q}$ as a function of $q$ for several fixed values of $Q$ from 0 to 1.25 fm${}^{-1}$.