### Nuclear density

#### What is special about nuclear density

It seems remarkable, but if we filled this teaspoon with the stuff that neutron stars are made from then the mass would be much more than, for example, the mass of all of the half million elephants in Africa.

We know from Rutherford’s alpha particle scattering experiment that atoms are mostly empty space, if you want to check out what Rutherford’s experiment is then there is a link here. Rutherford’s alpha scattering experiment.

To show how much empty space that is imagine a single atom in the middle of the worlds largest sport stadium. If we magnified that atom up until it filled the stadium then the nucleus would be **about the size of a fly**. Almost all of the mass is the nucleus, the electrons have hardly any mass.

In a neutron star, formed by the collapse of a giant sun, the pressure has caused the atoms themselves to collapse. The protons to absorb the electrons so that the star is a sphere of closely packed neutrons of unbelievable density.

Because an atom nucleus is composed of close packed neutrons and protons, the density of a nucleus seems likely to be similar to that of a neutron star. We can calculate the density of the nucleus of an atom.

#### Calculating nuclear density

Scattering experiments have shown that the size of a nucleus is directly proportional to the nucleon number (the atomic mass) and it turns out that the densities of all nuclei are about the same – no surprise really since they are made up from the same proton and neutron building blocks. It seems that the nucleons are arranged much like close packed marbles

Since the volume of a sphere is proportional to the cube of the radius the radius of a nucleus is proportional to the cube root of the number of nucleons. So we get the equation:

r = r_{o} A^{1/3}

where r_{o} is the constant of proportionality = 1.2 x 10^{-15}m

To justify the claim about a teaspoon holding millions of tons let us take an example of a carbon atom, atomic mass 12.

the radius would be 1.2 x 10^{-15} x12^{1/3} and since volume is 4/3 π r^{3}

that is 4/3 π (1.2 x 10^{-15} x12^{1/3} )^{3}

In one mole of carbon atoms, which is 12 grams there are about 6x 10^{23} atoms (Avogadro’s number)

so the mass of one atom is

12 x 10^{-3} / 6x 10^{23 } Kg almost all of which is the mass of the nucleus since electrons have a minute mass

Density = mass/volume so the density of our carbon nucleus will be:

12 x 10^{-3 }/ 6x 10^{23 }x^{ }4/3 π (1.2 x 10^{-15} x12^{1/3})^{3}

which is 2.3x 10^{17 } Kg m^{-3}

Going back to the teaspoon, it has a volume of 5ml which is 5 x 10^{-6 }m^{3 }From the density we calculated is 2.3 x 10^{17 }Kg m^{-3}

Mass is density x volume so the mass of these 5 ml of nuclear material would be 5 x 10^{-6 }x^{ }2.3 x 10^{17 }Kg

that is 11.5 x 10^{11 }Kg or 11.5 x 10^{8 }tonnes, so if our teaspoon was full of similar nuclear material that would be over one billion tonnes.

**Neutron Stars**

A **neutron star** is the collapsed core of a massive supergiant star.

Except for black holes and some hypothetical objects neutron stars are the smallest and densest stellar objects we know of. We think there are around one billion neutron stars in the Milky Way.

The pressure of collapse causes the electrons and protons to combine to produce neutrons.

Neutron stars that can be observed are very hot, typically around 600000 K rotating at up to several hundred times per second. They have a radius of about 10 kilometres (6.2 mi) and a mass of about 1.4 solar masses. They are so dense that a normal matchbox containing neutron-star material would have a mass of about 3 billion tonnes.