Bohr model - Knowledge (XXG)

435:. Lorentz explained that the Planck constant could be taken as determining the size of atoms, or that the size of atoms could be taken to determine the Planck 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 to 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. the Planck 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 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 5562: 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 5285:

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.

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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,

4859: = 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 306: 354:, was the best available. Thomson proposed a model with electrons rotating in coplanar rings within an atomic-sized, positively-charged, spherical volume. Thomson showed that this model was mechanically stable by lengthy calculations and electrodynamically stable under his original assumption of thousands of electrons per atom. Moreover he suggested that the particularly stable configurations of electrons in rings was connected to chemical properties of the atoms and he developed a formula for the scattering of 4883:

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

1650: 5668: 5329:"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. 3624:; 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. 4607:< 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 5418: 7938: 7948: 49: 5346:

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).

5598:. 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. 3789:

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.

552:. To avoid immediate collapse of this system he required that electrons come in pairs so the rotational acceleration of each electron was matched across the orbit. By 1913 Bohr had already shown from the analysis of alpha particle energy loss that hydrogen had only a single electron. Thus Bohr's atomic model would abandon classical electrodynamics. 250:. While the Rydberg formula had been known experimentally, it did not gain a theoretical basis until the Bohr model was introduced. Not only did the Bohr model explain the reasons for the structure of the Rydberg formula, it also provided a justification for the fundamental physical constants that make up the formula's empirical results. 3109: 520:." 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. 5338:, 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. 3783: 4960:

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

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published a model of the atom which would influence Bohr's model. Nicholson developed his model based on the analysis of astrophysical spectroscopy. He connected the observed spectral line frequencies with the orbits of electrons in his atoms. The connection he adopted associated the atomic electron

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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

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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

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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

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that seemed to match experimental results. However Thomson himself later showed that the atom had a factor of a thousand fewer electrons, challenging the stability argument and forcing the poorly understood positive sphere to have most of the atom's mass. Thomson was also unable to explain the many

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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

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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.

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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

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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

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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

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The other critical influence of Nicholson work was his detailed analysis of spectra. Before Nicholson's work Bohr thought the spectral data was not useful for understanding atoms. In comparing his work to Nicholson's, Bohr came to understand the spectral data and their value. When he then learned

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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

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However, in quantum mechanics, the quantization of angular momentum leads to discrete energy levels of the orbits, and the emitted frequencies are quantized according to the energy differences between these levels. This discrete nature of energy levels introduces a fundamental departure from the

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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

537:. Whereas Planck focused on a quantum of energy, Nicholson's angular momentum quantum relates to orbital frequency. This new concept gave Planck constant an atomic meaning for the first time. In his 1913 paper Bohr cites Nicholson as finding quantized angular momentum important for the atom. 2108:

In classical mechanics, if an electron is orbiting around an atom with period T, and if its coupling to the electromagnetic field is weak, so that the orbit doesn't decay very much in one cycle, it will emit electromagnetic radiation in a pattern repeating at every period, so that the Fourier

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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

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is the Rydberg constant, in terms of frequency equal to 3.28 x 10 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

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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

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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

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Well, yes," says Bohr. "But I can hardly imagine it will involve light quanta. Look, even if Einstein had found an unassailable proof of their existence and would want to inform me by telegram, this telegram would only reach me because of the existence and reality of radio

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Bohr considered circular orbits. Classically, these orbits must decay to smaller circles when photons are emitted. The energy level spacing between circular orbits can be calculated with the correspondence formula. For a hydrogen atom, the classical orbits have a period

4813:, 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 332:. As electrons in orbit are continuously accelerating, they would be mechanically unstable. Larmor noted that electromagnetic effect of multiple electrons suitable arranged would cancel each other, a restriction applied in the subsequent classical atomic models. 5257: 5435:, 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 1612:

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.

2449: 761:, of 0.0529 nm for hydrogen. 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 2899: 5654:

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

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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

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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

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developed increasingly accurate empirical formula matching measured atomic spectral lines. Critical for Bohr's later work, Rydberg expressed his formula in terms of wave-number, equivalent to frequency. These formula contained a constant,

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Nicholson's model of radiation was quantum but was attached to the orbits of the electrons. Bohr would adopt Nicholson's interpretation of frequency quantization, but associate it with differences in energy levels of his model of hydrogen.

7739:: 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 4100: 1809: 2625: 3388: 2733: 4895:

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."

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Therefore Bohr's theory gives the Rydberg formula and moreover the numerical value the Rydberg constant for hydrogen in terms of more fundamental constants of nature, including the electron's charge, the electron's mass, and the

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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

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developed a new scattering model, showing that the observed large angle scattering could be explained by a compact, highly charged mass at the center of the atom. Rutherford scattering did not involve the electrons and thus his

3474: 4887:). 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. 1925: 4969:, 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, 3983: 4843:

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

4848:. 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. 3992:

between orbital energy levels is able to explain these formula. For the hydrogen atom Bohr starts with his derived formula for the energy released as a free electron moves into a stable circular orbit indexed by

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has been widely used as a symbol for atoms and even for "atomic" energy (even though this is more properly considered nuclear energy). Examples of its use over the past century include but are not limited to:

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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

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Although Bohr's atomic model was superseded by quantum models in the 1920's, the visual image of electrons orbiting a nucleus has remained the popular concept of atoms. The concept of an atom as a tiny

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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

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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

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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.)

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Until the second decade of the 20th century, atomic models were generally speculative and even the concept of atoms let alone atoms with internal structure faced opposition from some scientists.

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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.

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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

5512: 2518: 5123: 3104:{\displaystyle E=-{\frac {Zk_{\mathrm {e} }e^{2}}{2r_{n}}}=-{\frac {Z^{2}(k_{\mathrm {e} }e^{2})^{2}m_{\mathrm {e} }}{2\hbar ^{2}n^{2}}}\approx {\frac {-13.6Z^{2}}{n^{2}}}~\mathrm {eV} .} 2340: 2096: 1518: 856: 644: 4389: 1304: 1420: 1362: 4975: 324:

In the late 1800's speculations on the possible structure of the atom included planetary models with orbiting charged electrons. These models faced a significant constraint. In 1897,

3143:, 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 3310: 1849:. 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: 269:. 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 4611:= 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:" 1249: 901:

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

3778:{\displaystyle m_{\text{red}}={\frac {m_{\mathrm {e} }m_{\mathrm {p} }}{m_{\mathrm {e} }+m_{\mathrm {p} }}}=m_{\mathrm {e} }{\frac {1}{1+m_{\mathrm {e} }/m_{\mathrm {p} }}}.} 735: 518: 1590: 773:. 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. 7632:

La théorie du rayonnement et les quanta : rapports et discussions de la réunion tenue à Bruxelles, du 30 octobre au 3 novembre 1911, sous les auspices de M.E. Solvay

3163: 2243: 694: 4400: 5650:, 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 2332: 1013: 5405:

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.

1200: 4965: − 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 + 4011: 1469: 1119:), 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 980: 456: 946: 926: 794: 1552: 1449: 1719: 3903: 2540: 1610: 1157: 1137: 1113: 1093: 1073: 1053: 1033: 876: 755: 476: 3912: 3321: 2640: 3480:

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

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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

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in the discussion of Planck's lecture raised the question of the composition of the atom based on Thomson's model with the quantum modification added by

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Electrons can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency

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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

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Bohr's derivation of the Rydberg constant, as well as the concomitant agreement of Bohr's formula with experimentally observed spectral lines of the

4255:{\displaystyle h\nu =W_{\tau _{2}}-W_{\tau _{1}}={\frac {2\pi ^{2}me^{4}}{h^{2}}}\left({\frac {1}{\tau _{2}^{2}}}-{\frac {1}{\tau _{1}^{2}}}\right)} 1854: 394:

was incomplete. Bohr begins his first paper on his atomic model by describing Rutherford's atom as consisting of a small, dense, positively charged

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Dahl, Jens Peder; Springborg, Michael (10 December 1982). "Wigner's phase space function and atomic structure: I. The hydrogen atom ground state".

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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

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The stationary orbits are attained at distances for which the angular momentum of the revolving electron is an integer multiple of the reduced

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On the Constitution of Atoms and Molecules ... Papers of 1913 reprinted from the Philosophical Magazine, with an introduction by L. Rosenfeld

6460: 6291: 6072: 6000: 5962: 2879:{\displaystyle r_{1}={\frac {\hbar ^{2}}{k_{\mathrm {e} }e^{2}m_{\mathrm {e} }}}\approx 5.29\times 10^{-11}~\mathrm {m} =52.9~\mathrm {pm} .} 4532:=3) series, and successful theoretical prediction of other lines not yet observed, was one reason that his model was immediately accepted. 1210:

condition: the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit:

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in spectral lines, which are known to be due to a variety of relativistic and subtle effects, as well as complications from electron spin.

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postulated the existence of "cells" which could each only contain two electrons each, and these were arranged in "equidistant layers".

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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.

7272: 5697: 4913:. Moseley's empiric formula was found to be derivable from Rydberg's formula and later Bohr's formula (Moseley actually mentions only 7727: 7708: 7678: 5802: 5436: 4577:

and the later discussion of the "Shell Model of the Atom" below). This was established empirically before Bohr presented his model.

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in 1910 but was rejected until the 1911 Solvay Congress where it was thoroughly discussed. The quantum theory of the period between

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from a friend about Balmer's compact formula for the spectral line data, Bohr quickly realized his model would match it in detail.

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Bohr also updated his model in 1922, assuming that certain numbers of electrons (for example, 2, 8, and 18) correspond to stable "

427:'s lecture ended with this remark: "... atoms or electrons subject to the molecular bond would obey the laws of quantum theory". 4903:

found an empirical relationship between the strongest X-ray line emitted by atoms under electron bombardment (then known as the

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Nicholson's model was based on classical electrodynamics with negative electron orbiting a positive nucleus along the lines of

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who corrected Bohr's work to show that electrons interacted through the outer rings, and Kossel called the rings: "shells".

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It is possible to determine the energy level spacings by recursively stepping down orbit by orbit, but there is a shortcut.

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The electron is able to revolve in certain stable orbits around the nucleus without radiating any energy, contrary to what

2006:, the energy is zero, corresponding to a motionless electron infinitely far from the proton. The total energy is half the 7372:

Schirrmacher, Arne (2009). "Bohr's Atomic Model". In Greenberger, Daniel M.; Hentschel, Klaus; Weinert, Friedel (eds.).

7047:(Interview). Interviewed by Thomas S. Kuhn; Leon Rosenfeld; Aage Petersen; Erik Rudinger. American Institute of Physics. 6385:(Interview). Interviewed by Thomas S. Kuhn; Leon Rosenfeld; Aage Petersen; Erik Rudinger. American Institute of Physics. 6367:(Interview). Interviewed by Thomas S. Kuhn; Leon Rosenfeld; Aage Petersen; Erik Rudinger. American Institute of Physics. 5789: 5395:; these are also due to more complicated quantum principles interacting with electron spin and orbital magnetic fields. 309:

Bohr model in 1921 after Sommerfeld expansion of 1913 model showing maximum electrons per shell with shells labeled in

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in that it considers electrons to have known orbits and locations, two things which cannot be measured simultaneously.

5252:{\displaystyle f=\nu =R_{\mathrm {v} }\left({\frac {3}{4}}\right)(Z-1)^{2}=(2.46\times 10^{15}~{\text{Hz}})(Z-1)^{2}.} 277:

before moving on to the more accurate, but more complex, valence shell atom. A related quantum model was proposed by

7808:—An interactive simulation to intuitively explain the quantization condition of standing waves in Bohr's atomic mode 7762:

Kragh, Helge (November 2011). "Conceptual objections to the Bohr atomic theory — do electrons have a 'free will'?".

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additional ad hoc assumptions are made. Emission spectra for atoms with a single outer-shell electron (atoms in the

2468: 2444:{\displaystyle \Delta E\propto {\frac {1}{(L+\Delta L)^{2}}}-{\frac {1}{L^{2}}}\approx -{\frac {2\Delta L}{L^{3}}}.} 955:

at integer multiples of this frequency. This result is obtained from the Bohr model for jumps between energy levels

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of the hydrogen atom using the broader and much more accurate quantum mechanics and thus may be considered to be an

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Bohr, N. (1913). "On the Constitution of Atoms and Molecules, Part II. Systems containing only a Single Nucleus".

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Next, Bohr was told by his friend, Hans Hansen, that the Balmer series is calculated using the Balmer formula, an

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rather form an essential part of the original constitution of the atom as determined by conditions of stability."

7991: 7864: 7805: 7135: 2054: 7044: 6958: 6382: 6364: 5981:. RePoSS: Research Publications on Science Studies 10. Aarhus: Centre for Science Studies, University of Aarhus. 5107:{\displaystyle E=h\nu =E_{i}-E_{f}=R_{\mathrm {E} }(Z-1)^{2}\left({\frac {1}{1^{2}}}-{\frac {1}{2^{2}}}\right),} 2454:

This is as desired for equally spaced angular momenta. If one kept track of the constants, the spacing would be

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Multi-electron atoms do not have energy levels predicted by the model. It does not work for (neutral) helium.

2002:. This means that it takes energy to pull the orbiting electron away from the proton. For infinite values of 235:

interpretation introduced by Haas and Nicholson, but forsaking any attempt to explain radiation according to

7981: 5579: 5555: 5333: 3391: 2010:, the difference being the kinetic energy of the electron. This is also true for noncircular orbits by the 1629: 262: 3276: 328:

showed that in classical electrodynamics an accelerating charge would radiate power, a result known as the

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In the history of atomic physics, it followed, and ultimately replaced, several earlier models, including

1163:, requiring quantum theory to agree with the classical theory only in the limit of large quantum numbers. 231:'s nuclear quantum model (1912). The improvement over the 1911 Rutherford model mainly concerned the new 5736: 5708: 5594:, the Bohr model and quantum mechanics make the same predictions for the spectral line splitting in the 5399: 769:

atoms and ions. These orbits are associated with definite energies and are also called energy shells or

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John L, Heilbron (1985). "Bohr's First Theories of the Atom". In French, A. P.; Kennedy, P. J. (eds.).

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The total kinetic energy is half what it would be for a single electron moving around a heavy nucleus.

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in 1885 that described wavelengths of some spectral lines of hydrogen. This was further generalized by

6062: 5366:). All these techniques essentially make use of Bohr's Newtonian energy-potential picture of the atom. 1649: 7771: 7579: 7538: 7505: 7472: 7439: 7326: 7204: 7092: 6928: 6847: 6808: 6613: 6155: 5869: 5843: 5778: 5711: 5583: 4814: 1633: 887: 83: 6601: 5667: 3256:{\displaystyle R_{\mathrm {E} }={\frac {(k_{\mathrm {e} }e^{2})^{2}m_{\mathrm {e} }}{2\hbar ^{2}}}.} 1159:), the radiation frequency has no unambiguous classical interpretation. This marks the birth of the 703: 587:

put forth three postulates to provide an electron model consistent with Rutherford's nuclear model:

90:). The orbits in which the electron may travel are shown as grey circles; their radius increases as 7996: 7081:

M.A.B. Whitaker (1999). "The Bohr–Moseley synthesis and a simple model for atomic x-ray energies".

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When Bohr began his work on a new atomic theory in the summer of 1912 the atomic model proposed by

7947: 4483:{\displaystyle {\frac {1}{\lambda }}=R\left({\frac {1}{n_{f}^{2}}}-{\frac {1}{n_{i}^{2}}}\right).} 1561: 7922: 7885: 7787: 7595: 7554: 7316: 7195: 7171: 7108: 6981: 6863: 6671: 6637: 6428: 6420: 6254: 6117: 6040: 5930: 5432: 2224: 1638: 649: 549: 351: 341: 287: 216: 158: 4535:

To apply to atoms with more than one electron, the Rydberg formula can be modified by replacing

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in 1911 on the subject of Radiation and Quanta, at which Bohr's mentor, Rutherford was present.

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The Life of Stars: The Controversial Inception and Emergence of the Theory of Stellar Structure

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this X-ray line came from a transition between energy levels with quantum numbers 1 and 2, and

4870:. One property was the size of atoms, which could be determined approximately by measuring the 3125:

less energy than a motionless electron infinitely far from the nucleus. The next energy level (

82:

and where an electron jumps between orbits, is accompanied by an emitted or absorbed amount of

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Despite many attempts, no theory of the atom could reproduce these relatively simple formula.

2314: 1827: 1707: 1625: 1617: 432: 420: 386: 278: 270: 236: 232: 150: 985: 7955: 7891: 7779: 7587: 7546: 7513: 7480: 7447: 7344: 7334: 7245: 7212: 7163: 7100: 7022: 6936: 6894: 6855: 6816: 6750: 6683: 6621: 6492: 6412: 6279: 6238: 6194: 6163: 6101: 6032: 5922: 5851: 5677: 5571: 5536: 4845: 4836:, after that the orbit is full, the next level would have to be used. This gives the atom a 4573:

is constant representing a screening effect due to the inner-shell and other electrons (see

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only to be replaced by the quantum atomic model in the 1920s. It consists of a small, dense

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For example, the lithium atom has two electrons in the lowest 1s orbit, and these orbit at

3996: 1804:{\displaystyle {\frac {m_{\mathrm {e} }v^{2}}{r}}={\frac {Zk_{\mathrm {e} }e^{2}}{r^{2}}},} 1454: 958: 438: 7878: 7736: 7006: 5773: 5690: 5611: 5575: 5363: 4954: 4824: 4264: 3867: 2620:{\displaystyle m_{\text{e}}{\sqrt {\dfrac {k_{\text{e}}Ze^{2}}{m_{\text{e}}r}}}r=n\hbar ,} 2113:

classical radiation law, giving rise to distinct spectral lines in the emitted radiation.

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occasionally scatter at large angles, a result inconsistent with Thomson's model. In 1911

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The combination of natural constants in the energy formula is called the Rydberg energy (

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The Bohr–Sommerfeld model was fundamentally inconsistent and led to many paradoxes. The

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when used in the formula for atoms heavier than hydrogen, should be diminished by 1, to

3383:{\displaystyle {\frac {k_{\mathrm {e} }e^{2}}{\hbar c}}=\alpha \approx {\frac {1}{137}}} 2728:{\displaystyle r_{n}={\frac {n^{2}\hbar ^{2}}{Zk_{\mathrm {e} }e^{2}m_{\mathrm {e} }}}.} 7951: 7941: 7916: 7667: 7349: 7304: 5603: 5567: 5392: 5377: 5342: 4949: 4820: 4810: 4586: 4574: 4520: 3888: 3587: 3580: 3579:

The actual energy levels cannot be solved analytically for more than one electron (see

3144: 2011: 1678: 1674: 1620:, in which Bohr's model of electrons traveling in quantized orbits was extended into a 1595: 1142: 1122: 1098: 1078: 1058: 1038: 1018: 861: 740: 461: 395: 382: 378: 329: 310: 285:(1900) and the advent of a mature quantum mechanics (1925) is often referred to as the 166: 130: 79: 1254:

According to de Broglie's hypothesis, matter particles such as the electron behave as

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McCormmach, Russell (1 January 1966). "The atomic theory of John William Nicholson".

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of any atom where one electron is far away from everything else. It can be used for

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Svidzinsky, Anatoly A.; Scully, Marlan O.; Herschbach, Dudley R. (23 August 2005).

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Bohr, N. (1985). "Rydberg's discovery of the spectral laws". In Kalckar, J. (ed.).

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these results can be expressed in terms of the wavelength of the photon given off:

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Lakhtakia, Akhlesh; Salpeter, Edwin E. (1996). "Models and Modelers of Hydrogen".

7783: 6754: 3469:{\displaystyle R_{\mathrm {E} }={\frac {1}{2}}(m_{\mathrm {e} }c^{2})\alpha ^{2}} 7857: 6283: 5614: 2750: 2109:

transform of the pattern will only have frequencies which are multiples of 1/T.

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