“Know how to solve every problem that has been solved.” “What I cannot create, I do not understand.” — Richard Feynman

What gets measured, what gets computed in nuclear physics

Nuclear Physics

The methods covered elsewhere on this site — the shell model, Hartree-Fock-Bogoliubov, Skyrme and relativistic mean field, ab-initio approaches like NCSM, GFMC, coupled cluster, and IMSRG — all produce numbers. The question, as in any other field, is which of those numbers nuclear physicists and spectroscopists actually use, and how those numbers correspond to what an instrument at GANIL, FRIB, RIKEN, or ISOLDE measures. This page is the dictionary between the two columns.

What an experiment measures

Four broad classes, in roughly increasing instrument complexity:

1. Mass and bulk structural data

Penning-trap mass spectrometry (LEBIT, ISOLTRAP, JYFLTRAP, MR-TOF instruments) measures atomic masses to relative precision $1 0^{- 10}$ – $1 0^{- 11}$ , giving binding energies via $B = Z m_{H} + N m_{n} - M (Z, N)$ . Electron elastic scattering (Mainz, JLab) maps the charge form factor $F_{ch} (q)$ , from which the proton density and the charge radius $⟨ r_{ch}^{2} ⟩^{1/2}$ are extracted. Optical isotope-shift spectroscopy and muonic-atom X-ray spectroscopy give charge radii too, often more precisely for short-lived isotopes. Interaction cross sections at radioactive-beam facilities give matter radii — the proton + neutron density — and the difference between matter and charge radii is the neutron skin thickness.

2. Spin, parity, and electromagnetic moments

Gamma-ray angular distributions from oriented nuclei give spin and parity assignments ( $J^{π}$ ). Magnetic dipole moments $μ$ come from β-NMR, hyperfine spectroscopy, level-mixing resonance, or perturbed angular correlations. Electric quadrupole moments $Q$ come from Coulomb excitation, hyperfine structure, and reorientation effect. Quadrupole moments are the workhorse observable for nuclear deformation — $Q \neq = 0$ means a non-spherical charge distribution.

3. Excited states, transition rates, decays

Gamma-ray spectroscopy with high-purity Ge detectors (GAMMASPHERE, AGATA, GRETINA) measures excitation energies $E_{x}$ to $\sim 0.1$ keV. Lifetimes $τ$ of excited states come from recoil-distance Doppler shift, fast timing, or Coulex feeding analysis. Lifetimes convert to reduced transition probabilities $B (E λ)$ and $B (M λ)$ , which are the standard measures of how collectively the nucleus is moving in that transition. Giant resonances appear as broad peaks (~5 MeV wide) in inelastic scattering and photonuclear cross sections: giant dipole resonance (GDR), giant monopole (GMR, "breathing mode"), giant quadrupole, isovector spin-flip — each is a coherent collective oscillation. Beta-decay half-lives and Q-values, combined with branching ratios, give the $lo g f t$ values that measure Fermi and Gamow-Teller matrix elements.

4. Reactions and cross sections

Elastic scattering probes the nucleus-nucleus optical potential. Inelastic scattering populates excited states. Transfer reactions ((d,p), (p,d), (³He,d), and exotic-beam analogs) populate specific single-particle states, and the cross section combined with DWBA analysis gives spectroscopic factors $S = ∣ ⟨ ψ_{A} ∣ a_{j}^{†} ∣ ψ_{A - 1} ⟩ ∣^{2}$ . Knockout reactions (with radioactive beams) do the same in inverse kinematics. Fusion / capture cross sections at low energies feed astrophysical S-factors — the input to stellar nucleosynthesis calculations. Photonuclear cross sections probe the giant dipole resonance and feed photo-disintegration rates relevant to r- and p-process nucleosynthesis.

What a calculation produces

Start from a Hamiltonian (chiral EFT, phenomenological NN+3N, or effective shell-model interaction) and a many-body method (shell model, HFB, NCSM, GFMC, CC, IMSRG, RPA, …) and you can compute the following.

Ground-state binding energy

The lowest eigenvalue of the many-body Hamiltonian $\hat{H}$ . In the shell model this is the lowest eigenvalue of a giant sparse matrix (often $1 0^{9}$ × $1 0^{9}$ ) found by Lanczos; in HFB it's the minimum of the energy functional under the BCS quasiparticle ansatz. Compared directly to experimental binding energies; AME 2020 tables are the reference.

Density distributions and radii

From a mean-field or many-body calculation: $ρ_{p} (r)$ , $ρ_{n} (r)$ , kinetic-energy density $τ (r)$ , spin density $s (r)$ , spin-orbit current $J (r)$ . Charge density follows by folding $ρ_{p}$ with the proton charge form factor. Radii come from second moments of these distributions; neutron skin is $⟨ r^{2} ⟩_{n}^{1/2} - ⟨ r^{2} ⟩_{p}^{1/2}$ .

Single-particle spectrum and pairing gap

From a self-consistent mean field: orbital energies $ε_{i}$ , occupation numbers $v_{i}^{2}$ (BCS coefficients), pairing gap $Δ$ . These connect to single-particle separation energies via Koopmans-like arguments, and to odd-even mass differences via $Δ$ .

Excitation spectrum and wavefunctions

The higher Lanczos eigenvalues give the excited-state energies $E_{n}$ and their wavefunctions in the configuration basis. The differences $E_{n} - E_{0}$ are the model's predicted excitation energies — compared directly to gamma-ray spectroscopy.

Multipole transition strengths

$B (E λ; i \to f) = (2 J_{i} + 1)^{- 1} ∣ ⟨ ψ_{f} ∥ \hat{M} (E λ) ∥ ψ_{i} ⟩ ∣^{2}$ and analogously $B (M λ)$ . Computed by sandwiching the multipole operator between Lanczos eigenvectors. Compared to experimental lifetimes + branching ratios.

Strength functions / giant resonances

The energy-weighted strength $S (E) = \sum_{n} ∣ ⟨ n ∣ \hat{O} ∣0 ⟩ ∣^{2} δ (E - E_{n} + E_{0})$ , where $\hat{O}$ is the relevant multipole operator. For a giant resonance this is a smooth peaked function over several MeV; computed efficiently by Lanczos+continued-fraction methods (same machinery as the Liouville-Lanczos page, applied to nuclear RPA / QRPA).

Spectroscopic factors

Overlap of an $A$ -body wavefunction with an $(A - 1)$ -body one with a nucleon removed: $S = ∣ ⟨ ψ_{A} ∣ a_{j}^{†} ∣ ψ_{A - 1} ⟩ ∣^{2}$ . Computed in the shell model basis. Combined with reaction theory (DWBA, ADWA, eikonal), gives predicted transfer-reaction cross sections.

Beta-decay matrix elements

Fermi matrix element $⟨ ψ_{f} ∣ \hat{T}_{\pm} ∣ ψ_{i} ⟩$ (isospin operator) and Gamow-Teller $⟨ ψ_{f} ∣ σ \hat{T}_{\pm} ∣ ψ_{i} ⟩$ . These determine $lo g f t$ values. The GT strength distribution as a function of daughter excitation energy governs r-process nucleosynthesis rates.

Reaction cross sections

Optical-model parameters fit to elastic scattering; DWBA / coupled-channels for inelastic and transfer reactions; R-matrix or microscopic reaction calculations for low-energy fusion; Hauser-Feshbach for compound-nucleus reactions and astrophysical rates. Each is a separate machinery on top of the structure calculation that provides the matrix elements.

The bridge

Row-by-row correspondence between experimental observable and model-side quantity:

Experimental observable	Model-side counterpart	Typical accuracy / notes
Atomic mass / binding energy (Penning trap)	Ground-state energy of $\hat{H}$ (Lanczos eigenvalue or HFB minimum)	~100 keV for mid-mass HFB; ~10 keV for ab-initio CC on closed shells
Charge radius $⟨ r_{ch}^{2} ⟩^{1/2}$ (e-scattering, isotope shift, muonic atoms)	Second moment of $ρ_{p} (r)$ folded with proton form factor	~0.02 fm for mean-field; isotope shifts are stricter test
Magnetic moment $μ$ (NMR, β-NMR, hyperfine)	$⟨ ψ ∣ \overset{μ}{^} ∣ ψ ⟩$ with effective $g_{s}$ factors	Sensitive to spin-orbit and meson-exchange currents; ~10% is typical
Quadrupole moment $Q$ (Coulex, hyperfine)	$⟨ ψ ∣ \hat{Q} ∣ ψ ⟩$ ; collective rotor formula for deformed nuclei	The standard observable for nuclear deformation $β_{2}$
Excitation energy $E_{x}$ (γ spectroscopy)	$E_{n} - E_{0}$ from successive Lanczos eigenvalues	Shell model: ~100 keV; ab-initio: ~few × 100 keV
$B (E 2)$ , $B (M 1)$ , etc. (lifetimes + branching)	$∣ ⟨ ψ_{f} ∥ \hat{M} (E λ / M λ) ∥ ψ_{i} ⟩ ∣^{2}$ with effective charges	Effective charges $(e_{p} \approx 1.5, e_{n} \approx 0.5)$ needed to absorb truncation
Giant resonance energy + strength (inelastic, photonuclear)	RPA / QRPA strength function via Lanczos+CF	GDR / GMR centroids to ~1 MeV; widths require coupling to 2p2h states
$lo g f t$ (β-decay half-life + Q-value)	Gamow-Teller / Fermi matrix elements in shell model	Need a "quenching factor" $g_{A}^{eff} / g_{A} \sim 0.7 - 0.8$ for shell-model GT
Spectroscopic factor (transfer reaction + DWBA)	$∣ ⟨ ψ_{A} ∣ a_{j}^{†} ∣ ψ_{A - 1} ⟩ ∣^{2}$	Reduced from sum-rule limit by correlations; consistent ~0.5–0.7 factor
Cross section $σ (E, θ)$ (scattering experiment)	S-matrix from optical model + DWBA / coupled-channels / R-matrix	Quality depends heavily on the reaction mechanism and energy regime
Neutron skin $Δ R_{n p}$ (PREX-style PV scattering)	$⟨ r^{2} ⟩_{n}^{1/2} - ⟨ r^{2} ⟩_{p}^{1/2}$ from mean-field densities	Constrains nuclear-matter symmetry energy and neutron-star radii
Astrophysical S-factor (low-energy fusion)	R-matrix or microscopic reaction calc + extrapolation to $E_{Gamow}$	Notoriously hard for sub-barrier resonances; nuclear-astro benchmark

Which method for which quantity

Nuclear physics has more many-body methods than most fields, because nuclei span an enormous range — from few-body $A = 3$ to heavy $A \sim 250$ — and no single method covers it all. Choose the method based on the quantity you need and the mass range:

Method	Best for	Doesn't handle	Cost / scope
Empirical shell model (NuShellX, BIGSTICK, KSHELL)	Spectroscopy within a single major shell (sd, pf, sdg); transition strengths; $lo g f t$	Cross-shell physics, halo nuclei, heavy deformed	Combinatorial in valence space; dimensions up to ~10¹⁰
HFB + Skyrme / Gogny	Bulk properties of medium-heavy nuclei; pairing; deformation	Spin-orbit fine structure; spectroscopic detail	$O (N^{3})$ per iteration; whole chart of nuclides feasible
Relativistic mean field (RMF / DD-ME / point-coupling)	Heavy nuclei; spin-orbit emerges naturally; neutron skin; isovector observables	Same scope as HFB	Similar cost to HFB
NCSM (no-core shell model)	Light nuclei ( $A ≲ 16$ ) from chiral EFT; benchmark spectra	Heavier nuclei; convergence in $N_{m a x}$ demanding	Combinatorial; cost grows with $N_{m a x}$ like the model space
GFMC / AFDMC	Light nuclei from chiral EFT; very accurate spectra	$A > 12$ ; complex spin-isospin operators slow MC	Exponential in $A$ without special tricks
Ab-initio coupled cluster (NCSM-CC, CCSD/CCSDT)	Medium-mass closed-shell nuclei ( $\leq^{132} Sn$ )	Strongly open-shell; deformed nuclei without extensions	$O (N^{6} - N^{7})$ ; routine for closed shells
IMSRG (in-medium SRG)	Closed-shell + valence-space derivations of effective interactions	Strongly deformed regimes	Similar to CC; the standard pipeline for ab-initio shell model
QRPA / RPA on top of HFB	Giant resonances; β-decay strength functions; transition densities	Strongly anharmonic modes; multi-phonon	$O (N^{2})$ eigenproblem; Lanczos+CF for large bases
Lattice EFT (NLEFT)	Light/medium nuclei from chiral EFT on a lattice	Continuum-limit extrapolations are subtle	MC; sign problem at finite isospin
R-matrix / DWBA / Hauser-Feshbach	Reaction cross sections; astrophysical rates	Direct reactions need DWBA; coherent scattering needs CC	Varies; statistical methods very fast for compound nucleus

One-sentence summary

A nuclear-physics calculation almost always answers "what will a spectrometer, scattering experiment, or β-NMR setup measure?" — and despite the field's heavier method zoo than quantum chemistry, the menu of observables is small enough to fit in two tables. Pick the experimental quantity, find its row in the bridge table, choose a method that handles that row and that mass range from the methods table, and run.

Related on this site

Nuclear shell model — the standard tool for spectroscopy within a major shell; Lanczos diagonalization of giant sparse matrices.
Hartree-Fock-Bogoliubov — mean-field with pairing; the workhorse for bulk properties of medium-heavy nuclei.
Multipole expansion in the shell model — how the multipole operators that compute $B (E λ)$ and friends are constructed.
Real-space vs second quantization — the two complementary representations underlying both shell model and mean-field calculations.
Liouville-Lanczos — the same Krylov+continued-fraction machinery used here for QRPA strength functions, applied in the TDDFT setting.
Same idea, quantum chemistry version and comp-neuro version — what gets measured and computed in those fields.