The Atomic Nucleus

To a chemist an atom is the smallest entity to have the chemical and physical properties of an element. The size of an atom is about 10-10 m and most all its space is taken up by negative electrons swirling chaotically about a very small positively charged nucleus which is 100,000 times smaller ( 10-15 m ) than the atom. The nucleus of the atom contains positively charged protons and neutral neutrons which make up most all the mass of the atom. The atomic number (symbol Z ) of the atom is equal to the number of positive protons in the nucleus as well as the number of negative electrons outside the nucleus. The name and symbol of the atom or its element is based on its number of protons. Or as Richard Feynman put it in his book “QED, The Strange Theory of Light and Matter”, “chemists have a complicated way of counting: instead of saying one, two, three, four, five protons, they say hydrogen, helium, lithium, beryllium, boron.” In other words, chemists know that the number of protons in the atomic nucleus is also the atomic number of Mendeleev.

Hydrogen (11H) is the only atom that does not have a neutron in the nucleus; it has only a single proton. All other atoms have both protons and neutrons in the nucleus. This includes a less abundant form of hydrogen called deuterium or heavy hydrogen, which has a proton and a neutron in its nucleus (12H). The proton and the neutron have about the same mass, which can be expressed in atomic mass units or in kilograms (1 amu, or 1.67 x 10-27 kg). The sum of the protons and neutrons in a nucleus is the mass number (p + n= A) of the nucleus.  The protons and neutrons are also called nucleons and every atom with a nucleus that has the same atomic number but a different mass number is an isotope or nuclide of the element. For example, sodium, element number eleven (Z=11) has a mass measured on a sensitive mass spectrograph of 22.989767 amu and rounded to a whole number is 23 or A=23. From A = p + n, 23 = 11 + n, or n = 23 – 11 = 12 neutrons. A better description is that sodium has only one stable isotope, 1123Na, with 11 protons, 11 electrons and 12 neutrons. Fluorine, Z=9, has only one stable isotope 919F with 9 protons, 9 electrons, and 10 neutrons. Magnesium, Z=12 has three stable isotopes with mass numbers of 24, 25, and 26 .  Here on earth magnesium is a mixture of 78,99% 24Mg, 10.00% 25Mg and 11.01% 26Mg for a weighted average of 24.3050 which is the average mass on the periodic table. In these designations for the isotopes of magnesium, the subscript for the number of protons has been dropped since (as Richard Feynman said) the symbol Mg means 12 protons and everyone who studies chemistry knows how to count protons.

There are about 3000 isotopes of the elements that have had one or more of their properties measured. The U. S. Government maintains a website of isotope or nucleon data at Brookhaven National Nuclear Center  (https://www.nndc.bnl.gov). Figure 1 is a chart of number of protons versus number of neutrons for all the known isotopes. All the small black squares up to Z= 83 are isotopes  that are generally considered or observed to be stable. The chart copied here is not interactive, but the interactive version of this chart can be used at https://www.nndc.bnl.gov/nudat2/ .

 

Figure 1: The Brookhaven National Nuclear Center chart of protons vs neutrons for all the known isotopes of the elements from: https://www.nndc.bnl.gov/nudat2/

Clicking on 14C at Brookhaven’s site gives the data for this unstable isotope of carbon:

E(level) (MeV)

Δ(MeV)

T1/2

Decay Modes

0.0

0+

3.0198

5700 y 30 

β- : 100.00 %


The Decay Modes entry tells us that carbon 14 is unstable and decomposes by eliminating a b
- particle from the nucleus.  A b- particle is actually an electron. On the other hand, carbon 11 is unstable and captures a 1s electron from outside the nucleus {electron capture} and becomes stable boron 11. Electron capture will be discussed in a later section. Carbon 14 has a half- life (T1/2) of 5700 years which means that only half the atoms of 14C decompose in 5700 years and half again in the next 5700 years. So starting with 100,000 atoms of 14C, 5700 years later only 50,000 atoms would remain and after 11,400 years (two half-lives) there would be 25,000 atoms etc. The half-lives of the isotopes are a measure of nuclear stability and are color coded in Figure 1. The isotopes represented by black dots are considered to be stable. The half-life by color is: blue 10,000,000 s, light blue 100s, green 10s, yellow 10-4s and pink 10-15s.  

The decomposition of unstable isotopes is accompanied by the elimination of high energy particles and is therefore dangerous. Each isotope or nuclide has its own unique half- life and particle or particles that they eliminate. The particles eliminated are helium nuclei called alpha particles (a, He2+), beta particles (b-, or electrons, e-) or (b+, positrons, e+), and gamma particles (high energy photons, g). Electron capture mentioned above is also a decay mode which is also accompanied by emission of g particles. The unstable nuclides that are above the dotted black line decay by positrons (e+, b+) emission, electron capture or both. Those below the black line emit electrons (e-, b-). Those isotopes with A > 83 emit alpha particles, (He2+)

The double horizontal lines and the vertical double lines in Figure 1 are the positions of the so-called magic numbers, lettered in blue for protons and black for neutrons. The magic numbers are nuclear states, very much like the noble gases, of unusual extra stability. The magic numbers are 2, 8, 20, 28, 50, 82, and 126, and for protons are represented by He, O, Ca, Ni, Sn, and Pb. These elements are very stable and more abundant in the solar system than their immediate neighbor elements (helium however is not as abundant as hydrogen). Also note that the magic numbers are all even numbers.

The most up to date version of the periodic table contains 118 elements. These elements are the basic building blocks of all matter here on Earth and the universe as we know it. Only 81 of the 118 elements have stable isotopes. Of the first 83, the elements technetium (Tc) element 43 and promethium (Pm) element 61 have no stable isotopes and are not part of the minerals that make-up the crust of the Earth.  Thorium (Th), radium (Ra) and uranium (U) are all unstable, alpha particle emitting elements that are present in the Earth’s crust in low concentrations (ppm range).

Of the 3000 measured nuclides or isotopes in Figure 1 there are only 254 that are stable and have not been observed to decay. Even numbers of nucleons favor stability. There are 148 stable isotopes with an even number of protons and an even number of neutrons. There are 53 stable isotopes with an even number of protons and an odd number of neutrons and 48 with an odd number of protons and an even number of neutrons. Only five stable isotopes have an odd number of both protons and neutrons ( 6Li, 10B, 14N, 50V and 180Ta). From this one concludes that pairing of like nucleons (p or n) contributes to nuclear stability.

Tin, with Z = 50, has ten stable isotopes, xenon has nine, and cadmium has eight. There are six elements with seven stable isotopes, eight with six, and ten with five. There are 25 elements that have only one stable isotope and they all have an odd number of protons. Since even numbers are favored, it turns out that the even Z elements are generally more abundant on earth and in the solar system. Lastly there are no isotopes known that have five or eight nucleons, that is A= 5 or 8.

The nucleons in the nucleus, both protons and neutrons, are in states very similar to the electron states in atoms and generally follow a level scheme given in Figure 2.

Figure 2,  Nuclear Shell level Diagram taken from Wikipedia

This is the accepted nuclear shell level diagram which won the 1963 Nobel Prize in Physics for Eugene WignerMaria Goeppert Mayer and J. Hans D. Jensen. This made Maria Goeppert Mayer only the second woman to win that prize after Marie Curie. It is similar to the electronic shell diagram except that there is no restriction on the orbital quantum number, l,  by the quantum number n. That is: 1p, 1d, 1f… etc. are possible. These nuclear states are for protons and, separately, for neutrons; that is protons and neutrons do not mix and have separate nuclear configurations. This level diagram was calculated using a distorted harmonic oscillator model with strong spin-orbit coupling which infers strong relativistic behavior which means that the protons and neutrons interact while moving close to the speed of light in a very small space. The Pauli principle, that no two protons or two neutrons may have the same quantum numbers, applies. The energy separations of the nuclear states are measured in megaelectron volts (MeV) and are about 106 times that of the electron states which are measured in electron volts (eV).

There is another measure of nuclear stability which compares the mass of the atom with the mass of the nucleons which compose the atom. The mass of the nucleons is always greater than the mass of the atom and the difference in the mass is the binding energy of all the nuclear particles. This uses the Einstein formula, E=mc2, to convert what is called the mass defect (or packing fraction) to energy and is recorded as the average binding energy per nucleon. A plot of the average binding energy per nucleon in MeV versus the number of nucleons in the nucleus from 1H to 238U has been taken from Wikipedia and posted as Figure 3. A version of this graph is what showed scientists in 1939 that powerful atomic weapons could be made with energies a million times the energy of chemical explosives. This gave rise to the Manhattan Project of World War II.

The graph shows that at about 56Fe the binding energy per nucleon reaches a maximum. The graph also suggests that perhaps we could take 56 1H atoms and fuse them into 56Fe and release about 56 X 8.8 = 490 MeV of energy. This is of course is what stars like our Sun are doing in a step-wise process. The process will end in about 10 billion years when all the hydrogen and helium in the sun is converted into iron. Stars that are more than10 times more massive than the Sun undergo this fusion process very fast and when they form a lot of iron they explode into a supernova and the elements are spread all over the cosmos. This is also the basis of the hydrogen bomb.

At the uranium side of the curve it is possible to split the heavy 235U isotope into elements closer to 56Fe . This was done by the nuclear reaction:

                                                     1n + 235U => 94Sr + 140Xe + 21n  

This process is called fission and is the basis of the Hiroshima atomic bomb and nuclear power reactors.

The curve also suggests that building up the light nuclei into isotopes with more particles in the nucleus enhances stability. But the building up reaches a point where increasing the number of protons increases the positive charges and the repulsive forces weaken the nucleus with each addition.

Figure 3  The Average Binding Energy per Nucleon versus the Number of Nucleons in the Nucleus, taken from Wikipedia.

The graph shows that at about 56Fe the binding energy per nucleon reaches a maximum. The graph also suggests that perhaps we could take 56 1H atoms and fuse them into 56Fe and release about 56 X 8.8 = 490 MeV of energy. This is of course is what stars like our Sun are doing in a step-wise process. The process will end in about 10 billion years when all the hydrogen and helium in the sun is converted into iron. Stars that are more than10 times more massive than the Sun undergo this fusion process very fast and when they form a lot of iron they explode into a supernova and the elements are spread all over the cosmos. This is also the basis of the hydrogen bomb.

At the uranium side of the curve it is possible to split the heavy 235U isotope into elements closer to 56Fe . This was done by the nuclear reaction:

                                                     1n + 235U => 94Sr + 140Xe + 21n  

This process is called fission and is the basis of the Hiroshima atomic bomb and nuclear power reactors.

The curve also suggests that building up the light nuclei into isotopes with more particles in the nucleus enhances stability. But the building up reaches a point where increasing the number of protons increases the positive charges and the repulsive forces weaken the nucleus with each addition.