Thursday, June 20, 2013

Hundred Years of the Bohr Atom. Part III: The road to Bohr’s Model

“ … children at school should not be introduced to quantum phenomena through any description of the atom which includes Bohr’s orbits” Mott, Contemporary Physics, 1962

With that qualification from the very reverend Rev Sir Neville Mott, Nobel Laureate 1977, it seems necessary for me to cite what Bohr himself said:- 
“Every sentence I utter must be understood not as an affirmation, but as a question.” — Bohr

It is the possible questions in Bohr’s postulates that we need to be interested in if not in the model itself.

This series of blogs was actually started with the intention of it being my beginner’s interpretation of the method behind Bohr’s model for the atom. Why it was necessary? How it evolved? In the next blog I hope to deal with how it impacted the development of quantum theory as applied now?

As dwelt with in my previous blog (Part II), quantum theory actually began with Planck (quantum of action) and was extended by Einstein (light quantum). The nature of Bohr’s contributions a hundred years ago is directly based on the work of these two giants. However, it has to be remembered that the impact of Bohr’s work was so great that the Nobel prize (1922) was awarded to him within nine years of the publication of his paper. Planck would have to wait 18 years after his 1900 publication; Einstein would have to wait 20 years. Despite this success, Bohr’s model has been the only one to have been almost completely ignored  (see left of figure below) in recent times mainly because of its failure to be generally applicable to atoms other than hydrogen-like atoms and its failure to be applicable to the simplest molecule, the hydrogen molecule. But then the Nobel committee has its own ways and Bohr’s success must be taken as a lesson in the way science is sometimes recognized. 

In the latter part of his life (right of above figure) Bohr worked mainly on the philosophical aspects of Quantum Mechanics most important of which perhaps is the “... postulation that the act of observing something affects the results of that observation”  as well as the wave particle duality. There is much to admire about Bohr and I would love to read Ottavani and Purvis’s illustrated comic book “Suspended in Language” from which two cartoons have been taken in the right of the figure above. As a Jew, he escaped from Nazi Germany in the bomber hold of an allied aircraft. He was also in the secret Manhattan project where he “...soon became a security concern, however, since his philosophical and globalist nature drove him to encourage open sharing of nuclear information with the Russians.”

Around a hundred years ago, Planck’s evidence for the quantum of action must have been the immediate scientific quantity seeking an application. Bohr indeed starts his first paper “On the Constitution of Atoms and Molecules” by noting the “... inadequacy of the classical electrodynamics in describing the behaviour of systems of atomic size...” and that “... it seems necessary to introduce in the laws in question a quantity foreign to the classical electrodynamics, i. e. Planck's constant, or as it often is called the elementary quantum of action … .“ Bohr acknowledges that the “... general importance of' Planck's theory for the discussion of the behaviour of atomic systems was originally pointed out by Einstein...”.

The pre-Bohr history on application of Planck’s constant to a model for the atom seems to be important in understanding the way Bohr applied his quantum model for the hydrogen atom even though he had many shoulders of others to stand upon.  It nevertheless marks the origin of the quantum chemical model for the structure of atoms which now stands in his name,

Early contributions to the development of Bohr’s model.

Bohr has acknowledged by name the contribution of Arthur Eric Haas, Bohr writes “The agreement as to the order of magnitude between values observed for the frequencies and dimensions of the atoms, and values for these quantities ... was first pointed out by Haas*, in an attempt to explain the meaning and the value of Planck's constant on the basis of J. J. Thomson's atom model.”  The remarkable aspect of Haas’s work in 1910 is that it was before Rutherford’s model.(1911). Haas’s model had a heavy positively charged nucleus with extranuclear electrons and was based on alpha-particle scattering.

Haas’s aim, it seems, was to produce a simple model for a Planck resonator and the one-electron case was the simplest. It need not have been the hydrogen atom. Haas actually changed Thomson’s plum pudding model where electrons are fixed in space and instead modelled a negatively charged electron as revolving around the surface of the atom with one energy quantum.

The important difference with Haas is that he considered the spatial dimensions of the atom to be fundamental, instead of the Planck’s constant.  Using the constant of Balmer’s equation, Haas also correctly derived the Rydberg constant from the action quantum h, the velocity of light c, and the fundamental magnitudes of the electron, e and m. He achieved this relation by a very formal second hypothesis, namely, that the frequency derived from his quantum rule corresponds with the constant of Balmer’s equation. From this size Haas obtained the value of the Rydberg constant within a numerical factor of eight. The value of Planck’s constant could then be expressed it in terms of the mass and radius of the hydrogen atom.

Arthur Erich Haas is a remarkable figure. Born as a Bohemian, he stumbled across Thomson’s earlier work while working on his thesis on history of science and the went on to write a treatise on Physics. Haas later went on to become a founding father of cosmology believing, like a true Christian --- and other seemingly simple folks --- the age of the universe to be finite. More interestingly (at least for me) Haas was among the first to propose that the total energy of the universe is zero having both positive and negative contributions.

Around this time, Bohr was in Copenhagen completing the defence of his doctoral thesis on the nature of electrons in metals. This interest was based on Thomson’s discovery that the spooky glow of cathode-rays emitted from metal electrodes were not disturbance of the aetherial medium as one expected, but were actually particles (which were later called electrons). Because of this interest in electrons in metals, Bohr would naturally use his award of a stipend from the Carlsberg foundation to join Thomson’s group in the prestigious Cavendish Laboratory at Cambridge. It has been well documented that Bohr was disappointed by Thomson’s lack of interest in Bohr’s interest in Thomson’s model.  Bohr became more attracted to the idea of joining Rutherford at Manchester.

When Bohr joined Rutherford’s group around 1912, phenomena associated with scattering of charged particles when impacted on matter was of natural interest, with the focus being naturally on what more information can be obtained on the constitution of matter from way Rutherford’s model of the atom. It is this interest that gave rise to powerful new results. These included, Geiger and Marsden’s crucial results with alpha particle scattering that confirmed Rutherford’s theory; Van den Broek’s results on atomic number, Z, and nuclear electrons; Moseley’s association of frequencies of X-ray radiation with atomic number; Soddy’s discovery of isotopes of elements and so on. All of these experiments are the jewels in Rutherford’s crown that followed directly from his model for the atom. The most shining theoretical jewel at that time was Bohr’s planetary model for the hydrogen atom, which precedes chronologically or is contemporaneous with the discoveries just mentioned.

It must have been “bliss in that dawn to be alive” in Rutherford’ laboratory. In Spangenbrg and Moser’s book “Niels Bohr: Atomic Theorist” we have this description
Each afternoon in the lab, work was set aside for tea. Rutherford would come in, sit down and talk. The lab group avidly discussed politics and sports and, of course, work. Ideas always were exchanged freely at these daily get-togethers. So much was happening in physics that no one was afraid that someone else would take his idea and publish it first. There were plenty of vital topics for everyone.

Bohr was little motivated by discussions with Rutherford. Instead, Bohr’s theoretical/mathematical background drew him to Charles Galton Darwin, an unusual man who later, at sixty, wrote a scientific treatise on the evolution in “The Next Million Years”. He was perhaps burdened by the reputation of his grandfather who had his treatise on the Origin of Species during the previous Million Years. We cannot really judge the merits of CGDarwin’s book until a million years have passed since it was written!

The Chapter on Material Conditions in this book, is perhaps the more easily judged from contemporary results. CGDarwin considered various environmental and energy issues that is relevant. He reveals his views on, what seems to be, the empiricism of theoreticians. He writes (about the various ages of the earth) “…theorists claim to have given an explanation on astronomical grounds for the recent four ages—but then if there had been five, might they not have discovered a different but equally cogent reason for there having been five?” This comment becomes relevant when one considers, say, the recent debate on the status of Pluto as a planet from both astronomical and astrological considerations. The more notorious recent example (perhaps not so recent) for the empiricism of theoreticians must be, of course, the theories for high-temperature superconductivity! During Darwin’s time also, there were various attempts to describe atomic structure that must have hampered his own research, since it could have depended on such knowledge.

Galton Darwin was set by Rutherford to make a theoretical model for the trajectory of α-particles when traversing matter. Darwin assumed that the a-particles lost velocity because of transfer of their kinetic energy to the electrons of the atoms during collisions of the atoms. The electrons were treated as free particles and the a-particles acted by forces varying inversely as the square of the distance apart. Consequently Darwin’s results depended on the size of the atom and the charge on the nucleus. Darwin was found that his formula would hold for the hydrogen atom if it had only one electron. There was no clear evidence for it at that time. 

Bohr’s first 1913 paper On the Theory of the Decrease of Velocity of Moving
Electrified Particles on passing through Matter in Philosophical Magazine built on CGDarwin’s work. He stressed the importance of forces “ by which the electrons are kept in their positions in the atoms. Under the influence of these forces the electrons will have a sort of vibratory motion if they are disturbed by an impulse from outside.” This seems to be the crucial point in Bohr’s modification of Darwin’s approach. Thus, as Mott writes, “…quantization applies to any movement of particles within a confined space, or any periodic motion, but not to unconfined motion such as that of an electron moving in free space or deflected by a magnetic field.”                              

The concept of Planck’s constant or the elementary quantum of action does not appear in Bohr’s first 2013 paper “On the theory of the decrease of velocity of moving electrified particles on passing through Matter although he borrowed concepts from Planck’s resonator. However, in the first of his 1913 papers “On the Constitution of Atoms and Molecules” Bohr, like Haas before him, would use Planck’s quantum of action:- “Now the essential point in Planck's theory of radiation is that the energy radiation from an atomic system does not take place in the continuous way assumed in the ordinary electrodynamics, but that it, on the contrary, takes place in distinctly separated emissions, the amount of energy radiated out from an atomic vibrator of frequency n in a single emission being equal to thn, where  t is an entire number, and h is a universal constant. Such forces “… will materially alter the motion of the electrons during the collision, and consequently the loss of energy of the particle, if the time of vibration of the electrons is of the same order of magnitude as the time …  the particle takes to travel through a distance of the same order of magnitude as the shortest distance apart of the electron from the path of the particle.” The decrease in velocity “… will depend purely on the frequency of the electrons and the velocity of the particles,”

Bohr acknowledged that such a theory borrows from the electromagnetic theory of dispersion when the frequency of the light is replaced “by the different times of collision of particles of different velocities” .Bohr realized that effects due to changes in the “frequency” (binding energy) of the “vibratory” electrons would be more rapid than the loss of velocity of “moving” (free) particles one could get “some more information about the internal structure of the atoms.” Bohr then concluded “If we adopt Rutherford's conception of the constitution of atoms, we see that the experiments on absorption of a-rays very strongly suggest, that a hydrogen atom contains only one electron outside the positively charged nucleus.”

Despite his association with Niels Bohr (see below), CGDarwin does not mention the word “quantum” in his future million years. CGDarwin writes that ordinarily “… scientific progress means the discovery of yet more exact effects produced by exact causes, and … that the cause-and-effect relation is the sole idea in the scientific method. He then emphasized that “… a very different new type of procedure is connected with the principle that the result of a great number of chances may be far more certain than the result of a few. … This newer type of reasoning is connected with the principle of probability. CGDarwin has the Boltzmann-Planck-Bose models in mind.

Planck’s statistical distribution model for the scattering of light would help Bohr improve upon Darwin’s model for the scattering of a-particles by matter from which he came to the very important conclusion of there being only one electron in the hydrogen atom. Once this was dealt with, the theoretical problem of the hydrogen atom had no electron-electron interactions to contend with. Further, this single electron character of the hydrogen atom allows Bohr to assume “that the orbit in question is circular … for systems containing only a single electron.” For those few working on the structure of atoms at that time this would have been a blessing. Bohr would be the first to realize this and the first to exploit this to account for a structural model for the one-electron hydrogen atom. 

From distinct allowed rotation to the idea of quantized angular momentum seems to be a logical step. It was Nicholson in 1912 who suggested that the angular momentum assumed values which were integral multiples of h/2π.
”If, therefore, the constant h of Planck has, as Sommerfeld has suggested, an atomic significance, it may mean that the angular momentum of an atom can only rise or fall by discrete amounts when electrons leave or return. It is readily seen that this view presents less difficulty to the mind than the more usual interpretation, which is believed to involve an atomic constitution of the energy itself.”
                                                                                 Nicholson (1912)

Mott, writes in his 1962-Contemporary-Physics article
Bohr-as every schoolboy knows-made the assumption that the angular momentum L should be given by L = nh/2p  where n is an integer. The factor 2p, turning up as it does, often seems arbitrary, and I do not think it can be explained without quantum mechanics. Bohr introduced it to obtain agreement with experiment for the energy levels of hydrogen.

Bohr’s first paper on the structure of the atom

One of the first lines of the first part of his paper in which he makes general considerations on the binding of electrons by positive nuclei, Bohr writes:-

Let us at first assume that there is no energy radiation. In this case the electron will describe stationary elliptical orbits.

It is the concept of stationary orbits and not quantization that seemed to be the most important step. In this Nobel lecture for “… a formulation of the principles of the quantum theory that could immediately account for the stability in atomic structure and the properties of the radiation” he states:-
(I). … there exist a number of so-called stationary states which, in spite of the fact that the motion of the particles in these states obeys the laws of classical mechanics …, possess a peculiar, mechanically unexplainable stability, of such a sort that every permanent change in the motion of the system must consist in a complete transition from one stationary state to another.
(2). … in contradiction to the classical electromagnetic …, a process of transition between two stationary states can be accompanied by the emission of electromagnetic radiation, which will have the same properties as that which would be sent out according to the classical theory from an electrified particle executing an harmonic vibration with constant frequency. This frequency v has, however, no simple relation to the motion of the particles of the atom, but is given by the relation hv = E’ E”, where h is Planck’s constant, and E’ and E” are the values of the energy of the atom in the two stationary states that form the initial and final state of the radiation process. Conversely, irradiation of the atom with electromagnetic waves of this frequency can lead to an absorption process, whereby the atom is transformed back from the latter stationary state to the former.

Although Bohr agreed with Nicholson that from Planck’s theory the radiation from an oscillator “is sent out in quanta” such a radiation cannot be “homogeneous … for, as soon as the emission of radiation is started, the energy and also the frequency of the system are altered.” This statement is perhaps a quantum equivalent of the classical argument that as electrons move about the nucleus it would lose energy and would spiral into the nucleus.

In order to obtain the binding energy of the hydrogen atom Bohr had to make two important assumptions. The first of these is that “… the stationary states can discussed by help of the ordinary mechanics …”. In a classical model, the electron’s acceleration in the hydrogen atom, is the centripetal acceleration, v2/r , and the only force (e2/4pe0r2) acting on the electron is the Coulomb attraction of the proton. From Newton’s second law mv2/r = (e2/4pe0r2). Since the kinetic energy, K, for a classical system is mv2/2 and the potential energy, U, for a negatively charged electron and a positively charged proton is – e2/4pe0r. we obtain from Newton’s second law, K + U/2 = 0. In classical mechanics this result is obtained from the virial theorem. The total energy E = K + U = -e2/4pe0r. In classical mechanics the virial theorem cannot have an absolute validity, but will
only hold in calculations of certain mean values of the motion of the electrons. What is necessary in order to account for spectroscopic facts is that “in obvious contrast to the ordinary ideas of electrodynamics” “one need not distinguish between the actual motions and their mean values.

The second point is that the transition between different stationary states cannot be treated classically. Here, the transition leads to emission of a homogeneous radiation following Planck with hv = E’ E”. In order to obtain these energies, one requires a knowledge of the radii, r’ and r”, of the two states between which the transitions take place. Bohr then used the term of a “permanent” state for the “the one among the stationary states during the formation of which the greatest amount of energy is emitted.” This “permanent” state is now referred to as the ground state. The calculation of the energy required a knowledge of the angular momentum.

Bohr acknowledges Nicholson’s effort in showing “… that the ratios between the wave-length of different sets of lines of the coronal spectrum can be accounted for with great accuracy by assuming that the ratio between the energy of the system and the frequency of rotation of the ring is equal to an entire multiple of Planck's constant.” Bohr would introduce quantization by stating that “… the angular momentum of the electron round the nucleus in a stationary state of the system is equal to an entire multiple of a universal value, independent of the charge on the nucleus.It may be argued that such a quantization of angular momentum is conceptually different from Planck’s quantization of energy. However, the quantization of angular momentum is equivalent to the quantization of energy.

The “entire multiple” (integer) by which the angular momentum is quantized was given the notation, t, by Bohr and it is now known as the principal quantum number. For the present we continue to use Bohr’s notation, t. The “permanent” or ground state would correspond to t = 1, which has the lowest energy and is mainly occupied at low temperatures. The radius of the “allowed” (t = integer) stationary states was obtained as what is now known as the Bohr radius. For the hydrogen atom, the Bohr radius, aH, for the ground state (t  = 1) is given by aH = (h/2p)2/me2. In general, aH(t) = t2(h/2p)2/me2 From the virial theorem we then obtain the energy E(t=1) for the ground state as E(t=1) = ­-e2/2aH. In general, Et = -me2/2aH(t) = -2p2me4/2t2h2,
If now we suppose that the radiation in question is homogeneous, and that the amount of energy emitted is equal to hn, where n is the frequency of the radiation, we get
Et2 - Et1 = 2p2me4/2h2(1/t22 -1/t12)
or the frequency, v, of the “homogeneous” radiation is given as
v = (Et2 - Et1)/h = 2p2me4/2h3(1/t22 -1/t12)

Bohr then concludes
We see that this expression accounts for the law connecting lines in the spectrum of hydrogen. If we put t2 = 2 and let t1 vary, we get the ordinary Balmer series. If we put t2 = 3, we get the series in the ultra-red observed by Paschen and previously suspected by Ritz. If we put t2 = 1 and t2 = 4, 5, . . , we get series respectively in the extreme ultra-violet and the extreme ultra-red, which are not observed, but the existence of which may be expected.”

Bohr commented also
It may be remarked that the fact, that it has not been possible to observe more than 12 lines of the Balmer series in experiments with vacuum tubes, while 33 lines are observed in the spectra of some celestial bodies, is just what we should expect from the above theory. According to the equation (for aH(t)) the diameter of the orbit of the electron in the different stationary states is proportional to t2. For t = 12 the diameter is equal to 1.6 x 10¯6 cm., or equal to the mean distance between the molecules in a gas at a pressure of about 7 mm. mercury; for t = 33 the diameter is equal to 1.2 x 10¯5 cm., corresponding to the mean distance of the molecules at a pressure of about 0.02 mm. mercury. According to the theory the necessary condition for the appearance of a great number of lines is therefore a very small density of the gas

All this would strongly suggest that the more important immediate impact of Bohr’s contributions has been in accounting for the spectral lines of the hydrogen atom. He was thought at that time to be a spectroscopist explaining spectral lines such as the Balmer lines. Even as a spectroscopist, Bohr was not the first to invoke quantization. In his Nobel lecture Bohr acknowledges Bjerrum’s contributions on spectra of rotating molecules in which Bjerrum “… emphasized the fact that the effect should not consist of a continuous widening of the lines such as might be expected from classical theory, which imposed no restrictions on the molecular rotations, but in accordance with the quantum theory he predicted that the lines should be split up into a number of components, corresponding to a sequence of distinct possibilities of rotation. … and the phenomenon may still be regarded as one of the most striking evidences of the reality of the quantum theory …” (From Bohr’s Nobel lecture).

Bohr’s derivation of the formula for the spectral lines of hydrogen atom is accurate. To date, it is the only quantum mechanical result that gives accurate values using first principle arguments. Later developments in wave mechanics through Schrödinger’s wave equation, do not give such striking agreements from completely ab-initio calculations. The formula is identical with that obtained by wave mechanics.                                        

Bohr is nevertheless historically associated with the initiation of a quantized approach to electronic structure of atoms that now dominates modern theoretical thinking, even if it is now in another Schrodinger’s-wave-function way. Bohr’s model would have many shortcomings. One of the earliest was when Rutherford would draw Bohr’s attention to the Stark effect which showed a splitting of lines in the presence of an electric field. This effect could not be explained, much as Bohr tried, using non-elliptical electron orbits that is imposed by Bohr’s use of t (later known as the principal quantum number, n) alone.

Sommerfield (around 1916) extended Bohr’s model to include elliptical orbits and magnetic fields he was able to model other quantum numbers such as the angular momentum quantum number, l (- 0, 1, 2, … (n-1)), and the (2l + 1) magnetic quantum number, ml (= -l, -(l -1), …(l-1), l), and Pauli added the spin quantum number, ms to complete the quantum numbers of the so-called old quantum theory. Such quantum numbers were mainly designed to explain the spectral lines of atoms, before Schrödinger’s wave equation (1926) became established as the new quantum mechanics.  What is important is that the use of such quantum numbers does not necessarily require Schrödinger’s wave equation as a pre-requisite.

The various quantum numbers are a consequence of the Bohr-Sommerfield model which, with its various quantum numbers, gives some validity to treating atoms like a microscopic solar system --- each electron keeping its place through its own unique identification card (its adhaar card, that we Indians like to complain about) consisting of various quantum numbers.

It was Fermi who first examined this aspect in detail. In the translation (arXiv:cond-mat/9912229v1 [cond-mat.stat-mech] 14 Dec 1999) of his Italian paper) of his 1926 paper On the Quantization of the Monoatomic Ideal Gas (Rend. Lincei, 3,145-149 (1926)) Fermi writes “It is necessary to admit that we must add some complements to Sommerfeld’s rules, in the case of systems, … in which the elements are not distinguishable from each other …like … atoms heavier than hydrogen … containing more than one electron. The fact that “…the K ring is already saturated when it contains two electrons, and in the same way the L ring is saturated when it contains 8 electrons, was interpreted by Stoner,[ E. C. Stoner, Phil. Mag. 48, 719 (1924)] and even more precisely by Pauli,[ W. Pauli, Zs. f. Phys. 31, 765 (1925)]. … To realize this fact, it is sufficient to assume that in the atom there can not be two electrons with the orbits described by the same quantum numbers; in other words it is required to admit that an electronic orbit is already ”occupied” when it contains only one electron.” It was this that gave the Pauli exclusion principle and then the complete Aufbau (building-up) principle from which the electronic configuration of atoms  could be obtained and the periodicity of the periodic table could be obtained. It is, however, to Bohr’s credit that it was he who first introduced the concept of an Aufbau principle based on two quantum numbers, principal and subordinate.

Bohr’s idea of stationary states as states --- such as those of a hydrogen atom --- is that in which an electron remained in the same orbital state for all time, in contrast to that in classical physics which would require the orbiting electron to radiate away its energy and spiral into the nucleus. Although Bohr’s original concept is now no longer appropriate, the notion of a stationary state remains a valid one. The stationary states of such quantum systems do not change in time in the sense that the probabilities of outcomes of a measurement of any property of the system is the same no matter at what time the measurement is made. The cartoon below on self-reference effect (“tendency of people to effectively recall information about themselves”) probably represents one view of this stationary state.

Bohr’s stationary states really follows Hamilton's principle of classical physics which means that of all the possible equations of motion of a mechanical system between two time intervals, the motion will occur along the curve that gives a stationary value (extreme value of a function where the derivative is zero) to the action integral which is the integral of the Lagrangian (difference between kinetic and potential energy) over the time interval. It is this Hamilton’s principle that was used by Schrödinger in developing his theory for “undulatory mechanics” (see An undulatory theory on the mechanics of atoms and molecules” Schrödinger, 1926, Phys. Rev.). The Hamilton principle comes from the “theory of propagation of light in a non-homogeneous medium, which... became the starting point for his (Hamilton’s) famous theories in pure mechanics.”

The philosophies on quantum mechanics that arose from Bohr’s model of stationary states follows in the next blog.