LOG#113. Bohr’s legacy (I).

Dedicated to Niels Bohr

and his atomic model

(1913-2013)

1st part: A centenary model

atomElement-117-discoverypurity

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.

Niels_Bohr

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.

A_New_System_of_Chemical_Philosophy_fp

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!).cathode-rays-formation

thomsonAndNagaokaModels

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.

Rutherford_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_transitionsBohr_atom_model_EnglishBohr_atombohrAndBalmer

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:

bohrPostulatesbohrmodelHypotheses

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

Please, check it yourself! :D.

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

Hydrogenic atoms

(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

HydrogenAtomSpectrumDiracEquationFirstLevelsor equivalently (I add comments from the slides)

HydrogenicAtomFirstLevelsDiracEq

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!):

bohrsommIdea

sod1dirac04onelectronSpectrum

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)…

13502276-green-atom-electron-llustration-on-black-background


LOG#094. Group theory(XIV).

Group theory and the issue of mass: Majorana fermions in 2D spacetime

We have studied in the previous posts that a mass term is “forbidden” in the bivector/sixtor approach and the Dirac-like equation due to the gauge invariance. In fact,

-i\overline{\Gamma}^\mu\partial_\mu

as an operator has an “issue of mass” over the pure Dirac equation i\Gamma^\mu\Psi=0 of fermionic fields. This pure Dirac equation provides

\overline{\Gamma_\mu}\Gamma_\nu\partial_\nu\partial_\mu\Psi=\partial_0^2\Psi+\nabla\times (\nabla\times \Psi)=\partial_0^2-\Delta\Psi+\nabla (\nabla\cdot \Psi)=0

Therefore, \Psi satisfies the wave equation

\square^2\Psi=\partial^\mu\partial_\mu\Psi=0

where \square^2=\Delta-\partial_0^2 if there are no charges or currents! If we introduce a mass by hand for the \Psi field, we obtain

(i\Gamma^\mu\partial_\mu-m )\Psi=0

and we observe that it would spoil the relativistic invariance of the field equation!!!!!!! That is a crucial observation!!!!

A more general ansatz involves the (anti)linear operator V:

i\Gamma^\mu\partial_\mu\Psi-mV\Psi=0

A plane wave solution is something like an exponential function \sim e^{\pm ipx} and it obeys:

p^2=p_\mu p^\mu=-m^2

If we square the Dirac-like equation in the next way

i\overline{\Psi}^\nu\partial_\nu (i\Gamma^\mu\partial_\mu\Psi)=-\square \Psi=-m^2\Psi=i\overline{\Psi}^\nu\partial_\nu (mV\Psi)

and a bivector transformation

i\overline{\Gamma}^\mu\partial_\mu (V\Psi)-m\Psi=0

V(i\overline{\Gamma}^\mu\partial_\mu (v\Psi))-mV\Psi=0

Vi\overline{\Gamma}^\mu\partial_\mu (V\Psi)=mV\Psi=i\Gamma^\mu \partial_\mu \Psi

from linearity we get

Vi\overline{\Gamma}^\mu=i\Gamma^\mu

V^2=I_3

V\tilde{S}_aV=-S_a

V\tilde{S}_aV^{-1}=-\tilde{S}_a

if a=1,2,3. But this is impossible! Why? The Lie structure constants are “stable” (or invariant) under similarity transformations. You can not change the structure constants with similarity transformations.

In fact, if V is an antilinear operator

V=\tilde{V}\kappa=iV where \kappa is a complex conjugation of the multiplication by the imaginary unit. Then, we would obtain

\tilde{V}\tilde{V}^*=-I_3

and

\tilde{V}\tilde{S}_a^*\tilde{V}^*=\tilde{S}_a

or equivalently

\tilde{V}\tilde{S}_a^*\tilde{V}^{-1}=-\tilde{S}_a

And again, this is impossible since we would obtain then

\det (V\tilde{V}^*)=\det (V)\det (\tilde{V}^*)=\det \tilde{V}\det \tilde{V}^*>0

and this contradicts the fact that \det (-I_3)=-1!!!

Remark: In 2d, with Pauli matrices defined by

\sigma=(\sigma_1,\sigma_2,\sigma_3)

and \epsilon\tilde{\sigma}^*\epsilon^{-1}=-\tilde{\sigma}

where

\epsilon=\begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix}

and we have

\epsilon^2=(\eta\epsilon)(\eta\epsilon)^*=-I_2

with

\det (\epsilon)=\det (-I_2) such that the so-called Majorana equation(s) (as a generalization of the Weyl equation(s) and the Lorentz invariance in 4d) provides a 2-component field equation describing a massive particle DOES exist:

i\sigma^\mu\partial_\mu\Psi-m\eta\epsilon\Psi^*=0

In fact, the Majorana fermions can exist only in certain spacetime dimensions beyond the 1+1=2d example above. In 2D (or 2d) spacetime you write

\boxed{i\sigma^\mu\partial_\mu\Psi-m\eta\epsilon\Psi^*=0}

and it is called the Majorana equation. It describes a massive fermion that is its own antiparticle, an option that can not be possible for electrons or quarks (or their heavy “cousins” and their antiparticles). The only Standard Model particle that could be a Majorana particle are the neutrinos. There, in the above equations,

\sigma^\mu=(I_2,\vec{\sigma}) and \eta is a “pure phase” often referred as the “Majorana phase”.

Gauge formalism and mass terms for field equations

Introduce some gauge potential like

A=A^R+iA^I=\begin{pmatrix}A_1^R+iA_1^I\\ A_2^R+iA_2^I\\ A_3^R+iA_3^I\end{pmatrix}

It is related to the massive bivector/sixtor field with the aid of the next equation

\Psi=i\overline{\Gamma}^\nu\partial_\nu A=(i\partial_0+\nabla\times)(A^R+iA^I)=-\dot{A}^O+\nabla\times A^R+i(\dot{A}^R+\nabla\times A^I)

It satisfies a massive wave equation

\square^2A+m^2A=0

This would mean that

i\Gamma^\mu\partial_\mu (i\overline{\Gamma}^\nu \partial_\nu A)=(-\partial_0^2-\nabla\times\nabla\times)A=(-\partial_0^2+\Delta-\nabla (\nabla\cdot))A=- m^2A

and then \nabla (\nabla\cdot A)=0. However, it would NOT BE a Lorentz invariant anymore!

Current couplings

From the Ampère’s law

\partial_t E=\nabla\times B-j

and where we have absorbed the multiplicative constant into the definition of the current j, we observe that i\Gamma^\mu\partial_\mu\Psi can NOT be interpreted as the Dirac form of the Maxwell equations since j=(j_1,j_2,j_3) have 3 spatial components of a charge current 4d density J=j^\mu=(j^0,\mathbf{j})=(j^0,j^1,j^2,j^3) so that

\partial_t\Psi=-i\nabla\times \Psi-\mathbf{j} and

\nabla\cdot (\partial_t \Psi)=\nabla\cdot (-i\nabla\times \Psi-\mathbf{j})

or

\nabla\cdot \dot{\Psi}=-i\nabla\cdot (\nabla\times \Psi)-\mbox{div}\mathbf{j}=\dot{\rho}

If the continuity equation \dot{\rho}+\mbox{div}\mathbf{j}=0 holds. In the absence of magnetic charges, this last equation is equivalent to \mbox{div} (\dot{E})=\dot{\rho} or \nabla\cdot E=\rho.

Remark: Even though the bivector/sixtor field couples to the spatial part of the 4D electromagnetic current, the charge distribution is encoded in the divergence of the field \Psi itself and it is NOT and independent dynamical variable as the current density (in 4D spacetime) is linked to the initial conditions for the charge distribution and it fixes the actual charge density (also known as divergence of \Psi at any time; \Psi is a bispinor/bivector and it is NOT a true spinor/vector).

Dirac spinors under Lorentz transformations

A Lorentz transformation is a map

X'=\Lambda X

A Dirac spinor field is certain “function” transforming under certain representation of the Lorentz group. In particular,

\Psi'(x')=Q_D(\Lambda)\Psi (x) for every Lorentz transformation belonging to the group SO(1,3)^+. Moreover,

x'_\mu=\Lambda^\mu_{\;\;\; \nu}

and Dirac spinor fields obey the so-called Dirac equation (no interactions are assumed in this point, only “free” fields):

i\gamma^\mu\partial_\mu\Psi-m\Psi=0

This Dirac equation is Lorentz invariant, and it means that it also hold in the “primed”/transformed coordinates since we have

i\gamma^\mu (\partial_{\mu '}\Psi' (x'))=i\gamma^\mu\partial_{\mu '}(Q_D\Psi (x))=mQ_D\Psi

and

i\gamma^\mu\Lambda^{\;\;\; \nu}_\mu\partial_\nu Q_D\Psi=mQ_D\Psi

Using that \Lambda^\alpha_{\;\;\; \nu}\Lambda^{\nu}_{\;\;\; \mu}=\delta^\alpha_{\;\;\; \mu}

we get the transformation law

\boxed{Q_D^{-1}\gamma^\alpha Q_D=\Lambda^\alpha_{\;\;\; \nu}\gamma^\nu}

Covariant Dirac equations are invariant under Lorentz transformations IFF the transformation of the spinor components is performed with suitable chosen operators Q_D. In fact,

Q^{-1}\Gamma^\alpha Q=\Lambda^\alpha_{\;\;\; \nu}\Gamma^\nu

Q^T\Gamma^\alpha Q=\Lambda^\alpha_{\;\;\; \nu}\Gamma\nu

Q^*\Gamma^\alpha Q=\Lambda^\alpha_{\;\;\; \nu}\Gamma\nu

DOES NOT hold for \Psi bispinors/bivectors. For bivector fields, you obtain

i\Gamma^\mu\partial_\mu\Psi=-i\mathbf{j}

and

i\Gamma^\mu_{ab}\partial_{\mu '}\Psi '(x')=-iJ'_a (x')

This last equation implies that

i\Gamma_{ab}^\mu\Lambda_{\mu}^{\;\;\; \nu}\partial_\nu Q_{bc}\Psi_c(x)=-i\Lambda^a_{\;\;\; \nu}j^\nu (x)=-i\Lambda^a_{0}\mbox{div} E(x)-i\Lambda^a_ cJ^c(x)

j^\mu=(\mbox{div}E,j^1,j^2,j^3)=(j^0,j^1,j^2,j^3)

with

\mbox{div} E=\delta^\nu_c\delta_\nu\Psi_c since \mbox{div} B=\nabla\cdot B=0 because there are no magnetic monopoles.

If \tilde{\Lambda} is the inverse of the 3d matric \Lambda^a_{\;\;\; b}, then we have

\tilde{\Lambda}^d_a\Lambda^a_b=\delta^d_b

In this case, we obtain that

i (\tilde{\Lambda}^d_a\Gamma^\mu_{ab}\Lambda^\nu_\mu Q_{bc}+\tilde{\Lambda}^d_a\Lambda^a_0\delta^\nu_c)\partial_\nu\Psi_c (x)=-i\tilde{\Lambda}^d_a\Lambda^a_cJ^c=-ij^d

so

\Gamma^\nu_{dc}=\tilde{\Lambda}^d_a\Gamma^\mu_{ab}\Lambda_{\mu}^\nu Q_{bc}+\tilde{\Lambda}^d_a\Lambda^a_0\delta^\nu_c

That is, for rotations we obtain that

\Lambda^a_{\;\;\; b}=Q_{ab} \tilde{\Lambda}^{-1}=Q^{-1} \Lambda^a_{\;\;\; 0}\;\;\forall a=1,2,3

and so

\boxed{\Gamma^\nu=\Lambda_{\mu}^{\;\;\; \nu}Q^{-1}\Gamma^\mu Q}

This means that for the case of pure rotations both bivector/bispinors and current densities transform as vectors under the group SO(3)!!!!

Conclusions of this blog post:

1st. A mass term analogue to the Marjorana or Dirac spinor equation does NOT arise in 4d electromagnetism due to the interplay of relativistic invariance and gauge symmetries. That is, bivector/bispinor fields in D=4 can NOT contain mass terms for group theoretical reasons: Lorentz invariance plus gauge symmetry.

2nd. The Dirac-like equation i\Gamma^\mu \partial_\mu \Psi=0 can NOT be interpreted as a Dirac equation in D=4 due to relativistic symmetry, but you can write that equation at formal level. However, you must be careful with the meaning of this equation!

3rd. In D=2 and other higher dimensions, Majorana “mass” terms arise and you can write a “Majorana mass” term without spoiling relativistic or gauge symmetries. Majorana fermions are particles that are their own antiparticles! Then, only neutrinos can be Majorana fermions in the SM (charged fermions can not be Majorana particles for technical reasons).

4th. The sixtor/bivector/bispinor formalism with F=E+iB has certain uses. For instance, it is used in the so-called Ungar’s formalism of special relativity, it helps to remember the electromagnetic main invariants and the most simple transformations between electric and magnetic fields, even with the most general non-parallel Lorentz transformation.