# 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)…

# LOG#070. Natural Units.

Happy New Year 2013 to everyone and everywhere!

Let me apologize, first of all, by my absence… I have been busy, trying to find my path and way in my field, and I am busy yet, but finally I could not resist without a new blog boost… After all, you should know the fact I have enough materials to write many new things.

So, what’s next? I will dedicate some blog posts to discuss a nice topic I began before, talking about a classic paper on the subject here:

https://thespectrumofriemannium.wordpress.com/2012/11/18/log054-barrow-units/

The topic is going to be pretty simple: natural units in Physics.

First of all, let me point out that the election of any system of units is, a priori, totally conventional. You are free to choose any kind of units for physical magnitudes. Of course, that is not very clever if you have to report data, so everyone can realize what you do and report. Scientists have some definitions and popular systems of units that make the process pretty simpler than in the daily life. Then, we need some general conventions about “units”. Indeed, the traditional wisdom is to use the international system of units, or S (Iabbreviated SI from French language: Le Système international d’unités). There, you can find seven fundamental magnitudes and seven fundamental (or “natural”) units:

1) Space: $\left[ L\right]=\mbox{meter}=m$

2) Time: $\left[ T\right]=\mbox{second}=s$

3) Mass: $\left[ M\right]=\mbox{kilogram}=kg$

4) Temperature: $\left[ t\right]=\mbox{Kelvin degree}= K$

5) Electric intensity: $\left[ I\right]=\mbox{ampere}=A$

6) Luminous intensity: $\left[ I_L\right]=\mbox{candela}=cd$

7) Amount of substance: $\left[ n\right]=\mbox{mole}=mol(e)$

The dependence between these 7 great units and even their definitions can be found here http://en.wikipedia.org/wiki/International_System_of_Units and references therein. I can not resist to show you the beautiful graph of the 7 wonderful units that this wikipedia article shows you about their “interdependence”:

In Physics, when you build a radical new theory, generally it has the power to introduce a relevant scale or system of units. Specially, the Special Theory of Relativity, and the Quantum Mechanics are such theories. General Relativity and Statistical Physics (Statistical Mechanics) have also intrinsic “universal constants”, or, likely, to be more precise, they allow the introduction of some “more convenient” system of units than those you have ever heard ( metric system, SI, MKS, cgs, …). When I spoke about Barrow units (see previous comment above) in this blog, we realized that dimensionality (both mathematical and “physical”), and fundamental theories are bound to the election of some “simpler” units. Those “simpler” units are what we usually call “natural units”. I am not a big fan of such terminology. It is confusing a little bit. Maybe, it would be more interesting and appropiate to call them “addapted X units” or “scaled X units”, where X denotes “relativistic, quantum,…”. Anyway, the name “natural” is popular and it is likely impossible to change the habits.

In fact, we have to distinguish several “kinds” of natural units. First of all, let me list “fundamental and universal” constants in different theories accepted at current time:

1. Boltzmann constant: $k_B$.

Essential in Statistical Mechanics, both classical and quantum. It measures “entropy”/”information”. The fundamental equation is:

$\boxed{S=k_B\ln \Omega}$

It provides a link between the microphysics and the macrophysics ( it is the code behind the equation above). It can be understood somehow as a measure of the “energetic content” of an individual particle or state at a given temperature. Common values for this constant are:

$k_B=1.3806488(13)\times 10^{-23}J/K = 8.6173324(78)\times 10^{-5}eV/K$

$k_B=1.3806488(13)\times 10^{-16}erg/K$

Statistical Physics states that there is a minimum unit of entropy or a minimal value of energy at any given temperature. Physical dimensions of this constant are thus entropy, or since $E=TS$, $\left[ k_B\right] =E/t=J/K$, where t denotes here dimension of temperature.

2. Speed of light.  $c$.

From classical electromagnetism:

$\boxed{c^2=\dfrac{1}{\sqrt{\varepsilon_0\mu_0}}}$

The speed of light, according to the postulates of special relativity, is a universal constant. It is frame INDEPENDENT. This fact is at the root of many of the surprising results of special relativity, and it took time to be understood. Moreover, it also connects space and time in a powerful unified formalism, so space and time merge into spacetime, as we do know and we have studied long ago in this blog. The spacetime interval in a D=3+1 dimensional space and two arbitrary events reads:

$\Delta s^2=\Delta x^2+\Delta y^2+\Delta z^2-c^2\Delta t^2$

In fact, you can observe that “c” is the conversion factor between time-like and space-like coordinates.  How big the speed of light is? Well, it is a relatively large number from our common and ordinary perception. It is exactly:

$\boxed{c=299,792,458m/s}$

although you often take it as $c\approx 3\cdot 10^{8}m/s=3\cdot 10^{10}cm/s$.  However, it is the speed of electromagnetic waves in vacuum, no matter where you are in this Universe/Polyverse. At least, experiments are consistent with such an statement. Moreover, it shows that $c$ is also the conversion factor between energy and momentum, since

$\mathbf{P}^2c^2-E^2=-m^2c^4$

and $c^2$ is the conversion factor between rest mass and pure energy, because, as everybody knows,  $E=mc^2$! According to the special theory of relativity, normal matter can never exceed the speed of light. Therefore, the speed of light is the maximum velocity in Nature, at least if specially relativity holds. Physical dimensions of c are $\left[c\right]=LT^{-1}$, where L denotes length dimension and T denotes time dimension (please, don’t confuse it with temperature despite the capital same letter for both symbols).

3. Planck’s constant. $h$ or generally rationalized $\hbar=h/2\pi$.

Planck’s constant (or its rationalized version), is the fundamental universal constant in Quantum Physics (Quantum Mechanics, Quantum Field Theory). It gives

$\boxed{E=h\nu=\hbar \omega}$

Indeed, quanta are the minimal units of energy. That is, you can not divide further a quantum of light, since it is indivisible by definition! Furthermore, the de Broglie relationship relates momentum and wavelength for any particle, and it emerges from the combination of special relativity and the quantum hypothesis:

$\lambda=\dfrac{h}{p}\leftrightarrow \bar{\lambda}=\dfrac{\hbar}{p}$

In the case of massive particles, it yields

$\lambda=\dfrac{h}{Mv}\leftrightarrow \bar{\lambda}=\dfrac{\hbar}{Mv}$

In the case of massless particles (photons, gluons, gravitons,…)

$\lambda=\dfrac{hc}{E}$ or $\bar{\lambda}=\dfrac{\hbar c}{E}$

Planck’s constant also appears to be essential to the uncertainty principle of Heisenberg:

$\boxed{\Delta x \Delta p\geq \hbar/2}$

$\boxed{\Delta E \Delta t\geq \hbar/2}$

$\boxed{\Delta A\Delta B\geq \hbar/2}$

Some particularly important values of this constant are:

$h=6.62606957(29)\times 10^{-34} J\cdot s$
$h=4.135667516(91)\times 10^{-15}eV\cdot s$
$h=6.62606957(29)\times 10^{-27} erg\cdot s$
$\hbar =1.054571726(47)\times 10^{-34} J\cdot s$
$\hbar =6.58211928(15)\times 10^{-16} eV\cdot s$
$\hbar= 1.054571726(47)\times 10^{-27}erg\cdot s$

It is also useful to know that
$hc=1.98644568\times 10^{-25}J\cdot m$
$hc=1.23984193 eV\cdot \mu m$

or

$\hbar c=0.1591549hc$ or $\hbar c=197.327 eV\cdot nm$

Planck constant has dimension of $\mbox{Energy}\times \mbox{Time}=\mbox{position}\times \mbox{momentum}=ML^2T^{-1}$. Physical dimensions of this constant coincide also with angular momentum (spin), i.e., with $L=mvr$.

4. Gravitational constant. $G_N$.

Apparently, it is not like the others but it can also define some particular scale when combined with Special Relativity. Without entering into further details (since I have not discussed General Relativity yet in this blog), we can calculate the escape velocity of a body moving at the speed of light

$\dfrac{1}{2}mv^2-G_N\dfrac{Mm}{R}=0$ with $v=c$ implies a new length scale where gravitational relativistic effects do appear, the so-called Schwarzschild radius $R_S$:

$\boxed{R_S=\dfrac{2G_NM}{c^2}=\dfrac{2G_NM_{\odot}}{c^2}\left(\dfrac{M}{M_{\odot}}\right)\approx 2.95\left(\dfrac{M}{M_{\odot}}\right)km}$

5. Electric fundamental charge. $e$.

It is generally chosen as fundamental charge the electric charge of the positron (positive charged “electron”). Its value is:

$e=1.602176565(35)\times 10^{-19}C$

where C denotes Coulomb. Of course, if you know about quarks with a fraction of this charge, you could ask why we prefer this one. Really, it is only a question of hystory of Science, since electrons were discovered first (and positrons). Quarks, with one third or two thirds of this amount of elementary charge, were discovered later, but you could define the fundamental unit of charge as multiple or entire fraction of this charge. Moreover, as far as we know, electrons are “elementary”/”fundamental” entities, so, we can use this charge as unit and we can define quark charges in terms of it too. Electric charge is not a fundamental unit in the SI system of units. Charge flow, or electric current, is.

An amazing property of the above 5 constants is that they are “universal”. And, for instance, energy is related with other magnitudes in theories where the above constants are present in a really wonderful and unified manner:

$\boxed{E=N\dfrac{k_BT}{2}=Mc^2=TS=Pc=N\dfrac{h\nu}{2}=N\dfrac{\hbar \omega}{2}=\dfrac{R_Sc^4}{2G_N}=\hbar c k=\dfrac{hc}{\lambda}}$

Caution: k is not the Boltzmann constant but the wave number.

There is a sixth “fundamental” constant related to electromagnetism, but it is also related to the speed of light, the electric charge and the Planck’s constant in a very sutble way. Let me introduce you it too…

6. Coulomb constant. $k_C$.

This is a second constant related to classical electromagnetism, like the speed of light in vacuum. Coulomb’s constant, the electric force constant, or the electrostatic constant (denoted $k_C$) is a proportionality factor that takes part in equations relating electric force between  point charges, and indirectly it also appears (depending on your system of units) in expressions for electric fields of charge distributions. Coulomb’s law reads

$F_C=k_C\dfrac{Qq}{r^2}$

Its experimental value is

$k_C=\dfrac{1}{4\pi \varepsilon_0}=\dfrac{c^2\mu_0}{4\pi}=c^2\cdot 10^{-7}H\cdot m^{-1}= 8.9875517873681764\cdot 10^9 Nm^2/C^2$

Generally, the Coulomb constant is dropped out and it is usually preferred to express everything using the electric permitivity of vacuum $\varepsilon_0$ and/or numerical factors depending on the pi number $\pi$ if you choose the gaussian system of units  (read this wikipedia article http://en.wikipedia.org/wiki/Gaussian_system_of_units ), the CGS system, or some hybrid units based on them.

## H.E.P. units

High Energy Physicists use to employ units in which the velocity is measured in fractions of the speed of light in vacuum, and the action/angular momentum is some multiple of the Planck’s constant. These conditions are equivalent to set

$\boxed{c=1_c=1}$ $\boxed{\hbar=1_\hbar=1}$

Complementarily, or not, depending on your tastes and preferences, you can also set the Boltzmann’s constant to the unit as well

$k_B=1_{k_B}=1$

and thus the complete HEP system is defined if you set

$\boxed{c=\hbar=k_B=1}$

This “natural” system of units is lacking yet a scale of energy. Then, it is generally added the electron-volt $eV$ as auxiliary quantity defining the reference energy scale. Despite the fact that this is not a “natural unit” in the proper sense because it is defined by a natural property, the electric charge,  and the anthropogenic unit of electric potential, the volt. The SI prefixes multiples of eV are used as well: keV, MeV, GeV, etc. Here, the eV is used as reference energy quantity, and with the above election of “elementary/natural units” (or any other auxiliary unit of energy), any quantity can be expressed. For example, a distance of 1 m can be expressed in terms of eV, in natural units, as

$1m=\dfrac{1m}{\hbar c}\approx 510eV^{-1}$

This system of units have remarkable conversion factors

A) $1 eV^{-1}$ of length is equal to $1.97\cdot 10^{-7}m =(1\text{eV}^{-1})\hbar c$

B) $1 eV$ of mass is equal to $1.78\cdot 10^{-36}kg=1\times \dfrac{eV}{c^2}$

C) $1 eV^{-1}$ of time is equal to $6.58\cdot 10^{-16}s=(1\text{eV}^{-1})\hbar$

D) $1 eV$ of temperature is equal to $1.16\cdot 10^4K=1eV/k_B$

E) $1 unit$ of electric charge in the Lorentz-Heaviside system of units is equal to $5.29\cdot 10^{-19}C=e/\sqrt{4\pi\alpha}$

F) $1 unit$ of electric charge in the Gaussian system of units is equal to $1.88\cdot 10^{-18}C=e/\sqrt{\alpha}$

This system of units, therefore, leaves free only the energy scale (generally it is chosen the electron-volt) and the electric measure of fundamentl charge. Every other unit can be related to energy/charge. It is truly remarkable than doing this (turning invisible the above three constants) you can “unify” different magnitudes due to the fact these conventions make them equivalent. For instance, with natural units:

1) Length=Time=1/Energy=1/Mass.

It is due to $x=ct$, $E=Mc^2$ and $E=hc/\lambda$ equations. Setting $c$ and $h$ or $\hbar$ provides

$x=t$, $E=M$ and $E=1/\lambda$.

Note that natural units turn invisible the units we set to the unit! That is the key of the procedure. It simplifies equations and expressions. Of course, you must be careful when you reintroduce constants!

2) Energy=Mass=Momemntum=Temperature.

It is due to $E=k_BT$, $E=Pc$ and $E=Mc^2$ again.

One extra bonus for theoretical physicists is that natural units allow to build and write proper lagrangians and hamiltonians (certain mathematical operators containing the dynamics of the system enconded in them), or equivalently the action functional, with only the energy or “mass” dimension as “free parameter”. Let me show how it works.

Natural units in HEP identify length and time dimensions. Thus $\left[L\right]=\left[T\right]$. Planck’s constant allows us to identify those 2 dimensions with 1/Energy (reciprocals of energy) physical dimensions. Therefore, in HEP units, we have

$\boxed{\left[L\right]=\left[T\right]=\left[E\right]^{-1}}$

The speed of light identifies energy and mass, and thus, we can often heard about “mass-dimension” of a lagrangian in the following sense. HEP units can be thought as defining “everything” in terms of energy, from the pure dimensional ground. That is, every physical dimension is (in HEP units) defined by a power of energy:

$\boxed{\left[E\right]^n}$

Thus, we can refer to any magnitude simply saying the power of such physical dimension (or you can think logarithmically to understand it easier if you wish). With this convention, and recalling that energy dimension is mass dimension, we have that

$\left[L\right]=\left[T\right]=-1$ and $\left[E\right]=\left[M\right]=1$

Using these arguments, the action functional is a pure dimensionless quantity, and thus, in D=4 spacetime dimensions, lagrangian densities must have dimension 4 ( or dimension D is a general spacetime).

$\displaystyle{S=\int d^4x \mathcal{L}\rightarrow \left[\mathcal{L}\right]=4}$

$\displaystyle{S=\int d^Dx \mathcal{L}\rightarrow \left[\mathcal{L}\right]=D}$

In D=4 spacetime dimensions, it can be easily showed that

$\left[\partial_\mu\right]=\left[\Phi\right]=\left[A^\mu\right]=1$

$\left[\Psi_D\right]=\left[\Psi_M\right]=\left[\chi\right]=\left[\eta\right]=\dfrac{3}{2}$

where $\Phi$ is a scalar field, $A^\mu$ is a vector field (like the electromagnetic or non-abelian vector gauge fields), and $\Psi_D, \Psi_M, \chi, \eta$ are a Dirac spinor, a Majorana spinor, and $\chi, \eta$ are Weyl spinors (of different chiralities). Supersymmetry (or SUSY) allows for anticommuting c-numbers (or Grassmann numbers) and it forces to introduce auxiliary parameters with mass dimension $-1/2$. They are the so-called SUSY transformation parameters $\zeta_{SUSY}=\epsilon$. There are some speculative spinors called ELKO fields that could be non-standandard spinor fields with mass dimension one! But it is an advanced topic I am not going to discuss here today. In general D spacetime dimensions a scalar (or vector) field would have mass dimension $(D-2)/2$, and a spinor/fermionic field in D dimensions has generally $(D-1)/2$ mass dimension (excepting the auxiliary SUSY grassmanian fields and the exotic idea of ELKO fields).  This dimensional analysis is very useful when theoretical physicists build up interacting lagrangians, since we can guess the structure of interaction looking at purely dimensional arguments every possible operator entering into the action/lagrangian density! In summary, therefore, for any D:

$\boxed{\left[\Phi\right]=\left[A_\mu\right]=\dfrac{D-2}{2}\equiv E^{\frac{D-2}{2}}=M^{\frac{D-2}{2}}}$

$\boxed{\left[\Psi\right]=\dfrac{D-1}{2}\equiv E^{\frac{D-1}{2}}=M^{\frac{D-1}{2}}}$

Remark (for QFT experts only): Don’t confuse mass dimension with the final transverse polarization degrees or “degrees of freedom” of a particular field, i.e., “components” minus “gauge constraints”. E.g.: a gauge vector field has $D-2$ degrees of freedom in D dimensions. They are different concepts (although both closely related to the spacetime dimension where the field “lives”).

In summary:

i) HEP units are based on QM (Quantum Mechanics), SR (Special Relativity) and Statistical Mechanics (Entropy and Thermodynamics).

ii) HEP units need to introduce a free energy scale, and it generally drives us to use the eV or electron-volt as auxiliary energy scale.

iii) HEP units are useful to dimensional analysis of lagrangians (and hamiltonians) up to “mass dimension”.

## Stoney Units

In Physics, the Stoney units form a alternative set of natural units named after the Irish physicist George Johnstone Stoney, who first introduced them as we know it today in 1881. However, he presented the idea in a lecture entitled “On the Physical Units of Nature” delivered to the British Association before that date, in 1874. They are the first historical example of natural units and “unification scale” somehow. Stoney units are rarely used in modern physics for calculations, but they are of historical interest but some people like Wilczek has written about them (see, e.g., http://arxiv.org/abs/0708.4361). These units of measurement were designed so that certain fundamental physical constants are taken as reference basis without the Planck scale being explicit, quite a remarkable fact! The set of constants that Stoney used as base units is the following:

A) Electric charge, $e=1_e$.

B) Speed of light in vacuum, $c=1_c$.

C) Gravitational constant, $G_N=1_{G_N}$.

D) The Reciprocal of Coulomb constant, $1/k_C=4\pi \varepsilon_0=1_{k_C^{-1}}=1_{4\pi \varepsilon_0}$.

Stony units are built when you set these four constants to the unit, i.e., equivalently, the Stoney System of Units (S) is determined by the assignments:

$\boxed{e=c=G_N=4\pi\varepsilon_0=1}$

Interestingly, in this system of units, the Planck constant is not equal to the unit and it is not “fundamental” (Wilczek remarked this fact here ) but:

$\hbar=\dfrac{1}{\alpha}\approx 137.035999679$

Today, Planck units are more popular Planck than Stoney units in modern physics, and even there are many physicists who don’t know about the Stoney Units! In fact, Stoney was one of the first scientists to understand that electric charge was quantized!; from this quantization he deduced the units that are now named after him.

The Stoney length and the Stoney energy are collectively called the Stoney scale, and they are not far from the Planck length and the Planck energy, the Planck scale. The Stoney scale and the Planck scale are the length and energy scales at which quantum processes and gravity occur together. At these scales, a unified theory of physics is thus likely required. The only notable attempt to construct such a theory from the Stoney scale was that of H. Weyl, who associated a gravitational unit of charge with the Stoney length and who appears to have inspired Dirac’s fascination with the large number hypothesis. Since then, the Stoney scale has been largely neglected in the development of modern physics, although it is occasionally discussed to this day. Wilczek likes to point out that, in Stoney Units, QM would be an emergent phenomenon/theory, since the Planck constant wouldn’t be present directly but as a combination of different constants. By the other hand, the Planck scale is valid for all known interactions, and does not give prominence to the electromagnetic interaction, as the Stoney scale does. That is, in Stoney Units, both gravitation and electromagnetism are on equal footing, unlike the Planck units, where only the speed of light is used and there is no more connections to electromagnetism, at least, in a clean way like the Stoney Units do. Be aware, sometimes, rarely though, Planck units are referred to as Planck-Stoney units.

What are the most interesting Stoney system values? Here you are the most remarkable results:

1) Stoney Length, $L_S$.

$\boxed{L_S=\sqrt{\dfrac{G_Ne^2}{(4\pi\varepsilon)c^4}}\approx 1.38\cdot 10^{-36}m}$

2) Stoney Mass, $M_S$.

$\boxed{M_S=\sqrt{\dfrac{e^2}{G_N(4\pi\varepsilon_0)}}\approx 1.86\cdot 10^{-9}kg}$

3) Stoney Energy, $E_S$.

$\boxed{E_S=M_Sc^2=\sqrt{\dfrac{e^2c^4}{G_N(4\pi\varepsilon_0)}}\approx 1.67\cdot 10^8 J=1.04\cdot 10^{18}GeV}$

4) Stoney Time, $t_S$.

$\boxed{t_S=\sqrt{\dfrac{G_Ne^2}{c^6(4\pi\varepsilon_0)}}\approx 4.61\cdot 10^{-45}s}$

5) Stoney Charge, $Q_S$.

$\boxed{Q_S=e\approx 1.60\cdot 10^{-19}C}$

6) Stoney Temperature, $T_S$.

$\boxed{T_S=E_S/k_B=\sqrt{\dfrac{e^2c^4}{G_Nk_B^2(4\pi\varepsilon_0)}}\approx 1.21\cdot 10^{31}K}$

## Planck Units

The reference constants to this natural system of units (generally denoted by P) are the following 4 constants:

1) Gravitational constant. $G_N$

2) Speed of light. $c$.

3) Planck constant or rationalized Planck constant. $\hbar$.

4) Boltzmann constant. $k_B$.

The Planck units are got when you set these 4 constants to the unit, i.e.,

$\boxed{G_N=c=\hbar=k_B=1}$

It is often said that Planck units are a system of natural units that is not defined in terms of properties of any prototype, physical object, or even features of any fundamental particle. They only refer to the basic structure of the laws of physics: c and G are part of the structure of classical spacetime in the relativistic theory of gravitation, also known as general relativity, and ℏ captures the relationship between energy and frequency which is at the foundation of elementary quantum mechanics. This is the reason why Planck units particularly useful and common in theories of quantum gravity, including string theory or loop quantum gravity.

This system defines some limit magnitudes, as follows:

1) Planck Length, $L_P$.

$\boxed{L_P=\sqrt{\dfrac{G_N\hbar}{c^3}}\approx 1.616\cdot 10^{-35}s}$

2) Planck Time, $t_P$.

$\boxed{t_P=L_P/c=\sqrt{\dfrac{G_N\hbar}{c^5}}\approx 5.391\cdot 10^{-44}s}$

3) Planck Mass, $M_P$.

$\boxed{M_P=\sqrt{\dfrac{\hbar c}{G_N}}\approx 2.176\cdot 10^{-8}kg}$

4) Planck Energy, $E_P$.

$\boxed{E_P=M_Pc^2=\sqrt{\dfrac{\hbar c^5}{G_N}}\approx 1.96\cdot 10^9J=1.22\cdot 10^{19}GeV}$

5) Planck charge, $Q_P$.

In Lorentz-Heaviside electromagnetic units

$\boxed{Q_P=\sqrt{\hbar c \varepsilon_0}=\dfrac{e}{\sqrt{4\pi\alpha}}\approx 5.291\cdot 10^{-19}C}$

In Gaussian electromagnetic units

$\boxed{Q_P=\sqrt{\hbar c (4\pi\varepsilon_0)}=\dfrac{e}{\sqrt{\alpha}}\approx 1.876\cdot 10^{-18}C}$

6) Planck temperature, $T_P$.

$\boxed{T_P=E_P/k_B=\sqrt{\dfrac{\hbar c^5}{G_Nk_B^2}}\approx 1.417\cdot 10^{32}K}$

From these “fundamental” magnitudes we can build many derived quantities in the Planck System:

1) Planck area.

$A_P=L_P^2=\dfrac{\hbar G_N}{c^3}\approx 2.612\cdot 10^{-70}m^2$

2) Planck volume.

$V_P=L_P^3=\left(\dfrac{\hbar G_N}{c^3}\right)^{3/2}\approx 4.22\cdot 10^{-105}m^3$

3) Planck momentum.

$P_P=M_Pc=\sqrt{\dfrac{\hbar c^3}{G_N}}\approx 6.52485 kgm/s$

A relatively “small” momentum!

4) Planck force.

$F_P=E_P/L_P=\dfrac{c^4}{G_N }\approx 1.21\cdot 10^{44}N$

It is independent from Planck constant! Moreover, the Planck acceleration is

$a_P=F_P/M_P=\sqrt{\dfrac{c^7}{G_N\hbar}}\approx 5.561\cdot 10^{51}m/s^2$

5) Planck Power.

$\mathcal{P}_P=\dfrac{c^5}{G_N}\approx 3.628\cdot 10^{52}W$

6) Planck density.

$\rho_P=\dfrac{c^5}{\hbar G_N^2}\approx 5.155\cdot 10^{96}kg/m^3$

Planck density energy would be equal to

$\rho_P c^2=\dfrac{c^7}{\hbar G_N^2}\approx 4.6331\cdot 10^{113}J/m^3$

7) Planck angular frequency.

$\omega_P=\sqrt{\dfrac{c^5}{\hbar G_N}}\approx 1.85487\cdot 10^{43}Hz$

8) Planck pressure.

$p_P=\dfrac{F_P}{A_P}=\dfrac{c^7}{G_N^2\hbar}=\rho_P c^2\approx 4.6331\cdot 10^{113}Pa$

Note that Planck pressure IS the Planck density energy!

9) Planck current.

$I_P=Q_P/t_P=\sqrt{\dfrac{4\pi\varepsilon_0 c^6}{G_N}}\approx 3.4789\cdot 10^{25}A$

10) Planck voltage.

$v_P=E_P/Q_P=\sqrt{\dfrac{c^4}{4\pi\varepsilon_0 G_N}}\approx 1.04295\cdot 10^{27}V$

11) Planck impedance.

$Z_P=v_P/I_P=\dfrac{\hbar^2}{Q_P}=\dfrac{1}{4\pi \varepsilon_0 c}\approx 29.979\Omega$

A relatively small impedance!

12) Planck capacitor.

$C_P=Q_P/v_P=4\pi\varepsilon_0\sqrt{\dfrac{\hbar G_N}{ c^3}} \approx 1.798\cdot 10^{-45}F$

Interestingly, it depends on the gravitational constant!

Some Planck units are suitable for measuring quantities that are familiar from daily experience. In particular:

1 Planck mass is about 22 micrograms.

1 Planck momentum is about 6.5 kg m/s

1 Planck energy is about 500kWh.

1 Planck charge is about 11 elementary (electronic) charges.

1 Planck impendance is almost 30 ohms.

Moreover:

i) A speed of 1 Planck length per Planck time is the speed of light, the maximum possible speed in special relativity.

ii) To understand the Planck Era and “before” (if it has sense), supposing QM holds yet there, we need a quantum theory of gravity to be available there. There is no such a theory though, right now. Therefore, we have to wait if these ideas are right or not.

iii) It is believed that at Planck temperature, the whole symmetry of the Universe was “perfect” in the sense the four fundamental foces were “unified” somehow. We have only some vague notios about how that theory of everything (TOE) would be.

The physical dimensions of the known Universe in terms of Planck units are “dramatic”:

i) Age of the Universe is about $t_U=8.0\cdot 10^{60} t_P$.

ii) Diameter of the observable Universe is about $d_U=5.4\cdot 10^{61}L_P$

iii) Current temperature of the Universe is about $1.9 \cdot 10^{-32}T_P$

iv) The observed cosmological constant is about $5.6\cdot 10^{-122}t_P^{-2}$

v) The mass of the Universe is about $10^{60}m_p$.

vi) The Hubble constant is $71km/s/Mpc\approx 1.23\cdot 10^{-61}t_P^{-1}$

## Schrödinger Units

The Schrödinger Units do not obviously contain the term c, the speed of light in a vacuum. However, within the term of the Permittivity of Free Space [i.e., electric constant or vacuum permittivity], and the speed of light plays a part in that particular computation. The vacuum permittivity results from the reciprocal of the speed of light squared times the magnetic constant. So, even though the speed of light is not apparent in the Schrödinger equations it does exist buried within its terms and therefore influences the decimal placement issue within square roots. The essence of Schrödinger units are the following constants:

A) Gravitational constant $G_N$.

B) Planck constant $\hbar$.

C) Boltzmann constant $k_B$.

D) Coulomb constant or equivalently the electric permitivity of free space/vacuum $k_C=1/4\pi\varepsilon_0$.

E) The electric charge of the positron $e$.

In this sistem $\psi$ we have

$\boxed{G_N=\hbar =k_B =k_C =1}$

1) Schrödinger Length $L_{Sch}$.

$L_\psi=\sqrt{\dfrac{\hbar^4 G_N(4\pi\varepsilon_0)^3}{e^6}}\approx 2.593\cdot 10^{-32}m$

2) Schrödinger time $t_{Sch}$.

$t_\psi=\sqrt{\dfrac{\hbar^6 G_N(4\pi\varepsilon_0)^5}{e^{10}}}\approx 1.185\cdot 10^{-38}s$

3) Schrödinger mass $M_{Sch}$.

$M_\psi=\sqrt{\dfrac{e^2}{G_N(4\pi\varepsilon_0)}}\approx 1.859\cdot 10^{-9}kg$

4) Schrödinger energy $E_{Sch}$.

$E_\psi=\sqrt{\dfrac{e^{10}}{\hbar^4(4\pi\varepsilon_0)^5G_N}}\approx 8890 J=5.55\cdot 10^{13}GeV$

5) Schrödinger charge $Q_{Sch}$.

$Q_\psi =e=1.602\cdot 10^{-19}C$

6) Schrödinger temperature $T_{Sch}$.

$T_\psi=E_\psi/k_B=\sqrt{\dfrac{e^{10}}{\hbar^4(4\pi\varepsilon_0)^5G_Nk_B^2}}\approx 6.445\cdot 10^{26}K$

## Atomic Units

There are two alternative systems of atomic units, closely related:

1) Hartree atomic units:

$\boxed{e=m_e=\hbar=k_B=1}$ and $\boxed{c=\alpha^{-1}}$

2) Rydberg atomic units:

$\boxed{\dfrac{e}{\sqrt{2}}=2m_e=\hbar=k_B=1}$ and $\boxed{c=2\alpha^{-1}}$

There, $m_e$ is the electron mass and $\alpha$ is the electromagnetic fine structure constant. These units are designed to simplify atomic and molecular physics and chemistry, especially the quantities related to the hydrogen atom, and they are widely used in these fields. The Hartree units were first proposed by Doublas Hartree, and they are more common than the Rydberg units.

The units are adapted to characterize the behavior of an electron in the ground state of a hydrogen atom. For example, using the Hartree convention, in the Böhr model of the hydrogen atom, an electron in the ground state has orbital velocity = 1, orbital radius = 1, angular momentum = 1, ionization energy equal to 1/2, and so on.

Some quantities in the Hartree system of units are:

1) Atomic Length (also called Böhr radius):

$L_A=a_0=\dfrac{\hbar^2 (4\pi\varepsilon_0)}{m_ee^2}\approx 5.292\cdot 10^{-11}m=0.5292\AA$

2) Atomic Time:

$t_A=\dfrac{\hbar^3(4\pi\varepsilon_0)^2}{m_ee^4}\approx 2.419\cdot 10^{-17}s$

3) Atomic Mass:

$M_A=m_e\approx 9.109\cdot 10^{-31}kg$

4) Atomic Energy:

$E_A=m_ec^2=\dfrac{m_ee^4}{\hbar^2(4\pi\varepsilon_0)^2} \approx 4.36\cdot 10^{ -18}J=27.2eV=2\times(13.6)eV=2Ry$

5) Atomic electric Charge:

$Q_A=q_e=e\approx 1.602\cdot 10^{-19}C$

6) Atomic temperature:

$T_A=E_A/k_B=\dfrac{m_ee^4}{\hbar^2(4\pi\varepsilon_0)^2k_B}\approx 3.158\cdot 10^5K$

The fundamental unit of energy is called the Hartree energy in the Hartree system and the Rydberg energy in the Rydberg system. They differ by a factor of 2. The speed of light is relatively large in atomic units (137 in Hartree or 274 in Rydberg), which comes from the fact that an electron in hydrogen tends to move much slower than the speed of light. The gravitational constant  is extremely small in atomic units (about 10−45), which comes from the fact that the gravitational force between two electrons is far weaker than the Coulomb force . The unit length, LA, is the so-called and well known Böhr radius, a0.

The values of c and e shown above imply that $e=\sqrt{\alpha \hbar c}$, as in Gaussian units, not Lorentz-Heaviside units. However, hybrids of the Gaussian and Lorentz–Heaviside units are sometimes used, leading to inconsistent conventions for magnetism-related units. Be aware of these issues!

## QCD Units

In the framework of Quantum Chromodynamics, a quantum field theory (QFT) we know as QCD, we can define the QCD system of units based on:

1) QCD Length $L_{QCD}$.

$L_{QCD}=\dfrac{\hbar}{m_pc}\approx 2.103\cdot 10^{-16}m$

and where $m_p$ is the proton mass (please, don’t confuse it with the Planck mass $M_P$).

2) QCD Time $t_{QCD}$.

$t_{QCD}=\dfrac{\hbar}{m_pc^2}\approx 7.015\cdot 10^{-25}s$

3) QCD Mass $M_{QCD}$.

$M_{QCD}=m_p\approx 1.673\cdot 10^{-27}kg$

4) QCD Energy $E_{QCD}$.

$E_{QCD}=M_{QCD}c^2=m_pc^2\approx 1.504\cdot 10^{-10}J=938.6MeV=0.9386GeV$

Thus, QCD energy is about 1 GeV!

5) QCD Temperature $T_{QCD}$.

$T_{QCD}=E_{QCD}/k_B=\dfrac{m_pc^2}{k_B}\approx 1.089\cdot 10^{13}K$

6) QCD Charge $Q_{QCD}$.

In Heaviside-Lorent units:

$Q_{QCD}=\dfrac{1}{\sqrt{4\pi\alpha}}e\approx 5.292\cdot 10^{-19}C$

In Gaussian units:

$Q_{QCD}=\dfrac{1}{\sqrt{\alpha}}e\approx 1.876\cdot 10^{-18}C$

## Geometrized Units

The geometrized unit system, used in general relativity, is not a completely defined system. In this system, the base physical units are chosen so that the speed of light and the gravitational constant are set equal to unity. Other units may be treated however desired. By normalizing appropriate other units, geometrized units become identical to Planck units. That is, we set:

$\boxed{G_N=c=1}$

and the remaining constants are set to the unit according to your needs and tastes.

## Conversion Factors

This table from wikipedia is very useful:

where:

i) $\alpha$ is the fine-structure constant, approximately 0.007297.

ii) $\alpha_G=\dfrac{m_e^2}{M_P^2}\approx 1.752\cdot 10^{-45}$ is the gravitational fine-structure constant.

Some conversion factors for geometrized units are also available:

Conversion from kg, s, C, K into m:

$G_N/c^2$  [m/kg]

$c$ [m/s]

$\sqrt{G_N/(4\pi\varepsilon_0)}/c^2$ [m/C]

$G_Nk_B/c^2$ [m/K]

Conversion from m, s, C, K into kg:

$c^2/G_N$ [kg/m]

$c^3/G_N$ [kg/s]

$1/\sqrt{G_N4\pi\varepsilon_0}$ [kg/C]

$k_B/c^2$[kg/K]

Conversion from m, kg, C, K into s

$1/c$ [s/m]

$G_N/c^3$[s/kg]

$\sqrt{\dfrac{G_N}{4\pi\varepsilon_0}}/c^3$ [s/C]

$G_Nk_B/c^5$ [s/K]

Conversion from m, kg, s, K into C

$c^2/\sqrt{\dfrac{G_N}{4\pi\varepsilon_0}}$[C/m]

$(G_N4\pi\varepsilon_0)^{1/2}$ [C/kg]

$c^3/(G_N/(4\pi\varepsilon_0))^{1/2}$[C/s]

$k_B\sqrt{G_N4\pi\varepsilon_0}/c^2$   [C/K]

Conversion from m, kg, s, C into K

$c^4/(G_Nk_B)$[K/m]

$c^2/k_B$ [K/kg]

$c^5/(G_Nk_B)$ [K/s]

$c^2/(k_B\sqrt{G_N4\pi\varepsilon_0})$ [K/C]

Or you can read off factors from this table as well:

and

Natural units have some advantages (“Pro”):

1) Equations and mathematical expressions are simpler in Natural Units.

2) Natural units allow for the match between apparently different physical magnitudes.

3) Some natural units are independent from “prototypes” or “external patterns” beyond some clever and trivial conventions.

4) They can help to unify different physical concetps.

However, natural units have also some disadvantages (“Cons”):

1) They generally provide less precise measurements or quantities.

2) They can be ill-defined/redundant and own some ambiguity. It is also caused by the fact that some natural units differ by numerical factors of pi and/or pure numbers, so they can not help us to understand the origin of some pure numbers (adimensional prefactors) in general.

Moreover, you must not forget that natural units are “human” in the sense you can addapt them to your own needs, and indeed,you can create your own particular system of natural units! However, said this, you can understand the main key point: fundamental theories are who finally hint what “numbers”/”magnitudes” determine a system of “natural units”.

Remark: the smart designer of a system of natural unit systems must choose a few of these constants to normalize (set equal to 1). It is not possible to normalize just any set of constants. For example, the mass of a proton and the mass of an electron cannot both be normalized: if the mass of an electron is defined to be 1, then the mass of a proton has to be $\approx 6\pi^5\approx 1936$. In a less trivial example, the fine-structure constant, α≈1/137, cannot be set to 1, because it is a dimensionless number. The fine-structure constant is related to other fundamental constants through a very known equation:

$\alpha=\dfrac{k_Ce^2}{\hbar c}$

where $k_C$ is the Coulomb constant, e is the positron electric charge (elementary charge), ℏ is the reduced Planck constant, and c is the again the speed of light in vaccuum. It is believed that in a normal theory is not possible to simultaneously normalize all four of the constants c, ℏ, e, and kC.

## Fritzsch-Xing  plot

Fritzsch and Xing have developed a very beautiful plot of the fundamental constants in Nature (those coming from gravitation and the Standard Model). I can not avoid to include it here in the 2 versions I have seen it. The first one is “serious”, with 29 “fundamental constants”:

However, I prefer the “fun version” of this plot. This second version is very cool and it includes 28 “fundamental constants”:

## The Okun Cube

Long ago, L.B. Okun provided a very interesting way to think about the Planck units and their meaning, at least from current knowledge of physics! He imagined a cube in 3d in which we have 3 different axis. Planck units are defined as we have seen above by 3 constants $c, \hbar, G_N$ plus the Boltzmann constant. Imagine we arrange one axis for c-Units, one axis for $\hbar$-units and one more for $G_N$-units. The result is a wonderful cube:

Or equivalently, sometimes it is seen as an equivalent sketch ( note the Planck constant is NOT rationalized in the next cube, but it does not matter for this graphical representation):

Classical physics (CP) corresponds to the vanishing of the 3 constants, i.e., to the origin $(0,0,0)$.

Newtonian mechanics (NM) , or more precisely newtonian gravity plus classical mechanics, corresponds to the “point” $(0,0,G_N)$.

Special relativity (SR) corresponds to the point $(0,1/c,0)$, i.e., to “points” where relativistic effects are important due to velocities close to the speed of light.

Quantum mechanics (QM) corresponds to the point $(h,0,0)$, i.e., to “points” where the action/angular momentum fundamental unit is important, like the photoelectric effect or the blackbody radiation.

Quantum Field Theory (QFT) corresponds to the point $(h,1/c,0)$, i.e, to “points” where both, SR and QM are important, that is, to situations where you can create/annihilate pairs, the “particle” number is not conserved (but the particle-antiparticle number IS), and subatomic particles manifest theirselves simultaneously with quantum and relativistic features.

Quantum Gravity (QG) would correspond to the point $(h,0,G_N)$ where gravity is quantum itself. We have no theory of quantum gravity yet, but some speculative trials are effective versions of (super)-string theory/M-theory, loop quantum gravity (LQG) and some others.

Finally, the Theory Of Everything (TOE) would be the theory in the last free corner, that arising in the vertex $(h,1/c,G_N)$. Superstring theories/M-theory are the only serious canditate to TOE so far. LQG does not generally introduce matter fields (some recent trials are pushing into that direction, though) so it is not a TOE candidate right now.

## Some final remarks and questions

1) Are fundamental “constants” really constant? Do they vary with energy or time?

2) How many fundamental constants are there? This questions has provided lots of discussions. One of the most famous was this one:

http://arxiv.org/abs/physics/0110060

The trialogue (or dialogue if you are precise with words) above discussed the opinions by 3 eminent physicists about the number of fundamental constants: Michael Duff suggested zero, Gabriel Veneziano argued that there are only 2 fundamental constants while L.B. Okun defended there are 3 fundamental constants

3) Should the cosmological constant be included as a new fundamental constant? The cosmological constant behaves as a constant from current cosmological measurements and cosmological data fits, but is it truly constant? It seems to be…But we are not sure. Quintessence models (some of them related to inflationary Universes) suggest that it could vary on cosmological scales very slowly. However, the data strongly suggest that

$P_\Lambda=-\rho c^2$

It is simple, but it is not understood the ultimate nature of such a “fluid” because we don’t know what kind of “stuff” (either particles or fields) can make the cosmological constant be so tiny and so abundant (about the 72% of the Universe is “dark energy”/cosmological constant) as it seems to be. We do know it can not be “known particles”. Dark energy behaves as a repulsive force, some kind of pressure/antigravitation on cosmological scales. We suspect it could be some kind of scalar field but there are many other alternatives that “mimic” a cosmological constant. If we identify the cosmological constant with the vacuum energy we obtain about 122 orders of magnitude of mismatch between theory and observations. A really bad “prediction”, one of the worst predictions in the history of physics!

Be natural and stay tuned!