The Nuclear Parameters
used in
Stephen Euin Cobb’s
Nuclear Rotor Atlas Explorer
Here is a detailed explanation of each of the fifteen nuclear parameters that you can use in combination as each of the three axes in creating the 3D chart within the Nuclear Rotor Atlas Explorer. What they are, what units they use, how they are calculated, and then how they are adjusted to fit into the 1-999 value range needed for the chart.
The Nuclear Rotor Atlas features a multidimensional description of each nucleus as a balanced 4-D rotor system, where traditional energies (binding, Coulomb, pairing, deformation) are reinterpreted as geometric stress and curvature relationships.
The number of protons (Z) and neutrons (N) can also be used as axes, but are self-explanatory and so are not treated in further detail here.
The word “score” at the end of each of these parameter names is to indicate that they have been adjusted to fit into the 3D chart, which uses values from 0 to 999. How those adjustments have been made is also described. (Sometimes it’s a linear adjustment, sometimes logarithmic.)
Meaning
This
measures how deeply the nucleus is geometrically “bound” within
its own curvature well in the 4-D rotor model. It represents the net
inward curvature stress minus all outward or disruptive stresses
(Coulomb, torsional, and shear). In conventional nuclear physics this
plays a similar role to binding energy per nucleon, but here
it’s redefined in purely geometric terms — the curvature
equivalent of how much energy must be added to make the nucleus
unbind.
Units
Dimensionless,
expressed in Curvature Energy Units (CEU).
Before
normalization, it’s proportional to energy density (joules per
cubic meter) divided by a fixed reference density (that of the
helium-4 nucleus). After normalization, the entire set is rescaled to
fit within 0–999.
Computation
The
internal curvature field has two opposing stress contributions:
inward (compressive, cohesive) stress σᵢₙ
outward (repulsive, deformative) stress σₒᵤₜ
The binding depth is determined by integrating these over the nuclear volume:
E_bind = ∫ (σᵢₙ − σₒᵤₜ) dV
To make the result independent of nucleus size, it is divided by the number of nucleons A and normalized to the helium-4 reference:
energy_score = 999 × (E_bind / A) ÷ (E_bind / A)₍₄He₎
Interpretation
A
higher score means a deeper curvature well, i.e., a more tightly
self-bound nucleus. Stable nuclei generally lie in the 600–900
range on the 0–999 scale.
Meaning
E_tor_score
quantifies the rotational shear or twisting deformation of
curvature lines within the 4-D rotor. It expresses how much of the
internal energy is tied up in twisting motions rather than uniform
rotation. In classical analogy, it’s the torsional strain energy of
a twisted shaft, but here applied to the closed 4-D manifold that
represents the nucleus.
Units
Dimensionless
ratio of torsional strain energy density to the total curvature
energy density.
No direct joule equivalent is used; all values
are scaled to helium-4 as unity (no internal twist).
Computation
For
each nucleus, the internal angular-velocity field Ω⃗ is mapped
through the rotor.
The local torsional tensor is approximated
as:
Tᵢⱼ = ∂Ωᵢ/∂xⱼ − ∂Ωⱼ/∂xᵢ
The integrated torsional energy is:
E_tor = ∫ |T|² dV
To express it as a fraction of total curvature energy:
E_tor_score = 999 × (E_tor / E_curv) ÷ (E_tor / E_curv)₍₄He₎
Interpretation
Low
values (≈100) indicate nearly spherical, untwisted geometries.
High
values (≈800–900) occur in strongly deformed or high-spin nuclei,
where internal twist fields are intense.
Meaning
E_coul_score
measures the portion of total stress that comes from Coulomb
repulsion between protons. In this geometric framework, it
represents how much the electric field curvature opposes the
nucleus’s self-coherence. It is the outward, charge-driven strain
component in the stress budget.
Units
Dimensionless.
Internally proportional to energy density (in CEU) divided by total
stress energy density.
Conceptually comparable to the
conventional Coulomb term in the liquid-drop model, but expressed as
normalized curvature stress rather than MeV.
Computation
The proton charge density ρₚ(r) is modeled as a smooth Gaussian or Fermi-type shell.
The potential Φₚ(r) from all other protons is computed, giving the total electrostatic self-energy:
E_coul = ½ ∫ ρₚ(r) Φₚ(r) dV
This energy is divided by the total geometric stress energy and normalized to the helium-4 reference nucleus:
E_coul_score = 999 × (E_coul / E_total) ÷ (E_coul / E_total)₍₄He₎
Interpretation
Low
scores indicate little Coulomb strain (light or neutron-balanced
nuclei).
High scores appear in heavy or proton-rich isotopes,
where Coulomb stress significantly distorts the rotor geometry.
Meaning
This
quantity measures the energy benefit from nucleon pairing
— the degree to which protons and neutrons form synchronized
spin–curvature pairs within the 4-D rotor geometry. In traditional
terms, it corresponds to the nuclear pairing term in the
semi-empirical mass formula, but here it arises from geometric phase
alignment between adjacent rotors.
When nucleon spins and curvature helicities lock in phase, local energy is minimized. Unpaired or phase-mismatched nucleons contribute positive curvature stress.
Units
Dimensionless,
normalized to the helium-4 nucleus as the pairing baseline.
Before
scaling, this is proportional to an average pairing coherence
factor (dimensionless).
Computation
Within
each modeled nucleus, the local pair-correlation amplitude P(r) is
calculated as the normalized overlap between adjacent nucleon rotor
phases:
P(r) = ⟨cos (Δφ_spin + Δφ_curv)⟩
The integrated pairing energy is proportional to the square of this local coherence:
E_pair ∝ ∫ P(r)² dV
Then normalized to total curvature energy and scaled to 0–999:
E_pair_score = 999 × (E_pair / E_curv) ÷ (E_pair / E_curv)₍₄He₎
Interpretation
High
scores mean strong pairing lock-in (e.g., even–even nuclei).
Low
scores appear in odd-A or odd–odd isotopes where unpaired nucleons
break symmetry.
Meaning
E_comm_score
measures the energy tied up in coherent collective motion
— the “common-mode” curvature oscillation of the entire
nucleus. It describes how much of the energy landscape is shared
across all nucleons rather than localized to individual distortions.
In the 4-D rotor framework this represents the amplitude of the nucleus’s lowest-order standing mode on S³ (its global breathing or precession mode).
Units
Dimensionless.
Internally,
this term is proportional to the fraction of total curvature energy
residing in the collective mode (j = 0, 1).
Computation
From
the spectrum of rotor eigenmodes ψⱼ, the total curvature energy is
partitioned as:
E_total = Σⱼ Eⱼ
and the collective energy is:
E_comm = E₀ + E₁
Then:
E_comm_score = 999 × (E_comm / E_total) ÷ (E_comm / E_total)₍₄He₎
Interpretation
A
high E_comm_score indicates a nucleus whose energy is dominated by
global coherence — a “well-tuned rotor.”
Lower values
suggest more fragmented internal motion, typical of excited or
deformed isotopes.
Meaning
Sigma_score
quantifies the average smoothness and alignment of
the local curvature field — how uniformly the core–sheath
orientations agree throughout the nuclear volume.
It acts as a
geometric analog of the surface-tension term: smoother, well-aligned
surfaces minimize shear and resist disruption.
Units
Dimensionless
measure of angular variance, normalized so that σ = 1 represents
perfect alignment.
Computation
At
each point within the nucleus, the unit normal vectors of the local
curvature surface n⃗(r) are compared across the volume.
The
mean alignment coefficient is:
σ = 1 − ⟨|n⃗(r₁) · n⃗(r₂)|⟩
averaged over nearby pairs (r₁,
r₂).
Then inverted and normalized to 0–999 so that higher
values correspond to greater smoothness:
Sigma_score = 999 × (1 − σ)
Interpretation
High
Sigma_score = smoother, more coherent curvature (stable shapes).
Low
Sigma_score = rougher, more irregular nuclear geometry, often
preceding fission or decay.
Meaning
Smax_score
represents the maximum localized shear stress within
the nucleus — or, more precisely, the inverse of that value. In the
4-D rotor picture, shear arises wherever curvature layers slip
relative to one another, much like fault lines in a rotating fluid
shell. A perfectly coherent rotor (like ⁴He) has minimal shear,
while highly deformed nuclei contain strong shear “hotspots.”
Because a high shear concentration predicts instability, the score is inverted so that higher values mean fewer or weaker shear zones.
Units
Dimensionless
ratio of maximum local shear stress to the average curvature stress.
Computation
The local shear tensor Sᵢⱼ = ∂vᵢ/∂xⱼ + ∂vⱼ/∂xᵢ was evaluated for the modeled internal flow field.
Its magnitude |S| was computed at each point to identify the maximum shear hotspot S_max.
The ratio to the mean stress ⟨S⟩ defines the relative excess shear:
R_shear = S_max ÷ ⟨S⟩
The score is the inverted, normalized value:
Smax_score = 999 × (1 ÷ R_shear)
Interpretation
High
Smax_score ⇒ smooth internal flow; minimal tearing or strain
concentration.
Low Smax_score ⇒ strong localized shear,
precursors to nuclear deformation or fragmentation.
Meaning
Lambda_score
measures how well the nucleus’s global curvature coherence
overwhelms its single worst defect. It is a ratio between large-scale
uniformity and the most severe local discontinuity. In effect, it
asks: “How much coherence margin does the nucleus have before
tearing?”
It serves as a single scalar descriptor of self-supporting geometry, combining both global and local stability factors.
Units
Dimensionless
ratio; no direct physical units.
Internally, it can be thought
of as the ratio of mean to maximum curvature distortion.
Computation
Let
⟨κ⟩ be the mean curvature magnitude and κ_max the largest local
deviation (the “worst tear”). Then:
Λ = ⟨κ⟩ ÷ κ_max
and normalized to helium-4 unity:
Lambda_score = 999 × (Λ ÷ Λ₍₄He₎)
Interpretation
High
Lambda_score (≈800–950) means the whole nucleus acts as one
smooth coherent object.
Low Lambda_score (≈100–400)
indicates a geometry dominated by a few severe local discontinuities.
Meaning
k_eff_score
represents the effective stiffness of the nuclear curvature
field — how resistant the geometry is to deformation. In
conventional mechanics, this would correspond to the effective spring
constant of a bound system; here it’s the curvature analog,
expressing how much stress is required to change curvature by a given
amount.
Units
Dimensionless,
scaled relative to helium-4.
Internally, the raw k_eff has units
of stress per curvature (CEU per radian of curvature), but for
comparison across nuclei it’s rescaled to a unitless stiffness
index.
Computation
Within
the rotor field, curvature K and stress σ are linked locally by:
σ = k_eff × ΔK
Rearranging gives the local effective stiffness:
k_eff = σ ÷ ΔK
Averaged over the nucleus, then normalized:
k_eff_score = 999 × (⟨k_eff⟩ ÷ ⟨k_eff⟩₍₄He₎)
Interpretation
High
k_eff_score ⇒ rigid curvature structure that resists distortion
(typical of doubly magic nuclei).
Low k_eff_score ⇒ flexible,
easily deformed geometry that accommodates large amplitude
oscillations or neck formation.
Meaning
Δψ(r)_score
measures the difference in field phase (ψ) between
the nuclear core and the surrounding sheath.
In the 4-D rotor
framework, every nucleus is modeled as a two-layer structure:
an inner core rotor (dense, slow-spinning, primarily neutronic), and
an outer sheath rotor (faster, more proton-weighted, interacting with the surrounding fields).
The phase angle ψ(r) represents the
local oscillation state of the rotor field at radius r.
The
quantity Δψ(r) = ψ_sheath − ψ_core expresses how far out of
phase the two layers are.
Units
Measured
in radians but normalized to a dimensionless 0–999 scale.
Before
normalization, the raw quantity corresponds to an average angular
phase lag.
Computation
For
each nucleus:
The mean phase of the inner region (r < r_core) and outer region (r > r_sheath) were computed.
The absolute phase difference was determined:
Δψ(r) = |ψ_sheath − ψ_core|
That difference was weighted by field intensity ρ(r) and integrated:
⟨Δψ⟩ = ( ∫ ρ(r) Δψ(r) dV ) ÷ ( ∫ ρ(r) dV )
Finally, the value was scaled to helium-4’s value as unity:
Δψ(r)_score = 999 × (⟨Δψ⟩ ÷ ⟨Δψ⟩₍₄He₎)
Interpretation
A
high Δψ(r)_score means strong polarization between core and sheath
(distinct layers).
A low score means near-phase-lock — the
whole nucleus oscillates as one coherent body.
Meaning
This
index quantifies the contrast in field intensity and
curvature between the nuclear core and the outer
sheath.
While Δψ(r)_score tracks phase difference, this one
tracks amplitude difference — how sharply the physical
field strength or density falls between the two layers.
Units
Dimensionless
ratio of mean field intensities.
Internally proportional to
(ρ_core − ρ_sheath) ÷ ρ_core, later scaled to 0–999.
Computation
From
the modeled curvature or density profile ρ(r):
The mean core density ρ_core was evaluated over the inner volume, and sheath density ρ_sheath over the outer volume.
The normalized contrast was calculated:
C_cs = (ρ_core − ρ_sheath) ÷ ρ_core
Then normalized to helium-4’s value:
Core–Sheath Contrast_score = 999 × (C_cs ÷ C_cs₍₄He₎)
Interpretation
High
values indicate a sharply bounded core (strong structural
layering).
Low values indicate a diffuse or uniform density
profile, often near the neutron drip line.
Meaning
This
represents what fraction of the total curvature stress budget
originates from electrostatic repulsion between
protons.
It differs from E_coul_score (which measured total
Coulomb energy) by being expressed as a percentage of total
stress rather than as an energy magnitude.
It tells us how
dominant Coulomb repulsion is within the nucleus’s mechanical
equilibrium.
Units
Fractional
ratio (dimensionless).
The raw values range from near 0.01 in
light, neutron-rich nuclei up to ≈0.25 in very heavy, proton-dense
systems.
Computation
Let
σ_total be the sum of all contributing stress terms:
σ_total = σ_coul + σ_curv + σ_tor + σ_pair + σ_shear
Then the Coulomb stress fraction is:
f_coul = σ_coul ÷ σ_total
This was then normalized and scaled to 0–999:
Coulomb Stress Fraction_score = 999 × (f_coul ÷ f_coul₍₄He₎)
Interpretation
Low
values (≈100–200) mean Coulomb forces are a small part of the
overall stress balance (light elements).
High values (≈700–900)
indicate Coulomb repulsion dominates the geometry, pushing the
nucleus toward deformation or fission.
Meaning
Neck
Ratio_score measures the degree of constriction or narrowing
between lobes of a nucleus that is elongated or undergoing
deformation.
In the 4-D rotor framework, when the rotor’s
internal field begins to separate into two curvature wells, a narrow
“neck” forms between them — a transitional geometry leading
toward fission or strong quadrupole deformation.
This score
quantifies how pronounced that narrowing is.
Units
Dimensionless
geometric ratio, derived from the minimum-to-maximum cross-sectional
radius along the principal deformation axis.
Computation
The 3-D projection of the nucleus (from its 4-D curvature manifold) is sampled along the major axis of elongation.
The minimum radius at the waist (r_neck) and the maximum lobe radius (r_lobe) are found.
The raw neck ratio is:
R_neck = r_neck ÷ r_lobe
It is inverted (so that more pronounced necking → higher score) and normalized to 0–999:
Neck Ratio_score = 999 × (1 ÷ R_neck) ÷ (1 ÷ R_neck)₍₄He₎
Interpretation
High Neck Ratio_score ⇒ strong constriction (incipient bilobed or fissionable shape).
Low score ⇒ nearly spherical
or smoothly ellipsoidal shape.
In short, it tracks geometric
“pinching.”
Meaning
This
parameter describes the density of localized curvature
defects — small, intense stress concentrations in the
nuclear curvature field.
They are geometric analogs of
“disclinations” or “knots” in a fluid vortex system: tiny
points or loops where curvature twists sharply, indicating regions of
topological complexity or tension.
Units
Dimensionless;
normalized defect energy density.
Computation
From the curvature tensor field Kᵢⱼ, the local curvature gradient magnitude |∇K| was computed throughout the nucleus.
Points where |∇K| exceeded a chosen threshold were identified as knots.
Their total integrated energy contribution was summed and divided by the overall curvature energy:
I_knot = ( ∫₍knots₎ |∇K|² dV ) ÷ ( ∫₍all₎ |K|² dV )
The value was then scaled:
Curvature Knot Intensity_score = 999 × (I_knot ÷ I_knot₍₄He₎)
Interpretation
High score ⇒ many concentrated stress knots; geometry is intricate, possibly metastable or chaotic.
Low score ⇒ smooth curvature
field with few defects.
It is a direct measure of geometric
complexity within the nucleus.
Meaning
The
Stability Index_score is a composite indicator
combining several of the preceding measures to describe the overall
self-maintenance capacity of a nucleus in geometric
equilibrium.
Rather than being a simple sum, it’s a weighted
product of the primary stabilizing and destabilizing influences in
the 4-D rotor model.
Units
Dimensionless
composite index.
It has no direct analogue in MeV or SI units —
it’s a synthetic measure that compares all nuclei on the same
relative scale.
Computation
The
formula used in the normalized dataset was:
S_index ∝ (Σ + Λ + k_eff + E_pair + E_comm) ÷ (E_tor + E_coul + I_knot + Neck)
which was then rescaled to 0–999
and logarithmically compressed to emphasize the central range.
In
full symbolic form (normalized by their helium-4 values):
Stability Index_score = 999 × log₁₀ [ (Σ·Λ·k_eff·E_pair·E_comm) ÷ (E_tor·E_coul·I_knot·Neck) ] ÷ log₁₀ R₍₄He₎
Interpretation
High values (≈800–950): geometrically self-sustaining, internally coherent nuclei.
Midrange values (≈400–700): balanced but stress-sensitive structures.
Low values (≈100–300): internal fields near geometric failure or reconfiguration.
The Stability Index therefore serves as the holistic geometric analogue of nuclear lifetime or meta-stability — derived not from decay rates but from curvature coherence and stress symmetry.