Editing Bohr model (section)

== Background ==
{{main| History of atomic theory}}
[[File:Atome bohr couches electroniques KLM.svg|thumb|Bohr model in 1921<ref name="Kragh1979">{{Cite journal |last=Kragh |first=Helge |date=1 January 1979 |title=Niels Bohr's Second Atomic Theory |journal=Historical Studies in the Physical Sciences |volume=10 |pages=123–186 |doi=10.2307/27757389 |jstor=27757389}}</ref> after Sommerfeld expansion of 1913 model showing maximum electrons per shell with shells labeled in [[X-ray notation]]]]
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.<ref name=KraghQuantumAtom2012/>{{rp|2}}

===Planetary models===
In the late 1800s speculations on the possible structure of the atom included planetary models with orbiting charged electrons.<ref name=Kragh2010>Helge Kragh (Oct. 2010). [https://css.au.dk/fileadmin/reposs/reposs-010.pdf Before Bohr: Theories of atomic structure 1850-1913]. RePoSS: Research Publications on Science Studies 10. Aarhus: Centre for Science Studies, University of Aarhus.</ref>{{rp|35}}
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 electromagnetic effect of multiple electrons, suitable arranged, would cancel each other. Thus subsequent atomic models based on classical electrodynamics needed to adopt such special multi-electron arrangements.<ref>{{Cite book |last=Wheaton |first=Bruce R. |title=The tiger and the shark: empirical roots of wave-particle dualism |date=1992 |publisher=Cambridge Univ. Press |isbn=978-0-521-35892-7 |edition=1. paperback ed., reprinted |location=Cambridge}}</ref>{{rp|113}}

===Thomson's atom model===
{{main| Plum pudding model}}
When Bohr began his work on a new atomic theory in the summer of 1912<ref name="Heilbron & Kuhn 1969"/>{{rp|237}} the atomic model proposed by [[J.&nbsp;J. Thomson]], now known as the plum pudding model, was the best available.<ref name="Heilbron1985">{{Cite book |last=John L |first=Heilbron |chapter-url=https://archive.org/details/nielsbohrcentena00fren/page/38/mode/2up |title=Niels Bohr: a centenary volume |date=1985 |publisher=Harvard University Press |isbn=978-0-674-62415-3 |editor-last=French |editor-first=A. P. |location=Cambridge, Mass |chapter=Bohr's First Theories of the Atom |editor-last2=Kennedy |editor-first2=P. J.}}</ref>{{rp|37}} 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 decay|beta particles]] that seemed to match experimental results.<ref name=Heilbron1985/>{{rp|38}}
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.<ref name=KraghQuantumAtom2012/>{{rp|18}}

===Rutherford nuclear model===
{{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 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>
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 [[electron]]s.<ref name="bohr1">{{Cite journal |last=Bohr |first=N. |date=July 1913 |title=I. On the constitution of atoms and molecules |url=https://zenodo.org/record/2493915 |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |volume=26 |issue=151 |pages=1–25 |doi=10.1080/14786441308634955|bibcode=1913PMag...26....1B }}</ref>

===Atomic spectra===
By the early twentieth century, it was expected that the atom would account for the many atomic spectral lines. These lines were summarized in empirical formula 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.

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.<ref name=PaisInwardBound/>{{rp|173|q=Ritz's principle was absolutely crucial to Bohr in his formulation of the quantum theory of spectra.}} 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.<ref name=Bohr1925>{{Cite journal |last=Bohr |first=N. |date=December 1925 |title=Atomic Theory and Mechanics1 |url=https://www.nature.com/articles/116845a0 |journal=Nature |language=en |volume=116 |issue=2927 |pages=845–852 |doi=10.1038/116845a0 |bibcode=1925Natur.116..845B |issn=0028-0836|url-access=subscription }}</ref>{{rp|847}}<ref name=Perovic2021>{{Cite book |last=Perović |first=Slobodan |title=From data to quanta: Niels Bohr's vision of physics |date=2021 |publisher=The University of Chicago press |isbn=978-0-226-79833-2 |location=Chicago London |chapter= Spectral Lines, Quantum States, and a Master Model of the Atom}}</ref><!-- Both Conway and Ritz discussed -->

=== 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}}
<math display="block">E_\text{pot}= \frac{- e^2}{a} = 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]]:
<math display="block">\frac{e^2}{a^2} = ma(2\pi f)^2</math>
where {{mvar|m}} is the mass of the electron. This combination relates the radius of the sphere to the Planck constant:
<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.
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, {{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 ===
The first [[Solvay Conference]], in 1911, was one of the first international physics conferences. Nine Nobel or future Nobel laureates attended, including
[[Ernest Rutherford]]<!-- second wikilink for key player -->, Bohr's mentor.<ref name=GilibertiLovisetti/>{{rp|271}} 
Bohr did not attend but he read the Solvay reports<ref name="aip.org">{{Cite interview |last=Bohr |first=Niels |subject-link=Niels Bohr |interviewer1=Thomas S. Kuhn |interviewer2=Leon Rosenfeld |interviewer3=Aage Petersen |interviewer4=Erik Rudinger |title=Niels Bohr – Session III |url=https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4517-3 |publisher=American Institute of Physics |date=7 November 1962}}</ref> and discussed them with Rutherford.<ref name="Heilbron & Kuhn 1969">{{Cite journal |last1=Heilbron |first1=John L. |last2=Kuhn |first2=Thomas S. |date=1969 |title=The Genesis of the Bohr Atom |journal=Historical Studies in the Physical Sciences |volume=1 |pages=vi–290 |doi=10.2307/27757291 |jstor=27757291}}</ref>{{rp|233}}

The subject of the conference was the theory of radiation and the energy quanta of [[Max Planck]]'s oscillators.<ref name="Heilbron2013">{{Cite journal |last=Heilbron |first=John L. |date=June 2013 |title=The path to the quantum atom |journal=Nature |volume=498 |issue=7452 |pages=27–30 |doi=10.1038/498027a |pmid=23739408 |s2cid=4355108}}</ref> 
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 atomic model|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.<ref name=GilibertiLovisetti>{{Cite book |last1=Giliberti |first1=Marco |chapter-url=https://link.springer.com/10.1007/978-3-031-57934-9_6 |title=Old Quantum Theory and Early Quantum Mechanics. Challenges in Physics Education. |last2=Lovisetti |first2=Luisa |date=2024 |publisher=Springer Nature Switzerland |isbn=978-3-031-57933-2 |location=Cham |language=en |chapter=Bohr's Hydrogen Atom |doi=10.1007/978-3-031-57934-9_6 }}</ref>{{rp|273}} 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.<ref name=GilibertiLovisetti/>{{rp|273}} 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.<ref name="Heilbron & Kuhn 1969"/>{{rp|244}} 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<!-- modern form is "the Planck constant", but this is a direct quote -->, or as it often is called the elementary quantum of action."<ref name="bohr1" /> Encouraged by the Solvay discussions, Bohr would  assume the atom was stable and abandon the efforts to stabilize classical models of the atom<ref name=PaisInwardBound/>{{rp|199}}

=== 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.<ref name="McCormmach1966">{{Cite journal |last=McCormmach |first=Russell |date=1 January 1966 |title=The atomic theory of John William Nicholson |journal=Archive for History of Exact Sciences |volume=3 |issue=2 |pages=160–184 |doi=10.1007/BF00357268 |jstor=41133258 |s2cid=120797894}}</ref>{{rp|169}}  In his 1913 paper Bohr cites Nicholson as finding quantized angular momentum important for the atom.<ref name="bohr1"/>

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 formula|Balmer's compact formula]] for the spectral line data, Bohr quickly realized his model would match it in detail.<ref name="McCormmach1966"/>{{rp|178}}

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.<ref name="McCormmach1966"/>{{rp|163}} 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.<ref name=PaisInwardBound/>{{rp|195}} Bohr's atomic model would abandon classical electrodynamics.

Nicholson's model of radiation was quantum but was attached to the orbits of the electrons.<ref name="Nicholson1912">{{Cite journal |last=Nicholson |first=J. W. |date=14 June 1912 |title=The Constitution of the Solar Corona. IL |journal=Monthly Notices of the Royal Astronomical Society |publisher=Oxford University Press |volume=72 |issue=8 |pages=677–693 |doi=10.1093/mnras/72.8.677 |issn=0035-8711|doi-access=free }}</ref><ref name=Heilbron2013/>{{rp|q=An oscillator could emit or absorb radiation when its natural frequency equalled that of the radiation (ν), and then only in energy increments (E = hν), or quanta}} 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.<ref name=PaisInwardBound/>{{rp|194}}

After his PhD, Bohr worked briefly in the lab of [[JJ Thomson]] before moving to Rutherford's lab in [[University of Manchester|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 where 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.<ref name=PaisInwardBound/>{{rp|195}}<!-- ref covers paragraph-->