How To Draw Atoms Using The Bohr

Diminutive model introduced by Niels Bohr in 1913

The cake model of the hydrogen atom ( Z = 1) or a hydrogen-similar ion ( Z > ane), where the negatively charged electron confined to an atomic shell encircles a pocket-sized, positively charged atomic nucleus and where an electron jumps between orbits, is accompanied past an emitted or absorbed corporeality of electromagnetic energy (hν).^[1] The orbits in which the electron may travel are shown as greyness circles; their radius increases equally n ², where n is the main quantum number. The iii → 2 transition depicted here produces the first line of the Balmer series, and for hydrogen ( Z = ane) it results in a photon of wavelength 656 nm (cherry calorie-free).

In atomic physics, the Bohr model or Rutherford–Bohr model, presented by Niels Bohr and Ernest Rutherford in 1913, is a organization consisting of a small, dense nucleus surrounded past orbiting electrons—like to the structure of the Solar System, merely with allure provided by electrostatic forces in place of gravity. After the solar system Joseph Larmor model (1897), the cubical model (1902), the Hantaro Nagaoka Saturnian model (1904), the plum pudding model (1904), the quantum Arthur Haas model (1910), the Rutherford model (1911), and the nuclear breakthrough John William Nicholson model (1912), came the Rutherford–Bohr model or just Bohr model for short (1913). The improvement over the 1911 Rutherford model mainly concerned the new quantum concrete estimation introduced past Haas and Nicholson, but forsaking whatever effort to align with classical physics radiations.

The model's key success lay in explaining the Rydberg formula for the spectral emission lines of atomic hydrogen. While the Rydberg formula had been known experimentally, information technology did not gain a theoretical underpinning until the Bohr model was introduced. Not only did the Bohr model explicate the reasons for the structure of the Rydberg formula, it also provided a justification for the key physical constants that make upwardly the formula'due south empirical results.

The Bohr model is a relatively archaic model of the hydrogen atom, compared to the valence shell cantlet model. As a theory, it tin 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 exist an obsolete scientific theory. However, because of its simplicity, and its right results for selected systems (see below for application), the Bohr model is notwithstanding commonly taught to introduce students to quantum mechanics or energy level diagrams before moving on to the more accurate, but more circuitous, valence shell atom. A related quantum model was originally proposed past Arthur Erich Haas in 1910 only was rejected until the 1911 Solvay Congress where it was thoroughly discussed.^[two] The quantum theory of the period between Planck's discovery of the quantum (1900) and the advent of a mature quantum mechanics (1925) is oft referred to as the old breakthrough theory.

Origin [edit]

Bohr model in 1921^[three] later on Sommerfeld expansion of 1913 model showing maximum electrons per shell with shells labeled in 10-ray notation

In the early 20th century, experiments past Ernest Rutherford established that atoms consisted of a diffuse cloud of negatively charged electrons surrounding a small, dense, positively charged nucleus.^[iv] Given this experimental data, Rutherford naturally considered a planetary model of the atom, the Rutherford model of 1911. This had electrons orbiting a solar nucleus, but involved a technical difficulty: the laws of classical mechanics (i.eastward. the Larmor formula) predict that the electron will release electromagnetic radiation while orbiting a nucleus. Because the electron would lose free energy, it would rapidly screw inwards, collapsing into the nucleus on a timescale of around sixteen picoseconds.^[5] Rutherford'due south atom model is disastrous because information technology predicts that all atoms are unstable.^[6] Also, as the electron spirals in, the emission would rapidly increase in frequency due to the orbital menstruation becoming shorter, resulting in electromagnetic radiation with a continuous spectrum. However, late 19th-century experiments with electric discharges had shown that atoms will only emit light (that is, electromagnetic radiations) at certain detached frequencies. By the early twentieth century, information technology 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 grade an essential part of the original constitution of the cantlet as determined by conditions of stability."^{[7] [8]}

The outline of Bohr's atom came during the proceedings of the start Solvay Conference in 1911 on the subject of Radiation and Quanta, at which Bohr'southward mentor, Rutherford was present. Max Planck's lecture ended with this remark: "… atoms or electrons subject to the molecular bond would obey the laws of quantum theory".^{[9] [10]} Hendrik Lorentz in the discussion of Planck'due south lecture raised the question of the limerick of the atom based on Thompson's model with a great portion of the discussion effectually the atomic model adult by Arthur Erich Haas. Lorentz explained that Planck'due south constant could be taken as determining the size of atoms, or that the size of atoms could exist taken to determine Planck's constant.^[xi] Lorentz included comments regarding the emission and assimilation of radiations terminal that "A stationary state volition be established in which the number of electrons entering their spheres is equal to the number of those leaving them."^[2] In the discussion of what could regulate energy differences between atoms, Max Planck simply stated: "The intermediaries could be the electrons."^[12] 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 society for an oscillator [molecule or cantlet] to exist able to provide radiation in accordance with the equation, it is necessary to introduce into the laws of its operation, as we accept already said at the outset of this Report, a item physical hypothesis which is, on a central betoken, in contradiction with classical Mechanics, explicitly or tacitly."^[xiii] Bohr'south first paper on his diminutive model quotes Planck about discussion for word, maxim: "Whatever the alteration in the laws of motility of the electrons may be, information technology seems necessary to introduce in the laws in question a quantity foreign to the classical electrodynamics, i. e. Planck'southward constant, or equally it often is called the unproblematic quantum of activeness." Bohr's footnote at the bottom of the page is to the French translation of the 1911 Solvay Congress proving he patterned his model straight on the proceedings and fundamental principles laid down past Planck, Lorentz, and the quantized Arthur Haas model of the cantlet which was mentioned seventeen times.^[4] Lorentz ended the discussion of Einstein's talk explaining: "The assumption that this free energy must be a multiple of ${\displaystyle h\nu }$ leads to the following formula, where ${\displaystyle n}$ is an integer: ${\displaystyle qv^{2}=nh\nu }$ ."^[14] Rutherford could have outlined these points to Bohr or given him a re-create of the proceedings since he quoted from them and used them as a reference.^[15] In a later on interview, Bohr said it was very interesting to hear Rutherford's remarks about the Solvay Congress.^[xvi] But Bohr said, "I saw the actual reports" of the Solvay Congress.^[17]

And so in 1912, Bohr came across the John William Nicholson theory of the cantlet model that quantized angular momentum as h/2π. Co-ordinate to a centennial celebration of the Bohr atom in Nature magazine, information technology was Nicholson who discovered that electrons radiate the spectral lines equally they descend towards the nucleus and his theory was both nuclear and quantum.^{[10] [xviii] [19]} Niels Bohr quoted him in his 1913 paper of the Bohr model of the atom.^[four] The importance of the piece of work of Nicholson'southward nuclear quantum atomic model on Bohr'south model has been emphasized by many historians.^{[20] [21] [nineteen] [22]}

Next, Bohr was told by his friend, Hans Hansen, that the Balmer series is calculated using the Balmer formula, an empirical equation discovered by Johann Balmer in 1885 that described wavelengths of some spectral lines of hydrogen.^{[23] [24]} This was further generalized by Johannes Rydberg in 1888 resulting in what is now known as the Rydberg formula. After this, Bohr declared, "everything became clear".^[24]

To overcome the problems of Rutherford'southward atom, in 1913 Niels Bohr put forth three postulates that sum up almost of his model:

1. The electron is able to revolve in sure stable orbits around the nucleus without radiating whatever energy, contrary to what classical electromagnetism suggests. These stable orbits are chosen stationary orbits and are attained at sure detached distances from the nucleus. The electron cannot have any other orbit in between the discrete ones.
2. 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: ${\displaystyle m_{\mathrm {east} }vr=n\hbar }$ , where n = 1, 2, 3, ... is called the principal breakthrough number, and ħ = h/twoπ . The everyman value of n is i; this gives the smallest possible orbital radius of 0.0529 nm known equally the Bohr radius. One time an electron is in this lowest orbit, information technology can become no closer to the proton. Starting from the angular momentum quantum rule every bit Bohr admits is previously given by Nicholson in his 1912 paper,^{[25] [10] [18] [19]} Bohr^[4] was able to summate the energies of the allowed orbits of the hydrogen atom and other hydrogen-like atoms and ions. These orbits are associated with definite energies and are also called energy shells or energy levels. In these orbits, the electron'south dispatch does not issue in radiation and energy loss. The Bohr model of an atom was based upon Planck's quantum theory of radiation.
3. Electrons can simply gain and lose free energy past jumping from ane allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency ν determined past the free energy departure of the levels according to the Planck relation: ${\displaystyle \Delta Due east=E_{ii}-E_{1}=h\nu }$ , where h is Planck's constant.

Other points are:

1. Similar Einstein's theory of the photoelectric effect, Bohr'due south formula assumes that during a quantum spring a discrete corporeality of energy is radiated. However, different 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 free energy levels; Bohr did not believe in the being of photons.^{[26] [27]}
2. 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 at integer multiples of this frequency. This result is obtained from the Bohr model for jumps between free energy levels E _n and Due east _n−k when m is much smaller than n. These jumps reproduce the frequency of the k-th harmonic of orbit due north. For sufficiently large values of north (and then-chosen Rydberg states), the two orbits involved in the emission process have most the aforementioned rotation frequency, so that the classical orbital frequency is not ambiguous. But for small-scale n (or large k), the radiation frequency has no unambiguous classical interpretation. This marks the birth of the correspondence principle, requiring quantum theory to concur with the classical theory merely in the limit of large breakthrough numbers.
3. 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 breakthrough jumps, with the conservation laws only holding on average.

Bohr's condition, that the athwart momentum is an integer multiple of ħ was later reinterpreted in 1924 past de Broglie equally 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:

${\displaystyle n\lambda =2\pi r.}$

According to de Broglie'due south hypothesis, matter particles such every bit the electron behave every bit waves. The de Broglie wavelength of an electron is

${\displaystyle \lambda ={\frac {h}{mv}},}$

which implies that

${\displaystyle {\frac {nh}{mv}}=two\pi r,}$

or

${\displaystyle {\frac {nh}{ii\pi }}=mvr,}$

where ${\displaystyle mvr}$ is the athwart momentum of the orbiting electron. Writing ${\displaystyle \ell }$ for this angular momentum, the previous equation becomes

${\displaystyle \ell ={\frac {nh}{2\pi }},}$

which is Bohr'south 2d postulate.

Bohr described angular momentum of the electron orbit as 1/2h while de Broglie's wavelength of λ = h/p 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 moving ridge interpretation. In 1913, the wave beliefs of matter particles such as the electron was not suspected.

In 1925, a new kind of mechanics was proposed, breakthrough mechanics, 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 past Werner Heisenberg. Another form of the aforementioned theory, moving ridge mechanics, was discovered past the Austrian physicist Erwin Schrödinger independently, and by different reasoning. Schrödinger employed de Broglie'south matter waves, but sought wave solutions of a iii-dimensional wave equation describing electrons that were constrained to motility nearly the nucleus of a hydrogen-like cantlet, by beingness trapped past the potential of the positive nuclear charge.

Electron energy levels [edit]

Models depicting electron energy levels in hydrogen, helium, lithium, and neon

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 lite. This not just involves one-electron systems such as the hydrogen atom, singly ionized helium, and doubly ionized lithium, just it includes positronium and Rydberg states of any atom where one electron is far away from everything else. It can exist used for K-line X-ray transition calculations if other assumptions are added (see Moseley's police force below). In loftier energy physics, it can exist used to calculate the masses of heavy quark mesons.

Adding 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.

${\displaystyle {\frac {m_{\mathrm {e} }5^{2}}{r}}={\frac {Zk_{\mathrm {due east} }e^{ii}}{r^{2}}},}$

where thousand _e is the electron's mass, due east is the accuse of the electron, k _e is the Coulomb constant and Z is the atom'due south atomic number. Information technology 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 whatever radius:

${\displaystyle v={\sqrt {\frac {Zk_{\mathrm {e} }east^{2}}{m_{\mathrm {e} }r}}}.}$

It as well determines the electron's total energy at any radius:

${\displaystyle E=-{\frac {one}{2}}m_{\mathrm {due east} }five^{ii}.}$

The total energy is negative and inversely proportional to r. This means that it takes free energy to pull the orbiting electron away from the proton. For infinite values of r, the energy is zero, respective to a motionless electron infinitely far from the proton. The total energy is half the potential energy, the difference being the kinetic free energy of the electron. This is also true for noncircular orbits by the virial theorem.

• A quantum dominion

The angular momentum L = one thousand _{due east} vr is an integer multiple of ħ:

${\displaystyle m_{\mathrm {e} }vr=n\hbar .}$

Derivation [edit]

If an electron in an atom is moving on an orbit with period T, classically the electromagnetic radiation volition echo 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 menstruum, so that the Fourier transform will have frequencies which are only multiples of 1/T. This is the classical radiations law: the frequencies emitted are integer multiples of 1/T.

In quantum mechanics, this emission must exist in quanta of light, of frequencies consisting of integer multiples of one/T, and then that classical mechanics is an approximate description at large quantum numbers. This means that the free energy level corresponding to a classical orbit of period 1/T must have nearby free energy levels which differ in energy by h/T, and they should be as spaced near that level,

${\displaystyle \Delta E_{north}={\frac {h}{T(E_{n})}}.}$

Bohr worried whether the energy spacing i/T should be best calculated with the period of the energy state ${\displaystyle E_{n}}$ , or ${\displaystyle E_{n+1}}$ , or some boilerplate—in retrospect, this model is only the leading semiclassical approximation.

Bohr considered circular orbits. Classically, these orbits must disuse 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 take a period T determined by Kepler's third police to scale every bit r ^three/2. The energy scales as 1/r, so the level spacing formula amounts to

${\displaystyle \Delta E\propto {\frac {1}{r^{3/2}}}\propto E^{3/2}.}$

It is possible to determine the energy levels by recursively stepping down orbit by orbit, but in that location is a shortcut.

The angular momentum L of the circular orbit scales as ${\displaystyle {\sqrt {r}}}$ . The free energy in terms of the angular momentum is then

${\displaystyle E\propto {\frac {1}{r}}\propto {\frac {1}{L^{two}}}.}$

Assuming, with Bohr, that quantized values of 50 are equally spaced, the spacing betwixt neighboring energies is

${\displaystyle \Delta Due east\propto {\frac {1}{(50+\hbar )^{2}}}-{\frac {1}{L^{ii}}}\approx -{\frac {2\hbar }{L^{3}}}\propto -E^{3/2}.}$

This is every bit desired for equally spaced angular momenta. If ane kept rail of the constants, the spacing would be ħ, and so the angular momentum should exist an integer multiple of ħ,

${\displaystyle L={\frac {nh}{2\pi }}=n\hbar .}$

This is how Bohr arrived at his model.

Substituting the expression for the velocity gives an equation for r in terms of due north:

${\displaystyle m_{\text{e}}{\sqrt {\dfrac {k_{\text{e}}Ze^{two}}{m_{\text{e}}r}}}r=due north\hbar ,}$

so that the immune orbit radius at whatever northward is

${\displaystyle r_{n}={\frac {n^{2}\hbar ^{2}}{Zk_{\mathrm {e} }eastward^{2}m_{\mathrm {e} }}}.}$

The smallest possible value of r in the hydrogen atom ( Z = 1) is chosen the Bohr radius and is equal to:

${\displaystyle r_{1}={\frac {\hbar ^{2}}{k_{\mathrm {due east} }e^{two}m_{\mathrm {e} }}}\approx v.29\times 10^{-11}~\mathrm {one thousand} .}$

The energy of the n-th level for whatsoever atom is determined by the radius and quantum number:

${\displaystyle E=-{\frac {Zk_{\mathrm {east} }due east^{two}}{2r_{north}}}=-{\frac {Z^{2}(k_{\mathrm {e} }eastward^{ii})^{two}m_{\mathrm {e} }}{2\hbar ^{ii}northward^{two}}}\approx {\frac {-thirteen.6Z^{ii}}{n^{2}}}~\mathrm {eV} .}$

An electron in the lowest energy level of hydrogen ( n = 1) therefore has about 13.6 eV less free energy than a motionless electron infinitely far from the nucleus. The adjacent free energy level ( n = two) is −iii.4 eV. The tertiary ( n = 3) is −1.51 eV, and 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 round orbit around the remainder of the cantlet. The hydrogen formula also coincides with the Wallis product.^[28]

The combination of natural constants in the energy formula is called the Rydberg free energy (R _E):

${\displaystyle R_{\mathrm {E} }={\frac {(k_{\mathrm {e} }due east^{two})^{ii}m_{\mathrm {e} }}{2\hbar ^{ii}}}.}$

This expression is clarified past interpreting it in combinations that course more natural units:

${\displaystyle m_{\mathrm {e} }c^{2}}$ is the rest mass energy of the electron (511 keV),

${\displaystyle {\frac {k_{\mathrm {e} }e^{2}}{\hbar c}}=\alpha \approx {\frac {i}{137}}}$ is the fine-structure constant,

${\displaystyle R_{\mathrm {Eastward} }={\frac {1}{ii}}(m_{\mathrm {e} }c^{2})\blastoff ^{2}}$ .

Since this derivation is with the supposition that the nucleus is orbited by one electron, we can generalize this result by letting the nucleus accept a accuse q = Ze , where Z is the atomic number. This will now give u.s.a. energy levels for hydrogenic (hydrogen-like) atoms, which can serve as a rough order-of-magnitude approximation of the bodily energy levels. Then for nuclei with Z protons, the free energy levels are (to a rough approximation):

${\displaystyle E_{n}=-{\frac {Z^{2}R_{\mathrm {E} }}{n^{2}}}.}$

The actual free energy levels cannot be solved analytically for more than than 1 electron (see n-body problem) because the electrons are not only afflicted by the nucleus but also interact with each other via the Coulomb Force.

When Z = 1/α ( Z ≈ 137), the motion becomes highly relativistic, and Z ² cancels the α ² in R; the orbit energy begins to be comparable to residual 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.^[29]

The Bohr formula properly uses the reduced mass of electron and proton in all situations, instead of the mass of the electron,

${\displaystyle m_{\text{red}}={\frac {m_{\mathrm {eastward} }m_{\mathrm {p} }}{m_{\mathrm {due east} }+m_{\mathrm {p} }}}=m_{\mathrm {east} }{\frac {1}{1+m_{\mathrm {eastward} }/m_{\mathrm {p} }}}.}$

However, these numbers are very most the same, due to the much larger mass of the proton, about 1836.1 times the mass of the electron, and then that the reduced mass in the arrangement is the mass of the electron multiplied past the constant 1836.1/(i+1836.1) = 0.99946. This fact was historically of import in disarming Rutherford of the importance of Bohr's model, for information technology explained the fact that the frequencies of lines in the spectra for singly ionized helium do non differ from those of hydrogen by a cistron of exactly 4, only rather by iv times the ratio of the reduced mass for the hydrogen vs. the helium systems, which was much closer to the experimental ratio than exactly iv.

For positronium, the formula uses the reduced mass also, but in this case, it is exactly the electron mass divided past 2. For any value of the radius, the electron and the positron are each moving at half the speed around their mutual centre of mass, and each has only one fourth the kinetic energy. The total kinetic energy is half what information technology would be for a single electron moving around a heavy nucleus.

${\displaystyle E_{n}={\frac {R_{\mathrm {E} }}{2n^{2}}}}$  (positronium).

Rydberg formula [edit]

The Rydberg formula, which was known empirically earlier Bohr'due south formula, is seen in Bohr'southward theory as describing the energies of transitions or quantum jumps betwixt orbital energy levels. Bohr'southward formula gives the numerical value of the already-known and measured the Rydberg constant, but in terms of more than fundamental constants of nature, including the electron'southward charge and the Planck constant.

When the electron gets moved from its original free energy level to a college one, it then jumps dorsum each level until information technology comes to the original position, which results in a photon being emitted. Using the derived formula for the different energy levels of hydrogen i may determine the wavelengths of light that a hydrogen atom tin can emit.

The energy of a photon emitted by a hydrogen atom is given by the difference of two hydrogen energy levels:

${\displaystyle E=E_{i}-E_{f}=R_{\text{E}}\left({\frac {ane}{n_{f}^{2}}}-{\frac {1}{n_{i}^{2}}}\right),}$

where due north _f is the final free energy level, and n _i is the initial free energy level.

Since the free energy of a photon is

${\displaystyle E={\frac {hc}{\lambda }},}$

the wavelength of the photon given off is given past

${\displaystyle {\frac {1}{\lambda }}=R\left({\frac {1}{n_{f}^{2}}}-{\frac {1}{n_{i}^{2}}}\right).}$

This is known as the Rydberg formula, and the Rydberg constant R is R _{Due east}/hc , or R _E/2π in natural units. This formula was known in the nineteenth century to scientists studying spectroscopy, but there was no theoretical caption for this course or a theoretical prediction for the value of R , until Bohr. In fact, Bohr'due south derivation of the Rydberg constant, every bit well as the concomitant understanding of Bohr's formula with experimentally observed spectral lines of the Lyman ( north_f =one), Balmer ( n_f =two), and Paschen ( north_f =3) serial, and successful theoretical prediction of other lines not nonetheless observed, was 1 reason that his model was immediately accustomed.

To apply to atoms with more than than one electron, the Rydberg formula tin be modified by replacing Z with Z −b or northward with n −b where b is abiding representing a screening consequence due to the inner-shell and other electrons (see Electron trounce and the later discussion of the "Vanquish Model of the Atom" beneath). This was established empirically earlier Bohr presented his model.

Trounce model (heavier atoms) [edit]

Bohr's original three papers in 1913 described mainly the electron configuration in lighter elements. Bohr called his electron shells, "rings" in 1913. Diminutive orbitals within shells did not exist at the fourth dimension of his planetary model. Bohr explains in Part iii 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 n_e unless n < eight." For smaller atoms, the electron shells would be filled equally 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 just be 2, 4, eight". Still, in larger atoms the innermost vanquish would comprise 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". See periodic table for more than about how Bohr congenital his 1913 model of electrons in elements

"From the to a higher place we are led to the following possible scheme for the arrangement of the electrons in light atoms:

Element	Electrons per shell
4	ii, 2
half dozen	2, 4
7	iv, three
eight	4, ii, ii
nine	4, 4, 1
10	8, 2
11	viii, 2, ane
16	8, four, 2, 2
18	viii, eight, 2

Periodic table of Bohr in 1913 showing electron configurations in his 2nd paper where he went to the 24th element.^{[30] [3] [xv]} In Bohr's tertiary 1913 paper Part Three called systems containing several nuclei, he says that ii atoms form molecules on a symmetrical plane and he reverts to describing Hydrogen.^[31] The 1913 Bohr model did non hash out college elements in detail and John William Nicholson was i of the first to bear witness in 1914 that it couldn't work for Lithium, simply was an attractive theory for Hydrogen and ionized helium.^{[15] [32]}

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 gauge model for heavier atoms. This gave a physical moving picture that reproduced many known atomic properties for the offset time although these properties were proposed contemporarily with the identical work of chemist Charles Rugeley Bury^{[3] [33]}

Bohr's partner in research during 1914 to 1916 was Walther Kossel who corrected Bohr's work to evidence that electrons interacted through the outer rings, and Kossel called the rings: "shells."^{[34] [35]} Irving Langmuir is credited with the first viable arrangement of electrons in shells with only 2 in the beginning shell and going up to eight in the next according to the octet dominion of 1904, although Kossel had already predicted a maximum of viii per crush in 1916.^[36] Heavier atoms have more protons in the nucleus, and more electrons to cancel the charge. Bohr took from these chemists the idea that each detached orbit could only hold a certain number of electrons. Per Kossel, after that the orbit is full, the next level would have to be used.^[iii] This gives the atom a beat out structure designed past Kossel, Langmuir, and Bury, in which each shell corresponds to a Bohr orbit.

This model is fifty-fifty more approximate than the model of hydrogen, because it treats the electrons in each crush as non-interacting. But the repulsions of electrons are taken into account somewhat by the miracle of screening. The electrons in outer orbits practice not just orbit the nucleus, only they also motility around the inner electrons, then the effective charge Z that they experience is reduced by the number of the electrons in the inner orbit.

For case, the lithium atom has two electrons in the lowest 1s orbit, and these orbit at Z = two. Each one sees the nuclear charge of Z = three minus the screening consequence 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 accuse by 2. This outer electron should be at about one Bohr radius from the nucleus. Because the electrons strongly repel each other, the constructive charge clarification is very judge; the constructive charge Z doesn't usually come out to exist an integer. But Moseley's law experimentally probes the innermost pair of electrons, and shows that they do encounter a nuclear charge of approximately Z − i, while the outermost electron in an atom or ion with simply one electron in the outermost vanquish orbits a cadre with constructive charge Z −one thousand where thousand is the full 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 tardily 19th century in the periodic tabular array of the elements. One property was the size of atoms, which could exist determined approximately by measuring the viscosity 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 terminal column of the table is chemically inert (noble gas).

In the crush model, this miracle is explained past vanquish-filling. Successive atoms get smaller because they are filling orbits of the aforementioned size, until the orbit is full, at which betoken the next atom in the tabular array has a loosely bound outer electron, causing it to expand. The first Bohr orbit is filled when it has ii electrons, which explains why helium is inert. The second orbit allows 8 electrons, and when it is full the cantlet is neon, again inert. The 3rd orbital contains eight again, except that in the more correct Sommerfeld handling (reproduced in modernistic quantum mechanics) at that place are extra "d" electrons. The third orbit may hold an actress 10 d electrons, but these positions are non filled until a few more orbitals from the next level are filled (filling the n=three d orbitals produces the x transition elements). The irregular filling pattern is an effect of interactions between electrons, which are non taken into account in either the Bohr or Sommerfeld models and which are difficult to calculate even in the modern treatment.

Moseley'south law and adding (K-alpha X-ray emission lines) [edit]

Niels Bohr said in 1962: "You lot see actually the Rutherford work was not taken seriously. We cannot understand today, just information technology was not taken seriously at all. There was no mention of it whatsoever place. The great change came from Moseley."^[37]

In 1913, Henry Moseley found an empirical relationship betwixt the strongest X-ray line emitted by atoms under electron bombardment (then known as the K-alpha line), and their atomic number Z . Moseley'southward empiric formula was found to be derivable from Rydberg'due south formula and later Bohr's formula (Moseley actually mentions but Ernest Rutherford and Antonius Van den Broek in terms of models as these had been published before Moseley's work and Moseley's 1913 newspaper was published the same calendar month as the showtime Bohr model paper).^[38] The ii additional assumptions that [1] this X-ray line came from a transition between energy levels with breakthrough numbers i and 2, and [ii], that the atomic number Z when used in the formula for atoms heavier than hydrogen, should be macerated by i, to (Z − one)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 to the lowest degree four electrons, non the two which would accept neatly explained the result. Then Moseley published his results without a theoretical explanation.

It was Walther Kossel in 1914 and in 1916 who explained that in the periodic table new elements would be created as electrons were added to the outer beat. In Kossel's paper, he writes: "This leads to the conclusion that the electrons, which are added further, should exist put into concentric rings or shells, on each of which ... only a certain number of electrons—namely, eight in our case—should be bundled. As presently as i ring or beat is completed, a new one has to be started for the side by side 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 trounce the chemical periodicity is repeated."^{[34] [35]} Later, chemist Langmuir realized that the upshot was caused by charge screening, with an inner shell containing but 2 electrons. In his 1919 paper, Irving Langmuir postulated the being of "cells" which could each but contain two electrons each, and these were arranged in "equidistant layers".

In the Moseley experiment, one of the innermost electrons in the cantlet is knocked out, leaving a vacancy in the everyman Bohr orbit, which contains a single remaining electron. This vacancy is so filled by an electron from the next orbit, which has n=2. But the n=2 electrons see an effective accuse of Z − 1, which is the value advisable for the charge of the nucleus, when a unmarried electron remains in the everyman Bohr orbit to screen the nuclear charge +Z, and lower information technology by −1 (due to the electron's negative charge screening the nuclear positive accuse). The free energy gained by an electron dropping from the 2nd crush to the outset gives Moseley's law for 1000-alpha lines,

${\displaystyle E=h\nu =E_{i}-E_{f}=R_{\mathrm {E} }(Z-1)^{2}\left({\frac {i}{1^{2}}}-{\frac {ane}{ii^{2}}}\right),}$

or

${\displaystyle f=\nu =R_{\mathrm {v} }\left({\frac {iii}{4}}\right)(Z-one)^{2}=(2.46\times 10^{15}~{\text{Hz}})(Z-1)^{two}.}$

Here, R _v = R _E/h is the Rydberg constant, in terms of frequency equal to iii.28 ten 10¹⁵ Hz. For values of Z betwixt 11 and 31 this latter relationship had been empirically derived past Moseley, in a simple (linear) plot of the square root of X-ray frequency against diminutive number (however, for silverish, Z = 47, the experimentally obtained screening term should be replaced by 0.4). However its restricted validity,^[39] Moseley's law not merely established the objective meaning of diminutive number, but equally Bohr noted, information technology also did more than than the Rydberg derivation to institute 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 Jan 1913 showing the periodic table was bundled according to accuse while Bohr's atomic model was not published until July 1913.^[40]

The 1000-alpha line of Moseley's time is now known to be a pair of close lines, written equally (Kα₁ and Kα₂ ) in Siegbahn note.

Shortcomings [edit]

The Bohr model gives an wrong value L=ħ for the ground land orbital athwart momentum: The angular momentum in the true basis land is known to be zero from experiment.^[41] Although mental pictures fail somewhat at these levels of scale, an electron in the lowest modern "orbital" with no orbital momentum, may exist thought of as not to rotate "around" the nucleus at all, just 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 similar Sommerfeld's. Even so, fifty-fifty the most sophisticated semiclassical model fails to explain the fact that the lowest free energy land is spherically symmetric – information technology doesn't point in any detail direction.

Nevertheless, in the modern fully quantum treatment in phase infinite, the proper deformation (conscientious full extension) of the semi-classical upshot adjusts the athwart momentum value to the right effective one.^[42] As a consequence, the physical ground state expression is obtained through a shift of the vanishing breakthrough athwart 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-abiding of probability-decay in hydrogen is equal to the inverse of the Bohr radius, just since Bohr worked with circular orbits, non zero area ellipses, the fact that these ii numbers exactly agree is considered a "coincidence". (However, many such coincidental agreements are constitute betwixt the semiclassical vs. full quantum mechanical treatment of the atom; these include identical energy levels in the hydrogen cantlet and the derivation of a fine-structure constant, which arises from the relativistic Bohr–Sommerfeld model (run into below) and which happens to be equal to an entirely different concept, in full modernistic quantum mechanics).

The Bohr model also has difficulty with, or else fails to explicate:

• Much of the spectra of larger atoms. At best, information technology can make predictions near the 1000-alpha and some Fifty-alpha 10-ray emission spectra for larger atoms, if two boosted advertisement hoc assumptions are made. Emission spectra for atoms with a single outer-shell electron (atoms in the lithium group) tin also be approximately predicted. Also, if the empiric electron–nuclear screening factors for many atoms are known, many other spectral lines can exist deduced from the information, in like atoms of differing elements, via the Ritz–Rydberg combination principles (encounter Rydberg formula). All these techniques essentially make use of Bohr'due south Newtonian energy-potential picture of the atom.
• the relative intensities of spectral lines; although in some simple cases, Bohr'south formula or modifications of it, was able to provide reasonable estimates (for case, calculations by Kramers for the Stark effect).
• The beingness 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 every bit complications from electron spin.
• The Zeeman effect – changes in spectral lines due to external magnetic fields; these are also due to more than complicated quantum principles interacting with electron spin and orbital magnetic fields.
• The model also violates the doubt principle in that it considers electrons to have known orbits and locations, ii 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 free energy levels should be very close together.
• Multi-electron atoms do not have energy levels predicted past the model. It does not work for (neutral) helium.

Refinements [edit]

Elliptical orbits with the same free energy and quantized angular momentum

Several enhancements to the Bohr model were proposed, about 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.^[1] This model supplemented the quantized angular momentum condition of the Bohr model with an boosted radial quantization status, the Wilson–Sommerfeld quantization condition^{[43] [44]}

${\displaystyle \int _{0}^{T}p_{r}\,dq_{r}=nh,}$

where p_r is the radial momentum canonically cohabit to the coordinate q, which is the radial position, and T is ane total orbital flow. The integral is the action of action-bending coordinates. This condition, suggested by the correspondence principle, is the merely one possible, since the quantum numbers are adiabatic invariants.

The Bohr–Sommerfeld model was fundamentally inconsistent and led to many paradoxes. The magnetic quantum number measured the tilt of the orbital plane relative to the xy airplane, and it could merely 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 brake. The Sommerfeld quantization can be performed in different canonical coordinates and sometimes gives different answers. The incorporation of radiation corrections was hard, because information technology required finding activeness-angle coordinates for a combined radiations/cantlet arrangement, which is difficult when the radiation is allowed to escape. The whole theory did not extend to not-integrable motions, which meant that many systems could not exist treated even in principle. In the terminate, 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 movie of the hydrogen atom is based on the atomic orbitals of wave mechanics, which Erwin Schrödinger developed in 1926.

Withal, 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, upward to offset-club perturbations, the Bohr model and quantum mechanics make the same predictions for the spectral line splitting in the Stark effect. At higher-social club perturbations, however, the Bohr model and quantum mechanics differ, and measurements of the Stark effect nether loftier field strengths helped confirm the definiteness of breakthrough mechanics over the Bohr model. The prevailing theory behind this divergence lies in the shapes of the orbitals of the electrons, which vary according to the free energy land of the electron.

The Bohr–Sommerfeld quantization conditions lead to questions in modern mathematics. Consistent semiclassical quantization status requires a certain blazon of structure on the phase space, which places topological limitations on the types of symplectic manifolds which tin can be quantized. In particular, the symplectic form should be the curvature form of a connexion of a Hermitian line bundle, which is called a prequantization.

Bohr also updated his model in 1922, bold that certain numbers of electrons (for example, two, viii, and xviii) correspond to stable "airtight shells".^[45]

Model of the chemical bond [edit]

Niels Bohr proposed a model of the atom and a model of the chemical bond. According to his model for a diatomic molecule, the electrons of the atoms of the molecule class a rotating ring whose plane is perpendicular to the axis of the molecule and equidistant from the diminutive nuclei. The dynamic equilibrium of the molecular system is accomplished through the balance of forces between the forces of attraction of nuclei to the plane of the band of electrons and the forces of mutual repulsion of the nuclei. The Bohr model of the chemic bond took into account the Coulomb repulsion – the electrons in the ring are at the maximum distance from each other.^{[46] [47]}

Encounter too [edit]

• 1913 in scientific discipline
• Balmer's Constant
• The Franck–Hertz experiment provided early back up 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 [edit]

Footnotes [edit]

1. ^ ^{a b} Lakhtakia, Akhlesh; Salpeter, Edwin Due east. (1996). "Models and Modelers of Hydrogen". American Periodical of Physics. 65 (9): 933. Bibcode:1997AmJPh..65..933L. doi:x.1119/ane.18691.
2. ^ ^{a b} de Broglie et al. 1912, pp. 122–123.
3. ^ ^{a b c d} Kragh, Helge (1 January 1979). "Niels Bohr'south Second Diminutive Theory". Historical Studies in the Concrete Sciences. 10: 123–186. doi:10.2307/27757389. JSTOR 27757389.
4. ^ ^{a b c d} Bohr, N. (July 1913). "I. On the constitution of atoms and molecules". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 26 (151): 1–25. doi:ten.1080/14786441308634955.
5. ^ Olsen, James D.; McDonald, Kirk T. (2005). "Classical lifetime of a bohr atom" (PDF). ^{[ self-published source? ]}
6. ^ "CK12 – Chemistry Flexbook Second Edition – The Bohr Model of the Cantlet". Retrieved 30 September 2014.
7. ^ Kragh, Helge (2012). Niels Bohr and the Breakthrough Cantlet: The Bohr Model of Diminutive Structure 1913-1925. OUP Oxford. p. 18. ISBN978-0-xix-163046-0.
8. ^ Rayleigh, Lord (Jan 1906). "Seven. On electric vibrations and the constitution of the cantlet". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 11 (61): 117–123. doi:10.1080/14786440609463428.
9. ^ de Broglie et al. 1912, p. 114.
10. ^ ^{a b c} Heilbron, John 50. (June 2013). "The path to the breakthrough cantlet". Nature. 498 (7452): 27–xxx. doi:ten.1038/498027a. PMID 23739408. S2CID 4355108.
11. ^ de Broglie et al. 1912, p. 124.
12. ^ de Broglie et al. 1912, p. 127.
13. ^ de Broglie et al. 1912, p. 109.
14. ^ de Broglie et al. 1912, p. 447.
15. ^ ^{a b c} Heilbron, John L.; Kuhn, Thomas S. (1969). "The Genesis of the Bohr Atom". Historical Studies in the Physical Sciences. 1: vi–290. doi:10.2307/27757291. JSTOR 27757291.
16. ^ Niels Bohr interview 1962 Session Three https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4517-iii
17. ^ Niels Bohr interview 1962 Session Two https://world wide web.aip.org/history-programs/niels-bohr-library/oral-histories/4517-2
18. ^ ^{a b} J. W. Nicholson, Month. Not. Roy. Astr. Soc. lxxii. pp. 49,130, 677, 693, 729 (1912).^{[ title missing ]}
19. ^ ^{a b c} McCormmach, Russell (i January 1966). "The diminutive theory of John William Nicholson". Archive for History of Exact Sciences. 3 (2): 160–184. doi:x.1007/BF00357268. JSTOR 41133258. S2CID 120797894.
20. ^ Hirosige, Tetu; Nisio, Sigeko (1964). "Formation of Bohr'southward theory of diminutive constitution". Japanese Studies in the History of Scientific discipline (3): 6–28. OCLC 1026682346.
21. ^ Heilbron, J. L. (1964). A History of Diminutive Models from the Discovery of the Electron to the Ancestry of Quantum Mechanics (Thesis).
22. ^ Wilson, William (Nov 1956). "John William Nicholson, 1881-1955". Biographical Memoirs of Fellows of the Imperial Society. 2: 209–214. doi:10.1098/rsbm.1956.0014.
23. ^ Niels Bohr interview 1962 Session Three https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4517-3
24. ^ ^{a b} Bohr, Niels; Rosenfeld, Léon Jacques Henri Abiding (1963). On the Constitution of Atoms and Molecules ... Papers of 1913 reprinted from the Philosophical Magazine, with an introduction by Fifty. Rosenfeld. Copenhagen; Due west.A. Benjamin: New York. OCLC 557599205. ^{[ folio needed ]}
25. ^ Niels Bohr interview 1962 Session 3 https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4517-3
26. ^ Stachel, John (2009). "Bohr and the Photon". Quantum Reality, Relativistic Causality, and Endmost the Epistemic Circle. Dordrecht: Springer. p. 79.
27. ^ Louisa Gilder, "The Age of Entanglement" The Arguments 1922 p. 55, "Well, yes," says Bohr. "But I tin hardly imagine it will involve light quanta. Expect, even if Einstein had plant an unassailable proof of their beingness and would want to inform me by telegram, this telegram would only reach me because of the beingness and reality of radio waves." 2009.
28. ^ "Revealing the subconscious connectedness between pi and Bohr's hydrogen model". Physics World (November 17, 2015).
29. ^ Müller, U.; de Reus, T.; Reinhardt, J.; Müller, B.; Greiner, West. (1988-03-01). "Positron production in crossed beams of bare uranium nuclei". Physical Review A. 37 (5): 1449–1455. Bibcode:1988PhRvA..37.1449M. doi:10.1103/PhysRevA.37.1449. PMID 9899816. S2CID 35364965.
30. ^ Bohr, N. (September 1913). "XXXVII. On the constitution of atoms and molecules". The London, Edinburgh, and Dublin Philosophical Magazine and Periodical of Scientific discipline. 26 (153): 476–502. doi:x.1080/14786441308634993.
31. ^ Bohr, N. (1 November 1913). "LXXIII. On the constitution of atoms and molecules". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 26 (155): 857–875. doi:10.1080/14786441308635031.
32. ^ Nicholson, J. West. (May 1914). "The Constitution of Atoms and Molecules". Nature. 93 (2324): 268–269. Bibcode:1914Natur..93..268N. doi:10.1038/093268a0. S2CID 3977652.
33. ^ Bury, Charles R. (July 1921). "Langmuir's Theory of the System of Electrons in Atoms and Molecules". Journal of the American Chemical Society. 43 (seven): 1602–1609. doi:x.1021/ja01440a023.
34. ^ ^{a b} Kossel, W. (1916). "Über Molekülbildung als Frage des Atombaus" [On molecular germination as a question of atomic structure]. Annalen der Physik (in German). 354 (3): 229–362. Bibcode:1916AnP...354..229K. doi:ten.1002/andp.19163540302.
35. ^ ^{a b} Kragh, Helge (2012). "Lars Vegard, atomic construction, and the periodic organisation" (PDF). Bulletin for the History of Chemistry. 37 (i): 42–49. OCLC 797965772. S2CID 53520045.
36. ^ Langmuir, Irving (June 1919). "The Organisation of Electrons in Atoms and Molecules". Journal of the American Chemical Society. 41 (half-dozen): 868–934. doi:10.1021/ja02227a002.
37. ^ "Interview of Niels Bohr past Thomas S. Kuhn, Leon Rosenfeld, Erik Rudinger, and Aage Petersen". Niels Bohr Library & Athenaeum, American Found of Physics. 31 October 1962. Retrieved 27 Mar 2019.
38. ^ Moseley, H.K.J. (1913). "The high-frequency spectra of the elements". Philosophical Mag. 6th serial. 26: 1024–1034.
39. ^ K.A.B. Whitaker (1999). "The Bohr–Moseley synthesis and a simple model for diminutive x-ray energies". European Journal of Physics. 20 (iii): 213–220. Bibcode:1999EJPh...20..213W. doi:x.1088/0143-0807/20/iii/312.
40. ^ A. van den Broek, "Die Radioelemente, das periodische Arrangement und die Konstitution der. Atome," Phys. Zeits, xiv, 32-41 (January. 1913).
41. ^ Smith, Brian. "Breakthrough Ideas: Calendar week ii" Lecture Notes, p.17. Academy of Oxford. Retrieved January. 23, 2015.
42. ^ Dahl, Jens Peder; Springborg, Michael (ten December 1982). "Wigner's phase space function and atomic structure: I. The hydrogen atom ground state". Molecular Physics. 47 (5): 1001–1019. doi:10.1080/00268978200100752.
43. ^ A. Sommerfeld (1916). "Zur Quantentheorie der Spektrallinien". Annalen der Physik (in German). 51 (17): 1–94. Bibcode:1916AnP...356....1S. doi:10.1002/andp.19163561702.
44. ^ W. Wilson (1915). "The quantum theory of radiation and line spectra". Philosophical Mag. 29 (174): 795–802. doi:10.1080/14786440608635362.
45. ^ Shaviv, Glora (2010). The Life of Stars: The Controversial Inception and Emergence of the Theory of Stellar Structure. Springer. p. 203. ISBN978-3642020872.
46. ^ Бор Н. (1970). Избранные научные труды (статьи 1909–1925). Vol. 1. М.: «Наука». p. 133.
47. ^ Svidzinsky, Anatoly A.; Scully, Marlan O.; Herschbach, Dudley R. (23 August 2005). "Bohr's 1913 molecular model revisited". Proceedings of the National Academy of Sciences of the Us. 102 (34): 11985–11988. arXiv:physics/0508161. Bibcode:2005PNAS..10211985S. doi:10.1073/pnas.0505778102. PMC1186029. PMID 16103360.

Primary sources [edit]

• Bohr, N. (July 1913). "I. On the constitution of atoms and molecules". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Scientific discipline. 26 (151): 1–25. doi:x.1080/14786441308634955.
• Bohr, Due north. (September 1913). "XXXVII. On the constitution of atoms and molecules". The London, Edinburgh, and Dublin Philosophical Mag and Journal of Science. 26 (153): 476–502. doi:10.1080/14786441308634993.
• Bohr, N. (1 November 1913). "LXXIII. On the constitution of atoms and molecules". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 26 (155): 857–875. doi:10.1080/14786441308635031.
• Bohr, N. (October 1913). "The Spectra of Helium and Hydrogen". Nature. 92 (2295): 231–232. Bibcode:1913Natur..92..231B. doi:10.1038/092231d0. S2CID 11988018.
• Bohr, Due north. (March 1921). "Atomic Structure". Nature. 107 (2682): 104–107. Bibcode:1921Natur.107..104B. doi:10.1038/107104a0. S2CID 4035652.
• A. Einstein (1917). "Zum Quantensatz von Sommerfeld und Epstein". Verhandlungen der Deutschen Physikalischen Gesellschaft. 19: 82–92. Reprinted in The Collected Papers of Albert Einstein, A. Engel translator, (1997) Princeton Academy Printing, Princeton. half-dozen p. 434. (provides an elegant reformulation of the Bohr–Sommerfeld quantization weather condition, also as an important insight into the quantization of non-integrable (cluttered) dynamical systems.)
• de Broglie, Maurice; Langevin, Paul; Solvay, Ernest; Einstein, Albert (1912). La théorie du rayonnement et les quanta : rapports et discussions de la réunion tenue à Bruxelles, du xxx octobre au 3 novembre 1911, sous les auspices de M.Eastward. Solvay (in French). Gauthier-Villars. OCLC 1048217622.

Further reading [edit]

• Linus Carl Pauling (1970). "Affiliate 5-i". Full general Chemistry (3rd ed.). San Francisco: Westward.H. Freeman & Co.
  • Reprint: Linus Pauling (1988). Full general Chemistry . New York: Dover Publications. ISBN0-486-65622-v.
• George Gamow (1985). "Chapter 2". Thirty Years That Shook Physics. Dover Publications.
• Walter J. Lehmann (1972). "Affiliate xviii". Diminutive and Molecular Structure: the development of our concepts. John Wiley and Sons. ISBN0-471-52440-nine.
• Paul Tipler and Ralph Llewellyn (2002). Modern Physics (4th ed.). W. H. Freeman. ISBN0-7167-4345-0.
• Klaus Hentschel: Elektronenbahnen, Quantensprünge und Spektren, in: Charlotte Bigg & Jochen Hennig (eds.) Atombilder. Ikonografien des Atoms in Wissenschaft und Öffentlichkeit des twenty. Jahrhunderts, Göttingen: Wallstein-Verlag 2009, pp. 51–61
• Steven and Susan Zumdahl (2010). "Chapter 7.4". Chemistry (eighth ed.). Brooks/Cole. ISBN978-0-495-82992-viii.
• Kragh, Helge (November 2011). "Conceptual objections to the Bohr atomic theory — practice electrons have a 'complimentary volition'?". The European Physical Journal H. 36 (iii): 327–352. Bibcode:2011EPJH...36..327K. doi:10.1140/epjh/e2011-20031-10. S2CID 120859582.

External links [edit]

• Continuing waves in Bohr'southward atomic model An interactive simulation to intuitively explicate the quantization status of standing waves in Bohr'due south atomic mode

Wikimedia Commons has media related to Bohr model.

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