Nucleus – Nuclear Density

The positive charge and the mass in every atom are concentrated very densely at the centre of the atom.  This concentration is termed the nucleus of an atom.  Its dimensions are way smaller when compared to the atom as a whole. Experiments demonstrated on the scattering of α-particles that the radius of an atom is bigger than the radius of the nucleus by a factor of approximately 104.  So, it can be said that an atom is mostly empty.  However, even if the nucleus is a very tiny part of the atom, it makes up more than 99.9% of the total mass of an atom. 

Nucleus

The very small place in an atom that comprises a positive charge and all the mass of the atom is called the nucleus.

The nucleus is made up of protons and neutrons that are called nucleons.

Concepts  about the nucleus:

• Atomic number – In an element, the number of protons in the nucleus of its atom is called the atomic number of that element.  It is denoted by the letter Z, which stands for ‘Zahl’, which in German means number.

• Mass number – In an element, the number of protons and neutrons together inside the nucleus of an atom is called the mass number of that element.  It is denoted by the letter A.

• Nuclear size – Scattering experiments have accurately measured the nuclei of various elements. In this experiment, instead of α – particles, are the projectiles.  It has been found through the experiment that the nucleus with a mass number A has a radius. 

R = R0A1/3

Where, R0 = 1.1 * 10-15 m is an empirical constant. 

• Nuclear density – The density of the nucleus is mass per volume. The density of the nucleus of an atom is, on average, about 2.3×1017 kg/m3.  

• Atomic mass unit – About 1/12th mass of a carbon nucleus defines the atomic mass unit.  It is denoted by the letter u and is abbreviated as amu. 

1 amu = 1.992678 * 10-26 / 12 kg

                          = 1.6 * 10-27 kg 

                          = 931 MeV

Nuclear Density 

The density of the nucleus of an atom is, on average, about 2.3×1017 kg/m3. This is called nuclear density.  It is the same for all the nuclei, as it does not depend upon the mass number or the size of the nucleus.

ρ = mass of nucleus / volume of nucleus ⇒ ρ = 3m / 4π R3o

where, m = average mass of a nucleon.

The nucleus size can help calculate the nuclear density of a typical nucleus. This calculation is based on the number of neutrons and protons within the nucleus. In terms of the number of nucleons, the radius of any typical nucleus is R = R0A1/3 . Here, A is the mass number and R0 is 1.25 fm. There is a typical deviation of about 0.2 fm from this value. 

Therefore, the number density of the nucleus is, 

n= A / 4/3πR3

As the density is constant in terms of mass number, then theoretically,

n = A / 4/3 π(A1/3R0)3

= 3/ 4π(1.25fm)3

 = 0.122 (fm)-3

= 1.22 ⋅10-44 m-3

The experimentally determined value for n is 0.16 fm−3, that is, 1.6·10-44 m−3.

Some Important Properties

• Isotopes – In an element, if the atoms have the same atomic number but different mass numbers, then they are called isotopes.  Example: Carbon-12, Carbon-13, Carbon-14 are three isotopes of the element carbon.  They have the mass numbers 12, 13 and 14, respectively. 

• Isobars – Isobars is the term given to atoms of different elements having the same mass numbers but different atomic numbers.  Example:  40S, 40Cl, 40Ar, 40K and 40Ca is a series of isobars.
• Isotones – When in an element, the atomic number and the mass number are different, but the number of neutrons remains the same, so they are called isotones.  Example:  6C14, 7N15 and boron-12, carbon-13.

• Isomers – When atoms differ in radioactive properties but have the same atomic number and mass number, then they are called isomers.  Example: ethylbenzene, m-xylene, p-xylene and o-xylene.

Nuclear force and binding energy

• Nuclear Force – The force between the nucleons acting inside the nucleus is called the nuclear force.  This force is very strong to bind both the protons and the neutrons together inside the nucleus. It is the strongest force in nature and is said to be 1038 times that of gravitational force.

• Nuclear binding energy – Protons and neutrons make up the nucleus, but the mass number is always less than the sum of individual masses of the protons and neutrons.  This difference measures the nuclear binding energy that holds the nucleus together.  

Binding energy is required to separate the nucleons from the nucleus.  The minimum energy required in this regard is called the Nuclear binding force. 

Nuclear binding energy per nucleon = Nuclear binding energy / Total number of nucleons

Binding energy, Eb = [Zmp + (A – Z) mn – mN]c2

Conclusion

The ability to understand the mechanism of this universe depends upon the studies and concepts related to physics.  It is important to learn the basics so as to widen the perspective and acquire more information in this field.  The protons and neutrons inside the nucleus of an atom alone have helped us understand the existence of isotopes, isobars and so on, upon which so many other aspects in an element can be categorised.