33

In the responses to Draw Bohr atomic model with electron shells in TeX?, there are nice drawings of atoms. However, the nuclei don't look very appealing or realistic.

I was wondering: Can anyone think of an algorithm to (semi-)automatically (for example in a randomized fashion) create large nuclei that look more realistic as for example the in the image attached?

enter image description here

It seems to be key that the balls are sufficiently spaced and that the spherical look requires more centered balls to be on top. Both requirements are not met with my code:

\documentclass{standalone}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}
\path (-2,-2) rectangle (2,2);
\pgfmathdeclarerandomlist{color}{{red}{white}}
\foreach \a in {1,...,200} {
    \pgfmathsetmacro{\r}{rnd}
    \pgfmathsetmacro{\a}{random(0,360)}
    \pgfmathrandomitem{\c}{color}
    \shade[ball color=\c] (\a:-\r) circle (5pt);
    }
\end{tikzpicture}
\end{document}

The result is: enter image description here

EDIT:

In case anyone is interested, here is what I am quite happy with: Based on the answer, I have defined slightly modified versions of the suggested nucleus in three different sizes and with the option to feed a random seed to get different species.

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{calc}

\begin{document}

\begin{tikzpicture}
\tikzset{
    pics/proton/.style={code={\shade[ball color=red] circle (3pt);}},
    pics/neutron/.style={code={\shade[ball color=white] circle (3pt);}},
    pics/nucleussmall/.style={code={%
        \pgfmathdeclarerandomlist{nucleon}{{proton}{proton}{neutron}{neutron}{neutron}}
        \pgfmathsetseed{#1+1}
        \foreach \A/\R in {8/0.2, 5/0.13, 1/0}{
        \pgfmathsetmacro{\S}{360/\A}
            \foreach \B in {0,\S,...,360}{
                \pgfmathrandomitem{\C}{nucleon}
                \pic at ($(\B+2*\A+5*rnd:\R)$) {\C}; } }} },
    pics/nucleusbig/.style={code={%
        \pgfmathdeclarerandomlist{nucleon}{{proton}{proton}{neutron}{neutron}{neutron}}
        \pgfmathsetseed{#1+1}
        \foreach \A/\R in {24/0.4, 24/0.3, 24/0.2, 13/0.35, 11/0.27, 6/0.15, 1/0}{
        \pgfmathsetmacro{\S}{360/\A}
            \foreach \B in {0,\S,...,360}{
                \pgfmathrandomitem{\C}{nucleon}
                \pic at ($(\B+2*\A+5*rnd:\R)$) {\C}; } }} },
    pics/nucleusbiggest/.style={code={%
        \pgfmathdeclarerandomlist{nucleon}{{proton}{proton}{neutron}{neutron}{neutron}}
        \pgfmathsetseed{#1+1}
        \foreach \A/\R in {24/0.5, 24/0.4, 24/0.3, 24/0.2, 13/0.47, 15/0.44, 13/0.37, 11/0.27, 6/0.15, 1/0}{
        \pgfmathsetmacro{\S}{360/\A}
            \foreach \B in {0,\S,...,360}{
                \pgfmathrandomitem{\C}{nucleon}
                \pic at ($(\B+2*\A+5*rnd:\R)$) {\C}; } }} },
    }
\pic at (0,0) {nucleussmall};
\pic at (2,0) {nucleusbig=1};
\pic at (4,0) {nucleusbiggest=1};    
\end{tikzpicture}
\end{document}

enter image description here

  • 2
    "More realistic" doesn't really make sense here. It's a quantum-mechanical object, and the wavefunctions all overlap. – Ben Crowell Sep 1 '18 at 19:44
  • 1
    Agreed, at least in terms of physics. What I was shooting for was spheres somehow bunched together to form a bigger sphere... – FlorianL Sep 1 '18 at 19:54
42

Here is a proposal that makes the nucleus look more like a compact ball. It works by building up circular rings starting from the outside in. By adjusting the number of protons/neutrons in each ring and its distance from the center, you can create a ball effect.

enter image description here

\documentclass{standalone}
\usepackage{tikz}
\usepackage[version=4]{mhchem}
\begin{document}
\begin{tikzpicture}
\path (-2,-2) rectangle (2,2);
\pgfmathdeclarerandomlist{color}{{red}{white}}
\pgfmathsetseed{1}
\foreach \A/\R in {25/1,12/0.9,15/0.8,20/0.7,12/0.5,7/0.3,1/0}{
      \pgfmathsetmacro{\S}{360/\A}
           \foreach \B in {0,\S,...,360}{
               \pgfmathrandomitem{\C}{color}
               \shade[ball color=\C] (\B+\A:\R) circle (5pt);
           }
}
\node at (-1,1.3) {\ce{^{226}_{88}Ra}};
\end{tikzpicture}
\end{document}
  • 4
    +1, again excellent work. – Sebastiano Sep 1 '18 at 1:40
13

Based on your code, I first draw protons/neutrons following a circular pattern three times, at radius 1, 0.5 and 0.2. I also draw random protons/neutrons in between.

\documentclass{standalone}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}
\path (-2,-2) rectangle (2,2);
\pgfmathdeclarerandomlist{color}{{red}{white}}

\foreach \a in {0,10,...,360}{
    \pgfmathrandomitem{\c}{color}
    \shade[ball color=\c] (\a:1) circle (5pt);
}

\foreach \a in {0,20,...,360}{
    \pgfmathrandomitem{\c}{color}
    \shade[ball color=\c] (\a:0.5) circle (5pt);
}

\foreach \a in {1,...,350} {
    \pgfmathsetmacro{\r}{rnd}
    \pgfmathsetmacro{\a}{random(0,360)}
    \pgfmathrandomitem{\c}{color}
    \shade[ball color=\c] (\a:\r) circle (5pt);
    }

\foreach \a in {0,60,...,360} {
    \pgfmathrandomitem{\c}{color}
    \shade[ball color=\c] (\a:0.2) circle (5pt);
}
\end{tikzpicture}
\end{document}

The results is:

Screenshot

7

Here is another version in which the spheres are put on the root lattice of A_3 and allowed to wiggle a bit. More explanations can be found here.

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{3d}
\tikzset{declare function={posx(\x,\y,\z)=\x-\y/2;
posy(\x,\y,\z)=\y/sqrt(2);
posz(\x,\y,\z)=-\y/2+\z;
}}
\newsavebox\Proton
\newsavebox\Neutron
\sbox\Proton{\tikz{\shade[ball color=red] circle({0.85/sqrt(2)});}}
\sbox\Neutron{\tikz{\shade[ball color=gray!20] circle({0.85/sqrt(2)});}}
\begin{document}
\xdef\Lst{{-1, 0, 2}, {-2, -1, 1}, 
 {0, 0, 2}, {-1, -1, 1}, 
 {-2, -2, 0}, {-1, 1, 2}, 
 {-2, 0, 1}, {1, 0, 2}, {0, -1, 1}, 
 {-1, -2, 0}, {-2, -3, -1}, 
 {0, 1, 2}, {-1, 0, 1}, 
 {-2, -1, 0}, {1, -1, 1}, 
 {0, -2, 0}, {-1, -3, -1}, 
 {1, 1, 2}, {0, 0, 1}, {-1, -1, 0}, 
 {-2, -2, -1}, {0, 2, 2}, 
 {-1, 1, 1}, {2, 1, 2}, {-2, 0, 0}, 
 {1, 0, 1}, {0, -1, 0}, 
 {-1, -2, -1}, {-2, -3, -2}, 
 {1, 2, 2}, {0, 1, 1}, {-1, 0, 0}, 
 {2, 0, 1}, {-2, -1, -1}, 
 {1, -1, 0}, {0, -2, -1}, 
 {-1, -3, -2}, {2, 2, 2}, 
 {1, 1, 1}, {0, 0, 0}, 
 {-1, -1, -1}, {-2, -2, -2}, 
 {1, 3, 2}, {0, 2, 1}, {-1, 1, 0}, 
 {2, 1, 1}, {-2, 0, -1}, {1, 0, 0}, 
 {0, -1, -1}, {-1, -2, -2}, 
 {2, 3, 2}, {1, 2, 1}, {0, 1, 0}, 
 {-1, 0, -1}, {2, 0, 0}, 
 {-2, -1, -2}, {1, -1, -1}, 
 {0, -2, -2}, {2, 2, 1}, {1, 1, 0}, 
 {0, 0, -1}, {-1, -1, -2}, 
 {1, 3, 1}, {0, 2, 0}, {-1, 1, -1}, 
 {2, 1, 0}, {1, 0, -1}, 
 {0, -1, -2}, {2, 3, 1}, {1, 2, 0}, 
 {0, 1, -1}, {-1, 0, -2}, 
 {2, 0, -1}, {1, -1, -2}, 
 {2, 2, 0}, {1, 1, -1}, {0, 0, -2}, 
 {2, 1, -1}, {1, 0, -2}}
\tdplotsetmaincoords{-90+109.471}{-90+70}
\foreach \X in {1,...,10}
{\begin{tikzpicture}
\path[use as bounding box] (-3.5,-3.5) rectangle (3.5,3.5);
\draw (0,0) circle ({1}); % /sqrt(2)
\begin{scope}[tdplot_main_coords]
 \draw[-latex] (0,0,0) coordinate (O) -- (1,0,0) node[right]{$\alpha_1$};
 \draw[-latex] (O) -- (-1/2,{1/sqrt(2)},-1/2) node[right]{$\alpha_2$};
 \draw[-latex] (O) -- (0,0,1) node[right]{$\alpha_3$};
 \draw[red,-latex] (O) -- (1/2,{1/sqrt(2)},1/2) node[right]{$-\theta$};
 \foreach \Z in \Lst
  {\pgfmathsetmacro{\myx}{{\Z}[0]}
  \pgfmathsetmacro{\myy}{{\Z}[1]}
  \pgfmathsetmacro{\myz}{{\Z}[2]}
  \pgfmathsetmacro{\mydeltax}{0.1*(rnd-0.5)}
  \pgfmathsetmacro{\mydeltay}{0.1*(rnd-0.5)}
  \pgfmathsetmacro{\mydeltaz}{0.1*(rnd-0.5)}
  \pgfmathtruncatemacro{\mycol}{int(2*rnd)}
  \ifnum\mycol=1
  \node at ({posx(\myx+\mydeltax,\myy+\mydeltay,\myz+\mydeltaz)},
  {posy(\myx+\mydeltax,\myy+\mydeltay,\myz+\mydeltaz)},
  {posz(\myx+\mydeltax,\myy+\mydeltay,\myz+\mydeltaz)}) {\usebox\Neutron};
  \else
  \node at ({posx(\myx+\mydeltax,\myy+\mydeltay,\myz+\mydeltaz)},
  {posy(\myx+\mydeltax,\myy+\mydeltay,\myz+\mydeltaz)},
  {posz(\myx+\mydeltax,\myy+\mydeltay,\myz+\mydeltaz)}) {\usebox\Proton};
  \fi}
\end{scope}
\end{tikzpicture}}
\end{document}

enter image description here

Just for fun: more nuclei. And no, it does not look like sphere, but like a set of sphere which are packed with maximum density. This is of course not the same as demanding that the nuclei should fill out a sphere. The latter might translate in the requirement that the sum of distances gets minimized or something like that, which obviously is not the same requirement as maximal packing. I do not know if there is a simple algorithm that minimizes the sum of distances while making sure the spheres do not overlap.

enter image description here

  • 1
    +1, but somehow this looks not exactly spherical… – TeXnician Sep 1 '18 at 15:56
  • @TeXnician It might be that I did something wrong but they are not supposed to look spherical. More precisely, they will approach a sphere in the limit of infinitely many spheres. Just do a 2D example and pack two circles. Does that look like a bigger circle? Definitely not. Same for any finite amount of circles. You will arrange them on a hexagonal lattice and put them as close as possible to a center, but the emerging shape is never a precise circle as long as you have a finite number of circles. This here is the 3D version of that (I hope). – marmot Sep 1 '18 at 16:02
  • 2
    Yes, I understand the approach which is interesting, I just understood the OP to reproduce a "spherical look", but who knows how realistic either of these representations is :) – TeXnician Sep 1 '18 at 16:06
  • @TeXnician Well, in reality nuclei are not spheres, and the interactions between them cannot be described by a spherical potential, rather there are the dominant strong interactions plus a bit of electromagnetic interactions. This lattice packing allows you to partly understand why there are these magical numbers. And I think that this approach comes close to the OP's version if I increase the number of spheres. Just fill a hexagonal lattice with a huge number of circles around a center, and it will approach a circle. If the number of circles is smaller, it will have "edges". – marmot Sep 1 '18 at 16:09
  • 1
    @FlorianL Well, you could define your own version: \tikzset{declare function={Veclen(\x,\y,\z)=sqrt(\x*\x+\y*\y+\z*z);}}. However, I am not sure if my result is more physical. Apart from the fact that nuclei are not spheres, my procedure produces something that has only discrete rotational symmetries, but I don't see a reason why a nucleus should have some preferred axes. I think all of these are just cartoons, and among the proposals so far, IMHO Milo's nice answer is the winner. It is nicer and TeX only. – marmot Sep 1 '18 at 20:57

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