2.6c Electron Arrangement in Atoms/Quantum/Laser

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2.6c: (Update 13 November 2021) Contents: In scroll-down  order: Planck's Quantum Theory,The Photon and the Dual Nature of Electromagnetic Radiation, Emission Spectrum of Hydrogen, Practical Application of Physics: The Laser, and the ending section on Quantum Mechanics.

Planck’s Quantum Theory

It is not surprising that while scientists were discovering the discontinuities of matter down to sub-atomic particle levels they still thought of energy as a seamless stuff one might turn on or off from a magic spigot and divide down infinitesimally. In 1900, a physicist, Max Planck, made an astounding discovery: Atoms and molecules emit energy only in whole numbers! It means there is a basic singularity in matter and energy that cannot be gotten inside of.  

Planck discovered that atoms and molecules could emit (or absorb) energy only in discrete whole-number quantities like small indivisible packets. A quantum (pl. quanta) is the name he gave the smallest, indivisible unit and it is emitted as electromagnetic wave or light-producing particle called photon. The energy of a quantum is proportional to its wave frequency (per second) and the higher the frequency the higher the energy. The Quantum Theory infers matter's basic discontinuity. The Quantum Theory says that the Universe at its smallest level is singularities. It brings us closer to a unifying theory of Nature because it brings energy with its wave motion into line with matter and its particle structure.

The Photon and the Dual Nature of Electromagnetic Radiation  Recall the previously described rainbow-color radiation wavelength (distance between successive wave-top tips), frequency (number of waves packed into one second) and amplitude (height of the wave), based on the vibration and alternating compression and relaxation on molecules and atoms in the wave peak and wave valley.

Electromagnetic radiation of which visible light is one part has been observed to behave as if it were at times particle and at times wave. The particle equivalent is the photon which derives from phot, in Greek, ‘light’. The photon acts as a basic quantum particle.

   Waves from each part of the electromagnetic spectrum differ in energy in proportion to the frequency of waves per second and as a function of Planck’s constant, h. For those who study physics further these relations provide the mathematics of energy.  

A distinctive emission spectrum can be obtained by heating or running electric current through a chemical element or compound. And the emission spectrum of each atom heated or electrified in gas phase is separated into thin lines so it is useful for classifying and identifying atoms; it is the basis of spectroscope analysis that identifies atoms and chemical compounds. (Used in a type of MRI called MRS to identify chemicals in the brain of schizophrenics and other disordered brain metabolism)

Emission Spectrum of Hydrogen:  Quantum theory gave a new understanding of atomic structure. It said that the electron in an H atom could be located only in certain orbitals of exactly specified distance from the center of the nucleus. To knock the electron into a further-out orbital would take an input of an exact integer (whole number) packet of energy. And, oppositely, when an electron falls from this further-out high-energy orbital into a nearer-to-nucleus orbital, it would release the energy integer packet. The quantum theory restriction is its dictate that electron orbitals are digitally determined. i.e., they are quantized, meaning an electron can be in an orbit in integer units 1 unit away from the nuclear center or 2 units away - but not a fraction between the integers.

   Note here, the analogy between “analog” and “digital” in computers; analog meaning smooth gradual increments that encompass fractional distances like dialing on a radio, while digital means quantum click or integer increments of 1, 2, 3, like digital punch buttons on a digitally tuned TV. This had important implications that explained the line spectra (plural of spectrum) -- the primary color lines produced when a chemical has been heated -- of H atoms subjected to electrical current. The quantum theory atom explained the radiation by energized H atom in line-spectra in terms of the heated or otherwise energized electron. Once the heat stopped being applied, the hot, cooling electron dropped from a higher energy orbital to a lower one and at that instant gave up (or "off") a certain number of quanta of energy in the form of photons of light at specific wavelengths which showed in the colors of the line spectrum.

In the diagram of a hydrogen atom, the n (the letter n, small case) stands for the principal quantum number and denotes an orbital with an electron at a set distance from center of the nucleus, and it may have values 1, 2, 3, …; (written as subscripts) the higher the number, the further out and more energized the electron. The least energized electron is n=1, closest to nucleus. It is said to be in the "ground state", and is the orbital an electron would go into if no outside energy were put into the system. With increasing energy input, the electron is knocked into a more distal orbital and its quantum number signifies the digitally increased energy state by increased integer number 2, 3, 4, … . The most energized electron is in a shell that is, theoretically, the distance from its nucleus where the electron has been knocked free of its atom – this is the so called “free electron.”

So the emission spectrum of an energized hydrogen H atom is explained by the fact that its electrically energized atom loses its potential energy in a series of quanta of photons as it “cools” (no longer receives the “flame” of electrical current or other energy source) and its energized electron falls back, digitally, integer step by step, like bouncing down a stair of equal size steps, one step at a time, to ground state condition (written n_g ). The different colors emitted by heated elements (like in Neon and other fluorescent lighting) relate to the n value of its electrons. (More energized electrons give line spectra of higher frequencies and shorter wavelength, the green, blue and violet, while less energized electrons give lower frequency, longer wavelength in red-orange) Practical application of this idea is seen today in neon or fluorescent lights and in the laser.

          Practical Application of Physics: The Laser

The dual understanding of light’s structure both as wave and particle is joined in understanding the Laser (Light Amplification by Stimulated Emission of Radiation). Laser is light produced by electronic stimulation of atoms that emit photons in a single wave phase and in a single direction, in contrast to usual light. There are many types of laser based on the different chemical compounds used to emit the photons. A rod of a chemical compound such as Ruby laser composed of the ions Al³⁺ and Cr³⁺ is surrounded by a coil of intense neon flashing light that when turned on excites and stimulates each ion’s outer electrons and, when the flashing turns off, the excited ions emit photons.

Explaining the figure above: A ruby laser apparatus: The ruby red is surrounded by flash lamp neon coil and at both ends are the reflecting mirror discs.  On your right, the partially reflecting mirror disc is also partial transmitting as a lens and the laser rays can be seen on the extreme right side emitting as the 3 waves in the same phase; thus the laser emits coherent light.

Each photon that is emitted bounces back and forth between the two end mirrors and as the photon travels it excites more ions to release more photons in chain reaction and at a certain point the bouncing photons produce enough pressure to pass through the mirror at the tube’s emitting end (Your right in the above Figure), and it exits as a coherent, focused ray of light. A coherent light beam is infinity-focused; it means each photon as a micro light ray moves in a parallel path to all the other photons. Laser light is useful because of 3 properties: it is intense, it has precise wavelength, and it is coherent. Its high intensity is useful in surgical operations, especially in the eye; its precise wavelength penetrates outer tissue without harm, and its coherence allows a laser light ray to travel immense distances without dissipating as in, for example, sending a laser light to bounce off the Moon; and also it allows laser light to be transmitted in flexible glass fibers in endoscopes that can twist through the intestinal tract without damaging it. The optical fibers used in laser are flexible glass fibers and can pass laser light in curvature and without heating, which metal fibers cannot, and so allow flexible searchlight instruments like the endoscope. The optical fibers have much greater capacity to pass laser digital messages which makes for their superior use in computers. 

Quantum Mechanics - The 4 Quantum Numbers

In quantum mechanics, 4 numbers describe the electron arrangement in atoms. As follows:

Quantum Number 1: Principal quantum number (n): The symbol n is already familiar from the many-energy-level quantum hydrogen atom 

The n tells two related things about an electron: its distance from its atom’s nucleus and its energy state, which, as energy is put into the system, increases digitally, 1 (Ground state,written n_g )., 2x, 3x, ... to a free electron state, written nx) as distance from nucleus increases.  

Quantum Number 2. The angular momentum quantum number, written ℓ, or L. The possible numbers of L for an n- level electron depend on the Quantum-1, n number. The numbers for L will range, from 0 to the quantity expressed as Quantum-1 number, n minus 1. Thus, in the ground-state hydrogen H atom, with a single electron at n=1, the L can only have an angular momentum number 0 (When n=1, n minus 1 = 0).  L = 0 means no angular momentum and gives only one geometric shape – the sphere so it tells that the shape of all innermost electron shells is a simple sphere whose diameter size depends on its energy state. If the electron is in a next higher energy state n=2; then, the two possible numbers for angular momentum are 0 or 1; if n=3, the possible numbers for L are 0, 1 or 2; and so on. These series of possible numbers for the angular momentum of an electron in orbital are notated by a series of letters for each set of orbitals of increasing angular momentum, which start with the 0 angular momentum and are given the letters s, p, d, f, g, h, and run as follows:

   Angular momentum.number:  0  1  2   3   4  5 

Letter symbol of orbital:            s  p  d  f  g  h

In the case of the atom hydrogen (H) with only one electron and in the ground state n=1, the angular momentum number of that electron can only have an L-value of 0 and the H atom can only contain its electron in the s- orbital, a sphere located at the closest ground state distance from the nucleus, labeled 1s. And the s orbital shell is satisfied by 2 electrons. The s, p, d, f, g, and h describe different electron orbital sets referred to as sub-shells. These electron arrangements characterize the structure of the atoms of specific elements and a sub-shell may contain more than one orbital but all the orbitals in one sub-shell have the same angular momentum or shape. A sub-shell contains the orbitals with the same n and same angular momentum within a shell. A shell with n=2 is at the next closest energized distance from the atomic nucleus after the shell of n=1 (And as 1 is to 2, it is an integral multiple), and may be composed of the two possible sub-shells, one sub-shell with angular momentum number 0 and the other with angular momentum number 1. These sub-shells are called the 2s sub-shell and the 2p sub-shell. (The 2 because the n=2) As we shall see when getting into the structure of the chemical element atoms, this understanding of sub-shells allows prediction of the structure of an element's atoms and its chemical reactivity.

Quantum Number 3: The Third, or Magnetic Quantum Number m_L (subscript L is used here for the more usually used cursive lower case, written ℓ) describes the orientation of the orbital in space. Within a sub-shell the value of m_L depends on the value of the angular momentum number, quantum number 2, or ℓ. For a certain value of ℓ there are (2ℓ+1) possible values of m_L but it has the limitation that the first value must be 0 and each successive value is respectively a plus (+) and a minus (–) 1 from the initial 0. Thus, when ℓ=0, 2ℓ + 1 = 1, meaning that at ℓ=0 there can only be one value for the magnetic quantum number m_L and it must be 0. But when ℓ=1, then 2ℓ + 1 = 3 possible values, and then, starting from an m_L of 0 we get two additional values +1 and –1. And so on. The number of m_L indicates the number of orbitals with a particular ℓ value in a sub-shell m_L. Consider an atom with electrons in n = 2 shell and ℓ = 1 sub-shell. The values of n and ℓ indicate that we have a 2p sub-shell. (2 because n=2; and p sub-shell because ℓ = 1, and the ℓ=1 gives the p sub-shell) And in this sub-shell are three 2p orbitals because of the three values of m_L given by +1, 0 and –1.

Quantum Number 4:The spin quantum number m_s which refers to the spin of a specific electron about its axis – either symbolized ↑ for counterclockwise spin like Earth around its North-South axis with N magnetic pole up ↑ or down↓. Proof of this is that when hydrogen gas, which is not naturally magnetic because it is a random combination of oppositely oriented spin H atoms in each H₂ molecule, is heated to very high temp in vacuum, its balanced spin, 2-atom H₂ molecule splits up into single H atoms. Each single H atom is magnetic because its one electron spins about the H nucleus axis to produce an unopposed, unbalanced magnetic field.

The electron spin of an atom can only have two values +1/2 (plus) or –1/2 (minus 1/2) and these are designated by upward or downward pointed arrows.

n        ℓ       m_ℓ    # of orbitals  Atomic orbitals

1       0        0                    1                  1s

2       0        0                    1                  2s

         1   –1, 0, 1               3              2p_x, 2p_y, 2p_z

3       0        0                    1                  3s

         1    –1, 0, 1              3             3p_x, 3p_y, 3p_z

         2   –2, –1, 0, 1 ,2   5             3d_xy, 3d_yz, 3d_xz,

                                                         3d_x²–y², 3d_z²

The above table shows how the knowledge of the quantum numbers allows the visualization of the structure of an atom and its surrounding electrons.

Note that all electrons in orbitals with a zero angular momentum, m_ℓ  = 0, have only s orbitals, whose surface is a sphere (the 1s, 2s, and 3s atomic orbitals in the chart above). These electrons are limited to a maximum 2 electrons per orbital and each such atom is paired in that each pair will have the same n, ℓ and m_ℓ but differ in the m_s, one being + 1/2 and the other in counter spin –1/2. As we shall see, the m_s of the orbital electrons determines the magnetism of an element. So think of an atom first with a central solid positive-charge nucleus, with one or more protons and with zero to many non-charged but equal weight neutrons. Then, imagine in the n=1 shell the spherical 1s orbital with space for two electrons. In a single hydrogen H atom there is only one electron, in its single 1s shell, and it has a counterclockwise spin (+1/2) causing its N magnetic pole (in fixed orientation) to point upward or downward (one or the other in random) in each H atom. Then, imagine the next larger atom, the He (Helium) atom, which has two 1s electrons filling the single-orbital sub-shell; one electron with m_{s (+1/2)} and the other m_s (-1/2), so Helium is not magnetic because its 1s electron pair's magnetic spins neutralize each other. In atoms heavier than Helium, since the 1s shell is already filled, the next new s shell electrons must be in 2s orbital or higher orbitals. The s shell never takes more than 2 electrons to fill; it is at once orbital, sub-shell and shell.

  The p Orbitals sets are sub-shells (All have the same angular momentum quantum number ℓ) at one or another principal quantum level (n=2 gives three 2p sub-shell, n=3, 3p sub-shells). At n=1 level there can be no p sub-shell because, at that level, ℓ can only have 0 value so there is only the 1s orbital as the orbital spherical shell at n=1. The p sub-shells are computed to look like a 3 dimensional figure-of-eight shape, like the symmetrical 2-jointed body of an ant, and in the case of the three 2p sub-shells, each one is oriented in one of the planes in 3 dimensions – the horizontal x, the 90⁰ horizontal y and the 90⁰ vertical z plane. Thus the x, y, z subscripts for the p orbitals.

  Then as n continues to increase integer by integer, each successive n shell produces the appropriately letter-labeled set of increasing sub-shells according to the (2ℓ+1) rule (n=3 gives five 3p sub-shells, ….)

  Concerning d orbitals and higher energy orbital sub-shells, when ℓ equals 2, there are five values of m_ℓ, which corresponds to five d orbitals. The lowest value of n for a d orbital is n=3. These five orbitals make up the 3d sub-shell of an element's atom and each has rather esoteric 3-dimensional geometric shape and is oriented in five different angular planes. They occupy the 3rd shell at an n=3 distance from the center of the atomic nucleus. And going through the 114 presently known elements, each element's atoms get built up composed of n number shells and s, p, d, f, g sub-shells, each sub-shell containing one or more electrons in orbital. 

  Summarizing from the H atom, its electron arrangement is notated as 1s¹. The front number 1 tells us the electron is in n=1, lowest energy, or ground state. The 1s tells that angular momentum = 0 because the s orbital is the one with ℓ = 0. The superscript 1s¹ tells that only one electron is in the orbital, which is also the case with the hydrogen (H) atom. If it was 1s², it would describe the helium (He) atom which has two electrons. To represent the electron spin we use combination of arrows to show particular spin in bracketed orbital.  In the case of H it is 1s¹(↑ ). In the case of He it would be 1s² (↑↓). In the He atom it tells a ground state n=1 with two electrons in the s orbital or shell (orbital and shell the same in a Hydrogen or Helium atom because no other than 1 or 2 electrons); also in the Helium atom the opposite arrows show a balanced electron spin – one electron spinning counter clockwise and the other clockwise - meaning it is not a magnetic element (not paramagnetic). This brings us to the Pauli Exclusion Principle, which states that no two electrons in an atom can have the four main quantum numbers the same. In the H atom this does not come up because there is only one electron but in the He atom with two n=1, ℓ=0 and m_L=0 electrons, it means the 2^nd electron must have a quantum spin number m_s opposite quantity and direction to its orbital partner (one, + 1/2, the other – 1/2). This becomes an important determinant of whether an element will be paramagnetic (attracted to an iron magnet) or not. Atoms with unbalanced electron spins (Hydrogen and all atoms with odd number of electrons in the s orbital or having odd atomic numbers) are paramagnetic (in the atomic state but not necessarily in the molecular because the magnetic fields from the extra odd electrons may balance each other). The condition of being paramagnetic is testable by attraction by an iron magnet. On the other hand, if the electron-spins in an orbital are paired, anti-parallel to each other, the magnetic effect cancels out and the atom and its substance are diamagnetic and not attracted to magnets; actually, they are slightly repelled by a magnet but, practically there seems no magnetic affect. If we experiment, we see that H hydrogen is paramagnetic (as single atomic hydrogen but not as molecular hydrogen which has 2 randomly paired opposite spin atoms in balance in each molecule) while He helium is diamagnetic.

  To further explain, take the next heavier atom, the lithium, Li atom, which has 3 electrons. The third electron cannot go into the 1s orbit because it would have the same 4 quantum numbers as one of the first 2 electrons and thus would contradict the Pauli Exclusion Principle. This third electron is “forced” outward (from the atom’s nucleus) into the next (energetically) higher orbital, which is the 2s orbital. Thus the electron configuration of Li, Lithium is 1s²2s¹, an orbital spin diagram (↑↓) for the 1s orbital and (↑) for the 2s orbital. So its electron spins are odd number and therefore it is paramagnetic and attracted by a magnet.

Electron Arrangements in Many-Electron Atoms: The hydrogen H atom is single-electron and so it is easy to figure its electron arrangement. But there are also at least 114 different-structure atoms of the chemical elements. In the building up of electron shells, sub-shells and electron orbital pairs or singlet in the "many-electron" atoms, I shall use the carbon C atom to demonstrate the Hund Rule.

  The electron configuration of carbon C is notated 1s²2s²2p². It means that the neutral carbon C atom has an atomic nucleus with 6 positive charge protons balanced in its electron orbitals by 6 negative charge electrons -- as an inner 1s shell filled with 2 electrons in orbital, a second, further-out n=2 shell consisting of a 2s sub-shell orbital filled with 2 electrons, and a 2p sub-shell that has 3 possible orbitals that can house up to 2 electrons in each orbital.

 The important question is how to label the carbon atom electrons in the 2p sub-shell. These 2p electrons comprise 3 orbitals with –1, 0 and +1 magnetic quantum numbers m_L, and each electron has either one or another direction electron spin (counter-clockwise vs. clockwise). The possibilities are shown below and note the 3 sets of triple brackets represent the 3 orbitals of a p sub-shell, each bracket an orbital with a place for up to 2 electron-spin arrows. The arrows up and down represent electrons with opposite spins – either up counterclockwise, north magnet pole up, or opposite clockwise, north magnet pole down. The 0 or 00 mean no electron is in the orbital space or spaces where an electron could be.

  So which of the three possibilities actually exists in nature?

(↓↑)(00)(00)    (↑0)(↓0)(00)       (↑0)(↑0)(00)

2p_x, 2p_y, 2p_z   or  2p_x, 2p_y, 2p_z   or  2p_x, 2p_y, 2p_z

(a)                             (b)                         (c)

If you inspect the above three possibilities for arrangement of electrons in the 2p sub-shell, you will see that each of the three possibilities satisfies Pauli’s Exclusion Principle in that none has all the same 4 quantum numbers as others of the remaining two. 

Actually, it is the 2p sub-shells shown in (c) which accord with the experimental evidence of carbon’s paramagnetism, which it would not show if (a) or (b) were the correct arrangement because in each of those cases there are no 2 electrons with parallel spins. The above also demonstrates Hund’s Rule, which states that the most stable arrangement of electrons in sub-shells is the one with the greatest number of parallel spins.

The general rules of atomic electron structure and in assigning electrons to atomic orbitals are:

 1) The atom’s principal quantum number determines the number of electron shells (e.g., n=1, one shell as in H and He; n=2, two shells as in Ne, or Neon; and etc.) and also determines the number of sub-shells which is based on the angular momentum (ℓ=n – 1) and magnetic quantum numbers (2ℓ+1)

2) The s sub-shells are always limited to a maximum 2 electrons and, in the case where the s sub-shell is filled by the 2 electrons, these are always spin paired (↑↓), or anti parallel and so have no magnetic moment.

 3) No two electrons in an atom can have the same 4 main quantum numbers. (Pauli Exclusion Principle)

 4) When all other rules are satisfied and there is still a question of electrons filling a sub-shell; then, that electron arrangement with the maximum number of parallel spins is the only correct one. (Hund's Rule)

 5) Atoms whose sum of electron spins are unbalanced – more (↑) than (↓) or vice versa – will prove paramagnetic (Attracted by iron magnet) while those with balanced electron spins will be diamagnetic (practically no attraction). Any atom with an odd number of electrons will be paramagnetic but among atoms with an even number of electrons the atom's magnetism will depend on the orbital balancing of spins within a total even number of electrons as determined by Hund’s Rule.

This ends the summary of essential physics and chemistry. It can allow you to understand the structure of each of our realities. But still, beneath it all, the mystery remains. For students the knowledge here could be invaluable for getting a free or low cost science education at University.

To supplement and for reference I advise Raymond Chang’s “Chemistry”, 12^th ed., 2016, (or more recent) by McGraw-Hill Publishers. A gem with multi-media extension to usual text format! A lifetime read! Also, do Seminars on Science and Math using this plus your own input as the core. 

 The Next  Section Continues with Einstein's Relativity. To read now, click 2.6d Einstein's Relativity/Time Machine/TOE (Theor...