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.  

Hundred Years of the Bohr atom . Part I: The pre-quantum years

This article has been prompted by personal reasons. I have been applying a non-planetary Feynmann adaptation of the planetary Bohr model to molecular structure (not dealt with here). I thought I should learn a little bit more about the historical antecedents/compulsions that drove Bohr to his model. The second reason is that being a Bengali (sometimes very proudly) 1913 is also the year that Rabindranath Tagore got his Nobel prize for Bengali literature! The first reason will form the main part of the blog.

Two 1913 Nobels

In July of 1913 Niels Bohr’s paper “On the Constitution of Atoms and Molecules” was published in the first twentyfive pages of Volume xxvi of the Philosophical magazine. This publication changed the way we look at small things --- the quantum theory for atoms --- when it was thought rhat “ … the laws of classical physics should be universally valid, even at the atomic level”. That laws of classical physics should prevail at all levels seemed to be a perfectly valid assumption considering that much of planetary physics, cosmology, particle physics continue to benefit from the laws of classical physics that were enunciated before the twentieth century.

In the same year Rabindranath Tagore was given the Nobel Prize for Literature ( ... the first non-European to win the prize) a matter of great intellectual pride for many Indian intellects who probably now felt at par with the European. Tagore himself (according to Wikepedia) descended from a shunned family of Brahmins because some of them were converted to Islam under the ruler Pir Ali (hence Pirali Brahmins). Tagore may have been re-embraced by the Brahmin community as an European although he was only penning folk thoughts inherited over tens of millennia. We will certainly celebrate Tagore’s Nobel centenary with more zeal than we will celebrate the Bohr model. We claim to have lost Tagore's Nobel medal, although we see o be sure it is likely to be tucked away carefully in some locker of a very less noble collector.

The Bohr’s model and Tagore’s Poems seemingly reflects a triumph of left- (analytic) and right- (synthetic) brain activities.  two mutually exclusive attitudes, somethimes. Tagore’s poems reflect ancient native thoughts that come from our experience/acceptance of the four elements. Bohr’s mind-function is perhaps more common among Europeans than it is among colonized Indians who have been trained by Europeans to carry out their instructions.

This contradiction is somewhat galling for us. In the western mind we Indians are appreciated for the thoughts of our right-brain past. Our left-brain activity in the present is rarely a matter of much competitive concern no matter how much we rave and rant about our post-independence II institutions (IITs and IISERs and IIMs). Quite simply put, we are not innovative enough (emphasis on enough).

When invited to speak at S’eminaire Poincaré  in 2007, the young (by my age standards) Jûrg Frôhlich made this opening remark:-

The 21st Century appears to witness a fairly strong decline in Society’s – the public’s, the politicians’, the media’s and the younger generations’ – interest in the hard sciences, including Physics, and, in particular, in fundamental theoretical science based on precise mathematical reasoning. It is hard to imagine that reports on a discovery like the deflection of light in the gravitational field of the sun and on the underlying theory, general relativity, along with a photograph of its creator, Albert Einstein, would make it onto the front pages of major daily newspapers, as it did in 1919

I couldn’t agree with him more. The young have their multimedia and their face-book/twitter/multi-media buzz to worry about. They have our mistakes to repeat. We cannot grudge them that, This blog is (hopefully) to put less trivium in trivial pursuits.

This indifference to matters scientific or even matters that are seen remotely intellectual pervades the Indian mind-set, including even that of our intellectual community (in a way). Part of our problem, is that we are not observative, inquisitive, nor creative – we are always derivative. (= copied, unoriginal, imitative, plagiaristic, lacking in originality; Thesaurus) without being integrative.

I can’t deduce why. I presume living is easy in this land of birds and bees and bovines; sustaining a reasonably enjoyable life is not a problem. When defending that life we are happy to be casted and scheduled and ruled.  We have not really learnt to mark out an intellectual kingdom, build upon it, preach, defend, impose and spread the kingdom and reap its benefits thereby.

I have not taught enough (emphasis on taught). It’s not the learning one needs to worry about. It’s about being innovative that one needs to worry our right-brain-activities about.

The idea in this blog, marking Bohr’s hundred, is mainly to record the atmosphere that led to  way Bohr’s “old” quantum mechanics.  I don’t expect that this topic will attract more readers than I usually do. At my age I don’t understand the indifference of the multi-mediated young minds to matters which I think are vitally important for a social outlook that comes from erudition and not from hip-hop-hype.

As a nation, we have the problem that we tend to think that all that is to be known is already mentioned in our scriptures even without a Hindutva saffron colour of the government at the centre. I happened to read a 1918 Nature  review on Hindu Achievement in Exact Science by a Benoy Kumar Sarkar. The review writes “... Apparently the author thinks that mere mention of a subject in ancient history proves that the person who has mentioned it discovered all about it. ... a considerable deficiency of the power of accurate scientific judgment, ... requires

practical evidence in the first place....” I will tend to agree with this assessment. There is life beyond vedic knowledge especially when we are not those who claim to have the same colour as the top of our Indian flag and, not coincidentally, the same ranking in the Indian community . 

In any case, one think that our Vedas do not definitely seem to have is any model for an atom (anu is not an atom... see my blog). 

This blog is to just stroll through the efforts that led to Bohr and his model. This will basically be my interpretation of the way good science is done without closure. It requires a society waiting for one step of the science to be climbed before the next step is seen for further progress step by step. We require a new breeding procedure for science that is not a step (or many steps) behind the frontiers but a new breed of scientists --- a new science gotra --- that creates frontiers.  

Pitfalls in Making Models

There are many scientists who are aware of experimental facts and theories that are said to explain them. For some reason, usually intuitive and rarely deductive to begin with, the scientist (let us say) would like to try out another theory ... just for the heck of it, say. So he (this includes the she) start turning things on their heads in his mind (Like the person in Thurber’s cartoon, Fig 1, left). He has to resort to many other conjectures and posits available to him to plug loopholes and prevent his, theoretical model from toppling all over him.  He hope by this complicated model he would eventually crown his feat with national recognition ... this usually requires positioning another scientist on the top balancing another hypothesis.  He does this until, his pet dog, say his mathematics, tickles his funny spot. The theory collapses, in good humour, and the next balancing act starts.

Its like a game of chess where the new scientific model is white, and black is the defending old model, Black’s stratetegies for defence takes all possible attacks on the model into account. Black thinks heis well fortified until he reaches a stage where any further defence move will only expose his king to a check-mate. This is a zugzwang in chess. Two problems from the internet of white mating in two moves (Fig 1, right) illustrates this state. In logic, I guess, zugzwang is the equivalent of reductio ab absurdum (“reduction to absurdity”  is a common form of argument which seeks to demonstrate that a statement is true by showing that a false, untenable, or absurd result follows from its denial; Wikipedia).

Models for Atom without Planck's constant

This blog is not required to inform the reader about the history of the Bohr Model. I have learnt from various articles available on the internet. These include Charles Baily’s Atomic Modeling in the Early 20th Century: 1904 – 1913 (referred to as Baily); Helge Kragh’s Before Bohr: theories of Atomic structure 1850-1913 (Kragh1) and The Early Reception of Bohr’s Atomic Theory (1913-1915)- A Preliminary Investigation (Kragh2) and The gallery of failed atomic models, 1903-1913 (http://skullsinthestars.com/2008/05/27/)  

Bohr’s model of the atom is the logo of the Atomic energy commission of the United States of America (Fig 2 left), “The persistence of the ubiquitous symbol reflects its ancestry: Bohr’s atomic theory imitates in part Newton’s planetary theory ....The “... curious blend of traditional and modern ideas that underlies Bohr’ theory has given it a remarkable vitality. The language and spirit of Bohr’s work have persisted in common thought long after the model  itself was commonly acknowledged (by Bohr among others) to have served its purpose. “

The logo of Kragh’s department (Fig 2, middle) at Aarhus University, Denmark, is meant to “combine scientific with the popular” which it does with an instantly recognizable depiction of what could be a Bohr-like atom asserting thereby the way the Bohr model is accepted by popular imagination. An interesting feature of the Aarhus logo is it combines another “popular imagination” fractal-like five-fold geometry. It reminds one of another popular five geometry obtained from Penrose-tiling (Fig 2, right). The evolution of logos follow the evolution in science..

Bohr’s July 1913 contribution saw an immediate (considering those days when all communication had to be by mail) national recognition in The Times newspaper in September 1913 when they referred “... to Jeans’s praise of the young Danish physicist and his new theory of the hydrogen spectrum...” The Bohr model is now thought to be terribly passé (out-of-date) because of its planetary two-dimensional nature, and most importantly, because of its failure to be applicable to the simplest molecule --- the hydrogen molecule. We will examine this aspect in the next blog. 

One sees virtues in the Bohr model in modern times, mainly because of its historical impact. As late as last year (Feb 2012), Jennifer Oulette, (who maintains a blog called Cocktail Party Physics in the  Scientific American) defended the Bohr model in her article “Don’t Be Dissin’ the Bohr Model!” because of its use as a tool to introduce the quantum model for the atoms:-

There’s ... shortcomings related to predictions about the spectra of larger atoms and the relative intensities of spectral lines, yadda, yadda, yadda, but we’re focusing on the most major objections for the sake of simplicity. ... To understand why physicists discarded the Bohr model, you’ve got to delve into the mind-bending intricacies of quantum mechanics, and explain all kinds of things the average person likely has never encountered in any real depth: wave functions, uncertainty, superposition of states, spectral lines, and so on. I prefer the Bohr model to introduce non-scientists to the basics of atomic structure

These comments do not have a gospel truth about them; they are meant to influence the very influential cocktail party crowd. To have a column on “Cocktail Party Physics” in the Scientific American is perhaps a sign of the times considering that there used to be a Martin Gardner (1914-2010) column --- among others --- on Mathematical Games (see Douglas Hopfstadter’s Martin Gardner: A Major Shaping Force in My Life, Scientific American, May 24 2010).

It is important to note that Bohr’s model for the hydrogen atom is  really a first long-lived stationary point of evolving models for the atom. Democritus’ atom as an indivisible aspect of matter which latter day Indian philosophers may have mistaken as the anu of adristam, (see Parts I and II of my blogs on God and God Particle) is quite different from the models of atom that came especially after the Thomson-Rutherford models. In Dalton’s atomic theory, atoms of different elements had nothing “substantial in common. The first indication that there is a commonality among the elements was perhaps due to William Prout around 1815 when he thought that all elements were made up of integral number of hydrogen atoms. This attempt was rejected primarily because of the observation of a definitely non-integral number of hydrogen atoms required to account for the atomic weight of chlorine.

When Bohr started his work on the structure of atoms there was not much interest in the physics community in the subject as a do-able problem. “Referring to the period about 1910, … for the average physicist of the time, speculations about atomic structure were something like speculations about life on Mars – very interesting for those who liked this kind of thing, but without much hope of support from convincing scientific evidence and without much bearing on scientific thought and development.”(Kragh1). It is probably the kind of work that will be equivalent in recent times to the work on teleportation in the context of H. G. Wells’ Time Machine.

Etched in the history of the Bohr model for the atom must, I suppose, have references to the men in Fig 3 --- Angström, Balmer, Thomson, Haas, Planck, Rutherford, Einstein and Bohr.(not necessarily in chronological order) and with some important exceptions such as Rydberg and Nagaoka (see later). The impact of Planck, Einstein and Bohr will be considered in the ext blog.

If one wants to go back to it all on the story of Bohr’s model, one must start perhaps with the prisms in fairs and the dispersion of light in rainbow colours. These prisms are associated with gypsies and their chandeliers. I think one can say with all fairness that one (even one of our know-all vedic scholars) does not know who discovered/invented the prism. One always remembers the elemental thrills when the dispersion of colour is seen from those simple triangulated glass objects. Yet, what a story these simple prisms had for science!!!

Fraunhopfer (early 1800s) is credited with the creation of the spectroscope and quantifying the solar spectrum as well as the positions of dark lines in the spectrum. The spectral lines were thought to be in chaotic positions and it was recognised that the solar spectrum had no heavenly single-god characteristic, Kirchoff and Bunsen (1850s) could identify these lines with elements. Anders Jonas Ångström was setting the foundations of spectroscopy from sparks and the sun from which he proved that the sun’s atmosphere is mainly hydrogen in 1862. He obtained and tabulated large number of spectral lines the inaccuracy of which was mainly due to the inaccuracy of the metre scale that he used for measurement. At that time such lines were of interest to astronomers who spent their time looking at the sky and worried about motions of stars and planets. It is therefore of little surprise that the model of the atom should evolve from the science of planets. The spectral lines of Angstorm (~1868) remained in Tables and charts for about twenty years. Little systematic was found although several attempts were made for the next twenty years. In particular, there was some hope of finding some regularity in the spectral lines of hydrogen.

It was at this time that Johann Balmer found the famous systematic that is now known as the Balmer formula for what is now known as the Balmer series of spectral lines in hydrogen. Balmer published this paper (the first of only two that he wrote) at the age of sixty, the age at which Angstorm passed away after spending a major part of his scientific life with spectral lines. Balmer took up mathematics for study and was a school teacher. He married late at 43 and between 43 and sixty he begat six children. When he finally found time from family-ing activities, he looked at the spectral lines of hydrogen and by looking at the position of four hydrogen lines in the visible spectrum he was able to conjecture up a formula. This formula correctly “predicted” another line in the near ultraviolet that he did not know existed.

The Balmer rule (1885) for the wavelength,  l  of the lines in the spectrum is

l_m = Bm²/(m² – 2²)

Where m > 2 is an integer and B is a constant. Balmer, it seems, did not give the method by which he got his formula in his first paper. He was able to look at a series of numbers and get a general formula for the series in a converse manner of Ramanujan who was able to look a series and get the limiting values.without really requiring proving it. Balmer could not get the correct positions for the lines of helium.  

Rydberg generalized (1888) Balmer’s formula to the with

1/l = R_H(1/2² – 1/n²) = (4/B)(1/2² – 1/m²)

where m is an integer with m = 3, 4, 5. This is a special case of the general expression (1889)

1/l = R_H(1/n² – 1/m²) (m > n  = 1, 2, .... m)

The lines for hydrogen has only n  = 2. The Lyman series for hydrogen lines discovered later (1906) has n  = 1, the Paschen series (1908) has n  = 3,  These three series were known to Bohr. The Brackett series (1922) and the Pfund series (1925) were n = 4 and n = 5, respectively corresponded to Rydberg’s formula, This is the formula that was first explained by Bohr.

Planetary Models and Saturn’s Rings

The early models for the atom did not attempt to explain the Balmer lines. Perhaps, the first planetary model is by Nagaoka who published a version in 1903 before publishing it in the Philosophical magazine in 1904. Nagaoka’s model drew heavily from James Clerk Maxwell’s treatment of Saturn’s rings.  Maxwell admits that he did this model purely for the love of it and not for any commercial reason.

But when we contemplate the Rings from a purely scientific point of view, they become the most remarkable bodies in the heavens, except, perhaps, those still less useful bodies—the spiral nebulae. When we have actually seen that great arch swung over the equator of the planet without any visible connexion, we cannot bring our minds to rest. We cannot simply admit that such is the case, and describe it as one of the observed facts in nature, not admitting or requiring explanation. We must either explain its motion on the principles of mechanics, or admit that, in the Saturnian realms, there can be motion regulated by laws which we are unable to explain.

The issue in Maxwell’s model (1856) is what would happen if a small planet collides with aother planet. The initial debris can coalesce together and form a Moon if the new debris/moon is outside a distance set by the Roche limit (1948), If it is within the Roche limit the gravitational tidal forces will disintegrate any moon bring formed under their own gravity. Inside the Roche limit, the dispersed orbiting material will form rings.

Maxwell’s model (1856) had dust particles under mutual gravitational interaction moving in closed orbits forced by Saturn’s gravitational field. The outer parts of Saturn’s rings orbited at a slower speed than the inner parts. This was proved by Doppler Shift measurements forty years later. The ring is actually made of millions of tiny moons orbiting Saturn and obeying Kepler’s third law (P²  proportional to a³, where P is the orbital period of the planet and a is the semi-major axis of the orbit). It is of interest to note that a poetic right-brain mind foresaw (1660) the presence of million’s of moons in Saturn’s rings before Maxwell proved them with his left-brain analytic mind. The poet was Jean Chapelain, a friend of Christian Huygens who first proposed (1655) that there is a ring surrounding Saturn after Galileo first observed (1610) first observed Saturn’s ring-like feature.

Although Maxwell did not look for an application of his model for Saturn’s rings his concept was new. I learnt from Kragh1 that the idea of atoms being structured like a planetary system was considered by Wilhelm Weber in the 1860-70s before electrons as negatively charged particles were considered.Weber’s credentials in electrical forces were solid by that time having unified electrostatics (Coulomb’s force, 1785) with electrodynamics (Amperes force, 1826)) with Faraday’s induction (1831) and having first considered the value, c, Weber, a close collaborator of Gauss, was expelled earlier from the university of Gottingen by Ernest Augustus I, Duke of Cumberland, along with Gauss and two Grimm brothers as part of the seven Gottigen Professors who refused to sign an oath In carrying the King's patent into effect, Weber proposed an electrical theory in which neutral ether consisted of positive and negative particles orbiting each other. For the atom, Weber had a massive negatively charged central atom with tiny positively charged particles revolving around them.

Maxwell’s model also perhaps prompted Jean Perrin in 1901 (Nobel laureate for Brownian motion, 1926) in a popular talk to think of a Saturn-like model in which atoms are “constituted” by “one or more” masses which are “very strongly charged with positive electricity” as well as  “by a multitude of corpuscles, in the manner of small negative planets” with “the total negative charge exactly equivalent to the total positive charge, in such a way that the atom is electrically neutral. (from Kragh1) .

Planetary Models for Atoms

Nagaoka’s model (Fig 4, left; these illustrations are taken from the internet and do not appear in the original publications) showed considerable insights into orbits of charged particles. It is perhaps better to quote the model from his paper He first describes his modification Maxwell’s model as consisting (from his abstract and quoted several times in the net) “ ... of a large number of particles of equal mass arranged in a circle at equal angular intervals and repelling each other with forces inversely proportional to the square of distance; at the centre of the circle, place a particle of large mass attracting the other particles according to the same law of force. If these repelling particles be revolving with nearly the same velocity about the attracting centre, the system will generally remain stable, for small disturbances, provided the attracting force be sufficiently great.” He then adds his model: “The present case will evidently be approximately realized if we replace these satellites by negative electrons and the attracting centre by a positively charged particle...” Nagaoka sought t make contact with experiments when he added in his abstract “Such an ideal atom will not be contradictory to the results of recent experiments on kathode rays, radioactivity, and other allied phenomena. Nagaoka abandoned his model four years later when it was impressed upon him that a ring of negative charges would be unstable due to “disruptive repulsion” of the electrons. Unlike Maxwell, Nagaoka had no Roche limit to stimulate thought on the stability of charges in a ring.

Thomson’s model has Nagoka’s planar ingredients:  “... a sphere of uniform positive electrification, and inside this sphere a number of corpuscles arranged in a series of parallel rings ... “. He also considers a three-dimensional partitioning in which the rings are “...so arranged that those which contain a large number of corpuscles are near the surface of the sphere while those in which there are a smaller number of corpuscles are more in the inside. ... they will arrange themselves in a series of concentric shells ... .” Thomson also addressed himself to a phenomenology that yields some of the characteristics of spectral lines thus: “... if all the corpuscles in one atom have the same angular velocity, the frequency of the vibrations produced by the rotation of the ring of corpuscles is proportional to the number of corpuscles in the ring; and thus in the spectrum of each element in the series there would be a series of frequencies bearing the same ratio to each other...”. Thomson’s model for the corpuscles essentially involves their relative positioning being static inside the ring.

Removing a negatively charged electron would leave an opposite charge behind, so that for the sake of neutrality one could consider each electron being bound to a positive charge (Fig 4, second from right), This is Lenard’s model. Lenard proposed that in an atom of a  heavy substance (he chose platinum) the “most impenetrable proper volume” of the atom would have a size which is about one-thousandth of the actual atom ... while the remainder is “as empty as the sky”. This empty space has “only fields of force” as in free ether. Lenard then proposed the existence of “electrically neutral” dynamids  as “the centres of their fields”  possessing same amounts of negative and positive electricity .  The negative electricity he identifies with that found in cathode rays while the “positive electricity, on the other hand, appears to be something much more specifically proper to the atoms of matter.”

Thomson’s so-called plum pudding model has incorporated in it the idea that an atom’s positive mass is spread out diffusely. Rutherford was fortunate to have skilled assistants such as Geiger and Marsden who set up all the instrumentation for studying alpha-particle scattering. Indeed, it seems, it was Marsden’s first attempts that led to the observation that a few of the alpha particles  did not pass through a gold foil (as expected) without changing angles (without scattering) but a few of them bounced back in an opposite direction as if they had struck a solid wall!

In his 1911 paper Rutherford gives credit to Geiger and Marsden. He has noted:

The observations, however, of Geiger and Marsden on the scattering of  a rays indicate that some of the a particles must suffer a deflexion of more than a right angle at a single encounter. They found, for example, that a small fraction of the incident alpha particles, about 1 in 20,000, were turned through an average angle of 90^o in passing through a layer of gold-foil about "00001/4 cm. thick, ... It seems reasonable to suppose that the deflexion through a large angle is due to a single atomic encounter ... The theory of Sir J. J. Thomson is based on the assumption that the scattering due to a single atomic encounter is small, and the particular structure assumed for the atom does not admit of a very large deflexion of an a particle in traversing a single atom, unless it be supposed that the diameter of the sphere of positive electricity is minute compared with the diameter or' the sphere of influence of the atom.”

It may be supposed that at the time Rutherford made his model, he must have been immersed in the best traditions of science and he did not propose any further than he could prove. He was therefore cautious to a fault (by present-day standards) when he writes

the deductions from the theory are independent of whether the central charge is supposed to be positive or negative. For convenience, the sign will be assumed to be positive. The question of the stability of the atom proposed need not be considered at this stage, for this will obviously depend upon the minute structure of the atom, and on the motion of the constituent charged parts. ... Considering the evidence as a whole, it seems simplest to suppose that the atom contains a central charge distributed through a very small volume, and that the large single deflexions are due to the central charge as a whole, and not to its constituents. At the same time, the experimental evidence is not precise enough to negative the possibility that a small fraction of the positive charge may be carried by satellites extending some distance from the centre.

Rutherford gives adequate credit to Nagaoka’s Saturnian model when he noted “From

the point of view considered in this paper, the chance of large deflexion would practically be unaltered, whether the atom is considered to be a disk or a sphere.” He further modestly adds

... it has not so far been found possible to obtain definite evidence to determine whether it (the central charge) be positive or negative. It may be possible to settle the question of sign by consideration of the difference of the laws of absorption of the b particle to be expected on the two hypotheses, for the effect of radiation in reducing the velocity of the b particle should be far more marked with a positive than with a negative centre

Rutherford’s 1911 paper spent considerable time on the experiments of Crowther with b rays that seemed to support Thomson;s model. In private correspondence with Bragg, Rutherford writes (fom Mansoor Niaz’s article, From Cathodes Rays to Alpha Particles to Quantum of Action...)  “...I have looked into Crowther’s scattering paper carefully, and the more I examine it the more I marvel at the way he made it fit (or thought he made it fit) J. J.’s theory . . . I am quite sure the numbers of the earlier part of the curve [Crowther’s] were fudged.”

To be fair, most of Rutherford’s paper was really reporting of the results of Geiger and Marsden. It does not look like a bold paper with assertions of a dramatic picture. In a 2011 article in the CERN courier Campbell writes that after his 7^th May 2011 lecture on the model of the atom, “... it created no great stir among scientists and the public at the time. Three nights after his announcement, Rutherford addressed the Society of Industrial Chemists on ‘Radium’. The nuclear atom was not mentioned by Sir William Ramsay in his opening address to that year’s meeting of the British Association ... Rutherford’s busy life continued as normal: accepting a Corresponding Membership of the Munich Academy of Sciences; giving talks on all manner of subjects but the nuclear atom --- motoring in the car recently purchased with the money that had accompanied his Nobel prize ...”

Is there a Roche-like limit in the Saturn model for the atom?

Although the Saturnian model played such an important role in the planetary model for the atom, I have not seen any mention of the relevance of  Roche-like limit for stabilizing an electron-ring, say, in Thomson's model. While perusing the internet I came across Fedele’s article “From Maxwell's theory of Saturn's rings to the negative mass instability” (Phil. Trans. R. Soc. A, vol. 366 pp 1717-1733 (2008)). I am not proficient at all in this area (as in many others) but it does not deter me from trying even if it may look like a cut-and-paste effort no matter how much I try. Fedele was writing about a mechanism by Nielsen and co-workers that was put forward a hundred after Maxwell. . There is an analogy between Saturn’s rings and the rings of particle accelerators in which the particles are under the action of electromagnetic forces (instead of the gravitational field of the planet). In this mechanism particles repel each other instead of attracting each other as in Maxwell’s mechanism.

Let us consider a relativistic charged particle beam travelling with velocity close to that of light in a circular accelerator. The synchronous particle does not change its energy or the radius of its circular orbit Because of repulsion between the particles one may think that the beam would be unstable in a ring.  Bunching of charged particles can also take place for charged particles undergoing circular motion in a particle accelerator. When particles are accelerated towards the speed of light they increase their momentum by having a larger radius of curvature. When a particle gains energy because of repulsion it could its rotation rate. Particles leading the bunch in energy therefore fall back into the bunch and the bunch is spontaneously retained. There is a transition energy (akin perhaps to a Roche limit) above which normally repelling particles begin attracting each other through what has been termed a negative mass effect since the particles are drawn in the opposite direction towards which the total force is acting, Maxwell’s gravitational mechanism for the formation of Saturn’s rings is the gravitational analogue for the bunching of charges. 

One wonders what Nagaoka/Thomson would have thought about their models if they knew about the particle accelerators and (what I think) the Roche-limit-like transition energy.

One of my most memorable technological events is the Voyager I mission which sent back pictures from distant planets after travelling several years and millions of miles. It has 1977 technology in it. Voyager I fires up my admiration like nothing else can about the spirit of man as an inquisitive and accurate experimentalist.  It is now at the edge of the solar system along a “magnetic highway” being powered by a plutonium battery. It is still being monitored and it is still sending data and still surprising scientists. . As its disrector for the last thirty five years or so, Edward Stone, has said. “The new region isn't what we expected, but we've come to expect the unexpected from Voyager.”The planetary community has learnt so much from Voyagers and the subsequent missions. The legend of Voyager and its service to mankind must be rememebered in some way; say, by the Nobel peace prize that will restore to it some of the pride after the recent disastrous awards.

Among my 1980 memories are the pictures (Fig 5, from internet) sent by Voyagers about Saturn’s outermost discrete F-ring. It is not important to record the chronology; what I remember is the wave-like nature of the rings and it reminding me of de Broglie waves for the atom’s rings. Later Cassini images (Fig 5 inset) and analyses  showed the role of the moons. A video that is taken in 2005 which I like is at http://www.youtube.com/watch?v=fdUlpeUFfxI . Prometheus and Pandora play an important role not only in holding together (shepherding) these rings in their place but also in making these rings the “most active in the Solar system”. Prometheus orbits Saturn faster than the material in the F-ring. The attractive gravitational influence of Prometheus carves out new periodic channels in the ring when it borrows some of the material. It then gives it back. This exchange of mass gives it the equivalent of a mass resonance”

I cannot help imagining that if I take the outermost F-ring of Saturn as its “valence” ring that is most sensitive to external influences, the quantum models for atoms could have taken a different shape.

Models are built by observations. The model lies in the experience of the observer. I am reminded of another Thurber cartoon.