“I have no special talents. I am only passionately curious.” Albert Einstein

Bhaskara was an Indian mathematician and astronomer, he was born in 1114. He made a large impact on the fields of arithmetic, algebra (Bijaganita), sphere(Goladhyaya) and planets(Grahaganita).

His legendary book in arithmetic reveals a predication about his daughter Lilavati. He studied Lilavati’s horoscope and predicted that her husband would die soon after the marriage if the marriage did not take place at an auspicious time.

He constructed a very simple time machine with bowl and vessel filled with water and sized a hole at the bottom so that bowl will sink at a right favorable time of her marriage, hole size was per his arithmetical calculation. He cautioned his daughter not to go near the device. But out of curiosity one day she went to check and unknowingly dropped her nose ring in the device. Obviously, the marriage took place at the wrong time and sadly, she became a widow soon after.

All pictures are Courtesy of Google

Please review Publication 9.

Spectroscopic Notation

Long before quantum mechanics, scientists in the field of the light spectrum found series of line spectra of atoms and assigned letters to the spectra. Quantum mechanics started to use this notation in terms of the quantum number as electron states, the notation is as follows:

Courtesy of Google

l represents orbital angular momentum. The higher numerical value of l represents higher energy. For the atomic shell, model n is the principal quantum number. Nuclear shell model does not use this number in the notation for energy states. It uses a number indicating energy, such as 1 is the lowest, next higher energy is 2, and so forth. Total angular momentum is the summation of spin and orbital angular momentum.

Courtesy of Google

Electron Shell Configuration – shorthand notation for showing what sublevels are filled-
Carbon atom has 6 electrons –1s²  2s² 2p² because 2+2+2=6
1s² (n=1 Principal quantum number), 2s² 2p² (n=2 Principal quantum number)
Silicon atom has 14 electrons – 1s² 2s² 2p⁶ 3s² 3p² because 2+2+6+2+2=14
By nature, electrons are filled according to lowest energy sublevels first.

Understanding of energy is the most important of all. Since the big bang and thereafter all kind of material are created. Scientists understood how atom worked, then how element formed out of those atoms. So we will look into the atomic structure in great detail. This will give us a better understanding of our evolution. Again, we will start by saying that there are four forces as scientists defined how elements dependent on the forces.

Nuclear Shell:

So far we discuss atomic shell structure and notation. Nuclear shell model differs from the above notation.

Carbon structure:

Notation for nuclear energy states (Nuclear Shell)

Spectroscopic notations (s,p,d,f ….) are very useful in quantum level, they also denote the energy level of the nuclear shell.

Electron spin is an intrinsic property of electrons. Electrons have intrinsic angular momentum characterized by quantum number s = 1/2.

The 1s spatial state can have two states of electron spin, which we usually just refer to as “spin-up” and “spin down”. With two electrons in the 1s state, it is filled and stable, forming the noble gas helium.

For particle, the spin part is S =1/2, or -1/2, and orbital angular momentum with l =2, the state is represented by “d”. So total angular momentum could be either l – S = 3/2 or l + S = 5/2. Thus two states would then be labeled:

1d_3/2 and 1d_5/2 where 1 is not the principal quantum number as defined for atomic shell model. It is just an indexing number of energy level. 1 is the lowest level. Higher levels with l=2  would be labeled as 2d_3/2, 2d_5/2, 3d_3/2, 3d_5/2, etc.

Stability is the ultimate goal of nature. Also, the buildup process starts to fill the lowest level to the highest level. A closed shell does not contribute to the total nuclear spin energy. But outside shell may contain more neutrons or fewer neutrons than protons so forming isotopes, thus it will contribute to the spin energy to the elements.

For example, ¹⁷O has one neutron outside a closed shell and its ground state would be designated (1d_5/2)¹.

¹⁷O: [(1s_1/2)²(1p_3/2)⁴(1p_1/2)²]…….(1d_5/2)¹
Closed shell plus one odd neutron

¹⁶O: [(1s_1/2)²(1p_3/2)⁴(1p_1/2)²]

The parity of the nuclear state is also an important characteristic and often listed as a part of the notation for the state.
The state of the ¹⁵O is sometimes written as 1p_1/2–
¹⁷O as 1d_5/2+ to indicate both the spin and parity of those nuclei in their ground states.

Shell Model of Nucleus

The “Pauli exclusion principle” constrains the entire buildup process of elements.
It is evident that even numbers of protons and neutrons should be stable than odd numbers. Physicists found that there are some magic numbers when nuclear stability is the most. These are as follows:

2, 8, 20, 28, 50, 82, 126

Nuclear shell:

The nuclear shell model of the atomic nucleus uses the “Pauli exclusion principle” to describe the structure of the nucleus in terms of energy levels.

The nucleus is tightly packed. In an atom, electron space is 1800 times more than nucleus space. Proton and neutron are not constantly colliding; its Stability comes due to the above magic number.

Proton and neutron are made of quark. The strong force, one of the fundamental forces holds the nucleus together, quark like gluon, pion and other quarks-antiquark plays the primary role to bind nucleus together.

This stability suggests that it does not behave like an electronic shell structure. Scientist determines nucleon moves in an effective potential well/box.

One very important concept of quantum energy states and its interaction is if there are no unfilled quantum states that are reachable by the available energy, then the interaction will not occur. So the interaction between proton and neutron are limited. They are not in constant collision.

Total angular momentum of a nucleus is denoted by I (nuclear spin).

In quantization of energy and its states indicate that multiple states can be identified indicating a dependence on potential well and the orbital quantum number with spin-orbit interaction.

With this set of identified nuclear states and the magic numbers, scientists calculate the nuclear spin of a nucleus, magnetic moment and the angular momentum of the nucleus.

Simple harmonic motion:

We all know from our early mechanics and physics about mass-spring harmonic oscillator and simple harmonic motion. Quantum physics due to extreme micro nature extends this classical idea in analysis of energy waveform. It is rather complex, so, for now, we accept nuclear spin in the conservation of momentum.

Please stay tuned for more about the nuclear shell, potential well, particle in a box and momentum.