# and The Periodic Table

Niels Bohr (1923) was the first to propose that the periodicity in the properties of the chemical elements might be explained by the electronic structure of the atom. In fact, his early proposals were based on his own “toy-model” (Bohr atom) for the hydrogen atom in which the electron shells were orbits at a fixed distance from the nucleus. Bohr’s original configurations would seem strange to a present-day chemist: the sulfur atom was given a shell structure of  (2,4,4,6)  instead of $1s^22s^22p^63s^23p^4$, the right structure being (2,8,6).

The following year, E.C.Stoner incorporated the Sommerfeld’s corrections to the electron configuration rules, and thus, incorporating the third quantum number into the description of electron shells, and this correctly predicted the shell structure of sulfur to be the now celebrated sulfur shell structure (2,8,6). However neither Bohr’s system nor Stoner’s could correctly describe the changes in atomic spectra in a magnetic field (known as the Zeeman effect). We had to wait to the complete Quantum Mechanics formalist to arise in order to give a description of this atomic phenomenon an many others (like the Stark’s effect, spectrum split due to an electric field).

Bohr was well aware of all this stuff. Indeed, he had written to his friend Wolfgang Pauli   to ask for his help in saving quantum theory (the system now known as “old quantum theory”). Pauli realized that the Zeeman effect could be due only to the outermost electrons of the atom, and was able to reproduce Stoner’s shell structure, but with the correct structure of subshells, by his inclusion of a fourth quantum number and his famous exclusion principle (for fermions like the electrons theirselves) around 1925. He said:

It should be forbidden for more than one electron with the same value of the main quantum number n to have the same value for the other three quantum numbers k [l], j [ml] and m [ms].

The next step was the Schrödinger equation. Firstly published by E. Schrödinger in 1926, it gave three of the four quantum numbers as a direct consequence of its solution for the hydrogen atom: his solution yields the (quantum mechanical) atomic orbitals which are shown today in textbooks of chemistry (and above). The careful study of atomic spectra allowed the electron configurations of atoms to be determined experimentally, and led to an empirical rule (known as Madelung’s rule (1936) for the order in which atomic orbitals are filled with electrons. The Madelung’s law is generally written as a formal sketch (picture):

## Shells and subshells versus orbitals

In the picture of the atom given by Quantum Mechanics, the notion of trajectory looses its meaning. The description of electrons in atoms are given by “orbitals”. Instead of orbits, orbitals arise as the zones where the probability of finding an electron is “maximum”. The classical world seems to vanish into the quantum realm. However, the electron configuration was first conceived of under the Bohr model of the (hydrogen) atom, and it is still common to speak of shells and subshells (imagine an onion!!!)  despite the advances in understanding of the quantum-mechanical nature of electrons (both, wave and particles, due to the de Broglie hypothesis). Any particle (e.g. an electron) does have wave and particle features. The de Broglie hypothesis says that to any particle with linear momentum $p=mv$ corresponds a wave length (or de Broglie wavelength) given by

$\lambda=\dfrac{h}{p}=\dfrac{h}{mv}$

Remark: this formula can be easily generalized to the relativistic domain by a simple shift from the classical momentum to the relativistic momentum $P=m\gamma v$, so

$\lambda =\dfrac{h\sqrt{1-\beta^2}}{mv}$ with $\beta=v/c$

An electron shell is the set of energetic allowed states that electrons may occupy which share the same principal quantum number   n (the number before the letter in the orbital label), and which gives the energy of the shell (or the orbital in the language of QM). An atom’s nth electron shell can accommodate $2n^2$ electrons, e.g. the first shell can accommodate 2 electrons, the second shell 8 electrons, and the third shell 18 electrons, the fourth 32, the fifth 50, the sixth 72, the seventh 92, the eighth 128, the ninth 162, the tenth 200, the eleventh 242, the twelfth 288 and so on. This sequence of “atomic numbers” is well known

$(2,8,32,50,72,92,128,162,200,242,288,...)$

In fact, I have to be more precise with the term “magic number”. Magic number (atomic or even nuclear physics), in the shell models of both atomic and nuclear structure, IS any of a series of numbers that connote stable structure.

The magic numbers for atoms are 2,10,18, 36, 54, and 86, 118, 168, 218, 290, 362,… They correspond to the total number of electrons in filled electron shells (having $ns^2np^6$ as electron configuration ). Electrons within a shell have very similar energies and are at similar distances from the nucleus, i.e., inert gases!

The factor of two above arises because the allowed states are doubled due to the electron spin —each atomic orbital admits up to two otherwise identical electrons with opposite spin, one with a spin +1/2 (usually noted by an up-arrow) and one with a spin −1/2 (with a down-arrow).

An atomic subshell is the set of states defined by a common secondary quantum number, also called azimutahl quantum number, ℓ, within a shell. The values ℓ = 0, 1, 2, 3 correspond to the spectroscopic values s, p, d, and f , respectively. The maximum number of electrons which can be placed in a subshell is given by 2(2ℓ + 1). This gives two electrons in an s subshell, six electrons in a p subshell, ten electrons in a d subshell and fourteen electrons in an f subshell. Therefore, subshells “close” after the addition of 2,8,10,18, 36,50,72,… electrons. That is, atomic shells close after we reach $ns^2np^6$, with n>1, i.e., shells close after reaching the inert gas electron configuration.

The numbers of electrons that can occupy each shell and each subshell arise from the equations of quantum mechanics,in particular the Pauli exclusion principle: no two electrons in the same atom can have the same values of the four quantum numbers stated above. The energy associated to an electron is that of its orbital. The energy of any electron configuration is often approximated as the sum of the energy of each electron, neglecting the electron-electron interactions. The configuration that corresponds to the lowest electronic energy is called the ground (a.k.a. fundamental) state.

## Aufbau principle and Madelung rule

The Aufbau principle (from the German word Aufbau, “building up, construction”) was an important part of Bohr’s original concept of electron configuration. It may be stated as:

a maximum of two electrons are put into orbitals in the order of increasing orbital energy: the lowest-energy orbitals are filled before electrons are placed in higher-energy orbitals.
The approximate order of filling of atomic orbitals, following the sketch given above arrows from 1s to 7p. After 7p the order includes orbitals outside the range of the diagram, starting with 8s.

The principle works very well (for the ground states of the atoms) for the first 18 elements, then decreasingly well for the following 100 elements. The modern form of the Aufbau principle describes an order of orbital energies given by Madelung’s rule (also referred as the Klechkowski’s rule). This rule was first stated by Charles Janet in 1929, rediscovered by E. Madelung in 1936, and later given a theoretical justification by V.M.Klechkowski. In modern words, it states that:

A) Orbitals are filled in the order of increasing n+l.

B) Where two orbitals have the same value of n+l, they are filled in order of increasing n.

This gives the following order for filling the orbitals:

1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p, (8s, 5g, 6f, 7d, 8p, and 9s)

In this list the orbitals in parentheses are not occupied in the ground state of the heaviest atom now known (circa 2013, July), the ununoctiom (Uuo), an atom with Z=118 protons in its nucleus and thus, 118 electrons in its ground state.

The Aufbau principle can be applied, in a modified form, to the protons and neutrons in the atomic nucleus, as in the atomic shell model. The nuclear shell model predicts the magic numbers at Z,N=2, 8, 20, 28, 50, 82, 126 (and Z,N=184 and 258 for spherical symmetry, but it does not seem to be the case for “deformed” nuclei at high values of Z and N).

## Shortcomings of the Aufbau principle

The Aufbau principle rests on a fundamental postulate that the order of orbital energies is fixed, both for a given element and between different elements; neither of these is true (although they are approximately true enough for the principle to be useful). It considers atomic orbitals as “boxes” of fixed energy into which can be placed two electrons and no more. However the energy of an electron “in” an atomic orbital depends on the energies of all the other electrons of the atom (or ion, or molecule, etc.). There are no “one-electron solutions” for systems of more than one electron, only a set of many-electron solutions which cannot be calculated exactly. The fact that the Aufbau principle is based on an approximation can be seen from the fact that there is an almost-fixed filling order at all, that, within a given shell, the s-orbital is always filled before the p-orbitals. In a hydrogenic (hydrogen-like) atoms , which only has one electron, the s-orbital and the p-orbitals of the same shell have exactly the same energy, to a very good approximation in the absence of external electromagnetic fields. (However, in a real hydrogen atom, the energy levels are slightly split by the magnetic field of the nucleus, and by the quantum electrodynamic effects like the Lamb shift).

There are several more exceptions to Madelung’s rule among the heavier elements, and it is more and more difficult to resort to simple explanations such as the stability of half-filled subshells. It is possible to predict most of the exceptions by Hartree–Fock calculations, which are an approximate method for taking account of the effect of the other electrons on orbital energies. For the heavier elements, it is also necessary to take account of the effects of Special Relativity on the energies of the atomic orbitals, as the inner-shell electrons are moving at speeds approaching the speed of light . In general, these relativistic effects tend to decrease the energy of the s-orbitals in relation to the other atomic orbitals. The electron-shell configuration of elements beyond rutherfodium (Z=104) has not yet been empirically verified, but they are expected to follow Madelung’s rule without exceptions until the element Ubn (Unbinillium, Z=120). Beyond that number, there is no accepted viewpoint (see below my discussion of Pykko’s model for the extended periodic table).

## from the Greeks to Mendeleiev and Seaborg

Atoms and their existence from Greeks to Mendeleiev have suffered historical evolution. In this section, I am going to give you a visual tour from the “ancient elements” until their current classifications via Periodic Tables (Mendeleiev’s being the first one!).

Some early elements and periodic tables:

Just for fun, Feng Shui elements are…

And you can also find today apps/games with elements as “key” pieces…Gamelogy! LOL…

Turning back to Chemistry…Or Alchemy (Modern Chemistry is an evolution from Alchemy in which we take the scientific method seriously, don’t forget it!)

After the chemical revolution in the 18th and 19th century, we also have these pictures (note the evolution of the chemical elements, their geometry and classification):

Some interesting pictures about “new tables” and geometries of some periodic tables and its “make-up” process:

The following one is just for fun (XD):

## and the island of stability

Seaborg conjectured that the 8th period elements were an interesting “laboratory” to test quantum mechanical and physical principles from relativity and quantum physics. He claimed that there could be possible that around some (high) values of Z, N (122, 126 in Z, and about 184 in N), some superheavy elements could be stable enough to be produced. This topic is yet controversial by the same reasons I mentioned in the previous post: finite size of the nucleus, relativistic effects make the nuclei to be deformed, and likely, some novel effects related to nonpertubative issues (like pair creation in strong fields, as Greiner et al. have remarked) should be taken into account. Anyway, the existence of the so-called island of stability is a hot topic in both theoretical chemistry and experimental chemistry (at the level of the synthesis of superheavy elements). It is also relevant for (quantum and relativistic) physics. However, we will have to wait to be able to find those elements in laboratories or even in the outer space!

Some extended periodic tables were proposed by theoretical chemists like Seaborg and many others:

## Pykko’s model and beyond

The finnish chemist Pekka Pykko has produced a beautiful modern extended periodic table from his numerical calculations. He has discovered that the Madelung’s law is modified and then, the likely correct superheavy element included Periodic Table should be something like this (with Z less or equal than 172):

You can visit P. Pykko homepage’s here http://www.chem.helsinki.fi/~pyykko/I urge to do it. He has really cool materials! The abstract of his periodic table paper deserves to be inserted here:

and some of his interesting results from it are the modified electron configurations with respect to the normal Madelung’s rule (as I remarked above):

Indeed, Pykko is able to calculate some “simple” and “stable” molecules made of superheavy elements!

It is interesting to compare Pykko’s table with other extended periodic tables out there, like this one:

and you can also watch a periodic table video by the most famous chemist in youtube talking about it here

We have already seen about the feynmanium in the last paper, but what is its electron configuration? It is not clear since we have up most theoretical predictions since NO atoms from E137 have been produced yet. Thus, Feynmanium’s electron configuration is assumed to be $\left[Ms\right] 5g^{17}8s^2$, but due to smearing of the orbitals due to the small separation between the orbitals, the electron configuration is believed to be $\left[Ms\right] 5g^{11}6f^{3}7d^18s^28p^2$. The hyperphysics web page also discusses this problem. It says:

“(…)Dirac showed that there are no stable electron orbits for more than 137 electrons, therefore the last chemical element on the periodic table will be untriseptium (137Uts) also known informally as feynmanium $_{137}Fy$. It’s full electron configuration would be something like …

1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6 6s2 4f14 5d10 6p6 7s2 5f14 6d10 7p6 8s2 5g17

or is it …

1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6 6s2 4f14 5d10 6p6 7s2 5f14 6d10 7p6 8s1 5g18 ?(…)”

What is the right electron configuration? Without a synthesized element, we do not know…

Even more, you can have fun with this page and references therein http://planetstar.wikia.com/wiki/Feynmanium

There, you can even find that there are proposals for almost every superheavy element (SHE) name! Let me remark that today, circa 2013, 10th July, we have named every chemical element till Z=112 (Copernicium), plus Z=114 (Flerovium) and Z=116 (Livermorium) “offitially”. Feynmanium, neutronium, and any other superheavy element name is not offitial. The IUPAC recommends to use a systematic name until the discoverers have proposed the name and it is “offitially” accepted. Thus, feynmanium should be called untriseptium until we can produce it!

More Periodic Table limits? What about a 0th element with Z=0? Sometimes it is called “neutronium” or “neutrium”. More details here

http://en.wikipedia.org/wiki/Neutronium

Of course it is an speculative idea or concept. Indeed, in japanese culture, the void is the 5th element! It is closer to the picture we get from particle physics today in which “elementary particles” are excitations from some vacuum for certain (spinorial, scalar, tensor,…) field. We could see the “voidium” (no, it is no the dalekenium! LOL) as the fundamental “element” for particle physics. And yet, we have that only a 5% of the known Universe are “radiation” and “known elements”. What a shock!

Just for fun, again, the anime Saint Seiya Omega uses 7 fundamental “elements” (yes, I am a geek, I recognize it!)

The Seaborg’s original proposal was something like the next table:

And you see, it is quite a different from the astrological first elements from myths and superstitions: And finally, let me show you the presently known elementary particles again, the smallest “elements” from which matter is believed to made of (till now, of course):

Remark: Chemistry is about atoms. High Energy Physics is about elementary particles.

Final questions:

1st. What is your favorite (theoretical or known to exist) chemical element?

2nd. What is your favorite elementary particle (theoretical or known to exist in the Standard Model)?

May The Chemical Elements and the Elementary Particles be with YOU!

# 1st part: A centenary model

This is a blog entry devoted to the memory of a great scientist, N. Bohr, one of the greatest master minds during the 20th century, one of the fathers of the current Quantum model of atoms and molecules.

One century ago, Bohr was the pioneer of the introduction of the “quantization” rules into the atomic realm, 8 years after the epic Annus Mirabilis of A. Einstein (1905). Please, don’t forget that Einstein himself was the first physicist to consider Planck hypothesis into “serious” physics problems, explaining the photoelectric effect in a simple way with the aid of “quanta of light” (a.k.a. photons!). Therefore, it is not correct to assest that N.Bohr was the “first” quantum physicist. Indeed, Einstein or Planck were the first. Said, this, Bohr was the first to apply the quantum hypothesis into the atomic domain, changing forever the naive picture of atoms coming from the “classical” physics.  I decided that this year I would be writting something in to honour the centenary of his atomic model (for the hydrogen atom).

I wish you will enjoy the next (short) thread…

## Atomic mysteries

When I was young, and I was explained and shown the Periodic Table (the ordered list or catalogue of elements) by the first time, I wondered how many elements could be in Nature. Are they 103? 118?Maybe 212? 1000? $10^{23}$? Or $10^{100}$? $\infty$, Infinity?

We must remember what an atom is…Atom is a greek word $\alpha\tau o\mu o\sigma$ meaning “with no parts”. That is, an atom is (at least from its original idea), something than can not be broken into smaller parts. Nice concept, isn’t it?

Greek philosophers thought millenia ago if there is a limit to the divisibility of matter, and if there is an “ultimate principle” or “arche” ruling the whole Universe (remarkably, this is not very different to the questions that theoretical physicists are trying to solve even now or the future!). Different schools and ideas arose. I am not very interested today into discussing Philosophy (even when it is interesting in its own way), so let me simplify the general mainstream ideas several thousands of years ago (!!!!):

1st. There is a well-defined ultimate “element”/”substance” and an ultimate “principle”. Matter is infinitely divisible. There are deep laws that govern the Universe and the physical Universe, in a cosmic harmony.

2nd. There is a well-defined ultimate “element”/”substance” and an ultimate “principle”. Matter is FINITELY divisible. There are deep laws that govern the Universe and the physical Universe, in a cosmic harmony.

3rd. There is no a well-defined ultimate “element”/”substance” or an ultimate principle. Chaos rules the Universe. Matter is infinitely divisible.

4th. There is no a well-defined ultimate “element”/”substance” or an ultimate principle. Chaos rules the Universe. Matter is finitely divisible.

Remark: Please, note the striking “similarity” with some of the current (yet) problems of Physics. The existence of a Theory Of Everything (TOE) is the analogue to the question of the first principle/fundamental element quest of ancient greek philosophers or any other philosophy in all over the world. S.W. Hawking himself provided in his Brief Story of Time the following (3!) alternative approaches

1st. There is not a TOE. There is only a chaotic pattern of regularities we call “physical laws”. But Nature itself is ultimately chaotic and the finite human mind can not understand its ultimate description.

2nd. There is no TOE. There are only an increasing number of theories more and more precise or/and more and more accurate without any limit. As we are finite beings, we can only try to guess better and better approximations to the ultimate reality (out of our imagination) and the TOE can not be reached in our whole lifetime or even in the our whole species/civilization lifetime.

3rd. There is a well defined TOE, with its own principles and consequences. We will find it if we are persistent enough and if we are clever enough. All the physical events could be derived from this theory. If we don’t find the “ultimate theory and its principles” is not because it is non-existent, it is only that we are not smart enough. Try harder (If you can…)!

If I added another (non Greek) philosophies, I could create some other combinations, but, as I told you above, I am not going to tell you Philosophy here, not at least more than necessary.

As you probably know, the atomic idea was mainly defended by Leucippus and Democritus, based on previous ideas by Anaxagoras. It is quite likely that Anaxagoras himself learned them from India (or even from China), but that is quite speculative… Well, the keypoint of the atomic idea is that you can not smash into smaller pieces forever smaller and smaller bits of matter. Somewhere, the process of breaking down the fundamental constituents of matter must end…But where? And mostly, how can we find an atom or “see” what an atom looks like? Obviously, ancient greeks had not idea of how to do that, or even knowing the “ground idea” of what a atom is, they had no experimental device to search for them. Thus, the atomic idea was put into the freezer until the 18th and 19th century, when the advances in experimental (and theoretical) Chemistry revived the concept and the whole theory. But Nature had many surprises ready for us…Let me continue this a bit later…

In the 19th century, with the discovery of the ponderal laws of Chemistry, Dalton and other chemists were stunned. Finally, Dalton  was the man who recovered the atomism into “real” theoretical Science. But their existence was controversial until the 20th century. However, Dalton concluded that there was a unique atom for each element, using Lavoisier’s definition of an element as a substance that could not be analyzed into something simpler. Thus, Dalton arrived to an important conclusion:

“(…)Chemical analysis and synthesis go no farther than to the separation of particles one from another, and to their reunion. No new creation or destruction of matter is within the reach of chemical agency. We might as well attempt to introduce a new planet into the solar system, or to annihilate one already in existence, as to create or destroy a particle of hydrogen. All the changes we can produce, consist in separating particles that are in a state of cohesion or combination, and joining those that were previously at a distance(…)”.

The reality of atoms was a highly debated topic during all the 19th century. It is worthy to remark that was Einstein himself (yes, he…agian) who went further and with his studies about the Brownian motion established their physical existence. It was a brillian contribution to this area, even when, in time, he turned against the (interpretation of) Quantum Mechanics…But that is a different story not to be told today.

Dalton’s atoms or Dalton atomic model was very simple.

Atoms had no parts and thus, they were truly indivisible particles. However, the electrical studies of matter and the electromagnetic theory put this naive atomic model into doubt. After the discovery of “the cathode” rays (1897) and the electron by J.J.Thomson (no, it is not J.J.Abrahams), it became clear that atoms were NOT indivisible after all! Surprising, isn’t it? It is! Chemical atoms are NOT indivisible. They do have PARTS.

Thomson’s model or “plum pudding” model, came into the rescue…Dalton believed that atoms were solid spheres, but J.J.Thomson was forced (due to the electron existence) to elaborate a “more complex” atomic model. He suggested that atoms were a spherical “fluid” mass with positive charge, and that electrons were placed into that sphere as in a “plum pudding” cake.  I have to admit that I were impressed by this model when I was 14…It seemed too ugly for me to be true, but anyway it has its virtues (it can explain the cathode ray experiment!).

The next big step was the Rutherford experiment! Thomson KNEW that electrons were smaller pieces inside the atom, but despite his efforts to find the positive particles (and you see there he had and pursued his own path since he discovered the reason of the canal rays), he could not find it (and they should be there since atoms were electrically neutrial particles). However, clever people were already investigating radioactivity and atomic structure with other ideas…In 1911, E. Rutherford, with the aid of his assistants, Geiger and Marsden, performed the celebrated gold foil experiment.

To his surprise (Rutherford’s), his assistants and collaborators provided a shocking set of results. To explain all the observations, the main consequences of the Rutherford’s experiment were the next set of hypotheses:

1st. Atoms are mostly vacuum space.

2nd. Atoms have a dense zone of positive charge, much smaller than the whole atom. It is the atomic nucleus!

3rd. Nuclei had positive charge, and electrons negative charge.

He (Rutherford) did not know from the beginning how was the charge arranged and distributed into the atom. He had to improve the analysis and perform additional experiment in order to propose his “Rutherford” solar atomic model and to get an estimate of the nuclei size (about 1fm or $10^{-15}m$). In fact, years before him, the japanase Nagaoka had proposed a “saturnian” atomic model with a similar looking. It was unstable, though, due to the electric repulsion of the electronic “rings” (previously there was even a “cubic” model of atom, but it was unsuccessful too to explain every atomic experiment) and it had been abandoned.

And this is the point where theory become “hard” again. Rutherford supposed that the electron orbits around nuclei were circular (or almost circular) and then electrons experimented centripetal forces due to the electrical forces of the nucleus. The classical electromagnetic theory said that any charged particle being accelerated (and you do have acceleration with a centripetal force) should emit electromagnetic waves, losing energy and, then, electrons should fall over the the nuclei (indeed, the time of the fall down was ridiculously small and tiny). We do not observe that, so something is wrong with our “classical” picture of atoms and radiation (it was also hinted with the photoelectric effect or the blackbody physics, so it was not too surprising but challenging to find the rules and “new mechanics” to explain the atomic stability of matter). Moeover, the atomic spectra was known to be discrete (not continuous) since the 19th century as well. To find out the new dynamics and its principles became one of the oustanding issues in the theoretical (and experimental) community. The first scientist to determine a semiclassical but almost “quantum” and realistic atomic spectrum (for the simpler atom, the hydrogen) was Niels Bohr. The Bohr model of the hydrogen atom is yet explained at schools not only due to its historical insterest, but to the no less important fact that it provides right answers (indeed, Quantum Mechanics reproduces its features) for the simplest atom and that its equations are useful and valid from a quantitative viewpotint (as I told you, Quantum Mechanics reproduces Bohr formulae). Of course, Bohr model does not explain the Stark effect, the Zeeman effect, or the hyperfine structure of the hydrogen atom and some other “quantum/relativistic” important effects, but it is a really useful toy model and analytical machine to think about the challenges and limits of Quantum Mechanics of atoms and molecules. Bohr model can not be applied to helium and other elements in the Periodic Table of the elements (its structure is described by Quantum Mechanics), so it can be very boring but, as we will see, it has many secrets and unexpected surprises in its core…

## Bohr model for the hydrogen atom

Bohr model hypotheses/postulates:

1st. Electrons describe circular orbits around the proton (in the hydrogen atom). The centripetal force is provided by the electrostatic force of the proton.

2nd. Electrons, while in “stationary” orbits with a fixed energy, do NOT radiate electromagnetic waves ( note that this postulate is againsts the classical theory of electromagnetics as it was known in the 19th century).

3rd. When a single electron passes from one energetic level to another, the energy transitions/energy differences satisfy the Planck law. That is, during level transitions, $\Delta E=hf$.

In summary, we have:

Firstly, we begin with the equality between the electron-proton electrostatic force and the centripetal force in the atom:

$\begin{pmatrix}\mbox{Centripetal}\\ \mbox{Force}\end{pmatrix}=\begin{pmatrix}\mbox{Electron-proton}\\ \mbox{electric force}\end{pmatrix}$

Mathematically speaking, this first postulate/ansatz requieres that $q_1=q_2=e$, where $e=1\mbox{.}602\cdot 10^{-19}C$ is the elementary electric charge of the electron (and equal in absolute value to the proton charge) and $m_e=9.11\cdot 10^{-31}kg$ is the electron mass:

$F_c=\dfrac{m_ev^2}{R}$ and $F_C=K_C\dfrac{q_1q_2}{R^2}=K_C\dfrac{e^2}{R^2}$ implies that

(1) $\boxed{F_c=F_{el,C}}\leftrightarrow \boxed{\dfrac{m_ev^2}{R}=\dfrac{K_Ce^2}{R^2}}\leftrightarrow \boxed{v^2=\left(\dfrac{K_C}{m_e}\right)\left(\dfrac{e^2}{R}\right)}$

Remark: Instead of having the electron mass, it would be more precise to use the “reduced” mass for this two body problem. The reduced mass is, by definition,

$\mu=m_{red}=\dfrac{m_1m_2}{m_1+m_2}=\dfrac{m_em_p}{m_e+m_p}$

However, it is easy to realize that the reduced mass is essentially the electron mass (since $m_p\approx 1836m_e$)

$\mu=\dfrac{m_e}{1+\left(\dfrac{m_e}{m_p}\right)}\approx m_e(1-\dfrac{m_e}{m_p}+\ldots)=m_e+\mathcal{O} \left(\dfrac{m_e^2}{m_p}\right)$

The second Bohr’s great idea was to quantize the angular momentum. Classically, angular momentum can take ANY value, Bohr great’s intuition suggested that it could only take multiple values of some fundamental constant, the Planck’s constant. In fact, assuming orbitar stationary orbits, the quantization rule provides

(2) $\boxed{L=m_ev(2\pi R)=nh}$ or $\boxed{L=m_evR=n\dfrac{h}{2\pi}=n\hbar}$ with $\hbar=\dfrac{h}{2\pi}$ and $n=1,2,3,\ldots,\infty$ a positive integer.

Remark: $h=6\mbox{.}63\cdot 10^{-34}Js$ and $\hbar=\dfrac{h}{2\pi}=1\mbox{.}055\cdot 10^{-34}Js$ are the Planck constant and the reduced Planck constant, respectively.

From this quantization rule (2), we can easily get

$vR=\left(\dfrac{n\hbar}{m_e}\right)$ and then $v^2R^2=\left(\dfrac{n\hbar}{m_e}\right)^2$

Thus, we have

$R^2=\left(\dfrac{n\hbar}{m_e}\right)^2\dfrac{1}{v^2}$

Using the result we got in (1) for the squared velocity of the electron in the circular orbit, we deduce the quantization rule for the orbits in the hydrogen atom according to Bohr’s hypotheses:

$R^2=\left(\dfrac{n\hbar}{m_e}\right)^2\left(\dfrac{m_eR}{K_Ce^2}\right)$

$R=\dfrac{n^2\hbar^2}{m_e^2}\dfrac{m_e}{K_Ce^2}$

(3) $\boxed{R_n=R(n)=\left(\dfrac{\hbar^2}{m_eK_Ce^2}\right)n^2}\leftrightarrow \boxed{R_n=a_Bn^2}$

where $n=1,2,3,\ldots,\infty$ again and the Bohr radius $a_B$ is defined to be

(4) $\boxed{a_B=\dfrac{\hbar^2}{m_eK_Ce^2}}$

Inserting values into (4), we obtain the celebrated value of the Bohr radius

$a_B\approx 0\mbox{.}53\AA=53pm=5\mbox{.}3\cdot 10^{-11}m$

The third important consequence in the spectrum of energy levels in the hydrogen atom. To obtain the energy spectrum, there is two equivalent paths (in fact, they are the same): use the virial theorem or use (1) into the total energy for the electron-proton system. The total energy of the hydrogen atom can be written

$E=\mbox{Kinetic Energy}+\mbox{(electrostatic) Potential Energy}$

$E=\dfrac{p^2}{2m_e}-\dfrac{K_Ce^2}{R}=\dfrac{m_ev^2}{2}-\dfrac{K_Ce^2}{R}$

Substituting (1) into this, we get exactly the expected expression for the virial theorem to a $1/r^2$ potential (i.e. $E=E_p/2$):

$E=\dfrac{m_ev^2}{2}-\dfrac{K_Ce^2}{R}=-K_C\dfrac{e^2}{2R}$

(5) $\boxed{E=-K_C\dfrac{e^2}{2R}}$

Inserting into (5) the quantized values of the orbit, we deduce the famous and well-known formula for the spectrum of the hydrogen atom (known to Balmer and the spectroscopists at the end of the 19th century and the beginning of the 20th century):

(6) $\boxed{E_n=E(n)=-\dfrac{m_eK_C^2e^4}{2\hbar^2n^2}=-\dfrac{m_e}{2}\left(\dfrac{K_Ce^2}{n\hbar}\right)^2=-\dfrac{\mbox{Ry}}{n^2}} \;\;\forall n=1,2,3,\ldots,\infty$

and where we have defined the Rydberg (constant) as

(7) $\boxed{\mbox{Ry}=\dfrac{m_e(K_Ce^2)^2}{2\hbar^2}=\dfrac{m_eK_C^2e^4}{2\hbar^2}=\dfrac{1}{2}\alpha^2 m_ec^2}$

Its value is $Ry=R_H=2.18\cdot 10^{-18}J=13\mbox{.}6eV$. Here, the electromagnetic fine structure constant (alpha) is

$\alpha=K_C\dfrac{e^2}{\hbar c}$

and $c$ is the speed of light. In fact, using the quantum relation

$E=\dfrac{hc}{\lambda}$

we can deduce that the Rydberg corresponds to a wavenumber

$k=1\mbox{.}097\cdot 10^{7}m^{-1}$

or a frequency

$f=\nu=3\mbox{.}29\cdot 10^{15}Hz$

and a wavelength

$\lambda =912\AA=91\mbox{.}2nm$

The above results allowed Bohr to explain the spectral series of the hydrogen atom. He won the Nobel Prize due to this wonderful achievement…

## (and positronium, muonium,…)

In fact, it is easily straightforward to extend all these results to “hydrogenic” (“hydrogenoid”) atoms, i.e., to atoms with only a single electron BUT a nucleus with charge equal to $Ze$, and $Z>1$ is an integer (atomic) number greater than one! The easiest way to obtain the results is not to repeat the deduction but to make a rescaling of the proton charge, i.e., you plug $q_2=Ze$ or/and make a rescaling of the electric charge $q_2=e\longrightarrow Ze$ (be aware of making the right scaling in the formulae). The final result for the radius and the energy spectrum is as follows:

A) From $R_n=\left(\dfrac{\hbar^2}{m_eK_Ce^2}\right)n^2$, with $e\longrightarrow Ze$, you get

(8) $\boxed{\bar{R}_n=\bar{R}(n)=\dfrac{\hbar^2}{m_eK_CZe^2}n^2=\dfrac{a_Bn^2}{Z}}$

B) From $E_n=-m_e\dfrac{(K_Ce^2)^2}{2\hbar^2n^2}$, with the rescaling $e\longrightarrow Ze$, you get

(9) $\boxed{\bar{E}_n=\bar{E}(n)=-m_e\dfrac{Z^2(K_Ce^2)^2}{2\hbar^2n^2}=-\dfrac{Z^2\alpha^2m_ec^2}{2n^2}=-\dfrac{Z^2Ry}{n^2}}$

Therefore, the consequence of the rescaling of the nuclear charge is that energy levels are “enlarged” by a factor $Z^2$ and that the orbits are “squeezed” or “contracted” by a factor $1/Z$.

Exercise: Can you obtain the energy levels and the radius for the positronium (an electron and positron system instead an electron a positron). What happens with the muonium (strange substance formed by electron orbiting and antimuon)?And the muonic atom (muon orbiting an proton)? And a muon orbiting an antimuon? And the tau particle orbiting an antitau or the electron orbiting an antitau or a tau orbiting a proton(supposing that it were possible of course, since the tau particle is unstable)? Calculate the “Bohr radius” and the “Rydberg” constant for the positronium, the muonium, the muonic atom (or the muon-antimuon atom) and the tauonium (or the tau-antitau atom). Hint: think about the reduced mass for the positronium and the muonium, then make a good mass/energy or radius rescaling.

Now, we can also calculate the velocity of an electron in the quantized orbits for the Bohr atom and the hydrogenic atom. Using (3) and (8),

$mvR=n\hbar\leftrightarrow mR=\dfrac{n\hbar}{m_e}\leftrightarrow v^2R^2=\dfrac{n^2\hbar^2}{m_e^2}$

or

$v^2=\left(\dfrac{n\hbar}{m_e}\right)^2\dfrac{1}{R^2}$

and inserting the quantized values of the orbit radius

$v_n^2=\dfrac{K_Ce^2}{m_eR_n}=\dfrac{m_e(K_Ce^2)^2}{m_en^2\hbar^2}$

so, for the Bohr atom (hydrogen)

(10) $\boxed{v_n=v(n)=\dfrac{K_Ce^2}{\hbar n}=\dfrac{\alpha c}{n}}$

In the case of hydrogenic atoms, the rescaling of the electric charge yields

(11) $\boxed{\bar{v}_n=\bar {v}(n)=\dfrac{ZK_Ce^2}{\hbar n}=\dfrac{Z\alpha c}{n}}$

so, the hydrogenic atoms have a “enlarged” electron velocity in the orbits, by a factor of $Z$.

## The feynmanium

This result for velocities is very interesting. Suppose we consider the fundamental level $n=1$ (or the orbital 1s in Quantum Mechanics, since, magically or not, Quantum Mechanics reproduces the results for the Bohr atom and the hydrogenic atoms we have seen here, plus other effects we will not discuss today relative to spin and some energy splitting for perturbed atoms). Then, the last formula yield, in the hydrogenic case,

$v_1=Z\alpha c$

Furthermore, suppose now in addition that we have some “superheavy” (hydrogenic) atom with, say, $Z>137$ (note that $\alpha\approx 1/137$ at ordinary energies), say $Z=138$ or greater than it. Then, the electron moves faster than the speed of light!!!!! That is, for hydrogenic atoms, with Z>137 and considering the fundalmental level, the electron would move with $v>c$. This fact is “surprising”. The element with Z=137 is called untriseptium (Uts) by the IUPAC rules, but it is often called the feynmanium (Fy), since R.P. Feynman often remarked the importance of this result and mystery. Of course, Special Relativity forbids this option. Therefore, something is wrong or Z=137 is the last element allowed by the Quantum Rules (or/and the Bohr atom). Obviously, we could claim that this result is “wrong” since we have not consider the relativistic quantum corrections or we have not made a good relativistic treatment of this system. It is not as simple as you can think or imagine, since using a “naive” relativistic treatment, e.g., using the Dirac equation , we obtain for the fundamental level of the hydrogenic atom the spectrum

(12) $\boxed{E_1=E=m_ec^2\sqrt{1-Z^2\alpha^2}}$. This result can be obtained from the Dirac equation spectrum for the hydrogen atom (in a Coulomb potential):

(13) $\boxed{E_{n,k;Z,\alpha}=E(n,k;Z,\alpha)=mc^2\left[1+\left(\dfrac{Z\alpha}{n-\vert k\vert+\sqrt{k^2-Z^2\alpha^2}}\right)^2\right]^{-1/2}}$

where n is a nonnegative integer number $n=N+\vert k\vert$ and $k^2=(j+\frac{1}{2})^2$. Putting these into numbers, we get

If you plug Z=138 or more into the above equation from the Dirac spectrum, you obtain an imaginary value of the energy, and thus an oscillating (unbound) system! Therefore, the problem for atoms with high Z even persist taking the relativistic corrections! What is the solution? Nobody is sure. Greiner et al. suggest that taking into account the finite (extended) size of the nuclei, the problem is “solved” until $Z\approx 172$. Beyond, i.e., with $Z>172$, you can not be sure that quantum fluctuations of strong fields introduce vacuum pair creation effects such as they make the nuclei and thus atoms to be unstable at those high values of Z. Some people believe that the issues arise even before, around Z=150 or even that strong field effects can make atoms even below of Z=137 to be non-existent. That is why the search for superheavy elements (SHE) is interesting not only from the chemical viewpoint but also to the fundamental physics viewpoint: it challenges our understanding of Quantum Mechanics and Special Relativity (and their combination!!!!).

Is the feynmanium (Z=137) the last element? This hypothetical element and other superheavy elements (SHE) seem to hint the end of the Periodic Table. Is it true? Options:

1st. The feynmanium (Fy) or Untriseptrium (Uts) is the last element of the Periodic Table.

2nd. Greiner et al. limit around Z=172. References:

(i) B Fricke, W Greiner and J T Waber,Theor. Chim. Acta, 1971, 21, 235.

(ii)W Greiner and J Reinhardt, Quantum Electrodynamics, 4th edn (Springer, Berlin, 2009).

3rd. Other predictions of an end to the periodic table include Z = 128 (John Emsley) and Z = 155 (Albert Khazan). Even Seaborg, from his knowledge and prediction of an island of stability around $Z,N= 126, 184,\ldots$, left this question open to interpretation and experimental search!

4th. There is no end of the Periodic Table. According to Greiner et al. in fact, even when superheavy nuclei can produce a challenge for Quantum Mechanics and Special Relativity, indeed, since there is always electrons in the orbitals (a condition to an element to be a well-defined object), there is no end of The Periodic Table (even when there are probabilities to a positron-electron pair to be produced for a superheavy nuclei, the presence of electrons does not allow for it; but strong field effects are important there, and it should be great to produce these elements and to know their properties, both quantum and relativistic!). Therefore, it would be very, very interesting to test the superheavy element “zone” of the Periodic Table, since it is a place where (strong) quantum effects and (non-negligible) relativistic effects both matter. Then, if both theories are right, superheavy elements are a beautiful and wonderful arena to understand how to combine together the two greatest theories and (unfinished?) revolutions of the 20th century. What awesome role for the “elementary” and “fundamental” superheavy (composite) elements!

Probably, there is no limit to the number of (chemical) elements in our Universe… But we DO NOT KNOW!

In conclusion: what will happen for superheavy elements with >173 (or Z>126, 128, 137, etc.) remains unresolved with our current knowledge. And it is one of the last greatest mysteries in theoretical Chemistry!

More about the fine structure constant, the Sommerfeld corrections and the Dirac equation+QED (Quantum ElectroDynamics) corrections to the hydrogen spectrum, in slides (think it yourself!):

Final remarks (for experts only): Some comments about the self-adjointness of the Dirac equation for high value of Z in Coulombian potentials. It is a well known fact that the Dirac operator for the hydrogen problem is essentially self-adjoint if Z<119. Therefore, it is valid for all the currently known elements (circa 2013, June, every element in the Periodic Table, for the 7th period, has been created and then, we know that chemical elements do exist at least up to Z=118 and we have tried to search for superheavy elements beyond that Z with negative results until now). However, for $119\leq Z\leq 137$ any “self-adjoint extension” requires a precise physical meaning. A good idea could be that the expectation value of every component of the Hamilton is finite in the selected basis. Indeed, the solution to the Coulombian potential for the hydrogenic atom using the Dirac equation makes use of hypergeometric functions that are well-posed for any $Z\leq 137$. If Z is greater than that critical value, we face the oscillating energy problem we discussed above. So, we have to consider the effect of the finite size of the nucleus and/or handle relativistic corrections more carefully. It is important to realize this and that we have to understand the main idea of all this crazy stuff. This means that the s states start to be destroyed above Z = 137, and that the p states begin being destroyed above Z = 274.  Note that this differs from the result of the Klein-Gordon equation, which predicts s states being destroyed above Z = 68 and p states destroyed above Z = 82. In summary, the superheavy elements are interesting because they challenge our knowledge of both Quantum Mechanics and Special Relativity. What a wonderful (final) fate for the chemical elements: the superheavy elements will test if the “marriage” between Quantum Mechanics or Special Relativity is going further or it ends into divorce!

Epilogue: What do you think about the following questions? This is a test for you, eager readers…

1) Is there an ultimate element?

2) Is there a theory of everything (TOE)?

3) Is there an ultimate chemical element?

4) Is there a single “ultimate” principle?

5) How many elements does the Periodic Table have?

6) Is the feynmanium the last element?

7) Are Quantum Mechanics/Special relativity consistent to each other?

8) Is Quantum Mechanics a fundamental and “ultimate” theory for atoms and molecules?

9) Is Special Relativity a fundamental and “ultimate” theory for “quick” particles?

10) Are the atomic shells and atomic structure completely explained by QM and SR?

11) Are the nuclei and their shell structure xompletely explained by QM and SR?

12) Do you think all this stuff is somehow important and relevant for Physics or Chemistry (or even for Mathematics)?

13) Will we find superheavy elements the next decade?

14) Will we find superheavy elements this century?

15) Will we find that there are some superheavy elements stable in the island of stability (Seaborg) with amazing properties and interesting applications?

16) Did you like/enjoy this post?

17) When you was a teenager, how many chemical elements did you know? How many chemical elements were known?

18) Did you learn/memorize the whole Periodic Table? In the case you did not, would you?

19) What is your favourite chemical element?

20) Did you know that every element in the 7th period of the Periodic table has been established to exist but th elements E113, E115,E117 and E118 are not named yet (circa, 2013, 30th June) and they keep their systematic (IUPAC) names ununtrium, ununpentium, ununseptium and ununoctium? By the way, the last named elements were the coperninicium (E112, Cn), the flerovium (Fl, E114) and the livermorium (Lv, E116)…