YOUR CALCULATION EQUILIBRIUM

Essentially, the atomic nucleus theory studies:

• what the nucleus consists of: protons and neutrons;
• What forces hold nucleons together?
• Why are some nuclei stable and others decay?
• how to describe excited states, reactions, division, and synthesis.

The nucleus is a quantum multi-particle system. It cannot be understood without:

• quantum mechanics,
• statistical physics,
• operator theory,
• differential equations,
• linear algebra,
• Theories of Groups.

The main physical idea. There are two competing things inside the kernel:

1. Strong interaction

It is short acting and attracts nucleons.

2. Coulomb Repulsion

Protons repel each other. Because of this:

• Small and medium nuclei are usually more stable.
• very heavy nuclei are more easily deformed and disintegrated,
• There is a limit to the number of protons and neutrons.

Basic Mathematics of Nuclear Theory

A. State Space

The state of the nucleus is described by a vector in Hilbert space:

|\Psi\rangle

For many nucleons, this is no longer a single function, but a complex antisymmetric wave function:

\Psi(\mathbf r_1, s_1, t_1; \mathbf r_2, s_2, t_2; \dots )

where:

• \mathbf r_i — coordinate,
• s_i — spin,
• t_i isospin.

B. Hamiltonian core. Basic model:

\hat H = \sum_{i=1}^{A} \hat T_i + \sum_{i<j} \hat V_{ij} + \sum_{i<j<k} \hat V_{ijk}, where:

• A is the mass number,
• \hat T_i is kinetic energy.
• \hat V_{ij} is a two-particle nucleon-nucleon interaction,
• \hat V_{ijk} is a three-nucleated force.

The Schrodinger equation is further solved:

\hat H |\Psi_n\rangle = E_n |\Psi_n\rangle

It is the E_n spectrum that gives the energy of the states of the nucleus.

C. Antisymmetry

Nucleons are fermions, so the wave function should change the sign when the same particles are rearranged:

\Psi(\dots,i,\dots,j,\dots) = -\Psi(\dots,j,\dots,i,\dots)

This leads to Slater's determinants and to the entire shell model.

Core Models

The drip model

The nucleus is regarded as a quantum charged liquid droplet. She explains well:

• the total energy of the connection,
• division of the nucleus,
• Dependence of resistance on A and Z.

Weizsacker's semi-empirical formula:

B(A,Z)=a_vA-a_sA^{2/3}-a_c\frac{Z(Z-1)}{A^{1/3}}-a_a\frac{(A-2Z)^2}{A}+\delta(A,Z)

where:

• B is the energy of communication,
• A is the number of nucleons,
• Z is the number of protons.

Meaning of Members:

• by volume,
• surface,
• Coulomb,
• Asymmetry,
• mating.

This is one of the most important formulas in nuclear physics.

Shell Model

Here each nucleon moves in the middle field of the others.

Main idea:

• Energy levels are discrete;
• There are magic numbers:
  2,\ 8,\ 20,\ 28,\ 50,\ 82,\ 126

They correspond to particularly stable nuclei.

Mathematically, these are problems with eigenvalues for the average potential:

\left[-\frac{\hbar^2}{2m}\nabla^2 + U(r) + U_{ls}\,\mathbf l\cdot\mathbf s \right]\psi = E\psi

Critically important spin-orbit member \mathbf l \cdot \mathbf s.

The collective model

This model describes:

• the core rotation,
• vibration of the core,
• deformation.

That is, the nucleus behaves not only as a set of independent nucleons, but also as a single object.

The energy of the rotational spectrum is often written as follows:

E_J \approx \frac{\hbar^2}{2\mathcal I}J(J+1)

where \mathcal I is the moment of inertia.

Model of interacting bosons

Paired nucleons are sometimes convenient to consider as effective bosons:

• s-bosons (L=0),
• d bosons (L=2).

This is already a strongly algebraic approach related to group theory.

The most important mathematical sections

Linear algebra

Needed for:

• operators,
• Hamiltonian matrices,
• Diagonalization,
• spectra.

Differential Equations

Needed for:

• Schredinger's equations,
• Radial functions,
• Dispersion,
• tunneling.

The theory of angular moment

Totally central section.

Used:

• spherical harmonics Y_{lm},
• Klebsch-Gordan coefficients,
• 3j, 6j, 9j-symbols.

For example, the total moment:

\mathbf J = \mathbf L + \mathbf S, and for one nucleon:

\mathbf j = \mathbf l + \mathbf s

Group Theory

Very important for symmetries:

• rotation group SO(3),
• SU(2) for back,
• sometimes SU(3), SU(4), symplectic and other algebras.

Symmetry helps to reduce the huge dimensions of the state space.

Variational methods

Often, an exact solution is impossible, so they look for an approximate state:

\delta \frac{\langle \Psi|\hat H|\Psi\rangle}{\langle \Psi|\Psi\rangle}=0

This is how they build:

• Hartree-Fock,
• Hartree-Fok-Godlyubov,
• Density functional methods.

Secondary quantization

Very convenient language for multi-part systems.

Hamiltonian is recorded through birth and destruction operators:

\hat H = \sum_{\alpha\beta} t_{\alpha\beta} a^\dagger_\alpha a_\beta + \frac{1}{4}\sum_{\alpha\beta\gamma\delta} \bar V_{\alpha\beta\gamma\delta} a^\dagger_\alpha a^\dagger_\beta a_\delta a_\gamma

This is the working language of the modern nuclear structure.

Nuclear decay and mathematics

Alpha decay

Quantum tunneling through a potential barrier.

Probability of passing:

P \sim e^{-2\int_{r_1}^{r_2} \kappa(r)\,dr}

where

\kappa(r)=\frac{\sqrt{2m(V(r)-E)}}{\hbar}

Beta decay

It is associated with weak interaction and matrix transition elements.

Transition speed:

\lambda \propto |M_{fi}|^2 \rho(E)

Gamma transitions

Described by multipole decomposition:

• E1, M1, E2, M2, etc.

The algebra of angular moments is especially important here.

Nuclear reactions

General view:

a + A \rightarrow b + B

Basic mathematical tools:

• theory of dispersion,
• S-matrix
• Differential sections,
• partial waves.

Differential section:

\frac{d\sigma}{d\Omega} = |f(\theta,\phi)|^2

Full section:

\sigma = \int \frac{d\sigma}{d\Omega}\, d\Omega

Modern directions

Nuclear theory is usually divided into several levels:

Phenomenological level

Fitting models for the experiment.

Microscopic level

Solution of the problem from nucleon-nucleon interactions.

Effective theories

For example, an effective field theory, where interactions are built on a hierarchy of scales.

Ab initio methods. An attempt to calculate the properties of nuclei almost "from the first principles." These include:

• No-Core Shell Model,
• Coupled Cluster,
• Green’s Function Monte Carlo,
• In-Medium SRG.

If you are very brief.

Core theory is the physics of a strongly bound quantum system of protons and neutrons. Core mathematics is the language by which this system is described:

• The operators,
• Spectral tasks,
• Symmetry,
• Variational methods,
• group theory,
• statistics,
• Numerical methods.

Minimum route studied

If you want to really get into the subject, the order is better than this:

• Quantum Mechanics
• The theory of angular moment
• Basics of Nuclear Physics
• Drip and sheath models
• Secondary quantization
• Multi-part theory
• Nuclear reactions
• Modern computational methods

One working formulation

Core Mathematics is a branch of mathematical physics in which the atomic nucleus is regarded as a quantum multiparticle fermi-system described by a Hamiltonian with strong interaction, angular momentum symmetries, antisymmetry of states, and spectral problems at eigenvalues.

State Space

|\Psi\rangle \in \mathcal{H}, \quad \mathcal{H} = \text{antisymmetric space of } A\text{ fermions}

Full wave function

\Psi = \Psi(\mathbf r_1, s_1, t_1; \dots ; \mathbf r_A, s_A, t_A)

Fermi-statistics condition:

\Psi(...i...j...) = -\Psi(...j...i...)

HAMILTONIAN NUCLEAR

\hat H = \sum_{i=1}^{A} \left( -\frac{\hbar^2}{2m}\nabla_i^2 \right) + \sum_{i<j} V_{ij} + \sum_{i<j<k} V_{ijk}

Own values (core theory)

\hat H \Psi_n = E_n \Psi_n

TWO-DAY REFERENCES

Operators

a^\dagger_\alpha,\quad a_\alpha

Anti-Commutation:

\{a_\alpha, a^\dagger_\beta\} = \delta_{\alpha\beta}

Hamiltonian

\hat H = \sum_{\alpha\beta} t_{\alpha\beta} a^\dagger_\alpha a_\beta + \frac{1}{4}\sum_{\alpha\beta\gamma\delta} \bar V_{\alpha\beta\gamma\delta} a^\dagger_\alpha a^\dagger_\beta a_\delta a_\gamma

SYMBOLS AND MOMENTS

Orbital + spin moment

\mathbf J = \mathbf L + \mathbf S

Square of Moment

\hat J^2 |JM\rangle = \hbar^2 J(J+1)|JM\rangle

Spin-orbit interaction

V_{ls} \propto \mathbf l \cdot \mathbf s

COVER MODEL

Nucleon motion equation

\left[ -\frac{\hbar^2}{2m}\nabla^2 + U(r) + V_{ls} \right]\psi = E\psi

Magic Numbers

2,\ 8,\ 20,\ 28,\ 50,\ 82,\ 126

CAPEL MODEL (LINK ENERGY)

B(A,Z)= a_vA -a_sA^{2/3} -a_c\frac{Z(Z-1)}{A^{1/3}} -a_a\frac{(A-2Z)^2}{A} +\delta

OPTIONAL PRINCIPLE

\delta \frac{\langle \Psi|H|\Psi\rangle}{\langle \Psi|\Psi\rangle} = 0

Hartree-Fock

\hat h \phi_i = \epsilon_i \phi_i

PAYMENTS

Single-particle density

\rho(\mathbf r) = \langle \Psi | \hat \psi^\dagger(\mathbf r)\hat \psi(\mathbf r)|\Psi\rangle

REPRESENTATIONS

1. Alpha decay (tunneling)

P \sim e^{-2\int_{r_1}^{r_2} \kappa(r)\,dr}

\kappa(r)=\frac{\sqrt{2m(V(r)-E)}}{\hbar}

Beta decay

\lambda \propto |M_{fi}|^2 \rho(E)

Gamma transitions

T(E\lambda),\quad T(M\lambda)

NUCLEAR REACTIONS

Differential section

\frac{d\sigma}{d\Omega} = |f(\theta)|^2

Full section

\sigma = \int \frac{d\sigma}{d\Omega} d\Omega

S-matrix

S = 1 + iT

COLLECTIVE DYNAMICS

Rotate Core

E_J = \frac{\hbar^2}{2\mathcal I}J(J+1)

Vibration

E_n = \hbar \omega (n + \tfrac12)

LEVEL SYSTEM (COMMON STRUCTURE)

You can put everything in one chain:

\boxed{ H \rightarrow \Psi \rightarrow E }

\boxed - E \rightarrow spectrum \rightarrow stability \rightarrow decays

\boxed-\Psi \rightarrow matrix elements \rightarrow transitions \rightarrow reaction ?

BEINGS (NUCLEAR MATHEMATICS)

As short as possible:

\boxed - \text - The kernel = is a problem on the eigenvalues of the multiparticle Hamiltonian

\boxed - \text - Physics = - Spectrum structure + - Transitions between states

IF YOU MAKE IT QUICK. It all comes down to 4 blocks:

• Hamiltonian
  H = T + V
• Status
  H\Psi = E\Psi
• Symmetry
  SU(2),\ SO(3)
• Transitions
  |M_{fi}|^2

Core Mathematics 2.0

Core architecture

The central idea

The core can be considered as a system of 5 related layers:

\boxed{ \text{Particles} \rightarrow \text{Interactions} \rightarrow \text{States} \rightarrow \text{Spectra} \rightarrow \text{Transitions} }

That is:

• There are nucleons;
• there is a law of communication - potential;
• there is a form of organization - a wave function;
• there is a structure of levels - spectrum;
• There are dynamics of changes - transitions, decays, reactions.

Five-layer core architecture

Layer I. Carriers

This is the elementary composition of the nucleus:

N = \text{neutrons}, \quad Z = \text{protons}, \quad A=N+Z

This is the layer:

• quantities,
• Composition,
• charge,
• The mass,
• back,
• isospina.

Basic parameters:

q_p = +e,\quad q_n = 0

s_p = s_n = \frac12

At this level, the nucleus is simply an ensemble of fermions.

Layer II. Field of interaction

This is how nucleons are related to each other.

Overall appearance: H = T + V_{NN} + V_{3N} + V_C

where:

• T is kinetic energy,
• V_{NN} - two-nucleon interaction,
• V_QQ3NQQ is a three-nucleon interaction.
• V_C - Coulomb interaction.

It is no longer just a set of particles, but a connected system.

Architecturally:

\boxed{ \text{Composition} + \text{Connection} = \text{Core capability} }

Layer III. State geometry

This is where the wave function is born:

\Psi(\mathbf r_1,s_1,t_1;\dots;\mathbf r_A,s_A,t_A)

This is the layer:

• configuration,
• Symmetry,
• antisymmetry,
• distribution of density,
• Correlation.

Condition of fermion:

\Psi(...i...j...)=-\Psi(...j...i...)

This means that the nucleus cannot be thought of as a chaotic mass.

It has a strict geometry of permissible states.

Layer IV. Energy structure

From the equation

H\Psi_n = E_n\Psi_n

The spectrum is:

E_0, E_1, E_2, \dots

This layer is responsible for:

• basic condition,
• Excited states,
• Sustainability,
• Magic numbers,
• deformations,
• collective regimes.

That is, the kernel receives an internal ladder of levels.

Layer V. Dynamics of transitions

When there is a spectrum, transitions appear:

\Psi_i \rightarrow \Psi_f

with probability determined by matrix element:

M_{fi}=\langle \Psi_f|\hat O|\Psi_i\rangle

This layer is responsible for:

• gamma transitions,
• beta decays,
• alpha decays,
• reactions,
• The division,
• Synthesis.

This is the layer of core life in time.

The main architectural formula

Everything can be put together in one nuclear circuit:

\boxed{ (N,Z) \rightarrow H \rightarrow \Psi \rightarrow E \rightarrow M_{fi} \rightarrow \text{evolution} }

Interpretation:

• (N,Z) - who makes up the nucleus;
• H - what forces are acting;
• \Psi - how the kernel is organized;
• E - what levels exist;
• M_{fi} - how the kernel changes state;
• evolution - how the nucleus lives, decays and interacts.

This is the kernel operating system.

 Three Core Organization Modes

Mode 1. Individual

Each nucleon moves almost independently in the middle field:

H \approx \sum_i h_i

This is the logic of the shell model.

Mode 2. Correlated

Nucleons are strongly tied to each other, and the independent movement is no longer working.

Then you need to take into account:

• pair correlations,
• collective excitement,
• Interaction of configurations.

Mode 3. Collective

The kernel behaves as a single object:

• and turn,
• vibrating,
• deformed.

Then:

E_J \approx \frac{\hbar^2}{2\mathcal I}J(J+1)

That is, the system moves from micrologic to macrologic.

Inner Core Axes

The kernel can be described by 4 basic axes.

Axle 1. Composition

(A,Z,N)

Shows what the core is made of.

Axle 2. Communications

B(A,Z)

Shows how much the kernel holds itself together.

Axle 3. Symmetry

J^\pi,\quad T

where:

• J is the full moment.
• \pi — parity,
• T isospin.

Shows the type of internal order.

Axle 4. Transition

\langle f|\hat O|i\rangle

Shows how much the nucleus is prone to change.

Core Structure Map

Now let's put together the core as a map.

The core has:

1. Massa

M(A,Z)

2. Communication Energy

B(A,Z)=\big[Zm_p + Nm_n - M(A,Z)\big]c^2

3. Radius

R = r_0 A^{1/3}

4. Density

\rho(r)

5. Spectrum

\{E_n\}

6. Channels of decay

\Gamma_\alpha,\ \Gamma_\beta,\ \Gamma_\gamma

7. Channels of reaction

\sigma(E)

This is a full-fledged passport core.

The core as a balance of power

There are 4 main balance sheets.

Balance 1. Attraction and repulsion

V_{\text{strong}} + V_C

Strong interaction collects, Coulomb breaks.

Balance 2. Pauli's Order and Principle

Even attractive nucleons can not sit down as they please:

\Psi_{\text{fermionic}} \Rightarrow \text{prohibition on overfilling states}

Balance 3. Locality and collectivity

Sometimes a separate nucleon is important, sometimes the whole nucleus as a whole.

Balance 4. Sustainability and Transition

Each core simultaneously:

• holding the form,
• It has the ability to change it.

The Core as a Computing System

Architecturally speaking, the core is like a computer.

Login:

(A,Z),\quad V

Calculating Core:

H\Psi = E\Psi

Output:

• spectrum,
• The radius,
• moments,
• Probability of transitions,
• section of reactions.

That is:

\boxed - \text - Core - = - \text - Quantum Computer of Eigenstates

The minimal metamodel. You can enter 6 basic entities:

\mathcal N = \text{nucleons}

\mathcal I = \text

\mathcal S = \text

\mathcal E = \text

\mathcal T = \text

\mathcal R = \text

Core architecture:

\boxed{ \mathcal N \xrightarrow{\mathcal I} \mathcal S \xrightarrow{} \mathcal E \xrightarrow{} \mathcal T \xrightarrow{} \mathcal R }

The Universal Nuclear Cycle

Each core goes through a cycle:

1. Formation

It is made up of nucleons.

2. Self-organization

Selects the acceptable state structure.

3. Stabilisation

It takes a minimum of energy.

4. Excitation

It gets energy from outside.

5. Transition

Move to another level.

Transformation

It breaks down or reacts.

Formulation:

\text{assembly} \rightarrow \Psi_0 \rightarrow \Psi_n \rightarrow \Psi_f

Translated into the language of “Nuclear Theory”

Then you can give such a definition:

Core theory is the theory of organization, stability and transformation of a quantum multiparticle system in which composition, interaction, symmetry and spectrum form a single contour of internal evolution.

Nucleus mathematics is a system of operators, symmetries, spectra, densities, and transitions that describe how a plurality of related fermions form a stable nucleus, allow excitation, and pass through transformation channels.

\boxed{ \text{Composition} \rightarrow \text{Connection} \rightarrow \text{Form} \rightarrow \text{Spectrum} \rightarrow \text{Transition} }

Or even harder:

\boxed{ \text{What exists} \rightarrow \text{How it is connected} \rightarrow \text{How it is structured} \rightarrow \text{How it lives} }

Frame for your project EQUILIBRIUM

If you transfer it to your architecture, you can already collect it 5 Modules EQUILIBRIUM- cores:

Module 1. Composition of the core

• Protons
• Neutrons
• Quantum Numbers
• Number of layers

Module 2. Core geometry

• Density
• radius
• Shell
• Deformity

Module 3. Core Energy

• Communication Energy
• spectrum
• Sustainability
• Thresholds of decay

Module 4. Core Dynamics

• Transitions
• Demolition
• Resonance
• reactions

Module 5. Metayadro

• Principles of Balance
• Symmetry
• self-organization
• Evolutionary Core Logic

SCHEME TABLE

Core Theory / Mathematics Nuclear / Formulas / Meanings / Architecture

• circular diagram (7 layers)
• Color indication
• Clickable levels

Option 2 — EQUILIBRIUM Map

• center → core
• Rings → modules
• Axles → Control

Option 3 - Control Panel

• Sliders:
• energy
• Communications
• Transitions

MAIN ADVANTAGE

You get:

\boxed - \text - VISIBILITY OF THE SYSTEM

Which means:

• Make decisions faster
• not cure the symptoms
• Working on structure

HONEST AND DIRECT

Here’s a key thought that can’t be ignored: if you don’t visualize the system, you don’t control it, you guess.

(screen, interface, prototype)

Drive your project through the map

and get:

• weak areas
• points of growth
• Specific actions