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In atomic physics, the Rutherford–Bohr model or Bohr model, introduced by Niels Bohr in 1913, depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar in structure to the solar system, but with attraction provided by electrostatic forces rather than gravity. After the cubic model (1902), the plum-pudding model (1904), the Saturnian model (1904), and the Rutherford model (1911) came the Rutherford–Bohr model or just ''Bohr model'' for short (1913). The improvement to the Rutherford model is mostly a quantum physical interpretation of it. The Bohr model has been superseded, but the quantum theory remains sound. 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, it also provided a justification for its empirical results in terms of fundamental physical constants. The Bohr model is a relatively primitive model of the hydrogen atom, compared to the valence shell 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 or energy level diagrams 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. == Origin == 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, Rutherford naturally considered a planetary-model atom, the Rutherford model of 1911 – electrons orbiting a solar nucleus – however, said planetary-model atom has a technical difficulty. The laws of classical mechanics (i.e. the Larmor formula), predict that the electron will release electromagnetic radiation while orbiting a nucleus. Because the electron would lose energy, it would rapidly spiral inwards, collapsing into the nucleus on a timescale of around 16 picoseconds.〔(Olsen and McDonald 2005 )〕 This atom model is disastrous, because it predicts that all atoms are unstable.〔(【引用サイトリンク】title=CK12 – Chemistry Flexbook Second Edition – The Bohr Model of the Atom )〕 Also, as the electron spirals inward, the emission would rapidly 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 have shown that atoms will only emit light (that is, electromagnetic radiation) at certain discrete frequencies. To overcome this difficulty, Niels Bohr proposed, in 1913, what is now called the ''Bohr model of the atom''. He suggested that electrons could only have certain ''classical'' motions: # Electrons in atoms orbit the nucleus. # The electrons can only orbit stably, without radiating, in certain orbits (called by Bohr the "stationary orbits") at a certain discrete set of distances from the nucleus. These orbits are associated with definite energies and are also called energy shells or energy levels. In these orbits, the electron's acceleration does not result in radiation and energy loss as required by classical electromagnetics. The Bohr model of an atom was based upon Planck's quantum theory of radiation. # Electrons can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency ''ν'' determined by the energy difference of the levels according to the Planck relation: where ''h'' is Planck's constant. The frequency of the radiation emitted at an orbit of period ''T'' is as it would be in classical mechanics; it is the reciprocal of the classical orbit period: The significance of the Bohr model is that the laws of classical mechanics apply to the motion of the electron about the nucleus ''only when restricted by a quantum rule''. Although Rule 3 is not completely well defined for small orbits, because the emission process involves two orbits with two different periods, Bohr could determine the energy spacing between levels using Rule 3 and come to an exactly correct quantum rule: the angular momentum ''L'' is restricted to be an integer multiple of a fixed unit: : where ''n'' = 1, 2, 3, ... is called the principal quantum number, and ''ħ'' = ''h''/2π. The lowest value of ''n'' is 1; this gives a smallest possible orbital radius of 0.0529 nm 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〔 was able to calculate the energies of the allowed orbits of the hydrogen atom and other hydrogen-like atoms and ions. Other points are: # Like Einstein's theory of the Photoelectric effect, Bohr's formula assumes that during a quantum jump a ''discrete'' amount of energy is radiated. However, unlike Einstein, Bohr stuck to the ''classical'' Maxwell 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. # 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 energy levels ''E''''n'' and ''E''''n''−''k'' when ''k'' is much smaller than ''n''. These jumps reproduce the frequency of the ''k''-th harmonic of orbit ''n''. For sufficiently large values of ''n'' (so-called Rydberg states), the two orbits involved in the emission process have nearly the same rotation frequency, so that the classical orbital frequency is not ambiguous. But for small ''n'' (or large ''k''), the radiation frequency has no unambiguous classical interpretation. This marks the birth of the correspondence principle, requiring quantum theory to agree with the classical theory only in the limit of large quantum numbers. # The Bohr-Kramers-Slater theory (BKS theory) is a failed attempt to extend the Bohr model, which violates the conservation of energy and momentum in quantum jumps, with the conservation laws only holding on average. Bohr's condition, that the angular momentum is an integer multiple of ''ħ'' was later reinterpreted in 1924 by de Broglie as a standing wave condition: the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit: : Substituting de Broglie's wavelength of ''λ'' = ''h''/''p'' reproduces Bohr's rule. 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 (i.e., matter waves) was not suspected. In 1925 a new kind of mechanics was proposed, quantum mechanics, in which Bohr's model of electrons traveling in quantized orbits was extended into a more accurate model of electron motion. The new theory was proposed by Werner Heisenberg. Another form of the same theory, wave mechanics, was discovered by the Austrian physicist Erwin Schrödinger independently, and by different reasoning. Schrödinger employed de Broglie's matter waves, but sought wave solutions of a three-dimensional wave equation describing electrons that were constrained to move about the nucleus of a hydrogen-like atom, by being trapped by the potential of the positive nuclear charge. ==Electron energy levels== The Bohr model gives almost exact results only for a system where two charged points orbit each other at speeds much less than that of light. This not only includes one-electron systems such as the hydrogen atom, singly ionized helium, doubly ionized lithium, but it includes positronium and Rydberg states of any atom where one electron is far away from everything else. It can be used for K-line X-ray transition calculations if other assumptions are added (see Moseley's law below). In high energy physics, it can be used to calculate the masses of heavy quark mesons. Calculation of the orbits requires two assumptions. *Classical mechanics :The electron is held in a circular orbit by electrostatic attraction. The centripetal force is equal to the Coulomb force. :: :where ''m''e is the electron's mass, ''e'' is the charge of the electron, ''k''e is Coulomb's constant and ''Z'' is the atom's atomic number. It is assumed here that the mass of the nucleus is much larger than the electron mass (which is a good assumption). This equation determines the electron's speed at any radius: :: : It also determines the electron's total energy at any radius: :: :The total energy is negative and inversely proportional to ''r''. This means that it takes energy to pull the orbiting electron away from the proton. For infinite values of ''r'', the energy is zero, corresponding to a motionless electron infinitely far from the proton. The total energy is half the potential energy, which is also true for noncircular orbits by the virial theorem. *A quantum rule :The angular momentum is an integer multiple of ''ħ'': :: , or some average—in hindsight, this model is only the leading semiclassical approximation. Bohr considered circular orbits. Classically, these orbits must decay to smaller circles when photons are emitted. The level spacing between circular orbits can be calculated with the correspondence formula. For a Hydrogen atom, the classical orbits have a period ''T'' determined by Kepler's third law to scale as ''r''3/2. The energy scales as 1/''r'', so the level spacing formula amounts to : It is possible to determine the energy levels by recursively stepping down orbit by orbit, but there is a shortcut. The angular momentum ''L'' of the circular orbit scales as √ . The energy in terms of the angular momentum is then :. Assuming, with Bohr, that quantized values of ''L'' are equally spaced, the spacing between neighboring energies is : This is as desired for equally spaced angular momenta. If one kept track of the constants, the spacing would be ''ħ'', so the angular momentum should be an integer multiple of ''ħ'', : This is how Bohr arrived at his model. |} :Substituting the expression for the velocity gives an equation for ''r'' in terms of n: :: :so that the allowed orbit radius at any n is: :: :The smallest possible value of ''r'' in the hydrogen atom (Z=1) is called the Bohr radius and is equal to: :: :The energy of the ''n''-th level for any atom is determined by the radius and quantum number: :: An electron in the lowest energy level of hydrogen () therefore has about 13.6 eV less energy than a motionless electron infinitely far from the nucleus. The next energy level () is −3.4 eV. The third (''n'' = 3) is −1.51 eV, and so on. For larger values of ''n'', these are also the binding energies of a highly excited atom with one electron in a large circular orbit around the rest of the atom. The combination of natural constants in the energy formula is called the Rydberg energy (''R''E): : This expression is clarified by interpreting it in combinations that form more natural units: : is the rest mass energy of the electron (511 keV) : is the fine structure constant : 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 ''q'' = ''Z e'' where ''Z'' is the atomic number. This will now give us energy levels for hydrogenic atoms, which can serve as a rough order-of-magnitude approximation of the actual energy levels. So for nuclei with ''Z'' protons, the energy levels are (to a rough approximation): : The actual energy levels cannot be solved analytically for more than one electron (see ''n''-body problem) because the electrons are not only affected by the nucleus 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 rest energy. Sufficiently large nuclei, if they were stable, would reduce their charge by creating a bound electron from the vacuum, ejecting the positron to infinity. This is the theoretical phenomenon of electromagnetic charge screening which predicts a maximum nuclear charge. Emission of such positrons has been observed in the collisions of heavy ions to create temporary super-heavy nuclei. The Bohr formula properly uses the reduced mass of electron and proton in all situations, instead of the mass of the electron: . However, these numbers are very nearly the same, due to the much larger mass of the proton, about 1836.1 times the mass of the electron, so that the reduced mass in the system is the mass of the electron multiplied by the constant 1836.1/(1+1836.1) = 0.99946. This fact was historically important in convincing Rutherford of the importance of Bohr's model, for it explained the fact that the frequencies of lines in the spectra for singly ionized helium do not differ from those of hydrogen by a factor of exactly 4, but rather by 4 times the ratio of the reduced mass for the hydrogen vs. the helium systems, which was much closer to the experimental ratio than exactly 4. For positronium, the formula uses the reduced mass also, but in this case, it is exactly the electron mass divided by 2. For any value of the radius, the electron and the positron are each moving at half the speed around their common center of mass, and each has only one fourth the kinetic energy. The total kinetic energy is half what it would be for a single electron moving around a heavy nucleus. : (positronium) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bohr model」の詳細全文を読む スポンサード リンク
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