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Encyclopedia > Bohr Model
The Bohr model of the hydrogen atom (Z = 1) or a hydrogen-like ion (Z > 1), where the negatively charged electron confined to an atomic shell encircles a small positively charged atomic nucleus, and an electron jump between orbits is accompanied by an emitted or absorbed amount of electromagnetic energy hν. The orbits that the electron may travel in are shown as grey circles; their radius increases as n2, where n is the principal quantum number. The $3 rightarrow 2$ transition depicted here produces the first line of the Balmer series, and for hydrogen (Z = 1) results in a photon of wavelength 656 nm (red).

Introduced by Niels Bohr in 1913, 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, it did not gain a theoretical underpinning until the Bohr model was introduced. Not only did the Bohr model explain the reason for the structure of the Rydberg formula, but it provided a justification for its empirical results in terms of fundamental physical constants. Niels Henrik David Bohr (October 7, 1885 â€“ November 18, 1962) was a Danish physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. ... The Rydberg formula (Rydberg-Ritz formula) is used in atomic physics for determining the full spectrum of light emission from hydrogen, later extended to be useful with any element by use of the Rydberg-Ritz combination principle. ... A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from an excess or deficiency of photons in a narrow frequency range, compared with the nearby frequencies. ... This article is about the chemistry of hydrogen. ...

The Bohr model is a primitive model of the hydrogen atom. As a theory, it can be derived as a first-order approximation of the hydrogen atom using the broader and much more accurate quantum mechanics, and thus may be considered to be an obsolete scientific theory. However, because of its simplicity, and its correct results for selected systems (see below for application), the Bohr model is still commonly taught to introduce students to quantum mechanics, before moving on to the more accurate but more complex valence shell atom. A related model was originally proposed by Arthur Erich Haas in 1910, but was rejected. The quantum theory of the period between Planck's discovery of the quantum (1900) and the advent of a full-blown quantum mechanics (1925) is often referred to as the `Old quantum theory'. For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... An obsolete scientific theory is a scientific theory that was once commonly accepted but (for whatever reason) is no longer considered the most complete description of reality by mainstream science; or a falsifiable theory which has been shown to be false. ... An atomic orbital is the description of the behavior of an electron in an atom according to quantum mechanics. ... Arthur Erich Haas (April 3, 1884, Brünn - 1941) was an Austrian physicist. ... This article is about Planck, the German physicist. ... For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ...

In the early 20th century, experiments by Ernest Rutherford established that atoms consisted of a diffuse cloud of negatively charged electrons surrounding a small, dense, positively charged nucleus. Given this experimental data, it was quite natural for Rutherford to consider a planetary model for the atom, the Rutherford model of 1911, with electrons orbiting a sun-like nucleus. However, the planetary model for the atom has a difficulty. The laws of classical mechanics, specifically the Larmor formula, predict that the electron will release electromagnetic radiation as it orbits a nucleus. Because the electron would be losing energy, it would gradually spiral inwards and collapse into the nucleus. This is a disaster, because it predicts that all matter is unstable. Ernest Rutherford, 1st Baron Rutherford of Nelson OM PC FRS (30 August 1871 â€“ 19 October 1937), widely referred to as Lord Rutherford, was a chemist (B.Sc. ... For other uses, see Atom (disambiguation). ... For other uses, see Electron (disambiguation). ... A stylised representation of the Rutherford model of a lithium atom (nuclear structure anachronistic) The Rutherford model or planetary model was a model of the atom devised by Ernest Rutherford. ... The Larmor formula is used to calculate the power radiated by a nonrelativistic electron as it accelerates. ... This box:      Electromagnetic (EM) radiation is a self-propagating wave in space with electric and magnetic components. ...

Also, as the electron spirals inward, the emission would gradually increase in frequency as the orbit got smaller and faster. This would produce a continuous smear, in frequency, of electromagnetic radiation. However, late 19th century experiments with electric discharges through various low-pressure gasses in evacuated glass tubes had shown that atoms will only emit light (that is, electromagnetic radiation) at certain discrete frequencies. a localised surplus of electrons migrates from the negative pole to the positive pole to overcome its unfavourable energetic condition. ... For other uses, see Gas (disambiguation). ...

To overcome this difficulty, Niels Bohr proposed, in 1913, what is now called the Bohr model of the atom. He suggested that electrons could only have certain classical motions: Niels Henrik David Bohr (October 7, 1885 â€“ November 18, 1962) was a Danish physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. ...

1. The electrons travel in circular orbits that have discrete (quantized) angular momenta, and therefore quantized energies. That is, not every circular orbit is possible but only certain specific ones, at certain specific distances from the nucleus and having specific energies.
2. The electrons do not continuously lose energy as they travel. They can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation at frequency ν determined by the energy difference ΔE = E2E1 of the levels according to Bohr's formula
$E_2 - E_1 =hnu,$
where h is Planck's constant.

The significance of the Bohr model is that the laws of classical mechanics apply to the motion of the electron about the nucleus only in a sense restricted by quantum rules like that for angular momentum L, restricting its value to The Bohr model of the hydrogen atom () or a hydrogen-like ion (), where the negatively charged electron confined to an atomic shell encircles a small positively charged atomic nucleus, and an electron jump between orbits is accompanied by an emitted or absorbed amount of electromagnetic energy . ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ...

$L = n cdot hbar = n cdot {h over 2pi}$

where n = 1,2,3,… and is called the principal quantum number. The lowest value of n is 1. This corresponds to a smallest possible radius of 0.0529 nm. This is known as the Bohr radius. Once an electron is in this lowest orbit, it can get no closer to the proton. Starting from the angular momentum quantum rule Bohr[1] was able to calculate the energies of the allowed orbits of the hydrogen atom and other hydrogenlike atoms and ions. In atomic physics, the principal quantum number symbolized as n is the first quantum number of an atomic orbital. ... In the Bohr model of the structure of an atom, put forward by Niels Bohr in 1913, electrons orbit a central nucleus. ... The Bohr model of the hydrogen atom () or a hydrogen-like ion (), where the negatively charged electron confined to an atomic shell encircles a small positively charged atomic nucleus, and an electron jump between orbits is accompanied by an emitted or absorbed amount of electromagnetic energy . ... The Bohr model of the hydrogen atom () or a hydrogen-like ion (), where the negatively charged electron confined to an atomic shell encircles a small positively charged atomic nucleus, and an electron jump between orbits is accompanied by an emitted or absorbed amount of electromagnetic energy . ...

Other points are:

1. Analogously to Einstein's theory of the Photoelectric effect it is assumed in Bohr's formula that on a quantum jump a discrete amount of energy is radiated. However, contrary to Einstein, Bohr stuck to the classical Maxwell theory of the electromagnetic field. Quantization of the electromagnetic field was explained by the discreteness of the atomic energy levels; Bohr did not believe in the existence of photons.
2. According to the Maxwell theory the frequency ν of the radiation is equal to the rotation frequency νrot of the electron in its orbit. This result is obtained to a good approximation from the Bohr model for jumps between energy levels En + 1 and En for sufficiently large values of n (so-called Rydberg states), the two orbits involved in emission for large values of n having nearly the same rotation frequency. However, in general the radiation frequencies are different from the rotation frequencies. This marks the birth of the correspondence principle, requiring quantum theory to yield agreement with the classical theory only in the limit of large quantum numbers.
3. The Bohr-Kramers-Slater (BKS) theory is an attempt to extend the Bohr model so as to account for conservation of energy and momentum in quantum jumps.

Bohr's condition, that the angular momentum is an integer multiple of $scriptstylehbar$ was later reinterpreted by DeBroglie as a standing wave condition: the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit: A diagram illustrating the emission of electrons from a metal plate, requiring energy gained from an incoming photon to be more than the work function of the material. ... For thermodynamic relations, see Maxwell relations. ... The Bohr model of the hydrogen atom () or a hydrogen-like ion (), where the negatively charged electron confined to an atomic shell encircles a small positively charged atomic nucleus, and an electron jump between orbits is accompanied by an emitted or absorbed amount of electromagnetic energy . ... In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ... The Rydberg states of an atom are electronically excited states with energies that follow the Rydberg formula as they converge on an ionic state with an ionization energy. ... In physics, the correspondence principle is a principle, first invoked by Niels Bohr in 1923, which states that the behavior of quantum mechanical systems reduce to classical physics in the limit of large quantum numbers. ... This article is about momentum in physics. ... Vibration and standing waves in a string, The fundamental and the first 6 overtones A standing wave, also known as a stationary wave, is a wave that remains in a constant position. ...

$n lambda = 2 pi r,$

Substituting DeBroglie's wavelength reproduces Bohr's rule. Bohr justified his rule by appealing to the correspondence principle, without providing a wave interpretation.

In 1925 rather a new kind of mechanics was proposed, viz. quantum 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 by Werner Heisenberg. Another form of the same theory, modern quantum mechanics, was discovered by the Austrian physicist Erwin Schrödinger independently and by different reasoning. For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. ... Werner Karl Heisenberg (December 5, 1901 â€“ February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics and acknowledged to be one of the most important physicists of the twentieth century. ... This box:      For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... SchrÃ¶dinger in 1933, when he was awarded the Nobel Prize in Physics Bust of SchrÃ¶dinger, in the courtyard arcade of the main building, University of Vienna, Austria. ...

## Electron energy levels

To calculate the orbits requires two assumptions:

1. Classical mechanics

The electron is held in a circular orbit by electrostatic attraction. The centripetal force is equal to the Coulomb force.
${m_e v^2over r} = {k e^2 over r^2}$
where me is the mass and e is the charge of the electron. This determines the speed at any radius:
$v = sqrt{ k e^2 over m_e r}$
It also determines the total energy at any radius:
$E= {1over 2} m_e v^2 - {k e^2 over r} = - { k e^2 over 2r}$
The total energy is negative and inversely proportional to r. This means that it takes energy to pull the orbiting electron away from the proton. For infinite values of r, the energy is zero, corresponding to a motionless electron infinitely far from the proton. The total energy is half the potential energy, which is true for non circular orbits too by the virial theorem. For larger nuclei, replace ke2 everywhere with Zke2 where Z is the number of protons. For positronium, replace me with the reduced mass me / 2.
Restricting ourselves to the hydrogen atom it follows from the expression for $scriptstyle v$ that angular momentum $scriptstyle L= m_evr$ is equal to
$L = sqrt{ k e^2 m_e r}$
Hence, for an arbitrary circular orbit we have
$E = -{ (k e^2)^2m_e over 2L^2}$
The rotation frequency of the electron in its orbit is
$nu_{rot} = {vover 2pi r} = { (k e^2)^2m_e over 2pi L^3} = 6.58 times 10^{15} operatorname{Hz}$
giving a period of $1.52 times 10^{-16} operatorname{s}$.

2. Quantum rule The centripetal force is the external force required to make a body follow a circular path at constant speed. ... Coulombs torsion balance In physics, Coulombs law is an inverse-square law indicating the magnitude and direction of electrostatic force that one stationary, electrically charged object of small dimensions (ideally, a point source) exerts on another. ... For other uses, see Electron (disambiguation). ... Potential energy can be thought of as energy stored within a physical system. ... In mechanics, the virial theorem provides a general equation relating the average total kinetic energy of a system with its average total potential energy , where angle brackets represent the average of the enclosed quantity. ... Reduced mass is an algebraic term of the form that simplifies an equation of the form The reduced mass is typically used as a relationship between two system elements in parallel, such as resistors; whether these be in the electrical, thermal, hydraulic, or mechanical domains. ...

Let the angular momentum $scriptstyle L$ of a circular orbit be an integer multiple of $scriptstyle hbar$,
$L = n frac{h}{2 pi} = n hbar$
n takes the values 1,2,3,... and is called the principal quantum number, h is Planck's constant.
This quantum rule gives the energy levels:
$E_n = - {1 over n^2} {(ke^2)^2 m_e over 2 hbar^2} = {-13.6 mathrm{eV} over n^2}$
So an electron in the lowest energy level of hydrogen (n = 1) has 13.606 eV less energy than a motionless electron infinitely far from the nucleus. The next energy level at (n = 2) is -3.4 eV. The third (n = 3) is -1.51 eV, and so on. For larger values of n, these are also the binding energies of a highly excited atom with one electron in a large circular orbit around the rest of the atom.
The radius of orbit number n is obtained from
$L = sqrt{k e^2 m_e r_n} = n hbar$
as
$r_n = n^2 {hbar^2 over k e^2 m_e}$
r1 is called the Bohr radius.
Expressing En in terms of the rotation frequency yields another quantum rule, viz.
$E_n = -{nhnu_{rot}over 2}$
used by Bohr as an alternative to the angular momentum quantum rule.

The combination of natural constants in the energy formula is called the Rydberg energy RE: This gyroscope remains upright while spinning due to its angular momentum. ... In atomic physics, the principal quantum number symbolized as n is the first quantum number of an atomic orbital. ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... The Bohr model of the hydrogen atom () or a hydrogen-like ion (), where the negatively charged electron confined to an atomic shell encircles a small positively charged atomic nucleus, and an electron jump between orbits is accompanied by an emitted or absorbed amount of electromagnetic energy . ... The electronvolt (symbol eV) is a unit of energy. ... In the Bohr model of the structure of an atom, put forward by Niels Bohr in 1913, electrons orbit a central nucleus. ...

$R_E = { (k e^2)^2 m_e over 2 hbar^2}$

This expression is clarified by interpreting it in combinations which form more natural units: In physics, natural units are physical units of measurement defined in terms of universal physical constants in such a manner that some chosen physical constants take on the numerical value of one when expressed in terms of a particular set of natural units. ...

$, m_e c^2$ : the rest energy of the electron (= 511 keV)
$, {k e^2 over hbar c} = alpha = {1over 137}$ : the fine structure constant
$, R_E = {1over 2} (m_e c^2) alpha^2$

For nuclei with Z protons, the energy levels are: The rest energy of a particle is its energy when it is not moving relative to a given inertial reference frame. ... The fine-structure constant or Sommerfeld fine-structure constant, usually denoted , is the fundamental physical constant characterizing the strength of the electromagnetic interaction. ...

$E_n = {Z^2 R_E over n^2}$ (Heavy Nuclei)

When Z is approximately 137 (about 1/α), the motion becomes highly relativistic. Then the Z2 cancels the α2 in R, so the orbit energy begins to be comparable to rest energy. Sufficiently large nuclei, if they were stable, would reduce their charge by creating a bound electron from the vacuum, ejecting the positron to infinity. This is the theoretical phenomenon of electromagnetic charge screening which predicts a maximum nuclear charge. Emission of such positrons has been observed in the collisions of heavy ions to create temporary super-heavy nuclei.

For positronium, the formula uses the reduced mass. For any value of the radius, the electron and the positron are each moving at half the speed around their common center of mass, and each has only one fourth the kinetic energy. The total kinetic energy is half what it would be for a single electron moving around a heavy nucleus.

$E_n = {R_E over 2 n^2 }$ (Positronium)

## Rydberg formula

The Rydberg formula, which was known empirically before Bohr's formula, is now in Bohr's theory seen as describing the energies of transitions or quantum jumps between one orbital energy level, and another. Bohr's formula gives the numerical value of the already-known and measured Rydberg's constant, but now in terms of more fundamental constants of nature, including the electron's charge and Planck's constant. The Rydberg formula (Rydberg-Ritz formula) is used in atomic physics for determining the full spectrum of light emission from hydrogen, later extended to be useful with any element by use of the Rydberg-Ritz combination principle. ... This article is about the physical phenomenon. ... The Bohr model of the hydrogen atom () or a hydrogen-like ion (), where the negatively charged electron confined to an atomic shell encircles a small positively charged atomic nucleus, and an electron jump between orbits is accompanied by an emitted or absorbed amount of electromagnetic energy . ... The Rydberg constant, named after physicist Johannes Rydberg, is a physical constant that appears in the Rydberg formula. ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ...

When the electron moves from one energy level to another, a photon is emitted. Using the derived formula for the different 'energy' levels of hydrogen one may determine the 'wavelengths' of light that a hydrogen atom can emit. In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ...

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

$E=E_i-E_f=R_E left( frac{1}{n_{f}^2} - frac{1}{n_{i}^2} right) ,$

where nf is the final energy level, and ni is the initial energy level.

Since the energy of a photon is In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ...

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

the wavelength of the photon given off is given by

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

This is known as the Rydberg formula, and the Rydberg constant R is RE / hc, or RE / 2π in natural units. This formula was known in the nineteenth century to scientists studying spectroscopy, but there was no theoretical explanation for this form or a theoretical prediction for the value of R, until Bohr. In fact, Bohr's derivation of the Rydberg constant, as well as the concomitant agreement of Bohr's formula with experimentally observed spectral lines of the Lyman (nf = 1), Balmer (nf = 2), and Paschen (nf = 3) series, and successful theoretical prediction of other lines not yet observed, was one reason that his model was immediately accepted. The Rydberg formula (Rydberg-Ritz formula) is used in atomic physics for determining the full spectrum of light emission from hydrogen, later extended to be useful with any element by use of the Rydberg-Ritz combination principle. ... In physics, natural units are physical units of measurement defined in terms of universal physical constants in such a manner that some chosen physical constants take on the numerical value of one when expressed in terms of a particular set of natural units. ... Animation of the dispersion of light as it travels through a triangular prism. ... The Bohr model of the hydrogen atom () or a hydrogen-like ion (), where the negatively charged electron confined to an atomic shell encircles a small positively charged atomic nucleus, and an electron jump between orbits is accompanied by an emitted or absorbed amount of electromagnetic energy . ... The Lyman series is the series of transitions and resulting emission lines of the hydrogen atom as an electron goes from n â‰¥ 2 to n = 1 (where n is the principal quantum number referring to the energy level of the electron). ... Two of the balmer lines (Î± and Î²) are clearly visible in this emission spectrum of a deuterium lamp. ... The Paschen series is the series of transitions and resulting emission lines of the hydrogen atom as an electron goes from n â‰¥ 4 to n = 3 (where n refers to the energy level of the electron). ...

## Shell model of the atom

Bohr extended the model of Hydrogen to give an approximate model for heavier atoms. This gave a physical picture which reproduced many known atomic properties for the first time.

Heavier atoms have more protons in the nucleus, and more electrons to cancel the charge. Bohr's idea was that each discrete orbit could only hold a certain number of electrons. After that orbit is full, the next level would have to be used. This gives the atom a shell structure, in which each shell corresponds to a Bohr orbit. In nuclear physics, the nuclear shell model is a model of the atomic nucleus. ...

This model is even more approximate than the model of hydrogen, because it treats the electrons in each shell as non-interacting. But the repulsions of electrons is taken into account somewhat by the phenomenon of screening. The electrons in outer orbits do not only orbit the nucleus, but they also orbit the inner electrons, so the effective charge Z that they feel is reduced by the number of the electrons in the inner orbit. The shielding effect or screening effect is an effect which occurs on a subatomic level between electrons occupying energy levels and is caused by repulsive forces of other electrons between it and the nucleus. ...

For example, the lithium atom has two electrons in the lowest 1S orbit, and these orbit at Z=2. Each one sees the nuclear charge of Z=3 minus the screening effect of the other, which crudely reduces the nuclear charge by 1 unit. This means that the innermost electrons orbit at approximately 1/4th the Bohr radius. The outermost electron in lithium orbits at roughly Z=1, since the two inner electrons reduce the nuclear charge by 2. This outer electron should be at nearly one Bohr radius from the nucleus. Because the electrons strongly repel each other, the effective charge description is very approximate, the effective charge Z doesn't usually come out to be an integer. But Moseley's law experimentally probes the innermost pair of electrons, and shows that they do see a nuclear charge of approximately Z-1, while the outermost electron in an atom or ion with only one electron in the outermost shell orbits a core with effective charge Z-k where k is the total number of electrons in the inner shells. Moseleys law is an empirical law concerning the characteristic x-rays that are emitted by atoms. ...

The shell model was able to qualitatively explain many of the mysterious properties of atoms which became codified in the late 19th century in the periodic table of the elements. One property was the size of atoms, which could be determined approximately by measuring the viscosity of gases and density of pure crystalline solids. Atoms tend to get smaller as you move to the right in the periodic table, becoming much bigger at the next line of the table. Atoms to the right of the table tend to gain electrons, while atoms to the left tend to lose them. Every element on the last column of the table is chemically inert (noble gas). The periodic table of the chemical elements, also called the Mendeleev periodic table, is a tabular display of the known chemical elements. ... For other uses, see Viscosity (disambiguation). ... This article is about the chemical series. ...

In the shell model, this phenomenon is explained by shell-filling. Successive atoms get smaller because they are filling orbits of the same size, until the orbit is full, at which point the next atom in the table has a loosely bound outer electron, causing it to expand. The first Bohr orbit is filled when it has two electrons, and this explains why helium is inert. The second orbit allows eight electrons, and when it is full the atom is neon, again inert. The third orbital contains eight again, except that in the more correct Sommerfeld treatment (reproduced in modern quantum mechanics) there are extra "d" electrons. The third orbit may hold an extra 10 d electrons, but these positions are not filled until a few more orbitals from the next level are filled (Filling the n=3 d orbitals produces the 10 transition elements). The irregular filling pattern is an effect of interactions between electrons, which are not taken into account in either the Bohr or Sommerfeld models, and which are difficult to calculate even in the modern treatment. This article is in need of attention. ...

## Moseley's law and calculation of K-alpha X-ray emission lines

Niels Bohr said in 1962, "You see actually the Rutherford work [the nuclear atom] was not taken seriously. We cannot understand today, but it was not taken seriously at all. There was no mention of it any place. The great change came from Moseley."

In 1913 Henry Moseley found an empirical relationship between the strongest X-ray line emitted by atoms under electron bombardment (then known as the K-alpha line), and their atomic number Z. Moseley's empiric formula was found to be derivable from Rydberg and Bohr's formula (Moseley actually mentions only Ernest Rutherford and Antonius Van den Broek in terms of models). The two additional assumptions that [1] this X-ray line came from a transition between energy levels with quantum numbers 1 and 2, and [2], that the atomic number Z when used in the formula for atoms heavier than hydrogen, should be diminished by 1, to (Z-1)². This article is about the physicist; for the naturalist see Henry Nottidge Moseley Henry Moseley at work. ... In X-ray spectroscopy, K-alpha emission or absorption lines result when an electron transitions from the innermost K shell (principal quantum number 1), to a 2p orbital of the second or L shell (with principal energy quantum number 2). ... Ernest Rutherford, 1st Baron Rutherford of Nelson OM PC FRS (30 August 1871 â€“ 19 October 1937), widely referred to as Lord Rutherford, was a chemist (B.Sc. ... Antonius Van den Broek (May 4, 1870 - October 25, 1926) was a Dutch amateur physicist (a real estate lawyer by training). ...

Moseley wrote to Bohr, puzzled about his results, but Bohr was not able to help. At that time, he thought that the postulated innermost "K" shell of electrons should have at least four electrons, not the two which would have neatly explained the result. So Moseley published his results without a theoretical explanation.

Later, people realized that the effect was caused by charge screening, with an inner shell containing only 2 electrons. In the experiment, one of the innermost electrons in the atom is knocked out, leaving a vacancy in the lowest Bohr orbit, which contains a single remaining electron. This vacancy is then filled by electrons in the next orbit, which has n=2. But the n=2 electrons see an effective charge of Z-1, which is the value appropriate for the charge of the nucleus, when a single electron remains in the lowest Bohr orbit to screen the nuclear charge +Z, and lower it by -1 (due to the electron's negative charge screening the nuclear positive charge). The energy gained by an electron dropping from the second shell to the first gives Moseley's law for K-alpha lines: Moseleys law is an empirical law concerning the characteristic x-rays that are emitted by atoms. ...

$E= hnu = E_i-E_f=R_E (Z-1)^2 left( frac{1}{1^2} - frac{1}{2^2} right) ,$

or

$f = nu = R_E/h = R_v left( frac{3}{4}right) (Z-1)^2 = (2.46 times 10^{15} operatorname{Hz})(Z-1)^2.$

Here, Rv is the Rydberg constant given in terms of frequency, or RE/h = 3.28 x 1015 Hz. This latter relationship had been empirically derived by Moseley, in a simple plot of the square root of X-ray frequency against atomic number. Moseley's law not only established the objective meaning of atomic number (see Henry Moseley for detail) but, as Bohr noted, it also did more than the Rydberg derivation to establish the validity of the Rutherford/Van den Broek/Bohr nuclear model of the atom, with atomic number as nuclear charge. This article is about the physicist; for the naturalist see Henry Nottidge Moseley Henry Moseley at work. ...

The K-alpha line of Moseley's time is now known to be a pair of close lines, written as (Kα1 and Kα2) in Siegbahn notation. In X-ray spectroscopy, K-alpha emission or absorption lines result when an electron transitions from the innermost K shell (principal quantum number 1), to a 2p orbital of the second or L shell (with principal energy quantum number 2). ... The Siegbahn notation is used to name the spectral lines that are characteristic to elements. ...

## Shortcomings

The Bohr model gives an incorrect value $scriptstyle mathbf{L} = hbar$ for the ground state orbital angular momentum. The angular momentum in the true ground state is known to be zero. Although mental pictures fail somewhat at these levels of scale, an electron in the lowest modern "orbital" with no orbital momentum, may be thought of as not to rotate "around" the nucleus at all, but merely to go tightly around it in an ellipse with zero area (this may be pictured as "back and forth", without striking or interacting with the nucleus). This is only reproduced in a more sophisticated semiclassical treatment like Sommerfeld's, but even in that case, the model fails to explain the empirically spherical nature of the orbital which represents the behavior of electrons with zero angular momentum.

In modern quantum mechanics, the electron in hydrogen is a spherical cloud of probability which grows more dense near the nucleus. The rate-constant of probability-decay in hydrogen is equal to the inverse of the Bohr radius, but since Bohr worked with circular orbits, not zero area ellipses, the fact that these two numbers exactly agree, is considered a "coincidence." (Though many such coincidenal agreements are found between the semi-classical vs. full quantum mechanial treatment of the atom; these include identical energy levels in the hydrogen atom, and the derivation of a fine structure constant, which arises from the relativistic Bohr-Sommerfield model (see below), and which happens to be equal to an entirely different concept, in full modern quantum mechanics). Electron cloud is a term used- if not originally coined- by the nobelaurate and acclaimed educator Richard Feynman in The Feynman Lectures on Physics, for discussing exactly what is an electron?. This intuitive model provides a simplified way of visualizing an electron as a solution of the SchrÃ¶dinger equation. ... The fine-structure constant or Sommerfeld fine-structure constant, usually denoted , is the fundamental physical constant characterizing the strength of the electromagnetic interaction. ...

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

• Much of the spectra of larger atoms. At best, it can make predictions about the K-alpha and some L-alpha X-ray emission spectra for larger atoms, if two additional ad hoc assumptions are made (see Moseley's law above). Emission spectra for atoms with a single outer-shell electron (atoms in the lithium group) can also be approximately predicted. Also, if the empiric electron-nuclear screening factors for many atoms are known, many other spectral lines can be deduced from the information, in similar atoms of differing elements, via the Ritz-Rydberg combination principles (see Rydberg formula). All these techniques essentially make use of Bohr's Newtonian energy-potential picture of the atom.
• The relative intensities of spectral lines; although in some simple cases, Bohr's formula or modifications of it, was able to provide reasonable estimates (for example, calculations by Kramers for the Stark effect).
• The existence of fine structure and hyperfine structure in spectral lines, which are known to be due to a variety of relativistic and subtle effects, as well as complications from electron spin.
• The Zeeman effect - changes in spectral lines due to external magnetic fields; these are also due to more complicated quantum principles interacting with electron spin and orbital magnetic fields.

In X-ray spectroscopy, K-alpha emission or absorption lines result when an electron transitions from the innermost K shell (principal quantum number 1), to a 2p orbital of the second or L shell (with principal energy quantum number 2). ... Moseleys law is an empirical law concerning the characteristic x-rays that are emitted by atoms. ... This article is about the chemical element. ... The Rydberg formula (Rydberg-Ritz formula) is used in atomic physics for determining the full spectrum of light emission from hydrogen, later extended to be useful with any element by use of the Rydberg-Ritz combination principle. ... The Stark effect is the splitting of a spectral line into several components in the presence of an electric field. ... In atomic physics, the fine structure describes the splitting of the spectral lines of atoms. ... In atomic physics, hyperfine structure is a small perturbation in the energy levels (or spectra) of atoms or molecules due to the magnetic dipole-dipole interaction, arising from the interaction of the nuclear magnetic dipole with the magnetic field of the electron. ... The Zeeman effect (IPA ) is the splitting of a spectral line into several components in the presence of a magnetic field. ... For the indie-pop band, see The Magnetic Fields. ...

## Refinements

Elliptical orbits with the same energy and quantized angular momentum

Several enhancements to the Bohr model were proposed; most notably the Sommerfeld model or Bohr-Sommerfeld model, which suggested that electrons travel in elliptical orbits around a nucleus instead of the Bohr model's circular orbits. This model supplemented the quantized angular momentum condition of the Bohr model with an additional radial quantization condition, the Sommerfeld-Wilson quantization condition Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ...

$int_0^T p_r dq_r = n h ,$

where p_r is the radial momentum canonically conjugate to the coordinate q which is the radial position and T is one full orbital period. The integral is the action of action-angle coordinates. This condition, suggested by the correspondence principle, is the only one possible, since the quantum numbers are adiabatic invariants. In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ... In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. ... In physics, the correspondence principle is a principle, first invoked by Niels Bohr in 1923, which states that the behavior of quantum mechanical systems reduce to classical physics in the limit of large quantum numbers. ... An adiabatic invariant is a property of a physical system which stays constant when changes are made slowly. ...

However, this is not to say that the Bohr model was without its successes. Calculations based on the Bohr-Sommerfeld model were able to accurately explain a number of more complex atomic spectral effects. For example, up to first-order perturbations, the Bohr model and quantum mechanics make the same predictions for the spectral line splitting in the Stark effect. At higher-order perturbations, however, the Bohr model and quantum mechanics differ, and measurements of the Stark effect under high field strengths helped confirm the correctness of quantum mechanics over the Bohr model. The prevailing theory behind this difference lies in the shapes of the orbitals of the electrons, which vary according to the energy state of the electron. Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem. ... The Stark effect is the splitting of a spectral line into several components in the presence of an electric field. ...

The Bohr-Sommerfeld quantization conditions lead to questions in modern mathematics. Consistent semiclassical quantization condition requires a certain type of structure on the phase space, which places topological limitations on the types of symplectic manifolds which can be quantized. In particular, the symplectic form should be the curvature form of a connection of a Hermitian line bundle, which is called a prequantization. In differential geometry, the curvature form describes curvature of principal bundle with connection. ... In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. ... Charles Hermite (pronounced in IPA, ) (December 24, 1822 â€“ January 14, 1901) was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. ... In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. ... In mathematical physics, geometric quantization is a mathematical approach to define a quantum theory corresponding to a given classical theory in such a way that certain analogies between the classical theory and the quantum theory remain manifest, for example the similarity between the Heisenberg equation in the Heisenberg picture of...

The Franck-Hertz experiment was a physics experiment that provided support for the Bohr model of the atom, a precursor to quantum mechanics. ... Moseleys law is an empirical law concerning the characteristic x-rays that are emitted by atoms. ... This article is about the physicist; for the naturalist see Henry Nottidge Moseley Henry Moseley at work. ... In physics and chemistry, the inert pair effect occurs when electrons are pulled closer to the nucleus, making them stabler and more difficult to ionise. ... The Lyman series is the series of transitions and resulting emission lines of the hydrogen atom as an electron goes from n â‰¥ 2 to n = 1 (where n is the principal quantum number referring to the energy level of the electron). ... This box:      For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... The theoretical and experimental justification for the SchrÃ¶dinger equation motivates the discovery of the SchrÃ¶dinger equation, the equation that describes the dynamics of nonrelativistic particles. ... Balmers Constant is used in chemistry to discern the frequency of light emitted when an atoms electron returns to the ground state. ... For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... See also: Other events of 1913 List of years in science . ...

## References

### Historical

• Niels Bohr (1913). "On the Constitution of Atoms and Molecules (Part 1 of 3)". Philosophical Magazine 26: 1-25.
• Niels Bohr (1913). "On the Constitution of Atoms and Molecules, Part II Systems Containing Only a Single Nucleus". Philosophical Magazine 26: 476-502.
• Niels Bohr (1913). "On the Constitution of Atoms and Molecules, Part III". Philosophical Magazine 26: 857-875.
• Niels Bohr (1914). "The spectra of helium and hydrogen". Nature 92: 231-232.
• Niels Bohr (1921). "Atomic Structure". Nature.
• 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 University Press, Princeton. 6 p.434. (Provides an elegant reformulation of the Bohr-Sommerfeld quantization conditions, as well as an important insight into the quantization of non-integrable (chaotic) dynamical systems.)

• Linus Pauling (1985). General Chemistry, Chapter 3 (3rd ed). Dover Publications.  A great explainer of Chemistry describes the Bohr model, appropriate for High School and College students.
• George Gamow (1985). Thirty years that shook Physics, Chapter 2. Dover Publications.  A popularizer of physics explains the Bohr model in the context of the development of quantum mechanics, appropriate for High School and College students
• Walter J. Lehmann (1972). Atomic and Molecular Structure: the development of our concepts, chapter 18. John Wiley and Sons.  Great explanations, appropriate for High School and College students
• Paul Tipler and Ralph Llewellyn (2002). Modern Physics (4th ed.). W. H. Freeman. ISBN 0-7167-4345-0.

Results from FactBites:

 The Bohr Model (0 words) The Bohr Model is probably familar as the "planetary model" of the atom illustrated in the adjacent figure that, for example, is used as a symbol for atomic energy (a bit of a misnomer, since the energy in "atomic energy" is actually the energy of the nucleus, rather than the entire atom). In the Bohr Model the neutrons and protons (symbolized by red and blue balls in the adjacent image) occupy a dense central region called the nucleus, and the electrons orbit the nucleus much like planets orbiting the Sun (but the orbits are not confined to a plane as is approximately true in the Solar System). This similarity between a planetary model and the Bohr Model of the atom ultimately arises because the attractive gravitational force in a solar system and the attractive Coulomb (electrical) force between the positively charged nucleus and the negatively charged electrons in an atom are mathematically of the same form.
 O=CHemBohrModel (365 words) The model bears his name because of his interpretation of the emission spectrum of hydrogen: If a small amount of hydrogen gas is confined within a glass tube and subjected to a high voltage, it emits light, some of which falls in the visible region of the electromagnetic spectrum. Bohr knew that the emission of light was the way the atoms released the energy they had absorbed when the high voltage was applied. Bohr's success in rationalizing the emission spectrum of hydrogen led to the general acceptance of the planetary model of the atom.
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