Computing nuclear properties in the FCC model​
​Adiba Shaikh
A face-centered cubic (FCC) structure for the arrangement of protons and neutrons within the nucleus of an atom was originally proposed by E. Wigner (1937). Computations by N. Cook (starting in the 1970s) tried to explain the data for low-energy nuclear reactions using the solid-phase FCC model. In this project, we implemented the FCC model for nuclear structure and computed some basic nuclear properties such as number density, quantum numbers, alpha particle structure and binding energy. This analysis is done for light nuclei, from Hydrogen-4 up to Oxygen-16. Comparison of the binding energy for various nuclei indicates their enhanced stability due to the presence of magic numbers of protons and neutrons. This work can be extended to heavier nuclei in the future.

Arranging nucleons in the FCC configuration

The required FCC configuration for the nucleon arrangement within the nucleus is done by importing the Crystallica [
0
] package.
In[]:=
Needs["Crystallica`"];
We use the function CrystalPlot from the above package for generating the nucleon arrangement. This function takes the position coordinates and number of protons and neutrons as arguments. The position coordinates required to build the FCC lattice configuration is taken from reference [
0
]. The isospin choice for the position coordinates need not be strictly followed and can be interchanged.
The position coordinates {x,y,z} for the nucleons are as follows, where the square bracket denotes all possible permutations of odd integer (including sign) to give alternate even and odd parity.​​n = 0 shell with [1,1,1] Absolute Sum = 3 Parity = Sign[x*y*z]: even (+1) ​4 coordinates: {1,1,1}, {1,-1,-1}, {-1,1,-1}, {-1,-1,1}.​​n = 1 shell with [1,1,3] ​Absolute Sum = 5 Parity = Sign[x*y*z]: odd (-1) ​12 coordinates: {1,1,-3}, {1,-1,3}, {-1,1,3}, {-1,-1,-3}, {1,3,-1}, {1,-3,1}, {-1,3,1}, {-1,-3,-1}, {3,1,-1}, {3,-1,1}, {-3,1,1}, {-3,-1,-1}.​​n = 2 shell with [1,3,3] , [1,1,5] Absolute Sum = 7Parity = Sign[x*y*z]: even (+1) ​24 coordinates: {1,3,3}, {1,-3,-3}, {-1,3,-3}, {-1,-3,3}, {3,1,3}, {3,-1,-3}, {-3,1,-3}, {-3,-1,3}, {3,3,1}, {3,-3,-1}, {-3,3,-1}, {-3,-3,1}, {1,1,5}, {1,-1,-5}, {-1,1,-5}, {-1,-1,5}, {1,5,1}, {1,-5,-1}, {-1,5,-1}, {-1,-5,1}, {5,1,1}, {5,-1,-1}, {-5,1,-1}, {-5,-1,1}.
In[]:=
Coord={{1,1,1},{-1,-1,1},{1,-1,-1},{-1,1,-1},{3,1,-1},{3,-1,1},{-3,1,1},{-3,-1,-1},{1,3,-1},{1,-3,1},{-1,3,1},{-1,-3,-1},{1,1,-3},{1,-1,3},{-1,1,3},{-1,-1,-3},{3,3,1},{-3,-3,1},{3,-3,-1},{-3,3,-1},{3,1,3},{-3,-1,3},{3,-1,-3},{-3,1,-3},{1,3,3},{1,-3,-3},{-1,-3,3},{-1,3,-3},{5,1,1},{-5,-1,1},{5,-1,-1},{-5,1,-1},{1,5,1},{-1,-5,1},{1,-5,-1},{-1,5,-1},{1,1,5},{-1,-1,5},{1,-1,-5},{-1,1,-5}};
The coordinates for proton and neutron can be classified based on its isospin quantum number which depend on the z-coordinate.
In[]:=
i[z_]:=
(z-1)/2
(-1)
​​QNos[x_,y_,z_]:={x,y,z}i[z]
Position coordinates for the nucleons associated with their isospin:
In[]:=
Association[QNos@@@Coord]
Out[]=
{1,1,1}1,{-1,-1,1}1,{1,-1,-1}-1,{-1,1,-1}-1,{3,1,-1}-1,{3,-1,1}1,{-3,1,1}1,{-3,-1,-1}-1,{1,3,-1}-1,{1,-3,1}1,{-1,3,1}1,{-1,-3,-1}-1,{1,1,-3}1,{1,-1,3}-1,{-1,1,3}-1,{-1,-1,-3}1,{3,3,1}1,{-3,-3,1}1,{3,-3,-1}-1,{-3,3,-1}-1,{3,1,3}-1,{-3,-1,3}-1,{3,-1,-3}1,{-3,1,-3}1,{1,3,3}-1,{1,-3,-3}1,{-1,-3,3}-1,{-1,3,-3}1,{5,1,1}1,{-5,-1,1}1,{5,-1,-1}-1,{-5,1,-1}-1,{1,5,1}1,{-1,-5,1}1,{1,-5,-1}-1,{-1,5,-1}-1,{1,1,5}1,{-1,-1,5}1,{1,-1,-5}-1,{-1,1,-5}-1
The coordinates for proton (1) and neutron (-1) based on isospin quantum number:
In[]:=
ProtonCoord=Keys@Select[Association[QNos@@@Coord],#==1&]
Out[]=
{{1,1,1},{-1,-1,1},{3,-1,1},{-3,1,1},{1,-3,1},{-1,3,1},{1,1,-3},{-1,-1,-3},{3,3,1},{-3,-3,1},{3,-1,-3},{-3,1,-3},{1,-3,-3},{-1,3,-3},{5,1,1},{-5,-1,1},{1,5,1},{-1,-5,1},{1,1,5},{-1,-1,5}}
In[]:=
NeutronCoord=Keys@Select[Association[QNos@@@Coord],#==-1&]
Out[]=
{{1,-1,-1},{-1,1,-1},{3,1,-1},{-3,-1,-1},{1,3,-1},{-1,-3,-1},{1,-1,3},{-1,1,3},{3,-3,-1},{-3,3,-1},{3,1,3},{-3,-1,3},{1,3,3},{-1,-3,3},{5,-1,-1},{-5,1,-1},{1,-5,-1},{-1,5,-1},{1,-1,-5},{-1,1,-5}}
In[]:=
Length/@{ProtonCoord,NeutronCoord}
Out[]=
{20,20}
In[]:=
ProtonCoordinates[_]:=Take[ProtonCoord,]​​NeutronCoordinates[_]:=Take[NeutronCoord,]​​Coordinates[_,_]:=Join[ProtonCoordinates[],NeutronCoordinates[]]
Specification of the nucleon type by mapping proton number () to type 1 in YellowGreen colour and neutron number () to type 2 in FireBrick colour:
In[]:=
Nucleons[_,_]:=Flatten[Join[ConstantArray[#,]&/@{1},ConstantArray[#,]&/@{2}]//Values]
Using the function CrystalPlot to create an image of the nucleus:
In[]:=
Nucleus[_,_]:=CrystalPlot[{{1,0,0},{0,1,0},{0,0,1}},Coordinates[,],Nucleons[,],AddQ->False,AtomCol->{"YellowGreen","Firebrick"},AtomRad->0.4,BondStyle->1,BondDist->4,MoveOrigin{0,0,0},CellLineStyle->False,Axes->False,AxesOrigin{0,0,0}]
Helium-4 (
4
​
2
He
) nucleus with
=2
and
=2
:
In[]:=
Framed[Show[Nucleus[2,2],SphericalRegionTrue]]
Out[]=
A tetrahedron for the above alpha-cluster can be created. The color of the stick connecting the nearest neighbouring nucleons indicates the type of the bond, i.e., proton-proton, neutron-neutron, proton-neutron:
In[]:=
[_,_]:=Graphics3D[{Cyan,Opacity[.3],Tetrahedron[Coordinates[,]]}]
In[]:=
Framed[Show[{Nucleus[2,2],[2,2]},SphericalRegionTrue]]
Out[]=
A circumsphere constructed
indicates the closed nucleon shell with magic number of nucleons:
In[]:=
[_,_]:=Graphics3D[{Magenta,Opacity[.2],Circumsphere[Coordinates[,]]}]
4
​
2
He
nucleus completely occupies the first shell corresponding to the radial quantum number n = 0.
Construction of n=0 circumsphere:
In[]:=
He4Spheren0=Graphics3D[{Blue,Opacity[0.3],Circumsphere[Coordinates[2,2]]}];
The center and radius for n=0 circumsphere:
In[]:=
Circumsphere[Flatten[{ProtonCoordinates[2],NeutronCoordinates[2]},1]]
Out[]=
Sphere{0,0,0},
3

Displaying n=0 circumsphere:
In[]:=
Framed[Show[{Nucleus[2,2],He4Spheren0},SphericalRegionTrue]]
Out[]=

Nuclear Properties

Various nuclear properties have been determined using the FCC structure of the nucleus as follows.

Quantum Numbers

Determining the quantum numbers from the position coordinates {x,y,z} of a nucleon:
Creating an association between the quantum numbers and coordinate points:

Nuclear Density

Each FCC unit consists of a total of 4 nucleons (8 corner nucleons contributing 1/8 part + 6 face-centered nucleons contributing 1/2).
Length of the side of FCC unit cell is the distance between proton positions 2 and 3:
The number density of the nucleus core is found to be close to that of the FCC lattice, indicating that it is a closed-packed structure.

Nearest Neighbour Nucleons

The nearest number of nucleons is determined graphically using the function NearestNeighborGraph.
Display up to third nearest neighbour:

Mean Electrostatic Repulsion per Nucleon

For the mean electrostatic (Coulomb) repulsion per proton (Q/A), we need to calculate the distance between individual proton pairs in the nucleus.
The function NuclearFCCbonds computes the number of bonds and distances between the first and second nearest neighbours for the nucleons with the same isospin (proton-proton, neutron-neutron) and opposite isospin (proton-neutron):
Averaging over all possible proton-proton pair distances in femtometer (scaled by a factor of 1/2 for double-counting), we get the mean electrostatic repulsion per proton within the nucleus:

Binding Energy per Nucleon

Binding energy per nucleon and comparison with the experimental data from IsotopeData function in Wolfram documentation:

Other Examples

Dataset:
Proton quantum numbers:
Neutron quantum numbers:
Mean electrostatic repulsion and binding energy per nucleon:
Dataset:
Proton quantum numbers:
Neutron quantum numbers:
Mean electrostatic repulsion and binding energy per nucleon:
The center and radius for n=1 circumsphere:
Displaying n = 0 and n = 1 circumspheres:
Dataset:
Proton quantum numbers:
Neutron quantum numbers:
Mean electrostatic repulsion and binding energy per nucleon:
Proton quantum numbers:
Neutron quantum numbers:
Mean electrostatic repulsion and binding energy per nucleon:

Concluding remarks

Keywords

◼
  • Face Centered Cubic
    • ◼
  • Nuclear structure
    • ◼
  • Quantum Numbers
    • ◼
  • Binding Energy per Nucleon

    • Acknowledgment

    I would like to acknowledge my mentor Paul Abbott for his support and encouragement during the course of this project. I appreciate his help in making me understand any difficulties by drafting notebooks with useful comments and examples. I also thank Rahul Sharma (TA) for being available for help whenever needed during the initial phase of the school and project work. Finally, I thank Tuseeta Banerjee, Mohammad Bahrami, Aravind Hanasoge and others for the wonderful organization of this school, and for giving me an opportunity to be a part of it.

    References

    ◼
  • [1] https://community.wolfram.com/groups/-/m/t/787142 and http://library.wolfram.com/infocenter/MathSource/9372/
    • ◼
  • [2] N. D. Cook, Computers in Physics 3, 73 (1989), https://aip.scitation.org/doi/pdf/10.1063/1.168319
    • ◼
  • [3] N. D. Cook, arXiv:1101.4251, https://arxiv.org/abs/1101.4251
    • ◼
  • [4] N. D. Cook and V. Dallacasa, Il Nuovo Cimento A (1965-1970) 97 (1987), https://link.springer.com/article/10.1007/BF02733847