In atomic physics, the Bohr model or Rutherford–Bohr model is an obsolete model of the atom that incorporated some early quantum concepts. Developed from 1911 to 1918 by Niels Bohr and building on Ernest Rutherford's discovery of the atom's nucleus, it supplanted the plum pudding model of J. J. Thomson only to be replaced by the quantum atomic model in the 1920s. It consists of a small, dense atomic nucleus surrounded by orbiting electrons. It is analogous to the structure of the Solar System, but with attraction provided by electrostatic force rather than gravity, and with the electron energies quantized (assuming only discrete values). In the history of atomic physics, it followed and ultimately replaced several earlier models, including Joseph Larmor's Solar System model (1897), Jean Perrin's model (1901), the cubical model (1902), Hantaro Nagaoka's Saturnian model (1904), the plum pudding model (1904), Arthur Haas's quantum model (1910), the Rutherford model (1911), and John William Nicholson's nuclear quantum model (1912). The improvement over the 1911 Rutherford model mainly concerned the new quantum mechanical interpretation introduced by Haas and Nicholson, but forsaking any attempt to explain radiation according to classical physics. The model's key success lies in explaining the Rydberg formula for hydrogen's spectral emission lines. While the Rydberg formula had been known experimentally, it did not gain a theoretical basis until the Bohr model was introduced. Not only did the Bohr model explain the reasons for the structure of the Rydberg formula, it also provided a justification for the fundamental physical constants that make up the formula's empirical results. The Bohr model is a relatively primitive model of the hydrogen atom, compared to the valence shell model. As a theory, it can be derived as a first-order approximation of the hydrogen atom using the broader and much more accurate quantum mechanics and thus may be considered to be an obsolete scientific theory. However, because of its simplicity, and its correct results for selected systems (see below for application), the Bohr model is still commonly taught to introduce students to quantum mechanics or energy level diagrams before moving on to the more accurate, but more complex, valence shell atom. A related quantum model was proposed by Arthur Erich Haas in 1910 but was rejected until the 1911 Solvay Congress where it was thoroughly discussed. The quantum theory of the period between Planck's discovery of the quantum (1900) and the advent of a mature quantum mechanics (1925) is often referred to as the old quantum theory.
Background
Until the second decade of the 20th century, atomic models were generally speculative. Even the concept of atoms, let alone atoms with internal structure, faced opposition from some scientists.
Planetary models In the late 1800s speculations on the possible structure of the atom included planetary models with orbiting charged electrons. These models faced a significant constraint. In 1897, Joseph Larmor showed that an accelerating charge would radiate power according to classical electrodynamics, a result known as the Larmor formula. Since electrons forced to remain in orbit are continuously accelerating, they would be mechanically unstable. Larmor noted that the electromagnetic effects of multiple electrons, suitably arranged, would cancel each other. Thus subsequent atomic models based on classical electrodynamics needed to adopt such special multi-electron arrangements.
Thomson's atom model
When Bohr began his work on a new atomic theory in the summer of 1912 the atomic model proposed by J. J. Thomson, now known as the plum pudding model, was the best available. Thomson proposed a model with electrons rotating in coplanar rings within an atomic-sized, positively-charged, spherical volume. Thomson showed that this model was mechanically stable by lengthy calculations and was electrodynamically stable under his original assumption of thousands of electrons per atom. Moreover, he suggested that the particularly stable configurations of electrons in rings was connected to chemical properties of the atoms. He developed a formula for the scattering of beta particles that seemed to match experimental results. However Thomson himself later showed that the atom had a factor of a thousand fewer electrons, challenging the stability argument and forcing the poorly understood positive sphere to have most of the atom's mass. Thomson was also unable to explain the many lines in atomic spectra.
Rutherford nuclear model
In 1908, Hans Geiger and Ernest Marsden demonstrated that alpha particles occasionally scatter at large angles, a result inconsistent with Thomson's model. In 1911 Ernest Rutherford developed a new scattering model, showing that the observed large angle scattering could be explained by a compact, highly charged mass at the center of the atom. Rutherford scattering did not involve the electrons and thus his model of the atom was incomplete. Bohr begins his first paper on his atomic model by describing Rutherford's atom as consisting of a small, dense, positively charged nucleus attracting negatively charged electrons.
Atomic spectra By the early twentieth century, it was expected that a successful model of the atom should account for the many atomic spectral lines. These lines were summarized in empirical formulas by Johann Balmer and Johannes Rydberg. In 1897, Lord Rayleigh showed that vibrations of electrical systems predicted spectral lines that depend on the square of the vibrational frequency, contradicting the empirical formula which depended directly on the frequency. In 1907 Arthur W. Conway showed that, rather than the entire atom vibrating, vibrations of only one of the electrons in the system described by Thomson might be sufficient to account for spectral series. Although Bohr's model would also rely on just the electron to explain the spectrum, he did not assume an electrodynamical model for the atom. The other important advance in the understanding of atomic spectra was the Rydberg–Ritz combination principle which related atomic spectral line frequencies to differences between 'terms', special frequencies characteristic of each element. Bohr would recognize the terms as energy levels of the atom divided by the Planck constant, leading to the modern view that the spectral lines result from energy differences.
Haas atomic model In 1910, Arthur Erich Haas proposed a model of the hydrogen atom with an electron circulating on the surface of a sphere of positive charge. The model resembled Thomson's plum pudding model, but Haas added a radical new twist: he constrained the magnitude of the electron's potential energy, | E pot | {\displaystyle |E_{\text{pot}}|} , on a sphere of radius a to equal the frequency, f, of the electron's orbit on the sphere times the Planck constant:
| E pot | = | − e 2 a | = h f {\displaystyle |E_{\text{pot}}|={\Big |}{\frac {-e^{2}}{a}}{\Big |}=hf}
where e represents the charge on the electron and the sphere. Haas combined this constraint with the balance-of-forces equation. The attractive force between the electron and the sphere balances the centrifugal force:
e 2 a 2 = m a ( 2 π f ) 2 {\displaystyle {\frac {e^{2}}{a^{2}}}=ma(2\pi f)^{2}}
where m is the mass of the electron. This combination relates the radius of the sphere to the Planck constant:
a = h 2 4 π 2 e 2 m {\displaystyle a={\frac {h^{2}}{4\pi ^{2}e^{2}m}}}
Haas solved for the Planck constant using the then-current value for the radius of the hydrogen atom. Three years later, Bohr would use similar equations with different interpretation. Bohr took the Planck constant as a given value and used the equations to predict a, the radius of the electron orbiting in the ground state of the hydrogen atom. This value is now called the Bohr radius.
Influence of the Solvay Conference The first Solvay Conference, in 1911, was one of the first international physics conferences. Nine Nobel or future Nobel laureates attended, including Ernest Rutherford, Bohr's mentor. Bohr did not attend but he read the Solvay reports and discussed them with Rutherford. The subject of the conference was the theory of radiation and the energy quanta of Max Planck's oscillators. Planck's lecture at the conference ended with comments about atoms and the discussion that followed it concerned atomic models. Hendrik Lorentz raised the question of the composition of the atom based on Haas's model, a form of Thomson's plum pudding model with a quantum modification. Lorentz explained that the size of atoms could be taken to determine the Planck constant as Haas had done or the Planck constant could be taken as determining the size of atoms. Bohr would adopt the second path. The discussions outlined the need for the quantum theory to be included in the atom. Planck explicitly mentions the failings of classical mechanics. While Bohr had already expressed a similar opinion in his PhD thesis, at Solvay the leading scientists of the day discussed a break with classical theories. Bohr's first paper on his atomic model cites the Solvay proceedings saying: "Whatever the alteration in the laws of motion of the electrons may be, 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." Encouraged by the Solvay discussions, Bohr would assume the atom was stable and abandon the efforts to stabilize classical models of the atom
Nicholson atom theory In 1911 John William Nicholson published a model of the atom which would influence Bohr's model. Nicholson developed his model based on the analysis of astrophysical spectroscopy. He connected the observed spectral line frequencies with the orbits of electrons in his atoms. The connection he adopted associated the atomic electron orbital angular momentum with the Planck constant. Whereas Planck focused on a quantum of energy, Nicholson's angular momentum quantum relates to orbital frequency. This new concept gave Planck constant an atomic meaning for the first time. In his 1913 paper Bohr cites Nicholson as finding quantized angular momentum important for the atom. The other critical influence of Nicholson work was his detailed analysis of spectra. Before Nicholson's work Bohr thought the spectral data was not useful for understanding atoms. In comparing his work to Nicholson's, Bohr came to understand the spectral data and their value. When he then learned from a friend about Balmer's compact formula for the spectral line data, Bohr quickly realized his model would match it in detail. Nicholson's model was based on classical electrodynamics along the lines of J.J. Thomson's plum pudding model but his negative electrons orbiting a positive nucleus rather than circulating in a sphere. To avoid immediate collapse of this system he required that electrons come in pairs so the rotational acceleration of each electron was matched across the orbit. By 1913 Bohr had already shown, from the analysis of alpha particle energy loss, that hydrogen had only a single electron not a matched pair. Bohr's atomic model would abandon classical electrodynamics. Nicholson's model of radiation was quantum but was attached to the orbits of the electrons. Bohr quantization would associate it with differences in energy levels of his model of hydrogen rather than the orbital frequency.
Bohr's previous work Bohr completed his PhD in 1911 with a thesis 'Studies on the Electron Theory of Metals', an application of the classical electron theory of Hendrik Lorentz. Bohr noted two deficits of the classical model. The first concerned the specific heat of metals which James Clerk Maxwell noted in 1875: every additional degree of freedom in a theory of metals, like subatomic electrons, cause more disagreement with experiment. The second, the classical theory could not explain magnetism. After his PhD, Bohr worked briefly in the lab of JJ Thomson before moving to Rutherford's lab in Manchester to study radioactivity. He arrived just after Rutherford completed his proposal of a compact nuclear core for atoms. Charles Galton Darwin, also at Manchester, had just completed an analysis of alpha particle energy loss in metals, concluding the electron collisions were the dominant cause of loss. Bohr showed in a subsequent paper that Darwin's results would improve by accounting for electron binding energy. Importantly this allowed Bohr to conclude that hydrogen atoms have a single electron.
Development Next, Bohr was told by his friend, Hans Hansen, that the Balmer series is calculated using the Balmer formula, an empirical equation discovered by Johann Balmer in 1885 that described wavelengths of some spectral lines of hydrogen. This was further generalized by Johannes Rydberg in 1888, resulting in what is now known as the Rydberg formula. After this, Bohr declared, "everything became clear". In 1913 Niels Bohr put forth five assumptions of postulates to provide an electron model consistent with Rutherford's nuclear model:
The atom cannot absorb nor radiate continuous energy, contrary to what classical electromagnetism suggests. Thus the electron maintains stable orbits, stationary orbits and the electron cannot have any other orbit in between the discrete ones. The electrons otherwise obey ordinary mechanics in the stationary orbits. The absorption or emission of electromagnetic radiation with a frequency ν {\displaystyle \nu } is related to an energy according to the Planck relation: E = h ν {\displaystyle E=h\nu } , where h {\displaystyle h} is the Planck constant. The stationary orbits are obtained by the condition that the ratio of the total energy emitted in forming the state and the frequency of the electron rotation about the positive nucleus is an integer multiple of h / 2 {\displaystyle h/2} . In atomic state corresponding to the maximum emitted energy, the stationary orbits are attained at distances for which the angular momentum of the revolving electron is equal to the reduced Planck constant, h / 2 π {\displaystyle h/2\pi } . The first two postulates formulate the experimental observations that absorption and emission of radiation for hydrogen do not obey the rules known at the time. The third postulate restates Planck's quantum theory of radiation which was already widely accepted by 1913. The fourth and fifth postulates are combined in modern discussions. In the combined form the stationary orbits are given by
m e v r = n ℏ {\displaystyle m_{\mathrm {e} }vr=n\hbar } , where n = 1 , 2 , 3 , . . . {\displaystyle n=1,2,3,...} is called the principal quantum number, and ℏ = h / 2 π {\displaystyle \hbar =h/2\pi } . The lowest value of n {\displaystyle n} is 1; this gives the smallest possible orbital radius, known as the Bohr radius, of 0.0529 nm for hydrogen. Once an electron is in this lowest orbit, it can get no closer to the nucleus. Bohr's condition, that the angular momentum be an integer multiple of ℏ , {\displaystyle \hbar ,} was later reinterpreted in 1924 by de Broglie as a standing wave condition. In this model, the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit:
n λ = 2 π r . {\displaystyle n\lambda =2\pi r.}
According to de Broglie's hypothesis, matter particles such as the electron behave as waves. The de Broglie wavelength of an electron is
λ = h m v , {\displaystyle \lambda ={\frac {h}{mv}},}
which implies that
n h m v = 2 π r , {\displaystyle {\frac {nh}{mv}}=2\pi r,}
or
n h 2 π = m v r , {\displaystyle {\frac {nh}{2\pi }}=mvr,}
where m v r {\displaystyle mvr} is the angular momentum of the orbiting electron. Writing ℓ {\displaystyle \ell } for this angular momentum, the previous equation becomes
ℓ = n h 2 π , {\displaystyle \ell ={\frac {nh}{2\pi }},}
which is the combined form of Bohr's fourth and fifth postulate. In 1932, these two postulates where further justified when C. V. Raman and Suri Bhagavantam experimentally demonstrated that a photon carries an angular momentum of ℏ {\displaystyle \hbar } . Bohr had assumed that electrons can transition between stationary orbits, during which a photon is either absorbed or emitted by the system. With the measurement of the photon angular momentum, the law of conservation of angular momentum predicts that the angular momentum of an electron on a stationary orbit must equal an integer multiple of ℏ {\displaystyle \hbar } . Starting from the angular momentum quantum rule as Bohr admits is previously given by Nicholson in his 1912 paper, Bohr was able to calculate the energies of the allowed orbits of the hydrogen atom and other hydrogen-like atoms and ions. These orbits are associated with definite energies and are also called energy shells or energy levels. In these orbits, the electron's acceleration does not result in radiation and energy loss.
Refinements
Several enhancements to the Bohr model were proposed, most notably the Sommerfeld or Bohr–Sommerfeld models, which suggested that electrons travel in elliptical orbits around a nucleus instead of the Bohr model's circular orbits. This model supplemented the quantized angular momentum condition of the Bohr model with an additional radial quantization condition, the Wilson–Sommerfeld quantization condition
∫ 0 T p r d q r = n h , {\displaystyle \int _{0}^{T}p_{\text{r}}\,dq_{\text{r}}=nh,}
where pr is the radial momentum canonically conjugate to the coordinate qr, which is the radial position, and T is one full orbital period. The integral is the action of action-angle coordinates. This condition, suggested by the correspondence principle, is the only one possible, since the quantum numbers are adiabatic invariants. The Bohr–Sommerfeld model was fundamentally inconsistent and led to many paradoxes. The magnetic quantum number measured the tilt of the orbital plane relative to the xy plane, and it could only take a few discrete values. This contradicted the obvious fact that an atom could have any orientation relative to the coordinates, without restriction. The Sommerfeld quantization can be performed in different canonical coordinates and sometimes gives different answers. The incorporation of radiation corrections was difficult, because it required finding action-angle coordinates for a combined radiation/atom system, which is difficult when the radiation is allowed to escape. The whole theory did not extend to non-integrable motions, which meant that many systems could not be treated even in principle. In the end, the model was replaced by the modern quantum-mechanical treatment of the hydrogen atom, which was first given by Wolfgang Pauli in 1925, using Heisenberg's matrix mechanics. The current picture of the hydrogen atom is based on the atomic orbitals of wave mechanics, which Erwin Schrödinger developed in 1926. However, this is not to say that the Bohr–Sommerfeld model was without its successes. Calculations based on the Bohr–Sommerfeld model were able to accurately explain a number of more complex atomic spectral effects. For example, up to first-order perturbations, the Bohr model and quantum mechanics make the same predictions for the spectral line splitting in the Stark effect. At higher-order perturbations, however, the Bohr model and quantum mechanics differ, and measurements of the Stark effect under high field strengths helped confirm the correctness of quantum mechanics over the Bohr model. The prevailing theory behind this difference lies in the shapes of the orbitals of the electrons, which vary according to the energy state of the electron. The Bohr–Sommerfeld quantization conditions lead to questions in modern mathematics. Consistent semiclassical quantization condition requires a certain type of structure on the phase space, which places topological limitations on the types of symplectic manifolds which can be quantized. In particular, the symplectic form should be the curvature form of a connection of a Hermitian line bundle, which is called a prequantization. Bohr also updated his model in 1922, assuming that certain numbers of electrons (for example, 2, 8, and 18) correspond to stable "closed shells".
Replacement In 1925, a new kind of mechanics was proposed, quantum mechanics, in which Bohr's model of electrons traveling in quantized orbits was extended into more accurate model of electron motion. The new framework of the theory was proposed by Werner Heisenberg in the Umdeutung paper that was later extended into matrix mechanics with Max Born and Pascual Jordan. Another form of the same theory, wave mechanics, was discovered by the Austrian physicist Erwin Schrödinger independently, and by different reasoning. Schrödinger employed de Broglie's matter waves, but sought wave solutions of a three-dimensional wave equation describing electrons that were constrained to move about the nucleus of a hydrogen-like atom, by being trapped by the potential of the positive nuclear charge.
Electron energy levels
The Bohr model gives almost exact results only for a system where two charged points orbit each other at speeds much less than that of light. This not only involves one-electron systems such as the hydrogen atom, singly ionized helium, and doubly ionized lithium, but it includes positronium and Rydberg states of any atom where one electron is far away from everything else. It can be used for K-line X-ray transition calculations if other assumptions are added (see Moseley's law below). In high energy physics, it can be used to calculate the masses of heavy quark mesons. Calculation of the orbits requires two assumptions.
Classical mechanics The electron is held in a circular orbit by electrostatic attraction. The centripetal force is equal to the Coulomb force.
m e v 2 r = Z k e e 2 r 2 , {\displaystyle {\frac {m_{\mathrm {e} }v^{2}}{r}}={\frac {Zk_{\mathrm {e} }e^{2}}{r^{2}}},}
where me is the electron's mass, e is the elementary charge, ke is the Coulomb constant and Z is the atom's atomic number. It is assumed here that the mass of the nucleus is much larger than the electron mass (which is a good assumption). This equation determines the electron's speed at any radius:
v = Z k e e 2 m e r . {\displaystyle v={\sqrt {\frac {Zk_{\mathrm {e} }e^{2}}{m_{\mathrm {e} }r}}}.}
It also determines the electron's total energy at any radius:
E = − 1 2 m e v 2 . {\displaystyle E=-{\frac {1}{2}}m_{\mathrm {e} }v^{2}.}
The total energy is negative and inversely proportional to r. This means that it takes energy to pull the orbiting electron away from the proton. For infinite values of r, the energy is zero, corresponding to a motionless electron infinitely far from the proton. The total energy is half the potential energy, the difference being the kinetic energy of the electron. This is also true for noncircular orbits by the virial theorem. A quantum rule The angular momentum L = mevr is an integer multiple of ħ:
m e v r = n ℏ . {\displaystyle m_{\mathrm {e} }vr=n\hbar .}
Derivation In classical mechanics, if an electron is orbiting around an atom with period T, and if its coupling to the electromagnetic field is weak, so that the orbit doesn't decay very much in one cycle, it will emit electromagnetic radiation in a pattern repeating at every period, so that the Fourier transform of the pattern will only have frequencies which are multiples of 1/T. However, in quantum mechanics, the quantization of angular momentum leads to discrete energy levels of the orbits, and the emitted frequencies are quantized according to the energy differences between these levels. This discrete nature of energy levels introduces a fundamental departure from the classical radiation law, giving rise to distinct spectral lines in the emitted radiation. Bohr assumes that the electron is circling the nucleus in an elliptical orbit obeying the rules of classical mechanics, but with no loss of radiation due to the Larmor formula. Denoting the total energy as E, the electron charge as −e, the nucleus charge as K = Ze, the electron mass as me, half the major axis of the ellipse as a, he starts with these equations:
E is assumed to be negative, because a positive energy is required to unbind the electron from the nucleus and put it at rest at an infinite distance. Eq. (1a) is obtained from equating the centripetal force to the Coulombian force acting between the nucleus and the electron, considering that E = T + U {\displaystyle E=T+U} (where T is the average kinetic energy and U the average electrostatic potential), and that for Kepler's second law, the average separation between the electron and the nucleus is a. Eq. (1b) is obtained from the same premises of eq. (1a) plus the virial theorem, stating that, for an elliptical orbit,
Then Bohr assumes that | E | {\displaystyle \vert E\vert } is an integer multiple of the energy of a quantum of light with half the frequency of the electron's revolution frequency, i.e.:
From eq. (1a, 1b, 2), it descends:
He further assumes that the orbit is circular, i.e. a = r {\displaystyle a=r} , and, denoting the angular momentum of the electron as L, introduces the equation:
Eq. (4) stems from the virial theorem, and from the classical mechanics relationships between the angular momentum, the kinetic energy and the frequency of revolution. From eq. (1c, 2, 4), it stems:
where:
that is:
This results states that the angular momentum of the electron is an integer multiple of the reduced Planck constant. Substituting the expression for the velocity gives an equation for r in terms of n:
m e k e Z e 2 m e r r = n ℏ , {\displaystyle m_{\text{e}}{\sqrt {\dfrac {k_{\text{e}}Ze^{2}}{m_{\text{e}}r}}}r=n\hbar ,}
so that the allowed orbit radius at any n is
r n = n 2 ℏ 2 Z k e e 2 m e . {\displaystyle r_{n}={\frac {n^{2}\hbar ^{2}}{Zk_{\mathrm {e} }e^{2}m_{\mathrm {e} }}}.}
The smallest possible value of r in the hydrogen atom (Z = 1) is called the Bohr radius and is equal to:
r 1 = ℏ 2 k e e 2 m e ≈ 5.29 × 10 − 11 m = 52.9 p m . {\displaystyle r_{1}={\frac {\hbar ^{2}}{k_{\mathrm {e} }e^{2}m_{\mathrm {e} }}}\approx 5.29\times 10^{-11}~\mathrm {m} =52.9~\mathrm {pm} .}
The energy of the n-th level for any atom is determined by the radius and quantum number:
E = − Z k e e 2 2 r n = − Z 2 ( k e e 2 ) 2 m e 2 ℏ 2 n 2 ≈ − 13.6 Z 2 n 2 e V . {\displaystyle E=-{\frac {Zk_{\mathrm {e} }e^{2}}{2r_{n}}}=-{\frac {Z^{2}(k_{\mathrm {e} }e^{2})^{2}m_{\mathrm {e} }}{2\hbar ^{2}n^{2}}}\approx {\frac {-13.6\ Z^{2}}{n^{2}}}~\mathrm {eV} .}
An electron in the lowest energy level of hydrogen (n = 1) therefore has about 13.6 eV less energy than a motionless electron infinitely far from the nucleus. The next energy level (n = 2) is −3.4 eV. The third (n = 3) is −1.51 eV, and so on. For larger values of n, these are also the binding energies of a highly excited atom with one electron in a large circular orbit around the rest of the atom. The hydrogen formula also coincides with the Wallis product. The combination of natural constants in the energy formula is called the Rydberg energy (RE):
R E = ( k e e 2 ) 2 m e 2 ℏ 2 . {\displaystyle R_{\mathrm {E} }={\frac {(k_{\mathrm {e} }e^{2})^{2}m_{\mathrm {e} }}{2\hbar ^{2}}}.}
This expression is clarified by interpreting it in combinations that form more natural units:
m e c 2 {\displaystyle m_{\mathrm {e} }c^{2}} is the rest mass energy of the electron (511 keV),
k e e 2 ℏ c = α ≈ 1 137 {\displaystyle {\frac {k_{\mathrm {e} }e^{2}}{\hbar c}}=\alpha \approx {\frac {1}{137}}} is the fine-structure constant,
R E = 1 2 ( m e c 2 ) α 2 {\displaystyle R_{\mathrm {E} }={\frac {1}{2}}(m_{\mathrm {e} }c^{2})\alpha ^{2}} . Since this derivation is with the assumption that the nucleus is orbited by one electron, we can generalize this result by letting the nucleus have a charge q = Ze, where Z is the atomic number. This will now give us energy levels for hydrogenic (hydrogen-like) atoms, which can serve as a rough order-of-magnitude approximation of the actual energy levels. So for nuclei with Z protons, the energy levels are (to a rough approximation):