In
atomic physics, the Bohr model or Rutherford–Bohr model, presented by
Niels Bohr
Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922 ...
and
Ernest Rutherford
Ernest Rutherford, 1st Baron Rutherford of Nelson, (30 August 1871 – 19 October 1937) was a New Zealand physicist who came to be known as the father of nuclear physics.
''Encyclopædia Britannica'' considers him to be the greatest ...
in 1913, is a system consisting of a small, dense nucleus surrounded by orbiting electrons—similar to the structure of the
Solar System
The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...
, but with attraction provided by
electrostatic force
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
s in place of
gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
. It came after the solar system
Joseph Larmor
Sir Joseph Larmor (11 July 1857 – 19 May 1942) was an Irish and British physicist and mathematician who made breakthroughs in the understanding of electricity, dynamics, thermodynamics, and the electron theory of matter. His most influent ...
model (1897), the solar system
Jean Perrin model (1901), the
cubical model (1902), the
Hantaro Nagaoka
was a Japanese physicist and a pioneer of Japanese physics during the Meiji period.
Life
Nagaoka was born in Nagasaki, Japan on August 19, 1865 and educated at the University of Tokyo. After graduating with a degree in physics in 1887, Naga ...
Saturnian model (1904), the
plum pudding model
The plum pudding model is one of several historical scientific models of the atom. First proposed by J. J. Thomson in 1904 soon after the discovery of the electron, but before the discovery of the atomic nucleus, the model tried to explain two pr ...
(1904), the quantum
Arthur Haas model (1910), the
Rutherford model
The Rutherford model was devised by the New Zealand-born physicist Ernest Rutherford to describe an atom. Rutherford directed the Geiger–Marsden experiment in 1909, which suggested, upon Rutherford's 1911 analysis, that J. J. Thomson's plum ...
(1911), and the nuclear quantum
John William Nicholson
John William Nicholson, FRS (1 November 1881 – 3 October 1955) was an English mathematician and physicist. Nicholson is noted as the first to create an atomic model that quantized angular momentum as h/2π. Nicholson was also the first to crea ...
model (1912). The improvement over the 1911 Rutherford model mainly concerned the new
quantum physical interpretation introduced by Haas and Nicholson, but forsaking any attempt to align with classical physics radiation.
The model's key success lay in explaining the
Rydberg formula
In atomic physics, the Rydberg formula calculates the wavelengths of a spectral line in many chemical elements. The formula was primarily presented as a generalization of the Balmer series for all atomic electron transitions of hydrogen. It wa ...
for the spectral
emission lines
A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to iden ...
of atomic
hydrogen
Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic ...
. 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 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 atom'' model. As a theory, it can be derived as a
first-order approximation
In science, engineering, and other quantitative disciplines, order of approximation refers to formal or informal expressions for how accurate an approximation is.
Usage in science and engineering
In formal expressions, the ordinal number used ...
of the hydrogen atom using the broader and much more accurate quantum mechanics and thus may be considered to be an
obsolete scientific theory
This list catalogs well-accepted theories in science and pre-scientific natural philosophy and natural history which have since been superseded by scientific theories. Many discarded explanations were once supported by a scientific consensus, b ...
. 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
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
or
energy level diagrams before moving on to the more accurate, but more complex,
valence shell
In chemistry and physics, a valence electron is an electron in the outer shell associated with an atom, and that can participate in the formation of a chemical bond if the outer shell is not closed. In a single covalent bond, a shared pair forms ...
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, ...
. A related quantum model was originally 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
The old quantum theory is a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics. The theory ...
.
Origin
![Atome bohr couches electroniques KLM](https://upload.wikimedia.org/wikipedia/commons/2/2c/Atome_bohr_couches_electroniques_KLM.svg)
In the early 20th century,
experiments by Ernest Rutherford established that
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, ...
s consisted of a diffuse cloud of negatively charged
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no ...
s surrounding a small, dense, positively charged
nucleus
Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to:
*Atomic nucleus, the very dense central region of an atom
* Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA
Nucl ...
.
Given this experimental data, Rutherford naturally considered a planetary model of the atom, the
Rutherford model
The Rutherford model was devised by the New Zealand-born physicist Ernest Rutherford to describe an atom. Rutherford directed the Geiger–Marsden experiment in 1909, which suggested, upon Rutherford's 1911 analysis, that J. J. Thomson's plum ...
of 1911. This had electrons orbiting a solar nucleus, but involved a technical difficulty: the laws of classical mechanics (i.e. the
Larmor formula
In electrodynamics, the Larmor formula is used to calculate the total power radiated by a nonrelativistic point charge as it accelerates. It was first derived by J. J. Larmor in 1897, in the context of the wave theory of light.
When any charged ...
) predict that the electron will release
electromagnetic radiation
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) li ...
while orbiting a nucleus. Because the electron would lose energy, it would rapidly spiral inwards, collapsing into the nucleus on a timescale of around 16
picoseconds. Rutherford's atom model is disastrous because it predicts that all atoms are unstable. Also, as the electron spirals inward, the emission would rapidly increase in frequency due to the
orbital period
The orbital period (also revolution period) is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting planets ...
becoming shorter, resulting in electromagnetic radiation with a continuous spectrum. However, late 19th-century experiments with
electric discharge
An electric discharge is the release and transmission of electricity in an applied electric field through a medium such as a gas (ie., an outgoing flow of electric current through a non-metal medium).American Geophysical Union, National Research C ...
s had shown that atoms will only emit light (that is, electromagnetic radiation) at certain discrete frequencies. By the early twentieth century, it was expected that the atom would account for the spectral lines. In 1897, Lord Rayleigh analyzed the problem. By 1906, Rayleigh said, “the frequencies observed in the spectrum may not be frequencies of disturbance or of oscillation in the ordinary sense at all, but rather form an essential part of the original constitution of the atom as determined by conditions of stability.”
The outline of Bohr's atom came during the proceedings of the first
Solvay Conference
The Solvay Conferences (french: Conseils Solvay) have been devoted to outstanding preeminent open problems in both physics and chemistry. They began with the historic invitation-only 1911 Solvay Conference on Physics, considered a turning point i ...
in 1911 on the subject of Radiation and Quanta, at which Bohr's mentor, Rutherford was present.
Max Planck
Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918.
Planck made many substantial contributions to theoretical p ...
’s lecture ended with this remark: “... atoms or electrons subject to the molecular bond would obey the laws of quantum theory”.
Hendrik Lorentz in the discussion of Planck's lecture raised the question of the composition of the atom based on Thomson's model with a great portion of the discussion around the atomic model developed by
Arthur Erich Haas. Lorentz explained that Planck's constant could be taken as determining the size of atoms, or that the size of atoms could be taken to determine Planck's constant. Lorentz included comments regarding the emission and absorption of radiation concluding that “A stationary state will be established in which the number of electrons entering their spheres is equal to the number of those leaving them.” In the discussion of what could regulate energy differences between atoms, Max Planck simply stated: “The intermediaries could be the electrons.” The discussions outlined the need for the quantum theory to be included in the atom and the difficulties in an atomic theory. Planck in his talk said explicitly: “In order for an oscillator
olecule or atomto be able to provide radiation in accordance with the equation, it is necessary to introduce into the laws of its operation, as we have already said at the beginning
of this Report, a particular physical hypothesis which is, on a fundamental point, in contradiction with classical Mechanics, explicitly or tacitly.” Bohr's first paper on his atomic model quotes Planck almost word for word, 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.” Bohr's footnote at the bottom of the page is to the French translation of the 1911 Solvay Congress proving he patterned his model directly on the proceedings and fundamental principles laid down by Planck, Lorentz, and the quantized
Arthur Haas model of the atom which was mentioned seventeen times.
Lorentz ended the discussion of Einstein's talk explaining: “The assumption that this energy must be a multiple of
leads to the following formula, where
is an integer:
.” Rutherford could have outlined these points to Bohr or given him a copy of the proceedings since he quoted from them and used them as a reference.
In a later interview, Bohr said it was very interesting to hear Rutherford's remarks about the Solvay Congress.
But Bohr said, “I saw the actual reports” of the Solvay Congress.
Then in 1912, Bohr came across the
John William Nicholson
John William Nicholson, FRS (1 November 1881 – 3 October 1955) was an English mathematician and physicist. Nicholson is noted as the first to create an atomic model that quantized angular momentum as h/2π. Nicholson was also the first to crea ...
theory of the atom model that quantized angular momentum as ''h''/2. According to a centennial celebration of the Bohr atom in Nature magazine, it was Nicholson who discovered that electrons radiate the spectral lines as they descend towards the nucleus and his theory was both nuclear and quantum.
Niels Bohr
Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922 ...
quoted him in his 1913 paper of the Bohr model of the atom. The importance of the work of Nicholson's nuclear quantum atomic model on Bohr's model has been emphasized by many historians.[
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
Johann Jakob Balmer (1 May 1825 – 12 March 1898) was a Swiss mathematician best known for his work in physics, the Balmer series of hydrogen atom.
Biography
Balmer was born in Lausen, Switzerland, the son of a chief justice also named Johan ...
in 1885 that described wavelengths of some spectral lines of hydrogen. This was further generalized by Johannes Rydberg
Johannes (Janne) Robert Rydberg (; 8 November 1854 – 28 December 1919) was a Swedish physicist mainly known for devising the Rydberg formula, in 1888, which is used to describe the wavelengths of photons (of visible light and other electrom ...
in 1888 resulting in what is now known as the Rydberg formula
In atomic physics, the Rydberg formula calculates the wavelengths of a spectral line in many chemical elements. The formula was primarily presented as a generalization of the Balmer series for all atomic electron transitions of hydrogen. It wa ...
.
After this, Bohr declared, “everything became clear”.[
To overcome the problems of Rutherford's atom, in 1913 ]Niels Bohr
Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922 ...
put forth three postulates that sum up most of his model:
# The electron is able to revolve in certain stable orbits around the nucleus without radiating any energy, contrary to what classical electromagnetism
Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fie ...
suggests. These stable orbits are called stationary orbits and are attained at certain discrete distances from the nucleus. The electron cannot have any other orbit in between the discrete ones.
# The stationary orbits are attained at distances for which the angular momentum of the revolving electron is an integer multiple of the reduced Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
: , where ''n'' = 1, 2, 3, ... is called the principal quantum number
In quantum mechanics, the principal quantum number (symbolized ''n'') is one of four quantum numbers assigned to each electron in an atom to describe that electron's state. Its values are natural numbers (from 1) making it a discrete variable.
A ...
, and . The lowest value of ''n'' is 1; this gives the smallest possible orbital radius of 0.0529 nm known as the Bohr radius
The Bohr radius (''a''0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an ...
. Once an electron is in this lowest orbit, it can get no closer to the nucleus. 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 level
A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The t ...
s. In these orbits, the electron's acceleration does not result in radiation and energy loss. The Bohr model of an atom was based upon Planck's quantum theory of radiation.
# Electrons can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency ''ν'' determined by the energy difference of the levels according to the Planck relation
The Planck relationFrench & Taylor (1978), pp. 24, 55.Cohen-Tannoudji, Diu & Laloë (1973/1977), pp. 10–11. (referred to as Planck's energy–frequency relation,Schwinger (2001), p. 203. the Planck relation, Planck equation, and Planck formula, ...
: , where ''h'' is Planck's constant.
Other points are:
# Like Einstein's theory of the photoelectric effect
The photoelectric effect is the emission of electrons when electromagnetic radiation, such as light, hits a material. Electrons emitted in this manner are called photoelectrons. The phenomenon is studied in condensed matter physics, and solid sta ...
, 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 theory of the electromagnetic field. Quantization of the electromagnetic field was explained by the discreteness of the atomic energy levels; Bohr did not believe in the existence of photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
s.
# According to the Maxwell theory the frequency ''ν'' of classical radiation is equal to the rotation frequency ''ν''rot of the electron in its orbit, with harmonics
A harmonic is a wave with a frequency that is a positive integer multiple of the '' fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', ...
at integer multiples of this frequency. This result is obtained from the Bohr model for jumps between energy levels ''E''''n'' and ''E''''n''−''k'' 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 ''n'' (so-called Rydberg state
The Rydberg states of an atom or molecule are electronically excited states with energies that follow the Rydberg formula as they converge on an ionic state with an ionization energy. Although the Rydberg formula was developed to describe atomic e ...
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 Bohr–Kramers–Slater theory (BKS theory) is a failed attempt to extend the Bohr model, which violates the conservation of energy and momentum in quantum jumps, with the conservation laws only holding on average.
Bohr's condition, that the angular momentum is an integer multiple of ''ħ'' was later reinterpreted in 1924 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:
:
According to de Broglie's hypothesis, matter particles such as the electron behave as waves
Waves most often refers to:
*Waves, oscillations accompanied by a transfer of energy that travel through space or mass.
*Wind waves, surface waves that occur on the free surface of bodies of water.
Waves may also refer to:
Music
*Waves (band) ...
. The de Broglie wavelength
Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave ...
of an electron is
:
which implies that
:
or
:
where is the angular momentum of the orbiting electron. Writing for this angular momentum, the previous equation becomes
:
which is Bohr's second postulate.
Bohr described angular momentum of the electron orbit as 1/2h while de Broglie's wavelength of described h divided by the electron momentum. In 1913, however, Bohr justified his rule by appealing to the correspondence principle, without providing any sort of wave interpretation. In 1913, the wave behavior of matter particles such as the electron was not suspected.
In 1925, a new kind of mechanics was proposed, quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, in which Bohr's model of electrons traveling in quantized orbits was extended into a more accurate model of electron motion. The new theory was proposed by Werner Heisenberg
Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent serie ...
. Another form of the same theory, wave mechanics, was discovered by the Austrian physicist Erwin Schrödinger
Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theo ...
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
A hydrogen-like atom (or hydrogenic atom) is any atom or ion with a single valence electron. These atoms are isoelectronic with hydrogen. Examples of hydrogen-like atoms include, but are not limited to, hydrogen itself, all alkali metals such a ...
, 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
Helium (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas and the first in the noble gas group in the periodic table. ...
, and doubly ionized lithium
Lithium (from el, λίθος, lithos, lit=stone) is a chemical element with the symbol Li and atomic number 3. It is a soft, silvery-white alkali metal. Under standard conditions, it is the least dense metal and the least dense solid ...
, but it includes positronium
Positronium (Ps) is a system consisting of an electron and its anti-particle, a positron, bound together into an exotic atom, specifically an onium. Unlike hydrogen, the system has no protons. The system is unstable: the two particles annih ...
and Rydberg states of any atom where one electron is far away from everything else. It can be used for K-line
K-Line Electric Trains is a brand name of O gauge and S gauge model railway locomotives, rolling stock, and buildings. Formerly the brand name under which Chapel Hill, North Carolina-based MDK Inc. sold its products, K-Line was then acquired ...
X-ray transition calculations if other assumptions are added (see Moseley's law
Moseley's law is an empirical law concerning the characteristic x-rays emitted by atoms. The law had been discovered and published by the English physicist Henry Moseley in 1913-1914. Until Moseley's work, "atomic number" was merely an element's ...
below). In high energy physics, it can be used to calculate the masses of heavy quark meson
In particle physics, a meson ( or ) is a type of hadronic subatomic particle composed of an equal number of quarks and antiquarks, usually one of each, bound together by the strong interaction. Because mesons are composed of quark subparticles, ...
s.
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.
::
:where ''m''e is the electron's mass, ''e'' is the elementary charge, ''k''e is the Coulomb constant and ''Z'' is the atom's atomic number
The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei, this is equal to the proton number (''n''p) or the number of protons found in the nucleus of every ...
. 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:
::
: It also determines the electron's total energy at any radius:
::
: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
In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by potential forces, with that of the total potential energy of the system. ...
.
* A quantum rule
:The angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
is an integer multiple of ''ħ'':
::
Derivation
If an electron in an atom is moving on an orbit with period ''T'', classically the electromagnetic radiation will repeat itself every orbital period. If the coupling to the electromagnetic field is weak, so that the orbit doesn't decay very much in one cycle, the radiation will be emitted in a pattern which repeats every period, so that the Fourier transform will have frequencies which are only multiples of 1/''T''. This is the classical radiation law: the frequencies emitted are integer multiples of 1/''T''.
In quantum mechanics, this emission must be in quanta of light, of frequencies consisting of integer multiples of 1/''T'', so that classical mechanics is an approximate description at large quantum numbers. This means that the energy level corresponding to a classical orbit of period 1/''T'' must have nearby energy levels which differ in energy by ''h''/''T'', and they should be equally spaced near that level,
:
Bohr worried whether the energy spacing 1/''T'' should be best calculated with the period of the energy state , or , or some average—in hindsight, this model is only the leading semiclassical approximation.
Bohr considered circular orbits. Classically, these orbits must decay to smaller circles when photons are emitted. The level spacing between circular orbits can be calculated with the correspondence formula. For a Hydrogen atom, the classical orbits have a period ''T'' determined by Kepler's third law
In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbi ...
to scale as ''r''3/2. The energy scales as 1/''r'', so the level spacing formula amounts to
:
It is possible to determine the energy levels by recursively stepping down orbit by orbit, but there is a shortcut.
The angular momentum ''L'' of the circular orbit scales as . The energy in terms of the angular momentum is then
:
Assuming, with Bohr, that quantized values of ''L'' are equally spaced, the spacing between neighboring energies is
:
This is as desired for equally spaced angular momenta. If one kept track of the constants, the spacing would be ''ħ'', so the angular momentum should be an integer multiple of ''ħ'',
:
This is how Bohr arrived at his model.
:Substituting the expression for the velocity gives an equation for ''r'' in terms of ''n'':
::
:so that the allowed orbit radius at any ''n'' is
::
:The smallest possible value of ''r'' in the hydrogen atom () is called the Bohr radius
The Bohr radius (''a''0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an ...
and is equal to:
::
:The energy of the ''n''-th level for any atom is determined by the radius and quantum number:
::
An electron in the lowest energy level of hydrogen () therefore has about 13.6 eV less energy than a motionless electron infinitely far from the nucleus. The next energy level () is −3.4 eV. The third (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
In mathematics, the Wallis product for , published in 1656 by John Wallis, states that
:\begin
\frac & = \prod_^ \frac = \prod_^ \left(\frac \cdot \frac\right) \\ pt& = \Big(\frac \cdot \frac\Big) \cdot \Big(\frac \cdot \frac\Big) \cdot \Big(\fr ...
.
The combination of natural constants in the energy formula is called the Rydberg energy (''R''E):
:
This expression is clarified by interpreting it in combinations that form more natural units
In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ma ...
:
: is the rest mass energy of the electron (511 keV),
: is the fine-structure constant
In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter ''alpha''), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between el ...
,
: .
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 , 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):
:
The actual energy levels cannot be solved analytically for more than one electron (see ''n''-body problem) because the electrons are not only affected by the nucleus
Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to:
*Atomic nucleus, the very dense central region of an atom
* Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA
Nucl ...
but also interact with each other via the Coulomb Force.
When ''Z'' = 1/''α'' (), the motion becomes highly relativistic, and ''Z''2 cancels the ''α''2 in ''R''; 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.
The Bohr formula properly uses the reduced mass
In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, however, that the mass ...
of electron and proton in all situations, instead of the mass of the electron,
:
However, these numbers are very nearly the same, due to the much larger mass of the proton, about 1836.1 times the mass of the electron, so that the reduced mass in the system is the mass of the electron multiplied by the constant 1836.1/(1+1836.1) = 0.99946. This fact was historically important in convincing Rutherford of the importance of Bohr's model, for it explained the fact that the frequencies of lines in the spectra for singly ionized helium do not differ from those of hydrogen by a factor of exactly 4, but rather by 4 times the ratio of the reduced mass for the hydrogen vs. the helium systems, which was much closer to the experimental ratio than exactly 4.
For positronium, the formula uses the reduced mass also, but in this case, it is exactly the electron mass divided by 2. 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.
: (positronium).
Rydberg formula
The Rydberg formula, which was known empirically before Bohr's formula, is seen in Bohr's theory as describing the energies of transitions or quantum jumps
Atomic electron transition is a change (or jump) of an electron from one energy level to another within an atom or artificial atom. It appears discontinuous as the electron "jumps" from one quantized energy level to another, typically in a few na ...
between orbital energy levels. Bohr's formula gives the numerical value of the already-known and measured the Rydberg constant
In spectroscopy, the Rydberg constant, symbol R_\infty for
heavy atoms or R_\text for hydrogen, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to the electromagnetic spectra of an atom. The constant first aro ...
, but in terms of more fundamental constants of nature, including the electron's charge and the Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
.
When the electron gets moved from its original energy level to a higher one, it then jumps back each level until it comes to the original position, which results in a photon being 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:
:
where is the final energy level, and is the initial energy level.
Since the energy of a photon is
:
the wavelength of the photon given off is given by
:
This is known as the Rydberg formula
In atomic physics, the Rydberg formula calculates the wavelengths of a spectral line in many chemical elements. The formula was primarily presented as a generalization of the Balmer series for all atomic electron transitions of hydrogen. It wa ...
, and the Rydberg constant is , or in natural units
In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ma ...
. 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 , until Bohr. In fact, Bohr's derivation of the Rydberg constant, as well as the concomitant agreement of Bohr's formula with experimentally observed spectral lines of the Lyman ( =1), Balmer ( =2), and Paschen ( =3) series, and successful theoretical prediction of other lines not yet observed, was one reason that his model was immediately accepted.
To apply to atoms with more than one electron, the Rydberg formula can be modified by replacing with or with where is constant representing a screening effect due to the inner-shell and other electrons (see Electron shell
In chemistry and atomic physics, an electron shell may be thought of as an orbit followed by electrons around an atom's nucleus. The closest shell to the nucleus is called the "1 shell" (also called the "K shell"), followed by the "2 shell" (or ...
and the later discussion of the "Shell Model of the Atom" below). This was established empirically before Bohr presented his model.
Shell model (heavier atoms)
Bohr's original three papers in 1913 described mainly the electron configuration in lighter elements. Bohr called his electron shells, “rings” in 1913. Atomic orbitals within shells did not exist at the time of his planetary model. Bohr explains in Part 3 of his famous 1913 paper that the maximum electrons in a shell is eight, writing: “We see, further, that a ring of ''n'' electrons cannot rotate in a single ring round a nucleus of charge ''ne'' unless ''n'' < 8.” For smaller atoms, the electron shells would be filled as follows: “rings of electrons will only join together if they contain equal numbers of electrons; and that accordingly the numbers of electrons on inner rings will only be 2, 4, 8”. However, in larger atoms the innermost shell would contain eight electrons, “on the other hand, the periodic system of the elements strongly suggests that already in neon ''N'' = 10 an inner ring of eight electrons will occur”. Bohr wrote "From the above we are led to the following possible scheme for the arrangement of the electrons in light atoms:"
In Bohr's third 1913 paper Part III called systems containing several nuclei, he says that two atoms form molecules on a symmetrical plane and he reverts to describing Hydrogen. The 1913 Bohr model did not discuss higher elements in detail and John William Nicholson
John William Nicholson, FRS (1 November 1881 – 3 October 1955) was an English mathematician and physicist. Nicholson is noted as the first to create an atomic model that quantized angular momentum as h/2π. Nicholson was also the first to crea ...
was one of the first to prove in 1914 that it couldn't work for Lithium, but was an attractive theory for Hydrogen and ionized helium.
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
Charles Rugeley Bury (29 June 1890 – 30 December 1968) was an English physical chemist who proposed an early model of the atom with the arrangement of electrons, which explained their chemical properties, alongside the more dominant model of Nie ...
[
Bohr's partner in research during 1914 to 1916 was ]Walther Kossel
Walther Ludwig Julius Kossel (4 January 1888 – 22 May 1956) was a German physicist known for his theory of the chemical bond (ionic bond/octet rule), Sommerfeld–Kossel displacement law of atomic spectra, the Kossel-Stranski model for crysta ...
who corrected Bohr's work to show that electrons interacted through the outer rings, and Kossel called the rings: “shells.” 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
The octet rule is a chemical rule of thumb that reflects the theory that main-group elements tend to bond in such a way that each atom has eight electrons in its valence shell, giving it the same electronic configuration as a noble gas. The rul ...
of 1904, although Kossel had already predicted a maximum of eight per shell in 1916. 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 Kossel is a surname. Notable people with the surname include:
*Albrecht Kossel (1853–1927), German biochemist and Nobel laureate
*Walther Kossel
Walther Ludwig Julius Kossel (4 January 1888 – 22 May 1956) was a German physicist known for his ...
, after that the orbit is full, the next level would have to be used.[ This gives the atom a 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 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.
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 1/2 the Bohr radius. The outermost electron in lithium orbits at roughly the Bohr radius, 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
Moseley's law is an empirical law concerning the characteristic x-rays emitted by atoms. The law had been discovered and published by the English physicist Henry Moseley in 1913-1914. Until Moseley's work, "atomic number" was merely an element's ...
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
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
of gases and density of pure crystalline solids. Atoms tend to get smaller toward the right in the periodic table, and become much larger 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
The noble gases (historically also the inert gases; sometimes referred to as aerogens) make up a class of chemical elements with similar properties; under standard conditions, they are all odorless, colorless, monatomic gases with very low ch ...
).
In the shell model, this phenomenon is explained by shell-filling. Successive atoms become 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, which 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
In chemistry, a transition metal (or transition element) is a chemical element in the d-block of the periodic table (groups 3 to 12), though the elements of group 12 (and less often group 3) are sometimes excluded. They are the elements that ca ...
). 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 (K-alpha X-ray emission lines)
Niels Bohr said in 1962: "You see actually the Rutherford work 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
Henry Gwyn Jeffreys Moseley (; 23 November 1887 – 10 August 1915) was an English physicist, whose contribution to the science of physics was the justification from physical laws of the previous empirical and chemical concept of the atomic num ...
found an empirical relationship between the strongest X-ray line emitted by atoms under electron bombardment (then known as the K-alpha Characteristic X-rays are emitted when outer- shell electrons fill a vacancy in the inner shell of an atom, releasing X-rays in a pattern that is "characteristic" to each element. Characteristic X-rays were discovered by Charles Glover Barkla in 190 ...
line), and their atomic number . Moseley's empiric formula was found to be derivable from Rydberg's formula and later Bohr's formula (Moseley actually mentions only Ernest Rutherford
Ernest Rutherford, 1st Baron Rutherford of Nelson, (30 August 1871 – 19 October 1937) was a New Zealand physicist who came to be known as the father of nuclear physics.
''Encyclopædia Britannica'' considers him to be the greatest ...
and Antonius Van den Broek in terms of models as these had been published before Moseley's work and Moseley's 1913 paper was published the same month as the first Bohr model paper). The two additional assumptions that '' this X-ray line came from a transition between energy levels with quantum numbers 1 and 2, and '', that the atomic number when used in the formula for atoms heavier than hydrogen, should be diminished by 1, to .
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.
It was Walther Kossel
Walther Ludwig Julius Kossel (4 January 1888 – 22 May 1956) was a German physicist known for his theory of the chemical bond (ionic bond/octet rule), Sommerfeld–Kossel displacement law of atomic spectra, the Kossel-Stranski model for crysta ...
in 1914 and in 1916 who explained that in the periodic table new elements would be created as electrons were added to the outer shell. In Kossel's paper, he writes: “This leads to the conclusion that the electrons, which are added further, should be put into concentric rings or shells, on each of which ... only a certain number of electrons—namely, eight in our case—should be arranged. As soon as one ring or shell is completed, a new one has to be started for the next element; the number of electrons, which are most easily accessible, and lie at the outermost periphery, increases again from element to element and, therefore, in the formation of each new shell the chemical periodicity is repeated.”[ Later, chemist Langmuir realized that the effect was caused by charge screening, with an inner shell containing only 2 electrons. In his 1919 paper, Irving Langmuir postulated the existence of "cells" which could each only contain two electrons each, and these were arranged in "equidistant layers”.
In the Moseley 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 an electron from 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,
:
or
:
Here, ''R''v = ''R''E/''h'' is the Rydberg constant, in terms of frequency equal to 3.28 x 1015 Hz. For values of Z between 11 and 31 this latter relationship had been empirically derived by Moseley, in a simple (linear) plot of the square root of X-ray frequency against atomic number (however, for silver, Z = 47, the experimentally obtained screening term should be replaced by 0.4). Notwithstanding its restricted validity, Moseley's law not only established the objective meaning of atomic number, 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 (place on the periodic table) standing for whole units of nuclear charge. Van den Broek had published his model in January 1913 showing the periodic table was arranged according to charge while Bohr's atomic model was not published until July 1913.
The ]K-alpha Characteristic X-rays are emitted when outer- shell electrons fill a vacancy in the inner shell of an atom, releasing X-rays in a pattern that is "characteristic" to each element. Characteristic X-rays were discovered by Charles Glover Barkla in 190 ...
line of Moseley's time is now known to be a pair of close lines, written as (Kα1 and Kα2) in Siegbahn notation The Siegbahn notation is used in X-ray spectroscopy to name the spectral lines that are characteristic to elements. It was introduced by Manne Siegbahn.
The characteristic lines in X-ray emission spectra correspond to atomic electronic transiti ...
.
Shortcomings
The Bohr model gives an incorrect value 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 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.
Nevertheless, in the modern ''fully quantum treatment in phase space'', the proper deformation (careful full extension) of the semi-classical result adjusts the angular momentum value to the correct effective one. As a consequence, the physical ground state expression is obtained through a shift of the vanishing quantum angular momentum expression, which corresponds to spherical symmetry.
In modern quantum mechanics, the electron in hydrogen is a 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).
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. Emission spectra for atoms with a single outer-shell electron (atoms in the lithium
Lithium (from el, λίθος, lithos, lit=stone) is a chemical element with the symbol Li and atomic number 3. It is a soft, silvery-white alkali metal. Under standard conditions, it is the least dense metal and the least dense solid ...
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
In atomic physics, the Rydberg formula calculates the wavelengths of a spectral line in many chemical elements. The formula was primarily presented as a generalization of the Balmer series for all atomic electron transitions of hydrogen. It wa ...
). 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
The Zeeman effect (; ) is the effect of splitting of a spectral line into several components in the presence of a static magnetic field. It is named after the Dutch physicist Pieter Zeeman, who discovered it in 1896 and received a Nobel priz ...
– changes in spectral lines due to external magnetic fields; these are also due to more complicated quantum principles interacting with electron spin and orbital magnetic fields.
* The model also violates the uncertainty principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
in that it considers electrons to have known orbits and locations, two things which cannot be measured simultaneously.
* 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.
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
:
where ''pr'' 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
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
of action-angle coordinates
In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequencies of oscillatory or rotational motion without solvin ...
. This condition, suggested by the correspondence principle, is the only one possible, since the quantum numbers are adiabatic invariant
A property of a physical system, such as the entropy of a gas, that stays approximately constant when changes occur slowly is called an adiabatic invariant. By this it is meant that if a system is varied between two end points, as the time for the ...
s.
The Bohr–Sommerfeld model was fundamentally inconsistent and led to many paradoxes. The magnetic quantum number
In atomic physics, the magnetic quantum number () is one of the four quantum numbers (the other three being the principal, azimuthal, and spin) which describe the unique quantum state of an electron. The magnetic quantum number distinguishes the ...
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 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 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 Schrödinger equation, wave mechanics, which Erwin Schrödinger
Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theo ...
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 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.
Bohr also updated his model in 1922, assuming that certain numbers of electrons (for example, 2, 8, and 18) correspond to stable "Electron configuration, closed shells".
Model of the chemical bond
Niels Bohr proposed a model of the atom and Bohr model of the chemical bond, a model of the chemical bond. According to his model for a diatomic molecule, the electrons of the atoms of the molecule form a rotating ring whose plane is perpendicular to the axis of the molecule and equidistant from the atomic nuclei. The dynamic equilibrium of the molecular system is achieved through the balance of forces between the forces of attraction of nuclei to the plane of the ring of electrons and the forces of mutual repulsion of the nuclei. The Bohr model of the chemical bond took into account the Coulomb repulsion – the electrons in the ring are at the maximum distance from each other.
See also
* 1913 in science
* Balmer's Constant
* Bohr–Sommerfeld model
* The Franck–Hertz experiment provided early support for the Bohr model.
* Free-fall atomic model
* The inert pair effect is adequately explained by means of the Bohr model.
* Introduction to quantum mechanics
* Theoretical and experimental justification for the Schrödinger equation
References
Footnotes
Primary sources
*
*
*
*
*
* 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
*
** Reprint:
*
*
*
* Klaus Hentschel: Elektronenbahnen, Quantensprünge und Spektren, in: Charlotte Bigg & Jochen Hennig (eds.) Atombilder. Ikonografien des Atoms in Wissenschaft und Öffentlichkeit des 20. Jahrhunderts, Göttingen: Wallstein-Verlag 2009, pp. 51–61
*
*
External links
Standing waves in Bohr’s atomic model
An interactive simulation to intuitively explain the quantization condition of standing waves in Bohr's atomic mode
{{DEFAULTSORT:Bohr Model
Atomic physics
Foundational quantum physics
Hydrogen physics
Niels Bohr
1913 in science