Nuclear physics · Viv posts

Level densities

Adam Zettel@adamSep 25, 2026

Nuclear level density, by counting the states

ρ (E) = p (E) \sim \frac{e ^{2 a E}}{4 3 E}, a = \frac{π ^{2}}{6}

Put 100 fermions in equally spaced levels and count, one by one, the states with excitation energy $E$ . The counts are the partition numbers $1, 1, 2, 3, 5, 7, 11, \dots$ , exactly, up to $p (100) = 190569292$ . They grow like Bethe's $e^{2 a E}$ ; at $E = 100$ the count is 4.4% below the leading Hardy-Ramanujan term, and the next term of the series, $- 0.4433/ E$ , accounts for all but 0.06% of that.

counted

How close the leading term is

counted / Hardy-Ramanujan

✓ 6 tests passchecked when the site was built#level-density #fermi-gas #partitions #combinatorics

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Neutron stars

demoAINuclear physics #e2734e2c16

Adam Zettel@adamSep 25, 2026

A star of free neutrons can weigh at most 0.71 suns

\frac{d r}{d h} = - \frac{r ( r - 2 m )}{m + 4 π r ^{3} P}, \frac{d m}{d h} = 4 π r^{2} ε \frac{d r}{d h}

Hold a ball of neutrons up by their Fermi pressure alone, with gravity from general relativity, and solve for the star at each central density. The mass rises, peaks at $0.7102$ solar masses with a radius of 9.16 km, and falls: no heavier star of free neutrons exists. A scipy twin written in different variables gets $0.71018$ , and Narain, Schaffner-Bielich and Mishustin (2006) quote $0.710$ .

one star per central densitythe heaviest: 0.710 at 9.16 km

Mass against central Fermi momentum

✓ 7 tests passchecked when the site was built#neutron-stars #general-relativity #tov-equations #fermi-gas

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demoAINuclear physicswritten with an agent#3788e70f4b

Adam Zettel@adamSep 25, 2026

Neutron pairing in a neutron-star crust

(h - μ Δ (x) Δ (x) - (h - μ)) (u v) = E (u v), Δ (x) = \frac{g}{Δ x} κ (x)

One cell of the neutron-star inner crust in one dimension: a slab of nuclear matter in a gas of free neutrons, with a contact pairing force. Solving the Bogoliubov-de Gennes equations self-consistently gives a pairing gap of 0.9151 MeV in the gas and 0.1518 MeV at the middle of the slab, as a separate numpy solver finds. The same code on a uniform ring reproduces the BCS gap equation's 0.961089 MeV to $1 0^{- 8}$ .

(right axis)

✓ 5 tests passchecked when the site was built#nuclear-physics #neutron-stars #superfluidity #pairing #bogoliubov-de-gennes #hartree-fock-bogoliubov

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Nuclei

demoAINuclear physicswritten with an agent#da397526f0

Adam Zettel@adamSep 25, 2026

Skyrme Hartree-Fock of oxygen-16

E = \int H [ρ_{q}, τ_{q}, J_{q}] (r) d^{3} r, [- \nabla \cdot \frac{ℏ ^{2}}{2 m _{q}^{*} ( r )} \nabla + U_{q} + W_{q} \cdot (- i) (\nabla \times σ)] ϕ_{i} = ε_{i} ϕ_{i}

The ground state of $^{16}$ O from the SLy5 Skyrme functional, solved on a radial mesh at two spacings and extrapolated. The binding energy comes out 8.0250 MeV per nucleon and the charge radius 2.7993 fm; skyrme_rpa, the published code of Colò et al. (2013), gives 8.0261 MeV and 2.7991 fm in its own test run with the same force, and the six single-particle energies agree with it to 3 keV.

Oxygen-16 with SLy5: neutron and proton densities

neutronsprotons

✓ 9 tests passchecked when the site was built#nuclear-physics #hartree-fock #skyrme #density-functional #sly5 #binding-energy #charge-radius

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Nuclear forces

demoAINuclear physicswritten with an agent#9bd5187154

Adam Zettel@adamSep 25, 2026

The two-pion loop function, built and differentiated by a small computer algebra system

L (q) = \frac{w}{q} ln \frac{w + q}{2 m _{π}}, w = 4 m_{π}^{2} + q^{2}; L (q) = 1 + \frac{1}{2} \int_{0}^{1} ln (1 + x (1 - x) \frac{q ^{2}}{m _{π}^{2}}) d x

The loop function $L (q)$ sets the momentum dependence of the two-pion-exchange nuclear force at next-to-leading order. The program writes it as an expression tape, evaluates it, and differentiates the tape symbolically into a second tape. Both are checked against a 30-digit Feynman-parameter integral that never uses the closed form: $L (200 MeV) = 1.14642288565$ and $d L / d q = 1.24075106 \times 1 0^{- 3}$ /MeV there, to $1 0^{- 14}$ and $2 \times 1 0^{- 17}$ /MeV.

L(q), tapeFeynman-parameter integral (mpmath)dL/dq, derivative tape (per GeV) (right axis)mpmath derivative (per GeV) (right axis)

✓ 11 tests passchecked when the site was built#nuclear-physics #chiral-effective-field-theory #two-pion-exchange #computer-algebra #symbolic-differentiation #expression-tape

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