MFGSB (ver. 1.0): Computer code for self-consistent mean-field calculations of atomic nuclei using Gaussian expansion method

The paper announces the availability of MFGSB, a computer code developed at Chiba University for performing self-consistent mean-field calculations of atomic nuclei using the Gaussian expansion method.

The Gist

Imagine you are trying to build a perfect model of a tiny, chaotic city inside an atom. This city is made of protons and neutrons (nucleons) that are constantly bumping into each other, dancing, and changing shapes. Physicists call this a "self-consistent mean-field calculation." It's a fancy way of saying: "Let's figure out exactly how every single particle moves based on how all the other particles are moving, and keep doing that until the whole picture makes sense."

The paper you shared introduces a new digital tool called MFGSB (version 1.0). Think of MFGSB as a super-smart, high-powered architectural blueprint generator specifically designed for these atomic cities.

Here is a breakdown of what this tool does, using some everyday analogies:

1. The "Magic Grid" (The Gaussian Expansion Method)

Most computer programs try to map a city by drawing a grid of tiny squares. But atomic particles don't like squares; they like curves and clouds.

• The Analogy: Imagine trying to paint a perfect circle using only square LEGO bricks. It looks jagged and wrong.
• The MFGSB Solution: Instead of squares, MFGSB uses Gaussian functions, which are like soft, fluffy clouds or smooth, round balloons. By stacking these "balloons" of different sizes, the code can perfectly mold the shape of the atomic nucleus, whether it's a perfect sphere or a squashed football. This is the "Gaussian Expansion Method."

2. The "Universal Kit" (No Tuning Required)

Usually, if you want to model a specific type of nucleus (like a heavy gold atom vs. a light carbon atom), you have to tweak the settings of your computer program for each one. It's like having to re-calibrate a telescope every time you look at a different star.

• The Analogy: MFGSB is like a universal remote control that works on every TV in the house without you ever pressing the "setup" button.
• Why? The "balloons" (parameters) it uses are so flexible that one single set of settings works for almost the entire "periodic table" of elements. You don't need to tune it for every new nucleus you study.

3. The "Heavy Lifting" (Memory and Size)

This tool is powerful, but it's also heavy.

• The Analogy: Think of the code as a giant library. The "books" inside are the rules of how particles interact.
• The Catch: The library is huge (16.6 GB!). Most of the space is taken up by the "interaction data" (the rules of how protons and neutrons hug or push each other).
• The Workaround: If your computer isn't a supercomputer, you can download a "light" version of the software and then download the heavy "rule books" separately, just like downloading a game engine and then the game assets later.

4. What Can It Actually Do? (The Menu of Options)

The code offers a menu of different ways to simulate the atomic city:

• The "Static" Mode: You can just look at the city under a fixed, pre-made map (Woods-Saxon potential) without letting the particles rearrange themselves.
• The "Dynamic" Modes (HF, HF+BCS, HFB): This is where the magic happens. The code lets the particles rearrange themselves until they find the most stable, comfortable shape.
  • HF (Hartree-Fock): Everyone finds their seat at the table.
  • HF+BCS / HFB: Now, some people at the table decide to hold hands and dance in pairs (superconductivity in nuclei). The code handles this pairing perfectly.

5. The "Rules of the Road" (Symmetry)

The code can simulate the nucleus in different shapes:

• Spherical: Like a perfect beach ball.
• Axial: Like a rugby ball or a cigar (stretched out).
• Asymmetric: Like a weirdly shaped potato.
  It can also add "constraints," which is like telling the code, "Keep this nucleus stretched out like a football," so you can see what happens when you squeeze it.

6. The "Engine" (How it Solves the Math)

To get the final answer, the code has to solve a massive math puzzle. It has two ways to do this:

• The "Iterative" Method: It guesses, checks, guesses again, and checks again, slowly getting closer to the truth (like refining a sketch).
• The "Conjugate Gradient" Method: It takes a more direct, mathematical shortcut to find the answer at the very end.

Why Should You Care?

This code is a free, open-source tool (available at Chiba University) for scientists to understand the fundamental building blocks of our universe. It helps them answer questions like:

• Why do some atoms exist and others don't?
• How do stars create heavy elements?
• What happens when we smash atoms together in a collider?

In short: MFGSB is a versatile, "plug-and-play" computer program that uses a clever "balloon" technique to simulate the complex dance of particles inside an atom, helping scientists map out the invisible architecture of matter.

Technical Summary

Based on the provided document, here is a detailed technical summary of the paper describing the MFGSB (ver. 1.0) computer code.

1. Problem Statement

The paper addresses the need for a robust, self-consistent mean-field (SCMF) computational tool for atomic nuclei that can handle complex nuclear interactions and various symmetry constraints without relying on excessive approximations. Traditional methods often struggle with:

• Handling specific interaction channels (like the tensor channel in Yukawa interactions) efficiently.
• Accurately describing the energy-dependent asymptotics of single-particle or quasiparticle wavefunctions.
• Requiring frequent re-tuning of basis function parameters for different nuclides or deformations.
• Approximating the center-of-mass (c.m.) motion or Coulomb exchange terms.

2. Methodology

The MFGSB code is written in FORTRAN77 and utilizes the Gaussian Expansion Method (GEM) as its core numerical framework.

• Core Algorithm: The code performs SCMF calculations by solving the Hartree-Fock (HF), Hartree-Fock-BCS (HF+BCS), and Hartree-Fock-Bogolyubov (HFB) equations.
• Basis Functions: It employs Gaussian expansion functions. A key feature of this methodology is that the parameters of these basis functions are insensitive to specific nuclides and deformation. This allows a single set of parameters to be applied across a wide range of the nuclear chart, eliminating the need for case-by-case tuning.
• Symmetry Constraints: The code supports calculations under three distinct symmetry assumptions imposed on one-body fields:
  • Spherical (J): Includes Parity (P) and Time-Reversal (T) conservation.
  • Axial (Jz): Includes P, T, and reflection symmetry with respect to the y-axis (R).
  • Jz only: Includes P and the product of R and T (RT).
• Convergence Methods: To attain self-consistency, the code offers two numerical approaches:
  • Iterative Method: Diagonalizes the HF or HFB Hamiltonian at every iteration step.
  • Conjugate Gradient Method: Diagonalizes the Hamiltonian only at the end of the process (more efficient for certain cases).
• Interaction Handling: The code supports various two-nucleon effective interactions. It uniquely handles the Yukawa interaction up to the tensor channel without additional approximations. It also treats the Coulomb exchange term and the two-body center-of-mass motion term exactly.

3. Key Contributions

The primary contribution of this work is the release of the MFGSB (ver. 1.0) software package, which offers several unique capabilities:

• Tensor Channel Support: It is identified as a unique SCMF code capable of utilizing Yukawa interactions including the tensor channel.
• Exact Treatment of c.m. and Coulomb Terms: Unlike many codes that approximate these terms, MFGSB handles the center-of-mass motion and Coulomb exchange terms without additional approximations.
• Universal Basis Parameters: The insensitivity of the Gaussian basis parameters to nuclear deformation and specific isotopes allows for a "one-size-fits-all" parameter set, significantly streamlining calculations across the nuclear chart.
• Efficient Asymptotics: The code efficiently describes the energy-dependent asymptotics of wavefunctions within moderate precision, a critical factor for unbound or weakly bound states.
• Flexibility: It supports non-self-consistent calculations (Woods-Saxon or harmonic oscillator potentials) as well as full self-consistent HF, HF+BCS, and HFB calculations.

4. Results and Technical Specifications

While the paper serves primarily as a software release and user guide rather than a presentation of new physical data, it outlines the following technical performance characteristics:

• Memory Requirements: Memory usage is directly tied to the cutoff parameter ($\ell_{cut}$).
  • For the default $\ell_{cut} = 7$, the code requires 8 GB of RAM.
  • Memory usage approximately doubles for every increment of 1 in the $\ell_{cut}$ value.
• Dependencies: The code requires the installation of BLAS and LAPACK libraries.
• Data Size: The full distribution (mfgsb.zip) is 16.6 GB, largely due to the storage of interaction data files. An alternative "light" version is available where interaction files can be downloaded separately.
• Availability: The code is hosted at the Chiba University Repository (DOI: 10.20776/900123722).

5. Significance

The release of MFGSB represents a significant advancement in computational nuclear physics for the following reasons:

• Precision in Interaction Modeling: By enabling the use of Yukawa interactions with tensor channels and exact c.m. corrections, it allows for more realistic modeling of nuclear forces, which is crucial for understanding nuclear structure and reactions.
• Computational Efficiency: The ability to use a single set of basis parameters for a wide range of nuclei reduces the computational overhead associated with parameter optimization, making large-scale surveys of the nuclear chart more feasible.
• Versatility: The code's ability to switch between different symmetry constraints (Spherical, Axial, Jz-only) and convergence methods (Iterative vs. Conjugate Gradient) makes it a versatile tool for studying diverse nuclear phenomena, from stable spherical nuclei to deformed and exotic systems.
• Reproducibility: By providing a standardized, open-access code with detailed documentation (UsersGuide, README), it facilitates reproducibility in SCMF research and allows other researchers to build upon established numerical methods.

In summary, MFGSB is a specialized, high-precision tool designed to overcome specific limitations in existing mean-field codes, particularly regarding interaction channels, basis function flexibility, and the exact treatment of kinematic terms.