Two-Cluster Distribution Functions

New

New: Green's function Monte Carlo results

We have computed momentum distributions and configuration space overlaps in a number of systems, including 3He, 4He, and 6Li, as discussed in Forest et al., Phys. Rev. C54, 646 (1996). The following are files of smoothed radial overlap functions, R0 and R2, for the S- and D-waves, illustrated in this paper:

dp in 3He (Fig.14)

dd in 4He (Fig.18)

αd in 6Li (Fig.21)

The next files are the actual outputs of our Monte Carlo codes, containing the relevant amplitudes in momentum space and the (unsmoothed) radial overlap functions, both with their Monte Carlo statistical errors.

dp in 3He

dd in 4He

tp in 4He

αd in 6Li

αt in 7Li

αα; in 8Be

Here is a figure of the 4+2, 4+3, and 4+4 overlaps in different excited states: cluster.eps

!!! UPDATE !!!

The following overlaps with 6Li have been made with VMC wave functions as of February 2014 using the Argonne v18 + Urbana X Hamiltonian. (Urbana X is intermediate between Urbana IX and Illinois-7.)

5He(3/2-) 14-Feb-14

5Li(3/2-) 14-Feb-14

!!! UPDATE !!!

All the 6He and 6Li files below have been updated using the latest (September 2009) variational Monte Carlo (VMC) wave functions for the Argonne v18 + Urbana IX Hamiltonian.

We have calculated the overlaps of 7Li (3/2-) ground state with 6He + proton where 6He can be in any of five states: (0+) ground state, (2+) excited state at 1.9 MeV, (2+) excited state at 3.8 MeV, (1+) excited state at 4.3 MeV, and (0+) excited state at 5.9 MeV (theoretical excitation energies). These correspond to states that are predominantly 1S0, 1D2, 3P2, 3P1, and 3P0 in character. The proton can be in p1/2, p3/2, f5/2, or f7/2 states (as permitted by angular momentum conservation). The output files with the momentum amplitudes are given below: after some header information, the amplitudes for the p1/2, p3/2, f5/2, and f7/2 are tabulated as a function of k, and then integrated spectroscopic factors are given at the end. Momentum distributions are determined by

N(k) = A(k)2/(4*pi)
and normalizations are such that
Spectroscopic factor = 1/(2*pi)3 Integral ( d3k N(k) ) .

Also included are r-space amplitudes with the normalization:

Spectroscopic factor = Integral ( A2(r) r2 dr) .

6He(0+) updated 18-Sep-09

6He(2+) updated 18-Sep-09

6He(2+)* updated 18-Sep-09

6He(1+) updated 18-Sep-09

6He(0+)* updated 18-Sep-09

We have also calculated the overlaps of 7Li (3/2-) ground state with 6Li + neutron where 6Li can be in any of five states: (1+) ground state, (3+) excited state at 3.1 MeV, (2+) excited state at 4.6 MeV, (1+) excited state at 6.0 MeV, and (1+) excited state at 11.1 MeV (theoretical excitation energies). These correspond to states that are predominantly 3S1, 3D3, 3D2, 3D1, and 1P1 in character. For the (0+;T=1) excited state at 3.9 MeV and the (2+;T=1) excited state at 5.9 MeV, one should use 1/2 of the corresponding 6He states above.

6Li(1+) updated 18-Sep-09

6Li(3+) updated 18-Sep-09

6Li(2+) updated 18-Sep-09

6Li(1+)* updated 18-Sep-09

6Li(1+)** updated 18-Sep-09

Here are some single-nucleon densities for these final states 6Li(1+,3+,0+)

New results for overlaps of 8Li (2+) ground state with 7Li + neutron where 7Li can be in a number of different states, starting with the (3/2-) ground state and (1/2-) first excited state at 0.48 MeV. The following files have been made with a charge-independent wave function, so they may also be used as a first approximation for 8B (2+) overlap with 7Be + proton. Output and normalization are as described above for the 6He files. Also included are r-space amplitudes with the normalization:

Spectroscopic factor = Integral ( A2(r) r2 dr) .

7Li(3/2-) ground state updated 10-Nov-09

7Li(1/2-) @ 0.48 MeV updated 10-Nov-09

7Li(7/2-) @ 4.65 MeV updated 09-Feb-07

7Li(5/2-) @ 6.60 MeV updated 09-Feb-07

7Li(5/2-) @ 7.45 MeV updated 10-Feb-07

7Li(3/2-) @ 8.75 MeV updated 10-Feb-07

7Li(1/2-) @ 9.09 MeV updated 10-Feb-07

7Li(7/2-) @ 9.57 MeV updated 11-Feb-07

Here are some single-nucleon densities for these final states 7Li(3/2-,1/2-)

Results for overlaps of 9Be (3/2-) ground state with 8Be(J) + neutron(j) for lowest-lying J states. Both momentum and configuration space overlaps are given, with normalizations as given above.

8Be(0+) updated 4/1/10

8Be(2+) updated 4/2/10

Results for overlaps of 9Be (3/2-) ground state with 8Li(J) + proton(j) for lowest-lying J states. Both momentum and configuration space overlaps are given, with normalizations as given above.

8Li(2+) updated 30-Dec-10

8Li(1+) updated 30-Dec-10

8Li(3+) updated 31-Dec-10

Results for overlap of 9Li(3/2-) ground state with 8Li(J) + neutron(j). Under charge symmetry, these would correspond to overlaps 9C(3/2-) with 8B(J) + proton(j):

8Li(2+) updated 16-Dec-10

8Li(1+) updated 17-Dec-10

8Li(3+) updated 18-Dec-10

!!! UPDATE !!!

8Li(0+;2) updated 24-Feb-14

and explicitly for 9C(3/2-) with 8B(J) + proton(j):

8B(0+;2) updated 25-Feb-14

!!! UPDATE !!!

Here are some single-nucleon densities for these final states 8Li(2+,1+,3+)

Results for overlaps of 9Li (3/2-) ground state with 8He(J) + proton(j) for lowest-lying J states. Both momentum and configuration space overlaps are given, with normalizations as given above.

8He(0+) updated 28-Dec-10

8He(2+) updated 28-Dec-10

8He(1+) updated 28-Dec-10

8He(0+)* updated 29-Dec-10

8He(2+)* updated 29-Dec-10

Results for overlap of 10Be(0+) ground state with 9Li(J) + proton(j). Under charge symmetry, this would correspond to overlap of 10C(0+) with 9C(J) + neutron(j):

9Li(3/2-) From 08-Jul-11

9Li(1/2-) From 08-Jul-11

Here are some single-nucleon densities for these final states 9Li(3/2-,1/2-)

Results for overlap of 10Be(0+) ground state with 9Be(J) + neutron(j). Under charge symmetry, this would correspond to overlap of 10C(0+) with 9B(J) + proton(j):

9Be(3/2-) From 16-Jul-11

Here is the single-nucleon density for the final state 9Be(3/2-)

Robert B. Wiringa
Last update 16 Jul 2011