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model of the [[hydrogen atom]] ({{nowrap|''Z'' {{=}} 1}}) or a hydrogen-like ion ({{nowrap|''Z'' > 1}}), where the negatively charged [[electron]] confined to an [[atomic shell]] encircles a small, positively charged [[atomic nucleus]] and where an electron jumps between orbits, is accompanied by an emitted or absorbed amount of [[electromagnetic wave|electromagnetic energy]] (''hν'').<ref name="Akhlesh Lakhtakia Ed. 1996">{{Cite journal |last1=Lakhtakia |first1=Akhlesh |last2=Salpeter |first2=Edwin E. |year=1996 |title=Models and Modelers of Hydrogen |journal=American Journal of Physics |volume=65 |issue=9 |pages=933 |bibcode=1997AmJPh..65..933L |doi=10.1119/1.18691}}</ref> The orbits in which the electron may travel are shown as grey circles; their radius increases as ''n''<sup>2</sup>, where ''n'' is the [[principal quantum number]]. The {{nowrap|3 → 2}} transition depicted here produces the first line of the [[Balmer series]], and for hydrogen ({{nowrap|''Z'' {{=}} 1}}) it results in a photon of [[wavelength]] 656 [[nanometre|nm]] (red light).]] | model of the [[hydrogen atom]] ({{nowrap|''Z'' {{=}} 1}}) or a hydrogen-like ion ({{nowrap|''Z'' > 1}}), where the negatively charged [[electron]] confined to an [[atomic shell]] encircles a small, positively charged [[atomic nucleus]] and where an electron jumps between orbits, is accompanied by an emitted or absorbed amount of [[electromagnetic wave|electromagnetic energy]] (''hν'').<ref name="Akhlesh Lakhtakia Ed. 1996">{{Cite journal |last1=Lakhtakia |first1=Akhlesh |last2=Salpeter |first2=Edwin E. |year=1996 |title=Models and Modelers of Hydrogen |journal=American Journal of Physics |volume=65 |issue=9 |pages=933 |bibcode=1997AmJPh..65..933L |doi=10.1119/1.18691}}</ref> The orbits in which the electron may travel are shown as grey circles; their radius increases as ''n''<sup>2</sup>, where ''n'' is the [[principal quantum number]]. The {{nowrap|3 → 2}} transition depicted here produces the first line of the [[Balmer series]], and for hydrogen ({{nowrap|''Z'' {{=}} 1}}) it results in a photon of [[wavelength]] 656 [[nanometre|nm]] (red light).]] | ||
In [[atomic physics]], the '''Bohr model''' or '''Rutherford–Bohr model''' | 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 (physics)|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 [[orbit]]ing [[electron]]s. It is [[analogy|analogous]] to the structure of the [[Solar System]], but with attraction provided by [[Coulomb's law|electrostatic force]] rather than [[gravity]], and with the electron energies quantized (assuming only discrete values). | ||
In the history of atomic physics, it followed | 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),<ref>{{Cite journal |last=Perrin |first=Jean |author-link=Jean Baptiste Perrin |year=1901 |title=Les Hypothèses moléculaires |url=https://fr.wikisource.org/wiki/Les_Hypoth%C3%A8ses_mol%C3%A9culaires |journal=La Revue scientifique |page=463}}</ref> the [[Cubical atom|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 mechanics|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 spectral series|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 model's key success lies in explaining the [[Rydberg formula]] for [[Hydrogen spectral series|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. | ||
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=== Rutherford nuclear model === | === Rutherford nuclear model === | ||
{{main| Rutherford atom| Rutherford scattering experiments}} | {{main| Rutherford atom| Rutherford scattering experiments}} | ||
In 1908, [[Hans Geiger]] and [[Ernest Marsden]] demonstrated that [[alpha particle]] occasionally scatter at large angles, a result inconsistent with Thomson's model. | In 1908, [[Hans Geiger]] and [[Ernest Marsden]] demonstrated that [[alpha particle]]s 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. | 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 [[Rutherford atom|model of the atom]] was incomplete.<ref name=Heilbron1968>{{Cite journal |last=Heilbron |first=John L. |date=1968 |title=The Scattering of α and β Particles and Rutherford's Atom |url=https://www.jstor.org/stable/41133273 |journal=Archive for History of Exact Sciences |volume=4 |issue=4 |pages=247–307 |doi=10.1007/BF00411591 |jstor=41133273 |issn=0003-9519|url-access=subscription }}</ref> | Rutherford scattering did not involve the electrons and thus his [[Rutherford atom|model of the atom]] was incomplete.<ref name=Heilbron1968>{{Cite journal |last=Heilbron |first=John L. |date=1968 |title=The Scattering of α and β Particles and Rutherford's Atom |url=https://www.jstor.org/stable/41133273 |journal=Archive for History of Exact Sciences |volume=4 |issue=4 |pages=247–307 |doi=10.1007/BF00411591 |jstor=41133273 |issn=0003-9519|url-access=subscription }}</ref> | ||
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=== Atomic spectra === | === Atomic spectra === | ||
By the early twentieth century, it was expected that the atom | 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 [[Balmer formula|Johann Balmer]] and [[Rydberg formula|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.<ref name=KraghQuantumAtom2012>{{Cite book |last=Kragh |first=Helge |title=Niels Bohr and the Quantum Atom: The Bohr Model of Atomic Structure 1913–1925 |date=2012 |publisher=Oxford University Press |isbn=978-0-19-163046-0}}</ref>{{rp|18|q=By the early twentieth century it was desirable for a candidate theory of atomic structure to account for line spectra and their regularities, but in fact none of the models available at the time were able to do so.}}<ref>{{Cite journal |last=Rayleigh |first=Lord |date=January 1906 |title=VII. On electrical vibrations and the constitution of the atom |url=https://zenodo.org/record/1837403 |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |volume=11 |issue=61 |pages=117–123 |doi=10.1080/14786440609463428}}</ref> | ||
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.<ref name="Whittaker">{{Cite book |last=Whittaker |first=Edmund T. |title=A history of the theories of aether & electricity. 2: The modern theories, 1900 - 1926 |date=1989 |publisher=Dover Publ |isbn=978-0-486-26126-3 |edition=Repr |location=New York}}</ref>{{rp|II:106}} 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. | 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.<ref name="Whittaker">{{Cite book |last=Whittaker |first=Edmund T. |title=A history of the theories of aether & electricity. 2: The modern theories, 1900 - 1926 |date=1989 |publisher=Dover Publ |isbn=978-0-486-26126-3 |edition=Repr |location=New York}}</ref>{{rp|II:106}} 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. | ||
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=== Haas atomic model<span class="anchor" id="Haas atomic model"></span> === | === Haas atomic model<span class="anchor" id="Haas atomic model"></span> === | ||
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 electron's potential energy, <math>E_\text{pot}</math>, on a sphere of radius {{mvar|a}} to equal the frequency, {{mvar|f}}, of the electron's orbit on the sphere times the [[Planck constant]]:<ref name=PaisInwardBound/>{{rp|197}} | 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, <math>|E_\text{pot}|</math>, on a sphere of radius {{mvar|a}} to equal the frequency, {{mvar|f}}, of the electron's orbit on the sphere times the [[Planck constant]]:<ref name=PaisInwardBound/>{{rp|197}} | ||
<math display="block">E_\text{pot}= \frac{- e^2}{a} = hf </math> | <math display="block">|E_\text{pot}|= \Big|\frac{- e^2}{a}\Big| = hf </math> | ||
where {{mvar|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]]: | where {{mvar|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]]: | ||
<math display="block">\frac{e^2}{a^2} = ma(2\pi f)^2</math> | <math display="block">\frac{e^2}{a^2} = ma(2\pi f)^2</math> | ||
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<math display="block">a = \frac{h^2}{4\pi^2e^2m}</math> | <math display="block">a = \frac{h^2}{4\pi^2e^2m}</math> | ||
Haas solved for the Planck constant using the then-current value for the radius of the hydrogen atom. | 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 given value and used the equations to predict | 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 {{mvar|a}}, the radius of the electron orbiting in the ground state of the hydrogen atom. This value is now called the [[Bohr radius]].<ref name=PaisInwardBound/>{{rp|197}} | ||
=== Influence of the Solvay Conference === | === Influence of the Solvay Conference === | ||
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<math> m_\mathrm{e} v r = n \hbar </math>, where <math>n= 1, 2, 3, ...</math> is called the [[principal quantum number]], and <math>\hbar = h/2\pi</math>. The lowest value of <math>n</math> 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. | <math> m_\mathrm{e} v r = n \hbar </math>, where <math>n= 1, 2, 3, ...</math> is called the [[principal quantum number]], and <math>\hbar = h/2\pi</math>. The lowest value of <math>n</math> 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 <math>\hbar,</math> was later reinterpreted in 1924 by [[de Broglie]] as a [[standing wave]] condition.<ref name="Pinke-2005">{{cite journal |last1=Pinke |first1=D. D.|last2=Ősz |first2=K. |last3=Lente |first3=G. |year=2025 |title=The origin of the postulates in the Bohr model of the hydrogen atom |journal=ChemTexts |volume=11 | | Bohr's condition, that the angular momentum be an integer multiple of <math>\hbar,</math> was later reinterpreted in 1924 by [[de Broglie]] as a [[standing wave]] condition.<ref name="Pinke-2005">{{cite journal |last1=Pinke |first1=D. D.|last2=Ősz |first2=K. |last3=Lente |first3=G. |year=2025 |title=The origin of the postulates in the Bohr model of the hydrogen atom |journal=ChemTexts |volume=11 |issue=3 |article-number=11 |doi=10.1007/s40828-025-00208-4 |bibcode=2025ChTxt..11...11P |doi-access=free }}</ref> 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: | ||
: <math>n \lambda = 2 \pi r.</math> | : <math>n \lambda = 2 \pi r.</math> | ||
According to de Broglie's hypothesis, matter particles such as the electron behave as [[Matter wave|waves]]. The | According to de Broglie's hypothesis, matter particles such as the electron behave as [[Matter wave|waves]]. The de Broglie wavelength of an electron is | ||
: <math>\lambda = \frac{h}{mv},</math> | : <math>\lambda = \frac{h}{mv},</math> | ||
which implies that | which implies that | ||
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which is the combined form of Bohr's fourth and fifth postulate.<ref name="Pinke-2005"/> | which is the combined form of Bohr's fourth and fifth postulate.<ref name="Pinke-2005"/> | ||
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 <math>\hbar</math>.<ref name="Raman-Bhagavantam">{{cite journal |last1=Raman |first1=C. V.|last2=Bhagavantam |first2=S. |year=1932 |title=Experimental Proof of the Spin of the Photon |journal=Nature |volume=129 |pages= | 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 <math>\hbar</math>.<ref name="Raman-Bhagavantam">{{cite journal |last1=Raman |first1=C. V.|last2=Bhagavantam |first2=S. |year=1932 |title=Experimental Proof of the Spin of the Photon |journal=Nature |volume=129 |issue=3244 |pages=22–23 |doi=10.1038/129022a0 |bibcode=1932Natur.129...22R |doi-access=free }}</ref><ref>{{Cite journal |last1=Ghatak |first1=Ajoy |last2=Mani |first2=H. S. |date=2024-09-18 |title=Angular Momentum of the Photon: Its Discovery and Heuristic Derivation |url=https://link.springer.com/10.1007/s12045-024-1177-z |journal=Resonance |language=en |volume=29 |issue=9 |pages=1177–1194 |doi=10.1007/s12045-024-1177-z |issn=0973-712X|url-access=subscription }}</ref> 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 <math>\hbar</math>.<ref name="Pinke-2005" /> | ||
Starting from the angular momentum quantum rule as Bohr admits is previously given by Nicholson in his 1912 paper,<ref name="aip.org" /><ref name=Heilbron2013/><ref name="Nicholson1912" /><ref name=McCormmach1966/> Bohr<ref name="bohr1" /> was able to calculate the [[#Electron energy levels|energies of the allowed orbits]] of the hydrogen atom and other [[#Shell model (heavier atoms)|hydrogen-like]] atoms and ions. These orbits are associated with definite energies and are also called energy shells or [[energy level]]s. In these orbits, the electron's acceleration does not result in radiation and energy loss. | Starting from the angular momentum quantum rule as Bohr admits is previously given by Nicholson in his 1912 paper,<ref name="aip.org" /><ref name=Heilbron2013/><ref name="Nicholson1912" /><ref name=McCormmach1966/> Bohr<ref name="bohr1" /> was able to calculate the [[#Electron energy levels|energies of the allowed orbits]] of the hydrogen atom and other [[#Shell model (heavier atoms)|hydrogen-like]] atoms and ions. These orbits are associated with definite energies and are also called energy shells or [[energy level]]s. In these orbits, the electron's acceleration does not result in radiation and energy loss. | ||
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== Replacement == | == 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 | 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]]. [[Schrödinger equation|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 == | == Electron energy levels == | ||
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*: It also determines the electron's total energy at any radius: | *: It also determines the electron's total energy at any radius: | ||
*:: <math> E = -\frac{1}{2} m_\mathrm{e} v^2.</math> | *:: <math> E = -\frac{1}{2} m_\mathrm{e} v^2.</math> | ||
*: 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]]. | *: 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''' | * '''A quantum rule''' | ||
*: The [[angular momentum]] {{nowrap|''L'' {{=}} ''m''<sub>e</sub>''vr''}} is an integer multiple of ''ħ'': | *: The [[angular momentum]] {{nowrap|''L'' {{=}} ''m''<sub>e</sub>''vr''}} is an integer multiple of ''ħ'': | ||
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=== Derivation === | === 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. | 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. | 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. | ||
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In 1921, following the work of chemists and others involved in work on the [[periodic table]], Bohr extended the model of hydrogen to give an approximate model for heavier atoms. This gave a physical picture that reproduced many known atomic properties for the first time although these properties were proposed contemporarily with the identical work of chemist [[Charles Rugeley Bury]]<ref name=Kragh1979/><ref>{{Cite journal |last=Bury |first=Charles R. |date=July 1921 |title=Langmuir's Theory of the Arrangement of Electrons in Atoms and Molecules |url=https://zenodo.org/record/1428812 |journal=Journal of the American Chemical Society |volume=43 |issue=7 |pages=1602–1609 |doi=10.1021/ja01440a023|bibcode=1921JAChS..43.1602B }}</ref> | In 1921, following the work of chemists and others involved in work on the [[periodic table]], Bohr extended the model of hydrogen to give an approximate model for heavier atoms. This gave a physical picture that reproduced many known atomic properties for the first time although these properties were proposed contemporarily with the identical work of chemist [[Charles Rugeley Bury]]<ref name=Kragh1979/><ref>{{Cite journal |last=Bury |first=Charles R. |date=July 1921 |title=Langmuir's Theory of the Arrangement of Electrons in Atoms and Molecules |url=https://zenodo.org/record/1428812 |journal=Journal of the American Chemical Society |volume=43 |issue=7 |pages=1602–1609 |doi=10.1021/ja01440a023|bibcode=1921JAChS..43.1602B }}</ref> | ||
Bohr's partner in research during 1914 to 1916 was [[Walther Kossel]] who corrected Bohr's work to show that electrons interacted through the outer rings, and Kossel called the rings: "shells".<ref name="Kossel1916">{{Cite journal |last=Kossel |first=W. |date=1916 |title=Über Molekülbildung als Frage des Atombaus |trans-title=On molecular formation as a question of atomic structure |url=https://zenodo.org/record/1447311 |journal=Annalen der Physik |language=de |volume=354 |issue=3 |pages=229–362 |bibcode=1916AnP...354..229K |doi=10.1002/andp.19163540302}}</ref><ref name="Kragh2012">{{Cite journal |last=Kragh |first=Helge |date=2012 |title=Lars Vegard, atomic structure, and the periodic system |url=http://acshist.scs.illinois.edu/bulletin_open_access/v37-1/v37-1%20p42-49.pdf |url-status=live |journal=Bulletin for the History of Chemistry |volume=37 |issue=1 |pages=42–49 |oclc=797965772 |archive-url=https://ghostarchive.org/archive/20221009/http://acshist.scs.illinois.edu/bulletin_open_access/v37-1/v37-1%20p42-49.pdf |archive-date=2022-10-09 |s2cid=53520045}}</ref> [[Irving Langmuir]] is credited with the first viable arrangement of electrons in shells with only two in the first shell and going up to eight in the next according to the [[octet rule]] of 1904, although Kossel had already predicted a maximum of eight per shell in 1916.<ref>{{Cite journal |last=Langmuir |first=Irving |author-link=Irving Langmuir |date=June 1919 |title=The Arrangement of Electrons in Atoms and Molecules |url=https://zenodo.org/record/1429026 |journal=Journal of the American Chemical Society |volume=41 |issue=6 |pages=868–934 |doi=10.1021/ja02227a002|bibcode=1919JAChS..41..868L }}</ref> Heavier atoms have more protons in the nucleus, and more electrons to cancel the charge. Bohr took from these chemists the idea that each discrete orbit could only hold a certain number of electrons. Per [[Kossel]], after that the orbit is full, the next level would have to be used.<ref name=Kragh1979/> This gives the atom a [[electron configuration|shell structure]] designed by Kossel, Langmuir, and Bury, in which each shell corresponds to a Bohr orbit. | Bohr's partner in research during 1914 to 1916 was [[Walther Kossel]] who corrected Bohr's work to show that electrons interacted through the outer rings, and Kossel called the rings: "shells".<ref name="Kossel1916">{{Cite journal |last=Kossel |first=W. |date=1916 |title=Über Molekülbildung als Frage des Atombaus |trans-title=On molecular formation as a question of atomic structure |url=https://zenodo.org/record/1447311 |journal=Annalen der Physik |language=de |volume=354 |issue=3 |pages=229–362 |bibcode=1916AnP...354..229K |doi=10.1002/andp.19163540302}}</ref><ref name="Kragh2012">{{Cite journal |last=Kragh |first=Helge |date=2012 |title=Lars Vegard, atomic structure, and the periodic system |url=http://acshist.scs.illinois.edu/bulletin_open_access/v37-1/v37-1%20p42-49.pdf |url-status=live |journal=Bulletin for the History of Chemistry |volume=37 |issue=1 |pages=42–49 |doi=10.70359/bhc2012v037p042 |oclc=797965772 |archive-url=https://ghostarchive.org/archive/20221009/http://acshist.scs.illinois.edu/bulletin_open_access/v37-1/v37-1%20p42-49.pdf |archive-date=2022-10-09 |s2cid=53520045}}</ref> [[Irving Langmuir]] is credited with the first viable arrangement of electrons in shells with only two in the first shell and going up to eight in the next according to the [[octet rule]] of 1904, although Kossel had already predicted a maximum of eight per shell in 1916.<ref>{{Cite journal |last=Langmuir |first=Irving |author-link=Irving Langmuir |date=June 1919 |title=The Arrangement of Electrons in Atoms and Molecules |url=https://zenodo.org/record/1429026 |journal=Journal of the American Chemical Society |volume=41 |issue=6 |pages=868–934 |doi=10.1021/ja02227a002|bibcode=1919JAChS..41..868L }}</ref> Heavier atoms have more protons in the nucleus, and more electrons to cancel the charge. Bohr took from these chemists the idea that each discrete orbit could only hold a certain number of electrons. Per [[Kossel]], after that the orbit is full, the next level would have to be used.<ref name=Kragh1979/> This gives the atom a [[electron configuration|shell structure]] designed by Kossel, Langmuir, and Bury, in which each shell corresponds to a Bohr orbit. | ||
This model is even more approximate than the model of hydrogen, because it treats the electrons in each shell as non-interacting. But the repulsions of electrons are taken into account somewhat by the phenomenon of [[Shielding effect|screening]]. The electrons in outer orbits do not only orbit the nucleus, but they also move around the inner electrons, so the effective charge Z that they feel is reduced by the number of the electrons in the inner orbit. | This model is even more approximate than the model of hydrogen, because it treats the electrons in each shell as non-interacting. But the repulsions of electrons are taken into account somewhat by the phenomenon of [[Shielding effect|screening]]. The electrons in outer orbits do not only orbit the nucleus, but they also move around the inner electrons, so the effective charge Z that they feel is reduced by the number of the electrons in the inner orbit. | ||
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== Shortcomings == | == Shortcomings == | ||
The Bohr model gives an incorrect value {{math|''L''{{=}}''ħ''}} for the ground state orbital angular momentum: The angular momentum in the true ground state is known to be zero from experiment. Although mental pictures fail somewhat at these levels of scale, an electron in the lowest modern "orbital" with no orbital momentum, may be thought of as not to revolve "around" the nucleus at all, but merely to go tightly around it in an ellipse with zero area (this may be pictured as "back and forth", without striking or interacting with the nucleus). This is only reproduced in a more sophisticated semiclassical treatment like Sommerfeld's. Still, even the most sophisticated semiclassical model fails to explain the fact that the lowest energy state is spherically symmetric – it | The Bohr model gives an incorrect value {{math|''L''{{=}}''ħ''}} for the ground state orbital angular momentum: The angular momentum in the true ground state is known to be zero from experiment. Although mental pictures fail somewhat at these levels of scale, an electron in the lowest modern "orbital" with no orbital momentum, may be thought of as not to revolve "around" the nucleus at all, but merely to go tightly around it in an ellipse with zero area (this may be pictured as "back and forth", without striking or interacting with the nucleus). This is only reproduced in a more sophisticated semiclassical treatment like Sommerfeld's. Still, even the most sophisticated semiclassical model fails to explain the fact that the lowest energy state is spherically symmetric – it does not point in any particular direction. | ||
In modern quantum mechanics, the electron in hydrogen is a [[Electron cloud|spherical cloud of probability]] that grows denser near the nucleus. The rate-constant of probability-decay in hydrogen is equal to the inverse of the Bohr radius, but since Bohr worked with circular orbits, not zero area ellipses, the fact that these two numbers exactly agree is considered a "coincidence". (However, many such coincidental agreements are found between the semiclassical vs. full quantum mechanical treatment of the atom; these include identical energy levels in the hydrogen atom and the derivation of a fine-structure constant, which arises from the relativistic Bohr–Sommerfeld model (see below) and which happens to be equal to an entirely different concept, in full modern quantum mechanics). | In modern quantum mechanics, the electron in hydrogen is a [[Electron cloud|spherical cloud of probability]] that grows denser near the nucleus. The rate-constant of probability-decay in hydrogen is equal to the inverse of the Bohr radius, but since Bohr worked with circular orbits, not zero area ellipses, the fact that these two numbers exactly agree is considered a "coincidence". (However, many such coincidental agreements are found between the semiclassical vs. full quantum mechanical treatment of the atom; these include identical energy levels in the hydrogen atom and the derivation of a fine-structure constant, which arises from the relativistic Bohr–Sommerfeld model (see below) and which happens to be equal to an entirely different concept, in full modern quantum mechanics). | ||
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* Doublets and triplets appear in the spectra of some atoms as very close pairs of lines. Bohr's model cannot say why some energy levels should be very close together. | * Doublets and triplets appear in the spectra of some atoms as very close pairs of lines. Bohr's model cannot say why some energy levels should be very close together. | ||
* Multi-electron atoms do not have energy levels predicted by the model. It does not work for (neutral) helium. | * Multi-electron atoms do not have energy levels predicted by the model. It does not work for (neutral) helium. | ||
Another issue with the Bohr model was that although an electron in a stationary state <math>n</math> is described in periodic motion, and therefore should be associated with a characteristic radiation frequency <math>\omega(n)</math>, no such frequency was experimentally measured. Heisenberg attempted to fix this issue in the [[Umdeutung paper]] by replacing the idea of a single orbital frequency associated with the electron's motion by the set of observable spectral frequencies <math>\omega(n,n-\alpha)</math> (for a quantum transition from stationary state <math>n</math> to <math>n-\alpha</math>). By this time, however, the electronic orbit was practically abandoned as physically meaningful.<ref>{{cite journal |last1=Heisenberg |first1=Werner |title=Development of concepts in the history of quantum theory |journal=American Journal of Physics |date=May 1975 |volume=43 |issue=5 |pages=389–394 |doi=10.1119/1.9833 |bibcode=1975AmJPh..43..389H |url=https://doi.org/10.1119/1.9833 |access-date=16 September 2025|url-access=subscription }}</ref> | |||
== Model of the chemical bond == | == Model of the chemical bond == | ||
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{{Commons category}} | {{Commons category}} | ||
* [http://www.lucamoroni.it/cdf-simulations/standing-waves-in-bohrs-atomic-model/ Standing waves in Bohr's atomic model]—An interactive simulation to intuitively explain the quantization condition of standing waves in Bohr's atomic mode | * [http://www.lucamoroni.it/cdf-simulations/standing-waves-in-bohrs-atomic-model/ Standing waves in Bohr's atomic model]—An interactive simulation to intuitively explain the quantization condition of standing waves in Bohr's atomic mode | ||
* [[wikiversity:De Broglie wavelength]] | |||
{{Atomic models}} | {{Atomic models}} | ||