Step 3 — Read the interaction
Nuclear Physics
Here is the part that surprises people: the entire nuclear force of this
calculation is a small text file. usdb.int holds the USDB
interaction (Brown & Richter, 2006) — three single-particle energies
(the cost of placing one particle in each orbital, with the frozen core's
influence folded in) and 63 two-body matrix elements
(TBMEs): numbers giving the
interaction energy between a pair of particles in orbitals
and a pair in . No potential wells, no
meson exchange — the force is this table, fit directly to hundreds
of measured sd-shell levels. The fit does double duty: as step 1 noted, it
also absorbs the influence of everything outside the valence space — that
is the "effective" in effective interaction.
Two labels on each matrix element need grounding. is the total angular momentum the pair is coupled to — a pair isn't described by "one particle in each orbital" but by how their angular momenta combine. is the pair's isospin: treat proton-vs-neutron as a two-valued label mathematically identical to spin-½, and a pair couples to (symmetric — includes nn and pp pairs) or (antisymmetric — proton-neutron only). One more wrinkle: USDB was fit for mass 18, and orbitals shrink as more valence particles pile in, so every TBME gets scaled by for a nucleus of mass . The header's trailing numbers encode exactly that — and BIGSTICK's input carries the same convention, which is how it was cross-checked.
The program
"""Build your own shell model -- step 3: read the interaction.
The physics of the valence space lives in one small text file: usdb.int, the
USDB interaction (Brown & Richter 2006), fit to hundreds of measured sd-shell
levels. The format (oxbash convention) is:
header line: -63 2.1117 -3.9257 -3.2079 16.0 18.0 0.3
| |________|________| | | |
| single-particle energies | | |
| for labels 1,2,3 (MeV): A_core A_ref power
| 1=d3/2 2=d5/2 3=s1/2
'-- line count (negative flags the mass scaling below)
then one line per two-body matrix element (TBME):
a b c d J T V
meaning <ab; JT| V |cd; JT> in MeV: the interaction energy between a PAIR
of particles put in orbitals (a,b) and a pair in (c,d), with the pair
coupled to total angular momentum J and isospin T. Isospin is the
proton/neutron label treated like a spin-1/2: T=1 pairs are the
symmetric combination (includes nn and pp), T=0 pairs antisymmetric
(proton-neutron only).
The mass scaling: USDB was fit at A = 18, but orbitals shrink as more valence
particles pile in, so every TBME is scaled by (18/A)^0.3 for nucleus mass A.
For 18O the factor is exactly 1; for 24Mg it is 0.917315 -- and BIGSTICK's
input file carries the same "1.0 18.0 24.0 0.3" line, which is how the
convention was cross-checked.
"""
def read_int(path):
spe, tbme, tail, n = {}, [], None, None
with open(path) as fh:
for raw in fh:
line = raw.strip()
if not line or line.startswith(('!', '#')):
continue
w = line.split()
if n is None:
n = int(w[0])
spe = {1: float(w[1]), 2: float(w[2]), 3: float(w[3])}
tail = [float(x) for x in w[4:]]
continue
a, b, c, d, J, T = (int(x) for x in w[:6])
tbme.append(((a, b, c, d, J, T), float(w[6])))
return spe, tbme, tail
NAMES = {1: 'd3/2', 2: 'd5/2', 3: 's1/2'}
spe, tbme, tail = read_int('usdb.int')
print("single-particle energies (the seat prices, MeV):")
for label in (2, 3, 1): # printed in energy order
print(f" {NAMES[label]:>5}: {spe[label]:+8.4f}")
print("d5/2 lowest -- filling it first is the cheap move; the interaction")
print("decides whether that naive filling survives.")
print(f"\n{len(tbme)} two-body matrix elements. The first few:")
print(f"{'a':>5} {'b':>5} {'c':>5} {'d':>5} {'J':>2} {'T':>2} {'V (MeV)':>9}")
for (key, V) in tbme[:5]:
a, b, c, d, J, T = key
print(f"{NAMES[a]:>5} {NAMES[b]:>5} {NAMES[c]:>5} {NAMES[d]:>5} {J:>2} {T:>2} {V:>9.4f}")
a_core, a_ref, power = tail[-3], tail[-2], tail[-1]
for A in (18, 20, 24):
scale = (a_ref / A) ** power
print(f"mass scaling for A = {A}: ({a_ref:.0f}/{A})^{power} = {scale:.6f}")
print("\nEvery number the final calculation produces is built from these")
print(f"{len(tbme)} matrix elements, 3 seat prices, and antisymmetry. Nothing else.")0.917315 for ²⁴Mg. The closing line is the
point: everything the final calculation produces is built from 63 matrix
elements, 3 energies, and antisymmetry. Nothing else.