Bohr model
4831
225379665
2008-07-13T11:06:05Z
Swedish fusilier
5072303
[[WP:UNDO|Undid]] revision 225379554 by [[Special:Contributions/59.94.112.12|59.94.112.12]] ([[User talk:59.94.112.12|talk]])
[[Image:Bohr-atom-PAR.svg|thumb|right|310px|The '''Rutherford Bohr model''' of the [[hydrogen atom]] (<math>Z = 1</math>) or a hydrogen-like ion (<math>Z > 1</math>), where the negatively charged [[electron]] confined to an [[atomic shell]] encircles a small positively charged [[atomic nucleus]], and an electron jump between orbits is accompanied by an emitted or absorbed amount of [[electromagnetic energy]] <math>h \nu</math>. The orbits that the electron may travel in are shown as grey circles; their radius increases as <math>n^2</math>, where <math>n</math> is the [[principal quantum number]]. The <math>3 \rightarrow 2</math> transition depicted here produces the first line of the [[Balmer series]], and for hydrogen (<math>Z = 1</math>) results in a photon of [[wavelength]] 656 nm (red).]]
In [[atomic physics]], the '''Bohr model''' created by [[Niels Bohr]] depicts the [[atom]] as a small, positively charged [[atomic nucleus|nucleus]] surrounded by [[electron]]s that travel in circular orbits around the nucleus—similar in structure to the [[solar system]], but with [[electrostatic force]]s providing attraction, rather than [[gravity]]. This was an improvement on the earlier [[cubic atoms| cubic model]] (1902), the [[plum-pudding model]] (1904), the [[Saturnian model]] (1904), and the [[Rutherford model]] (1911). Since the Bohr model is a quantum physics-based modification of the Rutherford model, many sources combine the two, referring to the '''Rutherford-Bohr model'''.
Introduced by [[Niels Bohr]] in 1913, the model's key success lay in explaining the [[Rydberg formula]] for the spectral [[emission line]]s of atomic [[hydrogen]]; while the Rydberg formula had been known experimentally, it did not gain a theoretical underpinning until the Bohr model was introduced. Not only did the Bohr model explain the reason for the structure of the Rydberg formula, but it provided a justification for its empirical results in terms of fundamental physical constants.
The Bohr model is a primitive model of the hydrogen atom. 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, before moving on to the more accurate but more complex [[atomic orbitals|valence shell atom]]. A related model was originally proposed by [[Arthur Erich Haas]] in 1910, but was rejected. The quantum theory of the period between [[Planck#Black-body_radiation|Planck's discovery of the quantum]] (1900) and the advent of a full-blown [[quantum mechanics]] (1925) is often referred to as the [[Old quantum theory]].
==History==
In the early 20th century, experiments by [[Ernest Rutherford]] established that [[atom]]s consisted of a diffuse cloud of negatively charged [[electron]]s surrounding a small, dense, positively charged nucleus. Given this experimental data, it was quite natural for Rutherford to consider a planetary model for the atom, the [[Rutherford model]] of 1911, with electrons orbiting a sun-like nucleus. However, the planetary model for the atom has a difficulty. The laws of classical mechanics, specifically the [[Larmor formula]], predict that the electron will release [[electromagnetic radiation]] as it orbits a nucleus. Because the electron would be losing energy, it would gradually spiral inwards and collapse into the nucleus. This is a disaster, because it predicts that all matter is unstable.
Also, as the electron spirals inward, the emission would gradually increase in frequency as the orbit got smaller and faster. This would produce a continuous smear, in frequency, of electromagnetic radiation. However, late 19th century experiments with [[electric discharge]]s through various low-pressure [[gas]]ses in evacuated glass tubes had shown that atoms will only emit light (that is, electromagnetic radiation) at certain discrete frequencies.
To overcome this difficulty, [[Niels Bohr]] proposed, in 1913, what is now called the '''Bohr model of the atom'''. He suggested that electrons could only have certain ''classical'' motions:
# The electrons can only travel in special orbits: at a certain discrete set of distances from the nucleus with specific energies.
# The electrons do not continuously lose energy as they travel. They can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency <math>\nu</math> determined by the energy difference <math>\Delta E = E_2 - E_1</math> of the [[Bohr model#Electron energy levels|levels]] according to '''Bohr's formula''' <math>E_2 - E_1 =h\nu\,</math> where ''h'' is [[Planck's constant]].
# the frequency of the radiation emitted at an orbit with period T is as it would be in classical mechanics--- it is the reciprocal of the classical orbit period:
:::<math> \nu = {1\over T}</math>
The significance of the Bohr model is that the laws of classical mechanics apply to the motion of the electron about the nucleus ''only when restricted by a quantum rule''. Although rule 3 is not completely well defined for small orbits, because the emission process involves two orbits with two different periods, Bohr could determine the energy spacing between levels using rule 3 and come to an exactly correct quantum rule: the angular momentum L is restricted to be an integer multiple of a fixed unit:
::<math> L = n \cdot \hbar = n \cdot {h \over 2\pi} </math>
where ''n'' = 1,2,3,… and is called the [[principal quantum number]]. The lowest value of ''n'' is 1. This gives a smallest possible orbital radius of 0.0529 nm. This is known as the [[Bohr radius]]. Once an electron is in this lowest orbit, it can get no closer to the proton. Starting from the angular momentum quantum rule Bohr<ref>N. Bohr, ''Philosophical Magazine'' '''26''', 1-25 (1913) (a link to this article is provided below)</ref> was able to calculate the [[Bohr model#Electron energy levels|energies of the allowed orbits]] of the hydrogen atom and other [[Bohr model#Shell model of the atom|hydrogenlike]] atoms and ions.
Other points are:
# Like Einstein's theory of the [[Photoelectric effect]], Bohr's formula assumes that during a quantum jump a ''discrete'' amount of energy is radiated. However, unlike Einstein, Bohr stuck to the ''classical'' [[Maxwell's equations|Maxwell theory]] of the electromagnetic field. Quantization of the electromagnetic field was explained by the discreteness of the atomic [[Bohr model#Electron energy levels|energy levels]]; Bohr did not believe in the existence of [[photon]]s.
# According to the Maxwell theory the frequency <math>\nu</math> of classical radiation is equal to the rotation frequency <math>\nu_{rot}</math> of the electron in its orbit, with [[harmonics]] at integer multiples of this frequency. This result is obtained from the Bohr model for jumps between energy levels <math>E_n</math> and <math>E_{n-k}</math> when k is much smaller than n. These jumps reproduce the frequency of the k-th harmonic of orbit n. For sufficiently large values of <math>n</math> (so-called [[Rydberg state]]s), the two orbits involved in the emission process have nearly the same rotation frequency, so that the classical orbital frequency is not ambiguous. But for small n or large k, the radiation frequency has no unambiguous classical interpretation. This marks the birth of the [[correspondence principle]], requiring quantum theory to agree with the classical theory only in the limit of large quantum numbers.
# The [[BKS theory|Bohr-Kramers-Slater (BKS) theory]] is a failed attempt to extend the Bohr model which violates the conservation of [[Conservation of energy|energy]] and [[Momentum#Conservation of linear momentum|momentum]] in quantum jumps, with the conservation laws only holding on average.
Bohr's condition, that the angular momentum is an integer multiple of <math>\scriptstyle\hbar</math> was later reinterpreted by [[de Broglie]] as a [[standing wave]] condition: 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>
Substituting de Broglie's wavelength reproduces Bohr's rule. Bohr justified his rule by appealing to the correspondence principle, without providing a wave interpretation.
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 a [[matrix mechanics|more accurate model]] of electron motion. The new theory was proposed by [[Werner Heisenberg]]. [[Schrödinger equation|Another form]] of the same theory, modern quantum mechanics, was discovered by the Austrian physicist [[Erwin Schrödinger]] independently and by different reasoning.
== 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 includes one-electron systems such as the [[hydrogen atom]], singly-ionized [[helium]], 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 (spectrometry)|K-line]] X-ray transition calculations if other assumptions are added (see [[#Moseley.27s_law_and_calculation_of_K-alpha_X-ray_emission_lines|Moseley's law]] below). In high energy physics,
it can be used to calculate the masses of [[quark|heavy quark]] [[meson]]s.
To calculate the orbits requires two assumptions:
1. '''Classical mechanics'''
:The electron is held in a circular orbit by electrostatic attraction. The [[centripetal force]] is equal to the [[Coulomb law|Coulomb force]].
:::<math> {m_e v^2\over r} = {k e^2 \over r^2} </math>
:where <math>m_e</math> is the mass and <math>e</math> is the charge of the [[electron]]. This determines the speed at any radius:
::: <math> v = \sqrt{ k e^2 \over m_e r} </math>
: It also determines the total energy at any radius:
::: <math> E= {1\over 2} m_e v^2 - {k e^2 \over r} = - { k e^2 \over 2r} </math>
:The total energy is negative and inversely proportional to <math>r</math>. This means that it takes energy to pull the orbiting electron away from the proton. For infinite values of <math>r</math>, the energy is zero, corresponding to a motionless electron infinitely far from the proton. The total energy is half the [[potential energy]], which is true for non circular orbits too by the [[virial theorem]].
:For larger nuclei, the only change is that <math>k e^2</math> is everywhere replaced by <math>Zk e^2</math> where <math>Z</math> is the number of protons. For positronium, <math>m_e</math> is replaced by the [[reduced mass]] <math>m_e/2</math>.
2. '''Quantum rule'''
:The [[angular momentum]] <math>\scriptstyle L = m_e v r </math> is an integer multiple of <math>\scriptstyle \hbar</math>.
:::<math> m_e v r = n \hbar </math>
:substituting the expression for the velocity gives an equation for r in terms of n:
:::<math> \sqrt{ke^2 m_e r} = n </math>
:so that the allowed orbit radius at any n is:
:::<math> r_n = {n^2\hbar^2\over ke^2 m_e} </math>
:The smallest possible value of r
::: <math>r_1 = {\hbar^2 \over (ke^2) m_e} = .510 \cdot 10^{-10} m </math>
:is called the [[Bohr radius]].
:The energy of the n-th level is determined by the radius:
:::<math> E = -{ke^2 \over 2r_n } = - { (ke^2)^2 m_e \over 2\hbar^2 n^2} = {-13.6 \mathrm{eV} \over n^2} </math>
:So an electron in the lowest energy level of hydrogen (''n'' = 1) has 13.606 [[electronvolt|eV]] less energy than a motionless electron infinitely far from the nucleus. The next energy level at (''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 combination of natural constants in the energy formula is called the Rydberg energy <math>R_E</math>:
:: <math> R_E = { (k e^2)^2 m_e \over 2 \hbar^2} </math>
This expression is clarified by interpreting it in combinations which form more [[natural units]]:
:: <math>\, m_e c^2 </math> : the [[rest energy]] of the electron (= 511 keV)
:: <math>\, {k e^2 \over \hbar c} = \alpha \approx {1\over 137} </math> : the [[fine structure constant]]
:: <math>\, R_E = {1\over 2} (m_e c^2) \alpha^2</math>
For nuclei with Z protons, the energy levels are:
::<math> E_n = -{Z^2 R_E \over n^2} </math> (Heavy Nuclei)
When Z is approximately 137 (about '''1/α'''), the motion becomes highly relativistic. Then the <math>Z^2</math> cancels the <math>\alpha^2</math> in R, so the orbit energy begins to be comparable to rest energy. Sufficiently large nuclei, if they were stable, would reduce their charge by creating a bound electron from the vacuum, ejecting the positron to infinity. This is the theoretical phenomenon of electromagnetic charge screening which predicts a maximum nuclear charge. Emission of such positrons has been observed in the collisions of heavy ions to create temporary super-heavy nuclei{{Fact|date=April 2007}}.
For positronium, the formula uses the reduced mass. For any value of the radius, the electron and the positron are each moving at half the speed around their common center of mass, and each has only one fourth the kinetic energy. The total kinetic energy is half what it would be for a single electron moving around a heavy nucleus.
::<math> E_n = {R_E \over 2 n^2 } </math> (Positronium)
==Rydberg formula==
The [[Rydberg formula]], which was known empirically before Bohr's formula, is now in Bohr's theory seen as describing the energies of transitions or [[quantum jump]]s between one orbital energy level, and another. [[Bohr model#History|Bohr's formula]] gives the numerical value of the already-known and measured [[Rydberg's constant]], but now in terms of more fundamental constants of nature, including the electron's charge and [[Planck's constant]].
When the electron moves from one energy level to another, a [[photon]] is emitted. Using the derived formula for the different 'energy' levels of hydrogen one may determine the 'wavelengths' of light that a hydrogen atom can emit.
The energy of a photon emitted by a hydrogen atom is given by the difference of two hydrogen energy levels:
::<math>E=E_i-E_f=R_E \left( \frac{1}{n_{f}^2} - \frac{1}{n_{i}^2} \right) \,</math>
where ''n''<sub>''f''</sub> is the final energy level, and ''n''<sub>''i''</sub> is the initial energy level.
Since the energy of a [[photon]] is
::<math>E=\frac{hc}{\lambda}, \,</math>
the wavelength of the photon given off is given by
::<math>\frac{1}{\lambda}=R \left( \frac{1}{n_{f}^2} - \frac{1}{n_{i}^2} \right). \,</math>
This is known as the [[Rydberg formula]], and the Rydberg constant R is <math>R_E/hc</math>, or <math>R_E/2\pi</math> in [[natural units]]. This formula was known in the nineteenth century to scientists studying [[spectroscopy]], but there was no theoretical explanation for this form or a theoretical prediction for the value of R, until Bohr. In fact, Bohr's derivation of the Rydberg constant, as well as the concomitant agreement of [[Bohr model#History|Bohr's formula]] with experimentally observed spectral lines of the [[Lyman series|Lyman]] (<math>n_f = 1</math>), [[Balmer series|Balmer]] (<math>n_f = 2</math>), and [[Paschen series|Paschen]] (<math>n_f = 3</math>) series, and successful theoretical prediction of other lines not yet observed, was one reason that his model was immediately accepted.
== Shell model of the atom ==
Bohr extended the model of Hydrogen to give an approximate model for heavier atoms. This gave a physical picture which reproduced many known atomic properties for the first time.
Heavier atoms have more protons in the nucleus, and more electrons to cancel the charge. Bohr's idea was that each discrete orbit could only hold a certain number of electrons. After that orbit is full, the next level would have to be used. This gives the atom a [[electron configuration|shell structure]], 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 is 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 orbit the inner electrons, so the effective charge Z that they feel is reduced by the number of the electrons in the inner orbit.
For example, the lithium atom has two electrons in the lowest 1S orbit, and these orbit at Z=2. Each one sees the nuclear charge of Z=3 minus the screening effect of the other, which crudely reduces the nuclear charge by 1 unit. This means that the innermost electrons orbit at approximately 1/4th the Bohr radius. The outermost electron in lithium orbits at roughly Z=1, since the two inner electrons reduce the nuclear charge by 2. This outer electron should be at nearly one Bohr radius from the nucleus. Because the electrons strongly repel each other, the effective charge description is very approximate, the effective charge Z doesn't usually come out to be an integer. But [[Moseley's law]] experimentally probes the innermost pair of electrons, and shows that they do see a nuclear charge of approximately Z-1, while the outermost electron in an atom or ion with only one electron in the outermost shell orbits a core with effective charge Z-k where k is the total number of electrons in the inner shells.
The shell model was able to qualitatively explain many of the mysterious properties of atoms which became codified in the late 19th century in the [[periodic table of the elements]]. One property was the size of atoms, which could be determined approximately by measuring the [[viscosity]] of gases and density of pure crystalline solids. Atoms tend to get smaller as you move to the right in the periodic table, becoming much bigger at the next line of the table. Atoms to the right of the table tend to gain electrons, while atoms to the left tend to lose them. Every element on the last column of the table is chemically inert ([[noble gas]]).
In the shell model, this phenomenon is explained by shell-filling. Successive atoms get smaller because they are filling orbits of the same size, until the orbit is full, at which point the next atom in the table has a loosely bound outer electron, causing it to expand. The first Bohr orbit is filled when it has two electrons, and this explains why helium is inert. The second orbit allows eight electrons, and when it is full the atom is neon, again inert. The third orbital contains eight again, except that in the more correct Sommerfeld treatment (reproduced in modern quantum mechanics) there are extra "d" electrons. The third orbit may hold an extra 10 d electrons, but these positions are not filled until a few more orbitals from the next level are filled (Filling the n=3 d orbitals produces the 10 [[transition elements]]). The irregular filling pattern is an effect of interactions between electrons, which are not taken into account in either the Bohr or Sommerfeld models, and which are difficult to calculate even in the modern treatment.
== Moseley's law and calculation of K-alpha X-ray emission lines ==
Niels Bohr said in 1962, "You see actually the Rutherford work [the nuclear atom] was not taken seriously. We cannot understand today, but it was not taken seriously at all. There was no mention of it any place. The great change came from Moseley."
In 1913 [[Henry Moseley]] found an empirical relationship between the strongest X-ray line emitted by atoms under electron bombardment (then known as the [[K-alpha]] line), and their atomic number Z. Moseley's empiric formula was found to be derivable from Rydberg and Bohr's formula (Moseley actually mentions only [[Ernest Rutherford]] and [[Antonius Van den Broek]] in terms of models). The two additional assumptions that '''[1]''' this X-ray line came from a transition between energy levels with quantum numbers 1 and 2, and '''[2]''', that the atomic number Z when used in the formula for atoms heavier than hydrogen, should be diminished by 1, to (Z-1)².
Moseley wrote to Bohr, puzzled about his results, but Bohr was not able to help. At that time, he thought that the postulated innermost "K" shell of electrons should have at least four electrons, not the two which would have neatly explained the result. So Moseley published his results without a theoretical explanation.
Later, people realized that the effect was caused by charge screening, with an inner shell containing only 2 electrons. In the experiment, one of the innermost electrons in the atom is knocked out, leaving a vacancy in the lowest Bohr orbit, which contains a single remaining electron. This vacancy is then filled by electrons in the next orbit, which has n=2. But the n=2 electrons see an effective charge of Z-1, which is the value appropriate for the charge of the nucleus, when a single electron remains in the lowest Bohr orbit to screen the nuclear charge +Z, and lower it by -1 (due to the electron's negative charge screening the nuclear positive charge). The energy gained by an electron dropping from the second shell to the first gives [[Moseley's law]] for K-alpha lines:
::<math>E= h\nu = E_i-E_f=R_E (Z-1)^2 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) \,</math>
or
::<math>f = \nu = R_E/h = R_v \left( \frac{3}{4}\right) (Z-1)^2 = (2.46 \times 10^{15} \operatorname{Hz})(Z-1)^2.</math>
Here, '''R<sub>v</sub>''' is the Rydberg constant given in terms of frequency, or '''R<sub>E</sub>/h''' = 3.28 x 10<sup>15</sup> Hz. This latter relationship had been empirically derived by Moseley, in a simple plot of the square root of X-ray frequency against atomic number. Moseley's law not only established the objective meaning of atomic number (see [[Henry Moseley]] for detail) but, as Bohr noted, it also did more than the Rydberg derivation to establish the validity of the Rutherford/Van den Broek/Bohr nuclear model of the atom, with atomic number as nuclear charge.
The [[K-alpha]] line of Moseley's time is now known to be a pair of close lines, written as ('''K<sub>α1</sub>''' and '''K<sub>α2</sub>''') in [[Siegbahn notation]].
==Shortcomings==
The Bohr model gives an incorrect value <math>\scriptstyle \mathbf{L} = \hbar </math> for the ground state orbital angular momentum. The angular momentum in the true ground state is known to be zero. 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 rotate "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 doesn't point in any particular direction.
In modern quantum mechanics, the electron in hydrogen is a [[electron cloud| spherical cloud of probability]] which grows more dense 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." (Though many such coincidental agreements are found between the semi-classical 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-Sommerfield model (see below), and which happens to be equal to an entirely different concept, in full modern quantum mechanics).
The Bohr model also has difficulty with, or else fails to explain:
* Much of the spectra of larger atoms. At best, it can make predictions about the [[K-alpha]] and some L-alpha X-ray emission spectra for larger atoms, if ''two'' additional ad hoc assumptions are made (see [[Moseley's law]] above). Emission spectra for atoms with a single outer-shell electron (atoms in the [[lithium]] group) can also be approximately predicted. Also, if the empiric electron-nuclear screening factors for many atoms are known, many other spectral lines can be deduced from the information, in similar atoms of differing elements, via the Ritz-Rydberg combination principles (see [[Rydberg formula]]). All these techniques essentially make use of Bohr's Newtonian energy-potential picture of the atom.
* The relative intensities of spectral lines; although in some simple cases, Bohr's formula or modifications of it, was able to provide reasonable estimates (for example, calculations by Kramers for the [[Stark effect]]).
* The existence of [[fine structure]] and [[hyperfine structure]] in spectral lines, which are known to be due to a variety of relativistic and subtle effects, as well as complications from electron spin.
* The [[Zeeman effect]] - changes in spectral lines due to external [[magnetic field]]s; these are also due to more complicated quantum principles interacting with electron spin and orbital magnetic fields.
==Refinements==
[[Image:Sommerfeld ellipses.svg|thumb|Elliptical orbits with the same energy and quantized angular momentum]]
Several enhancements to the Bohr model were proposed; most notably the '''[[Old quantum theory|Sommerfeld model]]''' or '''Bohr-Sommerfeld model''', 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 '''Sommerfeld-Wilson quantization condition'''
:<math>
\int_0^T p_r dq_r = n h
\,</math>
where ''p<sub>r</sub>'' is the radial momentum canonically conjugate to the coordinate ''q'' which is the radial position and ''T'' is one full orbital period. The integral is the [[action (physics)|action]] of [[action-angle coordinates]]. This condition, suggested by the [[correspondence principle]], is the only one possible, since the quantum numbers are [[adiabatic invariant]]s.
The Bohr-Sommerfeld model was fundamentally inconsistent and led to many paradoxes. The [[azimuthal quantum number]] measured the tilt of the orbital plane relative to the x-y plane, and it could only take a few discrete values. This contradicted the obvious fact that an atom could be turned this way and that relative to the coordinates without restriction. The Sommerfeld quantization can be performed in different canonical coordinates, and sometimes gives answers which are different. 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 the modern [[quantum mechanics|quantum mechanical]] treatment of the [[hydrogen atom]], which was first given by [[Wolfgang Pauli]] in 1925, using [[Werner Heisenberg|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 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 [[perturbation theory|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 (mathematics)|connection]] of a [[Charles Hermite|Hermitian]] [[line bundle]], which is called a [[geometric quantization|prequantization]].
==See also==
<div style="-moz-column-count:3; column-count:3;">
*[[Franck-Hertz experiment]] provided early support for the Bohr model.
*[[Moseley's law]] provided early support for the Bohr model. See also [[Henry Moseley]]
*[[Inert pair effect]] is adequately explained by means of the Bohr model.
*[[Lyman series]]
*[[Schrödinger equation]]
*[[Theoretical and experimental justification for the Schrödinger equation]]
*[[Balmer's Constant]]
*[[Old quantum theory]]
*[[Quantum Mechanics]]
*[[1913 in science]]
</div>
==References==
<references/>
===Historical===
*{{cite journal | author=Niels Bohr | title=[http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Bohr/Bohr-1913a.html On the Constitution of Atoms and Molecules (Part 1 of 3)] | journal=Philosophical Magazine | year=1913 | volume=26 | pages=1–25 | doi=<!--This comment stops DOI bot adding the incorrect DOI-->}}
*{{cite journal | author=Niels Bohr | title=On the Constitution of Atoms and Molecules, Part II Systems Containing Only a Single Nucleus | journal=Philosophical Magazine | year=1913 | volume=26 | pages=476–502}}
*{{cite journal | author=Niels Bohr | title=On the Constitution of Atoms and Molecules, Part III | journal=Philosophical Magazine | year=1913 | volume=26 | pages=857–875}}
*{{cite journal | author=Niels Bohr | title=The spectra of helium and hydrogen | journal=Nature | year=1914 | volume=92 | pages=231–232 | doi=10.1038/092231d0}}
*{{cite journal | author=Niels Bohr | title=[http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Bohr-Nature-1921.html Atomic Structure] | journal=Nature | year=1921 | volume= | pages=}}
*{{cite journal | author=A. Einstein |title=Zum Quantensatz von Sommerfeld und Epstein |journal=Verhandlungen der Deutschen Physikalischen Gesellschaft | year=1917 | volume=19 |pages=82–92}} ''Reprinted in ''The Collected Papers of Albert Einstein'', A. Engel translator, (1997) Princeton University Press, Princeton. '''6''' p.434.'' ''(Provides an elegant reformulation of the Bohr-Sommerfeld quantization conditions, as well as an important insight into the quantization of non-integrable (chaotic) dynamical systems.)''
===Further reading===
*{{cite book | author=Linus Pauling | title=General Chemistry, Chapter 3 (3rd ed) | publisher=Dover Publications | year=1985 }} ''A great explainer of Chemistry describes the Bohr model, appropriate for High School and College students.''
*{{cite book | author=George Gamow | title=Thirty years that shook Physics, Chapter 2 | publisher=Dover Publications | year=1985 }}'' A popularizer of physics explains the Bohr model in the context of the development of quantum mechanics, appropriate for High School and College students''
*{{cite book | author=Walter J. Lehmann | title= Atomic and Molecular Structure: the development of our concepts, chapter 18 |publisher= John Wiley and Sons| year=1972}}'' Great explanations, appropriate for High School and College students''
*{{cite book | author=Paul Tipler and Ralph Llewellyn | title=Modern Physics (4th ed.) | publisher=W. H. Freeman | year=2002 | id=ISBN 0-7167-4345-0}}
[[Category:Atomic physics]]
[[Category:Foundational quantum physics]]
[[Category:Fundamental physics concepts]]
[[Category:Hydrogen physics]]
[[ar:نموذج بور]]
[[an:Modelo atomico de Bohr]]
[[bs:Bohrov model atoma]]
[[ca:Model atòmic de Bohr]]
[[cs:Bohrův model atomu]]
[[da:Bohrs atommodel]]
[[de:Bohrsches Atommodell]]
[[el:Ατομικό πρότυπο του Μπορ]]
[[es:Modelo atómico de Bohr]]
[[eu:Bohren eredu atomikoa]]
[[fa:مدل بور]]
[[fr:Modèle de Bohr]]
[[kk:Бор постулаттары]]
[[ko:보어 모형]]
[[hr:Bohrov model atoma]]
[[id:Model Bohr]]
[[it:Modello atomico di Bohr]]
[[he:מודל האטום של בוהר]]
[[lv:Bora atoma struktūras modelis]]
[[hu:Bohr-féle atommodell]]
[[nl:Atoommodel van Bohr]]
[[ja:ボーアの原子模型]]
[[no:Skallmodellen]]
[[nn:Atommodell]]
[[pl:Model atomu Bohra]]
[[pt:Átomo de Bohr]]
[[ro:Modelul atomic Bohr]]
[[ru:Постулаты Бора]]
[[sr:Боров модел атома]]
[[fi:Bohrin malli]]
[[sv:Bohrs atommodell]]
[[vi:Mô hình Bohr]]
[[tr:Bohr Atom Modeli]]
[[uk:Постулати Бора]]
[[zh:玻尔模型]]