One Hundred Years of Bohr Model

GENERAL  ARTICLE One Hundred Years of Bohr Model

Avinash Khare

In this article I shall present a brief review of the hundred-year young Bohr model of the atom. In particular, I will ¯rst introduce the Thomson and the Rutherford models of atoms, their shortcomings and then discuss in some detail the development of the atomic model by Niels Bohr. Fur- ther, I will mention its re¯nements at the hand of Sommerfeld and also its shortcomings. Finally, I Avinash Khare is Raja will discuss the implication of this model in the Ramanna Fellow at IISER, Pune. His current interests development of quantum mechanics. are in the areas of low The `Bohr atom' has just completed one hundred years dimensional field theory, nonlinear dynamics and and it is worth recalling how it emerged, its salient fea- supersymmetric quantum tures, its shortcomings as well as the role it played in mechanics. Besides, he is the development of quantum mechanics. passionate about teaching an well as popularizing Thomson Model of Atoms science at school and college level. Till 1896, the popular view was that the atom was the basic constituent of matter. The ¯rst important clue regarding the internal structure of atoms came with the discovery of spontaneous radiation, ¯rst identi¯ed by Becquerel in 1896. The very existence of atomic radiation strongly suggested that atoms were not indivisible. With the discovery of the electron in 1897, J J Thomson was convinced that electrons must be fundamental con- stituents of matter and this led to his corpuscular theory of matter. This model was popularly known as `Plum Pudding Model'. In this model, the positive charge of the atom was assumed to be spread throughout the atom forming a kind of pudding in which negatively charged Keywords electrons were suspended like plums. Thomson showed Plum pudding model, Rutherford atom, Balmer series, Rydberg that his model had an amazing explanatory power for formula, star  Puppis, charac- the observed periodicity in the elements. Thomson later teristic X-rays, Bohr–Sommerfeld applied a modi¯ed version of this model to a variety of approach.

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phenomena such as dispersion of light by dilute gases and developed methods for estimating the actual number of electrons in an atom. By 1910, experiments had con¯rmed many of its predictions for the absorption and scattering of electrons in matter. However, this model was not suitable for predicting spectral lines which had already been seen by spectroscopists.

Figure 1. Thomson’s model Rutherford Model of Atoms of atom. In this ‘plum pudding’ model of atom devel- Rutherford wanted to test this Thomson picture of the oped byThomson and Kelvin atom. On his advice, Geiger and Marsden carried out in 1094, the electrons(plums) a series of experiments. Using their data, Rutherford are embedded in a sphere of in 1911 provided conclusive proof of the inadequacy of uniform positive charge (pud- the Thomson model. In these experiments, a collimated ding). 4 beam of alpha particles (i.e., He2 nucleus) from a ra- dium source strike a thin gold foil. To their surprise, they found that few alpha particles were even scattered at large angles. This would not be expected if the alpha particle had hit a much lighter particle like the electron. Rutherford argued that these experiments clearly showed that instead of being spread throughout the atom, the positive charge is in fact concentrated in a very small region at the center of the atom. This was one of the most important developments in atomic physics and was Figure 2. Rutherford’smodel the foundation of the subject of Nuclear Physics. We go of atom. In this model, the through the Rutherford argument here. atom consisted of a positively charged nucleus surrounded Let us suppose that a particle of mass M and velocity by negatively charged elec- v, hits a particle of mass m which is at rest. After the trons. collision, let us suppose that the particle of mass M continues along the same line with velocity v0, giving the target particle (with mass m) a velocity u (note, we are using the notation in which positive velocity is in the same direction as the incident particle of mass M, while a negative velocity will be in the opposite direction). Then the energy and momentum conservation equations are

1 2 1 2 2 Mv = mu + Mv0 ; Mv = (Mv0 + mu ) : (1) 2 2

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On eliminating u, we have a quadratic equation for v0 in terms of v

2 2 2 m(v v0 ) = M(v v0) ; (2) ¡ ¡ which has two solutions: either v0 = v or m M v0 = v ¡ : (3) ¡ µm + M ¶

The ¯rst solution v0 = v; u = 0 is a trivial solution. The interesting solution is the second one given by (3). It says that v0 can be negative (i.e., scattered particle recoils backwards) only if m > M (similarly, somewhat weaker limits on m can be inferred from scattering at any large angle). Thus it was clear to Rutherford that the experimental results of Geiger and Marsden could not be explained in terms of multiple encounters with a positively charged sphere of atomic dimensions, as was Thomson's view. The fact that a few alpha particles were observed to be scattered at large angles clearly showed that the alpha particles must be hitting something in the gold atom which is much heavier than the electron. As Rutherford later explained, \it was quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you ¯red a 15-inch shell at a piece of paper, and it comes back and hits you". Hence, Ruther- Rutherford ford concluded that the positive charge of the atom is concluded that the concentrated in a small heavy nucleus at the center of positive charge of the atom, around which the much lighter, negatively atom is concen- charged electrons circulate in orbits, like planets around trated in a small the sun. This is why the Rutherford nuclear model is heavy nucleus at often referred to as the planetary model of the atom. the centre of the This model, despite being successful, had one major atom, around which problem. It predicted that even light atoms like hy- the much lighter, drogen were unstable. The point was, according to the negatively charged classical electromagnetic theory, an electron revolving electrons circulate around a nucleus will radiate electromagnetic waves and in orbits.

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hence will deplete the electron's energy and it will eventually spiral inwards towards the nucleus. Thus an atom would rapidly collapse to nuclear dimensions (the col- 12 lapse time can be computed to be of order 10¡ sec!). Further, the continuous spectrum of the radiation that would be emitted in this process was not in agreement with the observed line spectrum. Bohr Model In the autumn of 1911, Bohr went to England for his post-doctoral research. He had already done interesting work on the electron theory of metals during his PhD thesis { it was so advanced that no one in Den- mark could evaluate it fully. Bohr ¯rst went to Cam- bridge University and worked for about a year with J J Thomson before being invited by Rutherford to work with him at University of Manchester. Using the data from the absorption of alpha-rays, and using Ruther- ford's model, Bohr showed that the hydrogen atom has only one electron outside the positively charged nucleus while the helium atom has two electrons outside the positively charged nucleus. It is worth noting here that till 1912, physicists were not sure about the number of electrons in the helium atom or even in the hydrogen atom. All this time, the problem which was really troubling Bohr was, however, the stability of the Rutherford atom and ¯nally he came up with a simple model of atomic structure. A key feature of this very successful model, proposed by Bohr in 1913, was the prediction of the line spectrum of radiation by atoms. So we shall digress here and describe what was known experimentally about A key feature of the atomic spectra at that time. Bohr model was the By 1900, the amount of information available about prediction of the atomic spectra was enormous. Spectroscopists had no- experimentally ticed that an atom can only absorb certain energies of observed line light (the absorption spectra) and once excited can only spectrum by atoms. release certain energies (the emission spectra), and these

888 RESONANCE October 2013 GENERAL  ARTICLE energies happen to be the same. Further, the spectra coming from di®erent atoms showed that each atom has its own characteristic spectrum, i.e., a characteristic set of wavelengths at which the lines of the spectrum are found. Amongst all the atoms, the spectrum of hydrogen is relatively simple. Since most of the universe consists of isolated hydrogen atoms, the hydrogen atom spectrum is of considerable importance. It was found that the hydrogen atom spectrum had a great regular- ity. This tempted several people to look for an empirical formula which would represent the wavelengths of the lines. Such a formula was discovered by Balmer, a Swiss school teacher, in 1885. He found the simple relation

n2 ¸ = 3646 ; n = 3; 4; 5; ::: ; (4) n2 4 ¡ where ¸ is the wavelength. Using this formula, he was able to predict the wavelengths of the ¯rst nine lines of the series to better than one part in 1000. This discovery initiated a search for similar empirical formulas that would apply to other series. Most of this work around 1890 was done by Rydberg, who found it convenient to deal with the reciprocals of the wavelengths of the lines, instead of their wavelengths. In terms of recip- rocal wavelength º, the Balmer formula can be written as 1 1 1 º = = RH( ) ; n = 3; 4; 5; ::: (5) ¸ 22 ¡ n2 where RH is the so-called Rydberg constant for hydrogen. It might be noted here that Balmer had already accurately calculated RH to one part in 10000. Balmer had In 1913, in 3 seminal papers [1] Bohr, who was then already accurately just 27 years old, presented his model for the atom and calculated the was successful in accurately explaining some of the spec- Rydberg constant troscopy data. Bohr's model can be said to be based on to one part in the following four postulates. 1000.

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The four postulates 1. Atomic electrons move in circular orbits about a of Bohr are an massive nucleus under the in°uence of the Coulomb unusual mixture of attraction between the electron and the nucleus, classical and obeying the laws of classical mechanics. nonclassical physics. 2. Instead of the in¯nity of orbits which would be possible in classical mechanics, an electron can in fact only move in an orbit for which its angular momentum L is quantized, i.e., it is an integral h multiple of ¹h = 2¼ , h being the Planck constant. 3. Even though it is constantly accelerating, an electron moving in such an allowed orbit does not radiate electromagnetic energy. 4. Electrons can only gain or lose energy by jump- ing from one allowed orbit to another, absorb- ing or emitting electromagnetic radiation with a frequency º, determined by the energy di®erence of the levels according to the Planck's relation ¢E = Ef Ei = hº. ¡ The ¯rst postulate of Bohr's model is based on the existence of the atomic nucleus. The second postulate in- troduces quantization. On the other hand, the third postulate removes the problem of stability of an electron moving in a circular orbit, due to the emission of electromagnetic radiation required of the electron by classical theory, by simply postulating that this feature of the classical theory is not valid for the case of an atomic electron. The fourth postulate is just Einstein's postulate that the frequency of a photon of electromagnetic radiation is equal to the energy carried by the photon divided by Planck's constant [2]. Clearly, these four postulates are an unusual mixture of classical and nonclassical physics. The electron moving in a circular orbit is assumed to obey classical physics, and yet the nonclassical idea of quantization of angular momentum is included. Further, while the electron

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is assumed to obey Coulomb's law of classical electro- Bohr was clear magnetic theory, yet it is assumed that it does not obey from the beginning the other feature of emission of radiation by an accel- that his model was erated charged particle. The postulate about transition in conflict with the between allowed orbits was in obvious con°ict with clas- classical theory. sical electrodynamics, but to Bohr it appeared to be necessary in order to account for the experimental data. Of course, Bohr was clear from the beginning that his model is in con°ict with the classical theory. That is why, he had already said that no attempt will be made at a mechanical formulation as it seems hopeless. I think this clearly shows Bohr's ability to entertain several con- °icting ideas. Blazing courage is one thing but tolerance of ambiguities is quite another. Predictions of Bohr Model Let us now derive the predictions that Bohr obtained using these postulates. Consider an atom with nuclear charge Ze and mass M and a single electron with mass m and charge e. Note, for hydrogen Z = 1. Following Bohr, we initia¡lly assume that the mass of the electron is completely negligible compared to the mass of the nucleus and consequently that the nucleus remains ¯xed in space. The condition of the mechanical stability of the electron (following from classical mechanics) comes by balancing the Coulomb force acting on the electron and the centripetal acceleration keeping the electron in a circular orbit, i.e., Ze2 mv2 = ; (6) r2 r where v is the speed of the electron in its orbit of radius r. Now applying the quantization condition on the angular momentum of the electron, L = mvr, we have Blazing courage is one thing but mvr = n¹h ; n = 1; 2; 3; ::: (7) tolerance of Using (6) and (7), we can solve for the radius of the ambiguities is circular orbit and the velocity of the electron in this quite another.

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Box 1. orbit. We get In the Rydberg formula, the n2h¹2 n¹h Ze2 r = ; v = = : (8) transitions to the ground mZe2 mr n¹h n n state (i.e., i = 1, f = 2, 3, 4, ...) are known as the The kinetic energy of the electron is then Lyman series. All lines in 1 Ze2 this series are in ultraviolet K:E: = mv2 = ; (9) region with wavelengths 2 2r ranging from 1216 to 912 Angstroms. On the other which is half of the potential energy, and hence the total hand, the transitions to the energy E is given by first excited state (i.e., n = i Ze2 mZ2e4 2, n = 3, 4, 5, ...) consti- f E = K:E: + P:E: = = 2 2 : (10) tute the Balmer series. Four ¡ 2r ¡ 2n h¹ of these lines are in the We thus see that the quantization of the angular mo- visible region with wave- mentum of the electron leads to a quantization of its to- lengths ranging from 6562 tal energy. Hence the frequency of the electromagnetic to 4101 Angstroms while radiation emitted when the electron makes a transition the other lines are in the ultraviolet region. Transi- from the quantum state nf to a state ni is given by tions to the second excited 2 1 1 state (i.e., n = 3, n = 4, 5, º = R Z ( 2 2 ) ; (11) i f 1 n ¡ n 6,...) constitute thePaschen i f series and all the wave- where ni; nf are integers with nf > ni and lengths are in the infrared region. me4 R = 3 : (12) 1 4¼¹h c

Equation (11) is the famous Rydberg formula obtained by him in 1890 (see Box 1). The essential predictions of the Bohr model are contained in (10) and (11). It is Bohr was able to worth pointing out that Bohr in fact derived this spec- derive the famous trum in three di®erent ways in his classic 1913 paper. Rydberg formula Further, in a footnote, Bohr mentioned that his results and hence the might also be obtained if one assumes that the line in- Balmer formula tegral p dl is an integral multiple of Planck's constant, which is a special i.e., R case of the pdl = nh ; (13) Rydberg formula. Z

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where p is the electron's momentum and dl is an element In one of his papers of length. Bohr had also put forward the It is interesting to note here that in his ¯rst paper, Bohr had also put forward the celebrated `Correspondence celebrated Principle' which states that the laws of quantum physics Correspondence must reduce to those of classical physics when quantum ‘Principle’ and using numbers such as n, as de¯ned above, become large. In it was able to derive fact one of the derivations given by Bohr for (10) and the famous Bohr (13) was by using this principle. The incredible thing formula (10). was that while the proof was given for large n, the ¯nal result as given by (10) and (13) is claimed to be true for any value of n! What luck that it could work!

For hydrogen (Z = 1), in case ni = 2, Bohr immediately recovered the Balmer formula provided R = RH. 1 Bohr evaluated R by using (12) and found that the resulting value wa1s in quite good agreement with the experimental value of RH. Later on, when Bohr made a correction for ¯nite nuclear mass, i.e., substituted m by mM the reduced mass ¹ = M+m , he found that the theoreti- cal and experimental values of RH agree to within three parts in 100,000! Another success of the Bohr model was about star ³ Puppis. Before Bohr, it had been wrongly interpreted as a new series of lines of hydrogen. It was another triumph for the Bohr model that it could explain these lines as those belonging to the spectra of ionized Helium. It may be noted here that till that time the spectral lines of ionized helium had not yet been observed in the laboratory. But as soon as this was done, the Bohr model was regarded as a great success. Yet another triumph was the explanation due to Mosley about the characteristic K lines of X-rays using Bohr's theory. ® Bohr successfully I might add here that while the most compelling of explained the Bohr's results was the derivation of the Balmer formula spectral lines from of 1885, Bohr claimed throughout his life that he was the star  Puppis unaware of the formula until he was already well along as due to ionized in the development of his theory. This appears rather Helium.

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Bohr was awarded strange given the fact that Balmer's work was exten- the Nobel Prize in sively discussed during major international physics con- Physics in 1922. ferences in 1890 as well as in subsequent years. How was the Bohr model received by the physics com- munity? Sommerfeld immediately wrote a letter to Bohr, complimenting him for calculating the empirical Ryd- berg constant in terms of the more fundamental con- stants, though he was skeptical about the atomic model in general. Once there was an explanation of the spectrum of star ³ Puppis, Bohr's theory received wide at- tention. Einstein too immediately recognized the importance of Bohr's theory saying it was a major development. It is fair to say that by and large people were unhappy with the two postulates of the Bohr model but were very impressed with its unprecedented success, which eventually led to its widespread adoption. And of course the crowning glory came when, for this work, Bohr was awarded the Nobel Prize in Physics in 1922. Bohr{Sommerfeld Approach The Bohr model was developed for circular orbits, but just as in the solar system, the generic orbit of a particle in a Coulomb ¯eld is not a circle but an ellipse. A gen- eralization of the Bohr quantization condition (7) was proposed by Sommerfeld and used to calculate energies of electrons in elliptical orbits. In particular, in addi- tion to the Bohr quantization of the azimuthal motion (angular momentum) of an electron around the nucleus, Sommerfeld quantized phase integrals for the radial motion (allowing for elliptical orbits), and the orientation of the orbital plane (spatial quantization). He replaced Sommerfeld equation (7) of Bohr by a system of three conditions generalized the Bohr p dr = n1h ; p dÁ = n2h ; p dµ = n3h : (14) quantization condition r Á µ Z Z Z and calculated the In this way, Sommerfeld made signi¯cant progress over energies of electrons the Bohr model. Further, Sommerfeld removed the de- in elliptical orbits. generacy in the hydrogen atom spectrum by treating

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the problem relativistically thereby explaining the ¯ne Box 2. structure observed experimentally (see Box 2). Note that the Bohr model was unable to explain this ¯ne Whereas the ground state structure observed in the spectrum of the hydrogen atom. energy of the hydrogen Further, Sommerfeld's approach could explain the Stark atom is 13.6 eV, the fine splitting between the 2P and the Zeeman e®ects in hydrogen. After reading these 3/2 and 2P states dueto spin– papers, Bohr wrote a letter to Sommerfeld saying \I do 1/2 orbit interaction is 4.5 not believe ever to have read anything with more joy  10–5 eV. On theother hand than your beautiful work". the Lamb shift (occurring What were the eventual failures of the Bohr{Sommerfeld mainly due to continual approach? While this approach was reasonably suc- emission and reabsorption cessful for atoms with one valence electron, it failed to of photons by the electron) between the 2S and 2P explain much of the spectra of atoms containing more 1/2 1/2 states is 4.35 10–6 eV. than one electron. Even for the hydrogen atom, the  Finally, the hyperfine split- Bohr model gives incorrect value for the orbital angu- ting (occurring due to the lar momentum of the ground state. Broadly speaking interaction between elec- the Bohr{Sommerfeld approach was fundamentally in- tron and proton spin) be- consistent and led to many paradoxes. The framework S S tween the 1 3/2 and 1 1/2 they proposed, a classical description of atoms to which states is 5.87  10–6 eV. quantization rules were added, was ¯nally rendered un- The radiative transition tenable. I might add here that Bohr himself had realized between these two states is that his model was not the ¯nal answer and he believed the famous 21 centimeter that a deeper revision of physics was required. line in the radio spectrum of hydrogen. Bohr Model and Development of Quantum Me- chanics Finally, what role did the Bohr atom play in the eventual development of quantum mechanics? It is fair to say that Bohr's model of the atom is simple, elegant, revolutionary but rather preliminary. It was undoubt- edly an essential step towards a correct theory of the Since Bohr atom atomic spectra. Since the Bohr model could well ex- could successfully plain the spectra of atoms with one valence electron, it explain the spectra had a domain of applicability. So it could not be entirely of atoms with one wrong and would have to correspond in some way with valence electron, it another, possibly more successful theory. It is fair to could not be say that the Bohr model has a special place in history, completely wrong.

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Bohr model has a a bridge between the older classical thinking and the special place in newer quantum thinking. It was a bridge over a chasm history, a bridge but one that collapsed after it was crossed. One may between the older wonder, how it could hold up so long and reach so far. classical thinking The reason probably is that with our present knowledge and the newer of quantum mechanics, we can look across the chasm quantum thinking. It from the other side and see that the semi-classical sit- uation works for two problems one of which is the hy- was important for drogen atom! The Bohr model was pathbreaking for being a tangible physics because it marked the transition between clas- break from the sical and quantum thinking. The model, though could priciples of classical not evolve continuously to modern quantum mechan- mechanics. ics. The Bohr model was important for being a tangible break from the principles of classical mechanics which were useless at explaining the quantum mechanical ef- fects in the atoms. Thanks to the Bohr model, physicists recognized this and insisted on building on what they had [3].

Address for Correspondence Suggested Reading Avinash Khare Raja Ramanna Fellow [1] N Bohr, Philos. Mag. Vol.26, pp.1–25, pp.476–50, pp.857–875, 1913. Indian Institute of Science [2] There are several simple accounts of the Bohr model; see for example, Education and Research R Eisberg and R Resnick, Quantum Physics of Atoms, Molecules, Solids, (IISER) Nuclei and Particles, Chapter 4, Wiley India, 2006. Pune 411 008, India. [3] There are some recent books exclusively discussing Bohr model of atom; Email:[email protected] see for example, H Kragh, Niels Bohr and the Quantum Atom: The Bohr Model of Atomic Structure, pp.1913–1925, Oxford Univ. Press, 2012.

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