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, a_H, for the ground state (t  = 1) is given by a_H = (h/2p)²/me². In general, a_H(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) = ­-e²/2a_H. In general, E_t = -me²/2a_H(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 a_H(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, m_l (= -l, -(l -1), …(l-1), l), and Pauli added the spin quantum number, m_s 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.

Hundred Years of the Bohr atom. Part 2: Planck’s Quantum Plank

In the beginning of the last century the gallery of leading scientists who contributed towards the development of principles of physics has been sketched below in the internet. The main players in this blog will necessarily have to include Planck, Einstein and Bohr (placed here along a principal diagonal) if we have to put in proper perspective the Bohr revolution.

During Bohr’s early times, the world of physics was being dominated by two people, Max Planck and Albert Einstein. As a bench-chemist I got it into my head in my early days that Niels Bohr is the genius that made the quantum leap for modelling the structure of atoms. This I learnt from various classrooms. This blog is to set right some of my --- as well as those of similar others --- early perspectives. One cannot get to Bohr without Planck’s quantum.  Planck’s plank was to assert from empirical evidence that radiation from a blackbody is best fitted by assuming that electromagnetic radiation from a black body is emitted by discrete portions or quanta and not continuously as assumed in classical physics. This bold step is now acknowledged to be the beginning of all quantum descriptions including that of Einstein in his famous year of 1905.

The daring of Planck forms the essence of this blog. The quantum postulate quintessentialy means that there are situations when changes are seen to happen by discrete steps rather than those which are described classically, say, by Newtonian mechanics. Bohr’s achievements are discussed later in this light. The blog includes at the end the contribution by S N Bose in understanding Planck’s law and the role of Einstein in interpreting Bose’s understanding that resulted in a prediction and eventual confirmation of a fifthe state of matter, the Bose-Einstein condensate.

Planck dared walk the quantum plank over a sea of classical physics knowing the dangers of making a false step. It is his daring that subsequently allowed Einstein’s first application of quantum physics to the solid state. It was also this that allowed Bohr to consolidate over earlier attempts to use Planck’s quantum of action to explain the positions of spectral lines of the hydrogen atom.

It is, I find, easy to pontificate on what constitutes good science and the way it should be done once the science has been done. It is a different matter to set an example by actually doing the good science

Mea Culpa anche.

I think part of the reason for writing this blog is to explain --- mainly to myself --- the way good science is done. It is not done by pursuing what is considered to be currently important. Rather it is to convince oneself of the veracity of one’s observation ---usually an empirical fit --- and then to assert one’s findings  and its consequences convincingly. Most importantly, perhaps, it is necessary to have a society that recognises the impending importance. In Planck’s case the society recognised the importance of understanding blackbody radiation if only to improve upon the more mundane behaviour of incandescent lamps,

Planck’s Plank. Resonance of a Musician.

At the end of the nineteenth century, classical theories such as the Newtonian laws of mechanics, Maxwell’s theory of electromagnetism and Boltzmann’s theory of statistical mechanics formed the highpoints of classical physics that could be used to explain most natural phenomena. Max Planck, looking for an area of research in the 1870s was told, as all young people are inevitably told, “...  almost everything is already discovered, and all that remains is to fill a few holes." Thus, the First Baron Kelvin, William Thomson, would comment in his famous lecture, Nineteenth-century clouds over the dynamical theory of heat and light, that the main problems in physics would be to measure known quantities to a great degree of precision so that two important clouds would clear. These were the failure of the Michelson-Morley experiment to detect a change in the speed of light in different directions as predicted by the theory of the luminous ether and the failure to understand the effect --- later termed by Ehrenfest as the “ultraviolet catastrophe” in 1911 --- which predicts that black body radiation will have infinite intensity at high (ultraviolet) frequencies. An ideal black body (usually referred to just as a black body) absorbs all radiation incident upon it without reflecting any of it independent of the wavelength or angle of incidence. Einstein, as we all seem to know, resolved the first problem of ether by his general theory of relativity. It was Planck who fitted the black body radiation spectrum over all wavelengths of light.

 

In his younger days Planck looked very much like an artist (Fig 1, left) when he started his research studies.Kuhn’s book on “Black-Body Theory and the Quantum Discontinuity, 1894-1912,” (University of Chicago Press, 1987) has it that Planck was a “… fine musician and used acoustical analogies in his work, may from the start have thought of the resonant response of a stretched string coupled by a spring or other continuous media to a driving source.” In particular, he must have been familiar with the notion of acoustic resonators in musical instruments that are used to produce sound waves of specific tones. The term resonator is most often used for a homogeneous object in which vibrations travel as waves, at an approximately constant velocity, bouncing back and forth between the sides of the resonator. Resonators can be viewed as being made of millions of coupled moving parts (such as atoms). Therefore they can have millions of resonant frequencies, although only a few may be used in practical resonators  

Planck could have been interested in the principles of Helmholtz resonances. The principle of such resonators is that when air is forced into a cavity, such as by blowing on top of a bottle, there will be a high pressure forcing air into the bottle. When this pressure is removed, the air molecules with higher pressure will move out causing the pressure in the bottle to be lower than outside and air will be drawn back in. The process repeats. Planck as a student took notes from Helmholtz’s lectures.

Planck’s interest in acoustic resonators must have drawn him to Maxwell’s electromagnetism and electric resonators in which a cavity radiation is oscillations of electromagnetic field. Such a field can be modelled in terms of a collection of harmonic oscillators. Planck must have also been familiar with Maxwell and his “demon” when conjecturing on the second law of thermodynamics: it is impossible in a system enclosed in an envelope ... to (spontaneously) produce any inequality of temperature or of pressure without the expenditure of work. The demon works as follows:- “… let us suppose that such a vessel is divided into two portions, A and B, by a division in which there is a small hole, and that a being, who can see the individual molecules, opens and closes this hole, so as to allow only the swifter molecules to pass from A to B, and only the slower molecules to pass from B to A. He will thus, without expenditure of work (by this Maxwell means that the observation and opening and closing of the particles does not involve any work) raise the temperature of B and lower that of A, in contradiction to the second law of thermodynamics...”. (from Maxwell’s Theory of Heat (New York: D. Appleton & Co., 1872), pp. 308-9).

From these Maxwell’s demon arguments Planck was interested in showing that “… irreversibility could be derived from consideration of purely conservative effects.” Of concern to Planck was “… that the … conversion of a plane wave, travelling in a single direction, to a spherical wave, travelling in all --- could not be reversed.”. Maxwell further continues “In dealing with masses of matter, while we do not perceive the individual molecules, we are compelled to adopt what I have described as the statistical method of calculation, and to abandon the strict dynamical method, in which we follow every motion by the calculus. It would be interesting to enquire how far those ideas about the nature and methods of science which have been derived from examples of scientific investigation in which the dynamical method is followed are applicable to our actual knowledge of concrete things, By this, Maxwell probably admitted to himself that the exact properties of discrete molecules, themselves, were not crucial to the exact understanding of bulk mater. The condition of irreversibility for radiation processes would determine the law for black body radiation. These considerations must have been important for the philosophy of his approach to blackbody radiation. It is Kuhn, I think, who drew attention to the importance of the difference between the ideal and the real. Thus while matter, as in gas particles, may be considered corpuscular, a theory for elasticity considers matter to be a continuum that fills whole of space. 

The study of black body radiation was important in the late nineteenth century in Western countries in order to understand, for illumination purposes, incandescence or luminosity or why a body glows when heated. Planck’ contribution was that the surface of black bodies had oscillators which absorb and emit radiation with energies which were integral multiples of some small universal value. The permitted energies were quantized! There is a distinction between collective, coherent or “ordered” motion and individual, incoherent or “disordered” motion. Boltzmann introduced statistical models based on microscopic disorder into thermodynamics instead of using classical deterministic continuum models that is suitable for coherent systems without disorder.  In particular, one required the Boltzmann factor, exp(-E_i/k_BT) for a given state, i, at energy E_i, where k_B is Boltzmann’s constant and T is the temperature. The Boltzmann statistics is used for degenerate states with more than one state a a particular energy. For a degeneracy g_i for states at energy E_i, the population, P_i, at energy E_i is given by

P_i = g_iexp(-E_i/k_BT).                                                                                 (1)

The partition function, Q_i, between the various states, i, is given by

Q_i = ∑_I g_i exp(-E_i /k_BT)                                                                             (2)                                                                   

The partition function provides a normalization factor so that population P(E_i) at energy E_i is given by P(E_i) = g_iexp(-E_i/k_BT)/Q. For a continuum of states the summation can be replaced by an integral. The average energy <E> is then obtained as

<E> = S_i E_iP(E_i)                                                                                     (3)

The familiar Boltzmann statistics that is applicable to a gas of molecules with the distribution of energy being treated as a continuous variable is obtained by integrating over all states.   

It is Planck who rephrased the second law of thermodynamics in terms of entropy or disorder. The entropy of a system which is in thermal and mechanical isolation, increases as it evolves towards thermodynamic equilibrium, Put more familiarly, entropy is a measure of disorder of the system, and should be a maximum at thermodynamic equilibrium.

The problem that Planck faced was that the blackbody radiation was fitted at short wavelngths by the Wien function and at long wavelengths by the Rayleigh function. Wien noted that the distribution of thermal radiation at various wavelengths is similar to Maxwell’s velocity distribution law. The Wien’s law for the energy, E_l, of a black body radiation at a wavelength l is given by E_l = Al^-5exp(-a/lT) where A and a are constants.  The Wien’s displacement law says that the wavelength, l_max (in angström), at which the blackbody radiation is maximum is inversely proportional to T in kelvin (l_max  = 2.897x10⁷ /T ). The Rayleigh-Jeans law for the radiation, B_l(T), from a black body at temperature T and wavelength l is given by B_l(T)= 2ck_BT/l⁴. The fit to the Wien’s function (in blue) and Rayleigh-Jeans function (in red) in terms of the frequency (inverse of wavelength) and In units of Radiance is shown in Fig 2.

What Planck did was to find a function that fitted (in yellow in Fig 2) the radiance for all wavelengths. This function is known as Planck’s formula. When one considers an oscillator it can have any frequency, v to oscillate in 3D space. The “ultraviolet catastrophe” comes when it is assumed that most oscillators have high v.  Planck’s epochal contribution comes when he suggested that the energy of each oscillator has integral multiples of some small universal value.

We consider, however – this is the most essential point of the whole calculation – E to be composed of a very definite number of equal parts and use thereto the constant of nature h = 6.55×10−27 erg · sec. This constant multiplied by the common frequency v of the resonators gives us the energy element e in erg, and dividing E by e we get the number P of energy elements which must be divided over the N resonators.

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Wien’s law was convincingly demonstrated by Lummer-Pringsheim-Kurlbaum-Reubens to be “not as generally valid, as many supposed to now”. Planck makes the bold assertion in his 1900 paper (M. Planck, Verhandl. Dtsch. phys. Ges., 2, 202 (1900); from “The Old Quantum Theory,” ed. by D. ter Haar, Pergamon Press, 1967, p. 79) that “Since I myself even in this Society have expressed the opinion that Wien’s law must be necessarily true, I may perhaps be permitted to explain briefly the relationship between the electromagnetic theory developed by me and the experimental data.” Using his knowledge of entropy (dS/dU = 1/T; U is vibrational energy) and Wien’s displacement law Planck showed in his 1900 paper that the expression

E = Cl^-5/[exp(c/lT) – 1)                                                                                (4)

which, as far as I can see at the moment, fits the  observational data, published upto now, as satisfactory as the best equations put forward for the spectrum … which I consider to be the simplest possible, apart from Wien’s expression from the point of view of the electromagnetic theory of radiation.

Planck’s 1900 equation above departs from Wien’s distribution only by the subtraction of unity from the denominator. Having found the law Planck’s anguish was to find an explanation for it.

Wien had used scaling-like arguments (say, fot electron localization) to suggest that at long wavelength matter can be considered to be continuous and a single vector may be used, whereas at shorter wavelengths the molecular constitution would be important. Wien was therefore reluctant to believe that a uniform radiation law for all wavelengths could be derived from processes at a molecular or atomic level. Planck who was a reductionist at heart, perhaps because of his prominent right-brain musical leanings, felt that such an attitude would contradict Maxwell’s electromagnetic theory which is applicable to all wavelengths.

In the paper (M. Planck, Verhandl. Dtsch. phys. Ges., 2, 237) in which Planck makes his announcement of his law, Planck writes (see http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Planck%20(1900),%20Distribution%20Law.pdf)

Entropy means disorder, and I thought that one should find this disorder in the irregularity with which even in a completely stationary radiation field the vibrations of the resonator change their amplitude and phase, as long as considers time intervals long compared to the period of one vibration, but short compared to the duration of a measurement.

Planck’s formulation of his equation that gave eqn 4 departed from convention by describing light as a wave phenomenon that is deterministically described by Maxwell’s equations, to a statistical description in terms of partitioning of quanta of energy á la Boltzmann’s new particle statistics. This was like a return to Newton’s corpuscular theory of light which must have caused Planck much agony because of wave-particle conflict. This agony, I think, is evident in Planck’s wild-ish image (Fig 1, middle) at that time.

Planck’s derivation of eqn 4 is simply based on the recognition that the expression for the average energy, <E>, of eqn 3 cannot be evaluated by treating the summation as an integral but as a summation involving a simple geometrical series. It is this summation that gives the [exp(c/lT) – 1] term in the denominator of eqn 4.

Such a description of the partitioning of quanta of energy marks a clear shift in calculating the “ways of distribution” of the energy or finding the equivalent of Boltzmann’s “complexion”s. The “distribution of P energy elements over N resonators can only take place in a finite, well–defined number of ways.”

The Planck function is then given by

B_l = 2phc²l^-5/[exp(hc/klT) – 1]                                                                   (5)

While deriving eqn 5, Planck came up with two parts of a theorem. The first part concerns the definition of the probability of a state so that the entropy is proportional to the logarithm of this probability. The second part of the theorem, which Planck says is the core of the whole theory.

Of this second part, Planck writes “…in the last resort its proof can only be given empirically.” This empirical proof is the heart of the success of his equation because from it he derives, Loschmidt’s number, L (the number of gas molecules in 1 cm³ at 0^o C and 1 atm), the Boltzmann-Drude constant, a (the average kinetic energy of an atom at the absolute temperature 1) and the elementary quantum of electricity e, (the electrical charge of a positive monovalent ion or of an electron).

The theoretical proof of Planck’s law was much less certain than the experimental proof. One theoretical proof of Planck’s equation led to a famous British-Indian sub-continental triumph --- the Bose-Einstein statistics. Part of this effort was due to Einstein. A crucial element in Einstein’s approach is the use of Planck’s lichtquantum (light quantum) in understanding the spectral lines of the hydrogen atom that has been consolidated and baptized as the Bohr model for the hydrogen atom.

Planck, Einstein and Bose

During the year 1905, when Einstein worked in an isolated atmosphere of the patent office in Berne, he had applied the quantum theory to solids, explained the photoelectric effect, worked on the Brownian motion and the theory of relativity in his “miracle” year of 1905.  Because of the disastrous effect of the Nuclear bomb over the unsuspecting people of Japan, Einstein became known as the genius he is thought to be more for his E = mc² formula than for anything else. The popular reputation of Einstein being infallible in all things he worked upon or said is reflected in the xkdc cartoon below.

After 1905, Einstein’s growing reputation forced him to work/comment on problems initiated by others. Unlike his patent office days, he became immersed among scholars in universities and worked on problems that were considered to be important and thereby lacked the original sparkle of his patent office days! They were nevertheless dazzling enough by common (“non-genius”) standards.

Einstein was interested in the way Planck’s formula for radiation distribution could be “obtained from the condition that the internal distribution of the molecules demanded by quantum theory should follow purely from an emission and absorption of radiation? It was basically a question that dealt with the consistency with which Bohr’s quantized description of spectral lines could be consistent with classical descriptions. In his 1917 paper “On the quantum theory of radiation” (Phys. Zs. 18, 121 (1917) Einstein begins with the sentence “The formal similarity between the chromatic distribution curve for thermal radiation and the Maxwell velocity distribution law is too striking to have remained hidden for long. In fact, it was this similarity that led Wien … to his displacement law (for the radiation density, r) r = v³/f(v/T) and the formula r = an³exp(-hv/kT).

The last sentence in this paper states “… a theory can only be regarded as justified when it is able to show that the impulses transmitted by the radiation field to matter lead to motions that are in accordance with the theory of heat.”

Einstein considered the probabilities for three processes A, B, and B’. Process A is the emission of radiation of energy from a state m to a state n. He considered this to be similar to that of a radioactive reaction with a time for decay that is negligibly small compared to the times spent at states m and n. The process A has a probability, dW, for decay with time as dW = A_mⁿdt. Under the influence of radiation density, r, Einstein considered two processes B and B’ that corresponded to a Planck resonator absorbing radiation energy to make a transition from n to m, with a probability dW = B_n^mrdt or to “liberate” energy with a probability dW = B_mⁿrdT. Einstein then looked for the the exchange of energy between radiation and molecules due to processes A, B, and B’ “ … such that the classical thermal distribution between various canonical states…”  W_n = p_nexp(-e_n/kT₎ is maintained. From these consideration Einstein was able to obtain Plank’s distribution law once it was assumed that the number of processes of type B is the same as that of the sum of processes A and B’ taken together. This derivation depended on Wien’s displacement law which gives A_mⁿ/B_mⁿ = an³ so that the distribution was obtained by Einstein as r= an³/{exp[(e_m - e_n)/kT -1} and from which Bohr’s relationship of (e_m - e_n) = hv followed. In the conclusion of the paper Einstein concedes that the “… weakness of the theory lies on the one hand that it does not get any closer to making the connection with wave theory; on the other, that it leaves the duration and direction of the elementary processes to ‘chance’.”

Einstein’s derivation of the {exp[(e_m - e_n)/kT -1} term in the denominator in Planck’s law basically depends on differences in the distinction of processes of emission (via radioactive-like decay) from processes involved in the absorption of radiation.

Our own S N Bose pointed out in his 1924 paper that Einstein’s derivation for Planck’s formula aimed at resolving the quantum/classical contradictions ended up using Wien’s displacement law which is based on classical theory and the high-temperature limit in which quantum theory agrees with classical theory. Bose wrote

… it appears to me that the derivations have insufficient logical foundation. In contrast, the combining of the light quanta hypothesis with statistical mechanics in the form adjusted by Planck to the needs of the quantum theory does appear to be sufficient for the derivation of the law, independent of any classical theory.

Bose invoked the notion of phase-space volume, h³, of a light quanta and the entropy obtained by the possible distribution of all the light quanta in these cells for a macroscopic light radiation. From this quantum condition all thermodynamic quantities can be calculated,

In the way Bose described his statistics he did not specifically mention he was considering statistic of indistinguishable particles of light quantum, which were yet to be known as photons at that time. Einstein, who had translated Bose’s paper into German so as to publish it in the prestigious Z. Physik, realized that The unique feature of the Bose distribution is the implications for the term {exp(-hv/kT) -1} in the denominator. For the blackbody radiation the number of photons emitted would be expected to decrease with temperature. Bose”s model for Planck’s radiation formula helped in describing the impending “catastrophe” when T tends to zero for a system with a finite number of particles. The number of particles occupying a given state could diverge to infinity! This happens for some particles and it is now known as the Bose-Einstein  Condensation. This Bose failed to point out specifically. Nevertheless, there would not have been an interpretation for a condensation, without the insights from Bose’s counting algorithm

It turns out that following Bose’s treatment for the Planck formula, there were several important development in theoretical physics in the years 1925-1927. These include including those by Pauli (spin quantum number, 1925), Fermi (statistics for indistinguishable particles following Pauli exclusion principle) Schrôdinger (wave equation, 1926) and Dirac (spinors, 1926) . These theories would bring out the full implications of the Bose-Einstein statistics. Such theories were not available to Bose and Einstein when they published their work on the Planck formula, so that Bose did not require specifying the nature of the indistinguishable particles or the different statistics when Pauli Exclusion Principle is applicable. Bose/Einstein could have built on their statistics later which they didn’t. Einstein probably had other matters to work on. In Bose’s defence, we may speculate that it is difficult for us Indians to be confined by an adhaar card-like exclusion system which makes one unique --- all gods are the same even if they are deified differently by different castes,

The physics of the Bose-Einstein condensation in a dilute gas of light-quantum-like particles --- now known as bosons --- initially was concerned with the technology of achieving extremely low temperatures (of the order of nano Kelvin or a billionth of one Kelvin). This meant isolating the system being studied. The details of this technology is given in Cornell and Wieman’s 2001 Nobel lecture “Bose-Einstein Condensation in a dilute gas; the first 70 years and some recent experiments”.

In the figures given below from the above article (Fig 3) we have highlighted two features (click on figure to expand). The one on the left shows the growth of the condensate at very low temperatures --- 50nK --- and the very small (~mesoscopic?) size of the condensate. Einstein describes this condensation in very dilute atomic gases by “A separation is effected; one part condenses, the rest remains a ‘saturated ideal gas’ ” at very low temperatures.

A macroscopic example of such a condensation is thought to be liquid helium, ⁴He. The superfluid state of ⁴He is thought to be a Bose condensate. ⁴He particles are bosons. The macroscopic quantum superfluid state of liquid He4 below its superfluid temperature (lambda point) is reflected in the absence of boiling in the superfluid state (see picture below). ³He particles are not bosons and therefore do not form superfluid (do not bose-condense)  in bulk or unrestricted geometries.

The diagram on the right of Fig 3 shows the collective nature of the excitations. A superfluid, being a fluid, is defined by its dynamical behavior. In the superfluid state any excitation is expected to be a collective coherent excitation at a particular frequency. A standing wave excitation of the density profile in the trap (for an angular momentum quantum number, m = 0) is shown after the system is allowed to evolve after some dwell time. These collective excitations suggest that under appropriate conditions one may move a macroscopic object in the same direction.

I am reminded of my reading an early article by Kohn and Sherrington (reviews of Modern Physics, 1970) on there being two kinds of bosons. The type I boson is the normal bose-condensing boson such as photon or He-4. Th type II boson is an elementary excitations such as phonons (vibrational excitation), excitons (fundamental quanta of electrical excitations consisting of an excited electron and hole bound together in a neutral pair), magnons (magnetic excitation such as misaligned spins in a ferromagnet or antiferromagnet). Bose–Einstein condensation of excitations can also happen by increase their density even at relatively high temperatures. There are several reports of bose condensation of magnetic excitations at room temperatures.

How far does such coherent condensation of excitations occur in nature is something one could worry about, I guess. If a thought is an excitation of a mind at rest, can we have a condensation such that several minds are thinking of the same thing. I am reminded of a cartoon by Thurber (Fig 4). An excitation (in the circle in red) makes a coherent mad rush of a pumped mobile population. Cartoon as it may be, it does portend dangerous mind-warping multi-media possibilities.

Epilogue: Bose in the Indian mind.

The phenomenon of Bose-Einstein condensation has been applauded by many of us Indians, some of whom are leading scientists and directors of National laboratories. The world of particles is now known to be divided into two categories of statistics: bosons (after an Indian) and Fermions (after an Italian, Fermi). It may be a proud matter for some of us. Maybe sometimes we overdo it. The cartoon below perhaps reflects this even if the so-called “god particle’ is a boson (Higgs’) and the bosons form the exciting Bose Einstein condensate.

Finally, we must remember that it is the genius of Fermi that was required to distinguish between different kinds of indistinguishable particle; it is the genius of Einstein that was required to obtain the full impact of Bose’s statistics. The differences perhaps lie in the kind of questions that we ask when we do our science.

Perhaps, we Indians get satisfied too easily.