Movatterモバイル変換

[0]ホーム
URL:

Jump to content
WikipediaThe Free Encyclopedia
Search

Bohr model

From Wikipedia, the free encyclopedia
(Redirected fromBohr model of the atom)
Atomic model introduced by Niels Bohr in 1913
"Bohr's law" redirects here. For other uses, seeBohr's law (disambiguation).
Not to be confused withBohr equation orBohr effect.
The Bohr model of thehydrogen atom (Z = 1) or a hydrogen-like ion (Z > 1), where the negatively chargedelectron confined to anatomic shell encircles a small, positively chargedatomic nucleus and where an electron jumps between orbits, is accompanied by an emitted or absorbed amount ofelectromagnetic energy (hν).[1] The orbits in which the electron may travel are shown as grey circles; their radius increases asn2, wheren is theprincipal quantum number. The3 → 2 transition depicted here produces the first line of theBalmer series, and for hydrogen (Z = 1) it results in a photon ofwavelength 656 nm (red light).

Inatomic physics, theBohr model orRutherford–Bohr model was the first successful model of theatom. Developed from 1911 to 1918 byNiels Bohr and building onErnest Rutherford's nuclearmodel, it supplanted theplum pudding model ofJ. J. Thomson only to be replaced by the quantum atomic model in the 1920s. It consists of a small, dense nucleus surrounded byorbiting electrons. It isanalogous to the structure of theSolar System, but with attraction provided byelectrostatic force rather thangravity, and with the electron energies quantized (assuming only discrete values).

In the history of atomic physics, it followed, and ultimately replaced, several earlier models, includingJoseph Larmor's Solar System model (1897),Jean Perrin's model (1901),[2] thecubical model (1902),Hantaro Nagaoka'sSaturnian model (1904), the plum pudding model (1904),Arthur Haas's quantum model (1910), theRutherford model (1911), andJohn William Nicholson's nuclear quantum model (1912). The improvement over the 1911 Rutherford model mainly concerned the newquantum mechanical interpretation introduced by Haas and Nicholson, but forsaking any attempt to explain radiation according toclassical physics.

The model's key success lies in explaining theRydberg formula forhydrogen's spectral emission lines. 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.

The Bohr model is a relatively primitive model of the hydrogen atom, compared to thevalence shell model. As a theory, it can be derived as afirst-order approximation of the hydrogen atom using the broader and much more accurate quantum mechanics and thus may be considered to be anobsolete 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 orenergy level diagrams before moving on to the more accurate, but more complex, valence shell atom. A related quantum model was proposed byArthur Erich Haas in 1910 but was rejected until the 1911 Solvay Congress where it was thoroughly discussed.[3] The quantum theory of the period betweenPlanck's discovery of the quantum (1900) and the advent of a mature quantum mechanics (1925) is often referred to as theold quantum theory.

Background

[edit]
Main article:History of atomic theory
Bohr model in 1921[4] after Sommerfeld expansion of 1913 model showing maximum electrons per shell with shells labeled inX-ray notation

Until the second decade of the 20th century, atomic models were generally speculative. Even the concept of atoms, let alone atoms with internal structure, faced opposition from some scientists.[5]: 2 

Planetary models

[edit]

In the late 1800s speculations on the possible structure of the atom included planetary models with orbiting charged electrons.[6]: 35 These models faced a significant constraint.In 1897,Joseph Larmor showed that an accelerating charge would radiate power according to classical electrodynamics, a result known as theLarmor formula. Since electrons forced to remain in orbit are continuously accelerating, they would be mechanically unstable. Larmor noted that electromagnetic effect of multiple electrons, suitable arranged, would cancel each other. Thus subsequent atomic models based on classical electrodynamics needed to adopt such special multi-electron arrangements.[7]: 113 

Thomson's atom model

[edit]
Main article:Plum pudding model

When Bohr began his work on a new atomic theory in the summer of 1912[8]: 237  the atomic model proposed byJ. J. Thomson, now known as the plum pudding model, was the best available.[9]: 37  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 was 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. He developed a formula for the scattering ofbeta particles that seemed to match experimental results.[9]: 38 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 lines in atomic spectra.[5]: 18 

Rutherford nuclear model

[edit]
Main articles:Rutherford atom andRutherford scattering experiments

In 1908,Hans Geiger andErnest Marsden demonstrated thatalpha particle occasionally scatter at large angles, a result inconsistent with Thomson's model.In 1911 Ernest Rutherford 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 hismodel of the atom was incomplete.[10]Bohr begins his first paper on his atomic model by describing Rutherford's atom as consisting of a small, dense, positively charged nucleus attracting negatively chargedelectrons.[11]

Atomic spectra

[edit]

By the early twentieth century, it was expected that the atom would account for the many atomic spectral lines. These lines were summarized in empirical formula byJohann Balmer andJohannes Rydberg. In 1897, Lord Rayleigh showed that vibrations of electrical systems predicted spectral lines that depend on the square of the vibrational frequency, contradicting the empirical formula which depended directly on the frequency.[5]: 18 [12]In 1907Arthur W. Conway showed that, rather than the entire atom vibrating, vibrations of only one of the electrons in the system described by Thomson might be sufficient to account for spectral series.[13]: II:106  Although Bohr's model would also rely on just the electron to explain the spectrum, he did not assume an electrodynamical model for the atom.

The other important advance in the understanding of atomic spectra was theRydberg–Ritz combination principle which related atomic spectral line frequencies to differences between 'terms', special frequencies characteristic of each element.[14]: 173  Bohr would recognize the terms as energy levels of the atom divided by the Planck constant, leading to the modern view that the spectral lines result from energy differences.[15]: 847 [16]

Haas atomic model

[edit]

In 1910,Arthur Erich Haas proposed a model of the hydrogen atom with an electron circulating on the surface of a sphere of positive charge. The model resembled Thomson's plum pudding model, but Haas added a radical new twist: he constrained the electron's potential energy,Epot{\displaystyle E_{\text{pot}}}, on a sphere of radiusa to equal the frequency,f, of the electron's orbit on the sphere times thePlanck constant:[14]: 197 Epot=−e2a=hf{\displaystyle E_{\text{pot}}={\frac {-e^{2}}{a}}=hf}wheree represents the charge on the electron and the sphere. Haas combined this constraint with the balance-of-forces equation. The attractive force between the electron and the sphere balances thecentrifugal force:e2a2=ma(2πf)2{\displaystyle {\frac {e^{2}}{a^{2}}}=ma(2\pi f)^{2}}wherem is the mass of the electron. This combination relates the radius of the sphere to the Planck constant:a=h24π2e2m{\displaystyle a={\frac {h^{2}}{4\pi ^{2}e^{2}m}}}Haas solved for the Planck constant using the then-current value for the radius of the hydrogen atom.Three years later, Bohr would use similar equations with different interpretation. Bohr took the Planck constant as given value and used the equations to predict,a, the radius of the electron orbiting in the ground state of the hydrogen atom. This value is now called theBohr radius.[14]: 197 

Influence of the Solvay Conference

[edit]

The firstSolvay Conference, in 1911, was one of the first international physics conferences. Nine Nobel or future Nobel laureates attended, includingErnest Rutherford, Bohr's mentor.[17]: 271  Bohr did not attend but he read the Solvay reports[18] and discussed them with Rutherford.[8]: 233 

The subject of the conference was the theory of radiation and the energy quanta ofMax Planck's oscillators.[19] Planck's lecture at the conference ended with comments about atoms and the discussion that followed it concerned atomic models.Hendrik Lorentz raised the question of the composition of the atom based onHaas's model, a form of Thomson's plum pudding model with a quantum modification. Lorentz explained that the size of atoms could be taken to determine the Planck constant as Haas had done or the Planck constant could be taken as determining the size of atoms.[17]: 273  Bohr would adopt the second path.

The discussions outlined the need for the quantum theory to be included in the atom. Planck explicitly mentions the failings of classical mechanics.[17]: 273  While Bohr had already expressed a similar opinion in his PhD thesis, at Solvay the leading scientists of the day discussed a break with classical theories.[8]: 244  Bohr's first paper on his atomic model cites the Solvay proceedings saying: "Whatever the alteration in the laws of motion of the electrons may be, it seems necessary to introduce in the laws in question a quantity foreign to the classical electrodynamics,i.e. Planck's constant, or as it often is called the elementary quantum of action."[11] Encouraged by the Solvay discussions, Bohr would assume the atom was stable and abandon the efforts to stabilize classical models of the atom[14]: 199 

Nicholson atom theory

[edit]

In 1911John William Nicholson 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 orbital angular momentum with the Planck constant.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.[20]: 169  In his 1913 paper Bohr cites Nicholson as finding quantized angular momentum important for the atom.[11]

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 from a friend aboutBalmer's compact formula for the spectral line data, Bohr quickly realized his model would match it in detail.[20]: 178 

Nicholson's model was based on classical electrodynamics along the lines ofJ.J. Thomson'splum pudding model but his negative electrons orbiting a positive nucleus rather than circulating in a sphere. 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.[20]: 163  By 1913 Bohr had already shown, from the analysis of alpha particle energy loss, that hydrogen had only a single electron not a matched pair.[14]: 195  Bohr's atomic model would abandon classical electrodynamics.

Nicholson's model of radiation was quantum but was attached to the orbits of the electrons.[21][19] Bohr quantization would associate it with differences in energy levels of his model of hydrogen rather than the orbital frequency.

Bohr's previous work

[edit]

Bohr completed his PhD in 1911 with a thesis 'Studies on the Electron Theory of Metals', an application of the classical electron theory ofHendrik Lorentz. Bohr noted two deficits of the classical model. The first concerned thespecific heat of metals whichJames Clerk Maxwell noted in 1875: every additional degree of freedom in a theory of metals, like subatomic electrons, cause more disagreement with experiment. The second, the classical theory could not explain magnetism.[14]: 194 

After his PhD, Bohr worked briefly in the lab ofJJ Thomson before moving to Rutherford's lab inManchester to study radioactivity. He arrived just after Rutherford completed his proposal of a compact nuclear core for atoms.Charles Galton Darwin, also at Manchester, had just completed an analysis of alpha particle energy loss in metals, concluding the electron collisions where the dominant cause of loss. Bohr showed in a subsequent paper that Darwin's results would improve by accounting for electron binding energy. Importantly this allowed Bohr to conclude that hydrogen atoms have a single electron.[14]: 195 

Development

[edit]

Next, Bohr was told by his friend, Hans Hansen, that the Balmer series is calculated using the Balmer formula, anempirical equation discovered byJohann Balmer in 1885 that described wavelengths of some spectral lines of hydrogen.[18][22] This was further generalized byJohannes Rydberg in 1888, resulting in what is now known as the Rydberg formula.After this, Bohr declared, "everything became clear".[22]

In 1913Niels Bohr put forth three postulates to provide an electron model consistent with Rutherford's nuclear model:

  1. The electron is able to revolve in certain stable orbits around the nucleus without radiating any energy, contrary to whatclassical electromagnetism 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.
  2. The stationary orbits are attained at distances for which the angular momentum of the revolving electron is an integer multiple of the reduced Planck constant:mevr=nℏ{\displaystyle m_{\mathrm {e} }vr=n\hbar }, wheren=1,2,3,...{\displaystyle n=1,2,3,...} is called theprincipal quantum number, andℏ=h/2π{\displaystyle \hbar =h/2\pi }. The lowest value ofn{\displaystyle n} is 1; this gives the smallest possible orbital radius, known as theBohr radius, 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,[18][19][21][20] Bohr[11] was able to calculate theenergies of the allowed orbits of the hydrogen atom and otherhydrogen-like atoms and ions. These orbits are associated with definite energies and are also called energy shells orenergy levels. 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.
  3. Electrons can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequencyν{\displaystyle \nu } determined by the energy difference of the levels according to thePlanck relation:ΔE=E2−E1=hν{\displaystyle \Delta E=E_{2}-E_{1}=h\nu }, whereh{\displaystyle h} is the Planck constant.

Other points are:

  1. Like Einstein's theory of thephotoelectric effect, Bohr's formula assumes that during a quantum jump adiscrete amount of energy is radiated. However, unlike Einstein, Bohr stuck to theclassicalMaxwell 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 ofphotons.[23][24]
  2. According to the Maxwell theory the frequencyν{\displaystyle \nu } of classical radiation is equal to the rotation frequencyν{\displaystyle \nu }rot of the electron in its orbit, withharmonics at integer multiples of this frequency. This result is obtained from the Bohr model for jumps between energy levelsEn{\displaystyle E_{n}} andEn−k{\displaystyle E_{n-k}} whenk{\displaystyle k} is much smaller thann{\displaystyle n}. These jumps reproduce the frequency of thek{\displaystyle k}-th harmonic of orbitn{\displaystyle n}. For sufficiently large values ofn{\displaystyle n} (so-calledRydberg 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 smalln{\displaystyle n} (or largek{\displaystyle k}), the radiation frequency has no unambiguous classical interpretation. This marks the birth of thecorrespondence principle, requiring quantum theory to agree with the classical theory only in the limit of large quantum numbers.
  3. TheBohr–Kramers–Slater theory (BKS theory) is a failed attempt to extend the Bohr model, which violates theconservation of energy andmomentum in quantum jumps, with the conservation laws only holding on average.

Bohr's condition, that the angular momentum be an integer multiple ofℏ{\displaystyle \hbar }, was later reinterpreted in 1924 byde Broglie as astanding 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:

nλ=2πr.{\displaystyle n\lambda =2\pi r.}

According to de Broglie's hypothesis, matter particles such as the electron behave aswaves. Thede Broglie wavelength of an electron is

λ=hmv,{\displaystyle \lambda ={\frac {h}{mv}},}

which implies that

nhmv=2πr,{\displaystyle {\frac {nh}{mv}}=2\pi r,}

or

nh2π=mvr,{\displaystyle {\frac {nh}{2\pi }}=mvr,}

wheremvr{\displaystyle mvr} is the angular momentum of the orbiting electron. Writingℓ{\displaystyle \ell } for this angular momentum, the previous equation becomes

ℓ=nh2π,{\displaystyle \ell ={\frac {nh}{2\pi }},}

which is Bohr's second postulate.

Bohr described angular momentum of the electron orbit as2/h{\displaystyle 2/h} whilede Broglie's wavelength ofλ=h/p{\displaystyle \lambda =h/p} describedh{\displaystyle h} divided by the electron momentum. In 1913, however, Bohr justified his rule by appealing to the correspondence principle, without providing any sort of wave interpretation. In 1913, the wave behavior of matter particles such as the electron was not suspected.

In 1925, a new kind of mechanics was proposed, quantum mechanics, in which Bohr's model of electrons traveling in quantized orbits was extended into amore accurate model of electron motion. The new theory was proposed byWerner Heisenberg.Another form of the same theory, wave mechanics, was discovered by the Austrian physicistErwin 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 ahydrogen-like atom, by being trapped by the potential of the positive nuclear charge.

Electron energy levels

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

The Bohr model gives almost exact results only for a system where two charged points orbit each other at speeds much less than that of light. This not only involves one-electron systems such as thehydrogen atom, singly ionizedhelium, and doubly ionizedlithium, but it includespositronium andRydberg states of any atom where one electron is far away from everything else. It can be used forK-line X-ray transition calculations if other assumptions are added (seeMoseley's law below). In high energy physics, it can be used to calculate the masses ofheavy quarkmesons.

Calculation of the orbits requires two assumptions.

  • Classical mechanics
The electron is held in a circular orbit by electrostatic attraction. Thecentripetal force is equal to theCoulomb force.
mev2r=Zkee2r2,{\displaystyle {\frac {m_{\mathrm {e} }v^{2}}{r}}={\frac {Zk_{\mathrm {e} }e^{2}}{r^{2}}},}
whereme is the electron's mass,e is theelementary charge,ke is theCoulomb constant andZ is the atom'satomic 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:
v=Zkee2mer.{\displaystyle v={\sqrt {\frac {Zk_{\mathrm {e} }e^{2}}{m_{\mathrm {e} }r}}}.}
It also determines the electron's total energy at any radius:
E=−12mev2.{\displaystyle E=-{\frac {1}{2}}m_{\mathrm {e} }v^{2}.}
The total energy is negative and inversely proportional tor. This means that it takes energy to pull the orbiting electron away from the proton. For infinite values ofr, the energy is zero, corresponding to a motionless electron infinitely far from the proton. The total energy is half thepotential energy, the difference being the kinetic energy of the electron. This is also true for noncircular orbits by thevirial theorem.
  • A quantum rule
Theangular momentumL =mevr is an integer multiple ofħ:
mevr=nℏ.{\displaystyle m_{\mathrm {e} }vr=n\hbar .}

Derivation

[edit]

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

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

Bohr assumes that the electron is circling the nucleus in an elliptical orbit obeying the rules of classical mechanics, but with no loss of radiation due to theLarmor formula.

Denoting the total energy asE, the negative electron charge ase, the positive nucleus charge asK=Z|e|, the electron mass asme, half the major axis of the ellipse asa, he starts with these equations:[11]: 3 

ν=2π|E|32|e|Kme   (1a){\displaystyle \nu ={\frac {\sqrt {2}}{\pi }}{\frac {\vert E\vert ^{\frac {3}{2}}}{\vert e\vert K{\sqrt {m}}_{e}}}\ \ \ (1a)}
2a=|e|K|E|   (1b){\displaystyle 2a={\frac {\vert e\vert K}{\vert E\vert }}\ \ \ (1b)}

E is assumed to be negative, because a positive energy is required to unbind the electron from the nucleus and put it at rest at an infinite distance.

Eq. (1a) is obtained from equating the centripetal force to the Coulombian force acting between the nucleus and the electron, considering thatE=T+U{\displaystyle E=T+U} (whereT is the average kinetic energy andU the average electrostatic potential), and that for Kepler's second law, the average separation between the electron and the nucleus isa.

Eq. (1b) is obtained from the same premises of eq. (1a) plus the virial theorem, stating that, for an elliptical orbit,

T=−12U   (1c).{\displaystyle T=-{\frac {1}{2}}U\ \ \ (1c).}

Then Bohr assumes that|E|{\displaystyle \vert E\vert } is an integer multiple of the energy of a quantum of light with half the frequency of the electron's revolution frequency,[11]: 4  i.e.:

|E|=nhν2   (2).{\displaystyle \vert E\vert =nh{\frac {\nu }{2}}\ \ \ (2).}

From eq. (1a,1b,2), it descends:

|E|=2π2mee2K2n2h2   (3a){\displaystyle \vert E\vert ={\frac {2\pi ^{2}m_{e}e^{2}K^{2}}{n^{2}h^{2}}}\ \ \ (3a)}
ν=4π2mee2K2n3h3   (3b){\displaystyle \nu ={\frac {4\pi ^{2}m_{e}e^{2}K^{2}}{n^{3}h^{3}}}\ \ \ (3b)}
2a=n2h22π2me|e|K   (3c).{\displaystyle 2a={\frac {n^{2}h^{2}}{2\pi ^{2}m_{e}\vert e\vert K}}\ \ \ (3c).}

He further assumes that the orbit is circular, i.e.a=r{\displaystyle a=r}, and, denoting the angular momentum of the electron asL, introduces the equation:

πL=Tν   (4).{\displaystyle \pi L={\frac {T}{\nu }}\ \ \ (4).}

Eq. (4) stems from thevirial theorem, and from the classical mechanics relationships between the angular momentum, the kinetic energy and the frequency of revolution.

From eq. (1c,2,4), it stems:

L=nLo,{\displaystyle L=nL_{o},}

where:

L0=h2π=ℏ,{\displaystyle L_{0}={\frac {h}{2\pi }}=\hbar ,}

that is:

L=nℏ.{\displaystyle L=n\hbar .}

This results states that the angular momentum of the electron is an integer multiple of the reduced Planck constant.[11]: 15 

Substituting the expression for the velocity gives an equation forr in terms ofn:

mekeZe2merr=nℏ,{\displaystyle m_{\text{e}}{\sqrt {\dfrac {k_{\text{e}}Ze^{2}}{m_{\text{e}}r}}}r=n\hbar ,}

so that the allowed orbit radius at anyn is

rn=n2ℏ2Zkee2me.{\displaystyle r_{n}={\frac {n^{2}\hbar ^{2}}{Zk_{\mathrm {e} }e^{2}m_{\mathrm {e} }}}.}

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

r1=ℏ2kee2me≈5.29×10−11 m=52.9 pm.{\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} .}

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

E=−Zkee22rn=−Z2(kee2)2me2ℏ2n2≈−13.6Z2n2 eV.{\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} .}

An electron in the lowest energy level of hydrogen (n = 1) therefore has about 13.6 eV less energy than a motionless electron infinitely far from the nucleus. The next energy level (n = 2) is −3.4 eV. The third (n =3) is −1.51 eV, and so on. For larger values ofn, 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 theWallis product.[25]

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

RE=(kee2)2me2ℏ2.{\displaystyle R_{\mathrm {E} }={\frac {(k_{\mathrm {e} }e^{2})^{2}m_{\mathrm {e} }}{2\hbar ^{2}}}.}

This expression is clarified by interpreting it in combinations that form morenatural units:

mec2{\displaystyle m_{\mathrm {e} }c^{2}} is therest mass energy of the electron (511 keV),
kee2ℏc=α≈1137{\displaystyle {\frac {k_{\mathrm {e} }e^{2}}{\hbar c}}=\alpha \approx {\frac {1}{137}}} is thefine-structure constant,
RE=12(mec2)α2{\displaystyle R_{\mathrm {E} }={\frac {1}{2}}(m_{\mathrm {e} }c^{2})\alpha ^{2}}.

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 chargeq =Ze, whereZ 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 withZ protons, the energy levels are (to a rough approximation):

En=−Z2REn2.{\displaystyle E_{n}=-{\frac {Z^{2}R_{\mathrm {E} }}{n^{2}}}.}

The actual energy levels cannot be solved analytically for more than one electron (seen-body problem) because the electrons are not only affected by thenucleus but also interact with each other via theCoulomb force.

WhenZ = 1/α (Z ≈ 137), the motion becomes highly relativistic, andZ2 cancels theα2 inR; 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.[26]

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

mred=mempme+mp=me11+me/mp.{\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} }}}.}

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.

En=RE2n2{\displaystyle E_{n}={\frac {R_{\mathrm {E} }}{2n^{2}}}} (positronium).

Rydberg formula

[edit]
Main article:Rydberg formula

Beginning in late 1860s,Johann Balmer and laterJohannes Rydberg andWalther Ritz 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.[27] These formula contained a constant,R{\displaystyle R}, now known theRydberg constant and a pair of integers indexing the lines:[14]: 247 ν=R(1m2−1n2).{\displaystyle \nu =R\left({\frac {1}{m^{2}}}-{\frac {1}{n^{2}}}\right).}Despite many attempts, no theory of the atom could reproduce these relatively simple formula.[14]: 169 

In Bohr's theory describing the energies of transitions orquantum jumps 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 byτ{\displaystyle \tau }:[28]Wτ=2π2me4h2τ2{\displaystyle W_{\tau }={\frac {2\pi ^{2}me^{4}}{h^{2}\tau ^{2}}}}The energy difference between two such levels is then:hν=Wτ2−Wτ1=2π2me4h2(1τ22−1τ12){\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)}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 thePlanck constant:[29]: 31 [30]cRH=2π2me4h3.{\displaystyle cR_{H}={\frac {2\pi ^{2}me^{4}}{h^{3}}}.}

Since the energy of a photon is

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

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

1λ=R(1nf2−1ni2).{\displaystyle {\frac {1}{\lambda }}=R\left({\frac {1}{n_{f}^{2}}}-{\frac {1}{n_{i}^{2}}}\right).}

Bohr's derivation of the Rydberg constant, as well as the concomitant agreement of Bohr's formula with experimentally observed spectral lines of theLyman (nf =1),Balmer (nf =2), andPaschen (nf =3) series, and successful theoretical prediction of other lines not yet observed, was one reason that his model was immediately accepted.[30]: 34 

To apply to atoms with more than one electron, the Rydberg formula can be modified by replacingZ withZ − b orn withn − b whereb is constant representing a screening effect due to the inner-shell and other electrons (seeElectron shell and the later discussion of the "Shell Model of the Atom" below). This was established empirically before Bohr presented his model.

Shell model (heavier atoms)

[edit]
Main article:Electron shell

Bohr's original three papers in 1913 described mainly the electron configuration in lighter elements. Bohr called his electron shells, "rings" in 1913. Atomic orbitals within shells did not exist at the time of his planetary model. Bohr explains in Part 3 of his famous 1913 paper that the maximum electrons in a shell is eight, writing: "We see, further, that a ring ofn electrons cannot rotate in a single ring round a nucleus of chargene unlessn < 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 neonN = 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:"[31][4][8]

Bohr's 1913 proposed configurations
ElementElectrons per shellElementElectrons per shellElementElectrons per shell
1194, 4, 1178, 4, 4, 1
22108, 2188, 8, 2
32, 1118, 2, 1198, 8, 2, 1
42, 2128, 2, 2208, 8, 2, 2
52, 3138, 2, 3218, 8, 2, 3
62, 4148, 2, 4228, 8, 2, 4
74, 3158, 4, 3238, 8, 4, 3
84, 2, 2168, 4, 2, 2248, 8, 4, 2, 2

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.[32] The 1913 Bohr model did not discuss higher elements in detail and John William Nicholson 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.[8][33]

In 1921, following the work of chemists and others involved in work on theperiodic table, Bohr extended the model of hydrogen to give an approximate model for heavier atoms. This gave a physical picture that reproduced many known atomic properties for the first time although these properties were proposed contemporarily with the identical work of chemistCharles Rugeley Bury[4][34]

Bohr's partner in research during 1914 to 1916 wasWalther Kossel who corrected Bohr's work to show that electrons interacted through the outer rings, and Kossel called the rings: "shells".[35][36]Irving Langmuir is credited with the first viable arrangement of electrons in shells with only two in the first shell and going up to eight in the next according to theoctet rule of 1904, although Kossel had already predicted a maximum of eight per shell in 1916.[37] 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. PerKossel, after that the orbit is full, the next level would have to be used.[4] This gives the atom ashell structure designed by Kossel, Langmuir, and Bury, in which each shell corresponds to a Bohr orbit.

This model is even more approximate than the model of hydrogen, because it treats the electrons in each shell as non-interacting. But the repulsions of electrons are taken into account somewhat by the phenomenon ofscreening. The electrons in outer orbits do not only orbit the nucleus, but they also move around the inner electrons, so the effective charge Z that they feel is reduced by the number of the electrons in the inner orbit.

For example, the lithium atom has two electrons in the lowest 1s orbit, and these orbit atZ = 2. Each one sees the nuclear charge ofZ = 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 chargeZ doesn't usually come out to be an integer.

The shell model was able to qualitatively explain many of the mysterious properties of atoms which became codified in the late 19th century in theperiodic table of the elements. One property was the size of atoms, which could be determined approximately by measuring theviscosity of gases and density of pure crystalline solids. Atoms tend to get smaller toward the right in the periodic table, and become much larger at the next line of the table. Atoms to the right of the table tend to gain electrons, while atoms to the left tend to lose them. Every element on the last column of the table is chemically inert (noble gas).

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

Moseley's law and calculation (K-alpha X-ray emission lines)

[edit]

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."[38]

In 1913,Henry Moseley found an empirical relationship between the strongest X-ray line emitted by atoms under electron bombardment (then known as theK-alpha line), and their atomic numberZ. Moseley's empiric formula was found to be derivable from Rydberg's formula and later Bohr's formula (Moseley actually mentions only Ernest Rutherford andAntonius Van den Broek in terms of models as these had been published before Moseley's work and Moseley's 1913 paper was published the same month as the first Bohr model paper).[39] 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 numberZ when used in the formula for atoms heavier than hydrogen, should be diminished by 1, to(Z − 1)2.

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

It wasWalther Kossel in 1914 and in 1916 who explained that in the periodic table new elements would be created as electrons were added to the outer shell. In Kossel's paper, he writes: "This leads to the conclusion that the electrons, which are added further, should be put into concentric rings or shells, on each of which ... only a certain number of electrons—namely, eight in our case—should be arranged. As soon as one ring or shell is completed, a new one has to be started for the next element; the number of electrons, which are most easily accessible, and lie at the outermost periphery, increases again from element to element and, therefore, in the formation of each new shell the chemical periodicity is repeated."[35][36] Later, chemist Langmuir realized that the effect was caused by charge screening, with an inner shell containing only 2 electrons. In his 1919 paper,Irving Langmuir postulated the existence of "cells" which could each only contain two electrons each, and these were arranged in "equidistant layers".

In the Moseley experiment, one of the innermost electrons in the atom is knocked out, leaving a vacancy in the lowest Bohr orbit, which contains a single remaining electron. This vacancy is then filled by an electron from the next orbit, which has n=2. But the n=2 electrons see an effective charge ofZ − 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,

E=hν=Ei−Ef=RE(Z−1)2(112−122),{\displaystyle E=h\nu =E_{i}-E_{f}=R_{\mathrm {E} }(Z-1)^{2}\left({\frac {1}{1^{2}}}-{\frac {1}{2^{2}}}\right),}

or

f=ν=Rv(34)(Z−1)2=(2.46×1015 Hz)(Z−1)2.{\displaystyle f=\nu =R_{\mathrm {v} }\left({\frac {3}{4}}\right)(Z-1)^{2}=(2.46\times 10^{15}~{\text{Hz}})(Z-1)^{2}.}

Here,Rv =RE/h is the Rydberg constant, in terms of frequency equal to 3.28 x 1015 Hz. For values of Z between 11 and 31 this latter relationship had been empirically derived by Moseley, in a simple (linear) plot of the square root of X-ray frequency against atomic number (however, for silver, Z = 47, the experimentally obtained screening term should be replaced by 0.4). Notwithstanding its restricted validity,[40] 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.[41]

TheK-alpha line of Moseley's time is now known to be a pair of close lines, written as (Kα1 andKα2) inSiegbahn notation.

Shortcomings

[edit]

The Bohr model gives an incorrect valueL=ħ 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 revolve "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.

In modern quantum mechanics, the electron in hydrogen is aspherical cloud of probability that grows denser near the nucleus. The rate-constant of probability-decay in hydrogen is equal to the inverse of the Bohr radius, but since Bohr worked with circular orbits, not zero area ellipses, the fact that these two numbers exactly agree is considered a "coincidence". (However, many such coincidental agreements are found between the semiclassical vs. full quantum mechanical treatment of the atom; these include identical energy levels in the hydrogen atom and the derivation of a fine-structure constant, which arises from the relativistic Bohr–Sommerfeld model (see below) and which happens to be equal to an entirely different concept, in full modern quantum mechanics).

The Bohr model also failed 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, iftwo additional ad hoc assumptions are made. Emission spectra for atoms with a single outer-shell electron (atoms in thelithium 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 (seeRydberg 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 theStark effect).
  • The existence offine structure andhyperfine 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.
  • TheZeeman effect – changes in spectral lines due to externalmagnetic fields; these are also due to more complicated quantum principles interacting with electron spin and orbital magnetic fields.
  • Doublets and triplets appear in the spectra of some atoms as very close pairs of lines. Bohr's model cannot say why some energy levels should be very close together.
  • Multi-electron atoms do not have energy levels predicted by the model. It does not work for (neutral) helium.

Refinements

[edit]
Elliptical orbits with the same energy and quantized angular momentum
Main article:Bohr–Sommerfeld model

Several enhancements to the Bohr model were proposed, most notably theSommerfeld or Bohr–Sommerfeld models, which suggested that electrons travel in elliptical orbits around a nucleus instead of the Bohr model's circular orbits.[1] This model supplemented the quantized angular momentum condition of the Bohr model with an additional radial quantization condition, theWilson–Sommerfeld quantization condition[42][43]

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

wherepr is the radial momentum canonically conjugate to the coordinateqr, which is the radial position, andT is one full orbital period. The integral is theaction ofaction-angle coordinates. This condition, suggested by thecorrespondence principle, is the only one possible, since the quantum numbers areadiabatic invariants.

The Bohr–Sommerfeld model was fundamentally inconsistent and led to many paradoxes. Themagnetic quantum number measured the tilt of the orbital plane relative to thexy plane, and it could only take a few discrete values. This contradicted the obvious fact that an atom could have any orientation relative to the coordinates, without restriction. The Sommerfeld quantization can be performed in different canonical coordinates and sometimes gives different answers. The incorporation of radiation corrections was difficult, because it required finding action-angle coordinates for a combined radiation/atom system, which is difficult when the radiation is allowed to escape. The whole theory did not extend to non-integrable motions, which meant that many systems could not be treated even in principle. In the end, the model was replaced by the modern quantum-mechanical treatment of the hydrogen atom, which was first given byWolfgang Pauli in 1925, using Heisenberg'smatrix mechanics. The current picture of the hydrogen atom is based on theatomic orbitals ofwave mechanics, whichErwin Schrödinger developed in 1926.

However, this is not to say that the Bohr–Sommerfeld model was without its successes. Calculations based on the Bohr–Sommerfeld model were able to accurately explain a number of more complex atomic spectral effects. For example, up to first-orderperturbations, the Bohr model and quantum mechanics make the same predictions for the spectral line splitting in theStark effect. At higher-order perturbations, however, the Bohr model and quantum mechanics differ, and measurements of the Stark effect under high field strengths helped confirm the correctness of quantum mechanics over the Bohr model. The prevailing theory behind this difference lies in the shapes of the orbitals of the electrons, which vary according to the energy state of the electron.

The Bohr–Sommerfeld quantization conditions lead to questions in modern mathematics. Consistent semiclassical quantization condition requires a certain type of structure on the phase space, which places topological limitations on the types of symplectic manifolds which can be quantized. In particular, the symplectic form should be thecurvature form of aconnection of aHermitianline bundle, which is called aprequantization.

Bohr also updated his model in 1922, assuming that certain numbers of electrons (for example, 2, 8, and 18) correspond to stable "closed shells".[44]

Model of the chemical bond

[edit]
Main article:Bohr model of the chemical bond

Niels Bohr proposed a model of the atom anda model of the chemical bond. According to his model for adiatomic molecule, the electrons of the atoms of the molecule form a rotating ring whose plane is perpendicular to the axis of the molecule and equidistant from the atomic nuclei. Thedynamic equilibrium of the molecular system is achieved through the balance of forces between the forces of attraction of nuclei to the plane of the ring of electrons and the forces of mutual repulsion of the nuclei. The Bohr model of the chemical bond took into account theCoulomb repulsion – the electrons in the ring are at the maximum distance from each other.[45][46]

Symbolism of planetary atomic models

[edit]
Shield of the U.S. Atomic Energy Commission

Although Bohr's atomic model was superseded by quantum models in the 1920s, the visual image of electrons orbiting a nucleus has remained the popular concept of atoms. The concept of an atom as a tinyplanetary system has been widely used as a symbol for atoms and even for "atomic" energy (even though this is more properly considered nuclear energy).[47]: 58  Examples of its use over the past century include but are not limited to:

See also

[edit]

References

[edit]

Footnotes

[edit]
  1. ^abLakhtakia, Akhlesh; Salpeter, Edwin E. (1996). "Models and Modelers of Hydrogen".American Journal of Physics.65 (9): 933.Bibcode:1997AmJPh..65..933L.doi:10.1119/1.18691.
  2. ^Perrin, Jean (1901)."Les Hypothèses moléculaires".La Revue scientifique: 463.
  3. ^de Broglie et al. 1912, pp. 122–123.
  4. ^abcdKragh, Helge (1 January 1979). "Niels Bohr's Second Atomic Theory".Historical Studies in the Physical Sciences.10:123–186.doi:10.2307/27757389.JSTOR 27757389.
  5. ^abcKragh, Helge (2012).Niels Bohr and the Quantum Atom: The Bohr Model of Atomic Structure 1913–1925. Oxford University Press.ISBN 978-0-19-163046-0.
  6. ^Helge Kragh (Oct. 2010).Before Bohr: Theories of atomic structure 1850-1913. RePoSS: Research Publications on Science Studies 10. Aarhus: Centre for Science Studies, University of Aarhus.
  7. ^Wheaton, Bruce R. (1992).The tiger and the shark: empirical roots of wave-particle dualism (1. paperback ed., reprinted ed.). Cambridge: Cambridge Univ. Press.ISBN 978-0-521-35892-7.
  8. ^abcdeHeilbron, John L.; Kuhn, Thomas S. (1969). "The Genesis of the Bohr Atom".Historical Studies in the Physical Sciences.1: vi–290.doi:10.2307/27757291.JSTOR 27757291.
  9. ^abJohn L, Heilbron (1985)."Bohr's First Theories of the Atom". In French, A. P.; Kennedy, P. J. (eds.).Niels Bohr: a centenary volume. Cambridge, Mass: Harvard University Press.ISBN 978-0-674-62415-3.
  10. ^Heilbron, John L. (1968)."The Scattering of α and β Particles and Rutherford's Atom".Archive for History of Exact Sciences.4 (4):247–307.doi:10.1007/BF00411591.ISSN 0003-9519.JSTOR 41133273.
  11. ^abcdefgBohr, N. (July 1913)."I. On the constitution of atoms and molecules".The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science.26 (151):1–25.Bibcode:1913PMag...26....1B.doi:10.1080/14786441308634955.
  12. ^Rayleigh, Lord (January 1906)."VII. On electrical vibrations and the constitution of the atom".The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science.11 (61):117–123.doi:10.1080/14786440609463428.
  13. ^Whittaker, Edmund T. (1989).A history of the theories of aether & electricity. 2: The modern theories, 1900 - 1926 (Repr ed.). New York: Dover Publ.ISBN 978-0-486-26126-3.
  14. ^abcdefghiPais, Abraham (2002).Inward bound: of matter and forces in the physical world (Reprint ed.). Oxford: Clarendon Press [u.a.]ISBN 978-0-19-851997-3.
  15. ^Bohr, N. (December 1925)."Atomic Theory and Mechanics1".Nature.116 (2927):845–852.Bibcode:1925Natur.116..845B.doi:10.1038/116845a0.ISSN 0028-0836.
  16. ^Perović, Slobodan (2021). "Spectral Lines, Quantum States, and a Master Model of the Atom".From data to quanta: Niels Bohr's vision of physics. Chicago London: The University of Chicago press.ISBN 978-0-226-79833-2.
  17. ^abcGiliberti, Marco; Lovisetti, Luisa (2024)."Bohr's Hydrogen Atom".Old Quantum Theory and Early Quantum Mechanics. Challenges in Physics Education. Cham: Springer Nature Switzerland.doi:10.1007/978-3-031-57934-9_6.ISBN 978-3-031-57933-2.
  18. ^abcBohr, Niels (7 November 1962)."Niels Bohr – Session III" (Interview). Interviewed by Thomas S. Kuhn; Leon Rosenfeld; Aage Petersen; Erik Rudinger. American Institute of Physics.
  19. ^abcHeilbron, John L. (June 2013). "The path to the quantum atom".Nature.498 (7452):27–30.doi:10.1038/498027a.PMID 23739408.S2CID 4355108.
  20. ^abcdMcCormmach, Russell (1 January 1966). "The atomic theory of John William Nicholson".Archive for History of Exact Sciences.3 (2):160–184.doi:10.1007/BF00357268.JSTOR 41133258.S2CID 120797894.
  21. ^abNicholson, J. W. (14 June 1912)."The Constitution of the Solar Corona. IL".Monthly Notices of the Royal Astronomical Society.72 (8). Oxford University Press:677–693.doi:10.1093/mnras/72.8.677.ISSN 0035-8711.
  22. ^abBohr, Niels; Rosenfeld, Léon Jacques Henri Constant (1963).On the Constitution of Atoms and Molecules ... Papers of 1913 reprinted from the Philosophical Magazine, with an introduction by L. Rosenfeld. Copenhagen; W.A. Benjamin: New York.OCLC 557599205.[page needed]
  23. ^Stachel, John (2009). "Bohr and the Photon".Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle. Dordrecht: Springer. p. 79.
  24. ^Gilder, Louisa (2009). "The Arguments 1909—1935".The Age of Entanglement. p. 55.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 waves.
  25. ^"Revealing the hidden connection between pi and Bohr's hydrogen model".Physics World. November 17, 2015.
  26. ^Müller, U.; de Reus, T.; Reinhardt, J.; Müller, B.; Greiner, W. (1988-03-01)."Positron production in crossed beams of bare uranium nuclei".Physical Review A.37 (5):1449–1455.Bibcode:1988PhRvA..37.1449M.doi:10.1103/PhysRevA.37.1449.PMID 9899816.S2CID 35364965.
  27. ^Bohr, N. (1985). "Rydberg's discovery of the spectral laws". In Kalckar, J. (ed.).Collected works. Vol. 10. Amsterdam: North-Holland Publ. Cy. pp. 373–379.
  28. ^Bohr, N. (July 1913)."I. On the constitution of atoms and molecules".The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science.26 (151):1–25.Bibcode:1913PMag...26....1B.doi:10.1080/14786441308634955.ISSN 1941-5982.
  29. ^Baggott, J. E. (2013).The quantum story: a history in 40 moments (Impression: 3 ed.). Oxford: Oxford Univ. Press.ISBN 978-0-19-965597-7.
  30. ^abBaily, C. (2013-01-01)."Early atomic models – from mechanical to quantum (1904–1913)".The European Physical Journal H.38 (1):1–38.arXiv:1208.5262.Bibcode:2013EPJH...38....1B.doi:10.1140/epjh/e2012-30009-7.ISSN 2102-6467.
  31. ^Bohr, N. (1913). "On the Constitution of Atoms and Molecules, Part II. Systems containing only a Single Nucleus".Philosophical Magazine.26:476–502.
  32. ^Bohr, N. (1 November 1913)."LXXIII. On the constitution of atoms and molecules".The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science.26 (155):857–875.Bibcode:1913PMag...26..857B.doi:10.1080/14786441308635031.
  33. ^Nicholson, J. W. (May 1914)."The Constitution of Atoms and Molecules".Nature.93 (2324):268–269.Bibcode:1914Natur..93..268N.doi:10.1038/093268a0.S2CID 3977652.
  34. ^Bury, Charles R. (July 1921)."Langmuir's Theory of the Arrangement of Electrons in Atoms and Molecules".Journal of the American Chemical Society.43 (7):1602–1609.Bibcode:1921JAChS..43.1602B.doi:10.1021/ja01440a023.
  35. ^abKossel, W. (1916)."Über Molekülbildung als Frage des Atombaus" [On molecular formation as a question of atomic structure].Annalen der Physik (in German).354 (3):229–362.Bibcode:1916AnP...354..229K.doi:10.1002/andp.19163540302.
  36. ^abKragh, Helge (2012)."Lars Vegard, atomic structure, and the periodic system"(PDF).Bulletin for the History of Chemistry.37 (1):42–49.OCLC 797965772.S2CID 53520045.Archived(PDF) from the original on 2022-10-09.
  37. ^Langmuir, Irving (June 1919)."The Arrangement of Electrons in Atoms and Molecules".Journal of the American Chemical Society.41 (6):868–934.Bibcode:1919JAChS..41..868L.doi:10.1021/ja02227a002.
  38. ^Bohr, Niels (31 October 1962)."Niels Bohr – Session I" (Interview). Interviewed by Thomas S. Kuhn; Leon Rosenfeld; Aage Petersen; Erik Rudinger. American Institute of Physics.
  39. ^Moseley, H.G.J. (1913)."The high-frequency spectra of the elements".Philosophical Magazine. 6th series.26:1024–1034.
  40. ^M.A.B. Whitaker (1999). "The Bohr–Moseley synthesis and a simple model for atomic x-ray energies".European Journal of Physics.20 (3):213–220.Bibcode:1999EJPh...20..213W.doi:10.1088/0143-0807/20/3/312.S2CID 250901403.
  41. ^van den Broek, Antonius (January 1913)."Die Radioelemente, das periodische System und die Konstitution der. Atome".Physikalische Zeitschrift (in German).14:32–41.
  42. ^A. Sommerfeld (1916)."Zur Quantentheorie der Spektrallinien".Annalen der Physik (in German).51 (17):1–94.Bibcode:1916AnP...356....1S.doi:10.1002/andp.19163561702.
  43. ^W. Wilson (1915)."The quantum theory of radiation and line spectra".Philosophical Magazine.29 (174):795–802.doi:10.1080/14786440608635362.
  44. ^Shaviv, Glora (2010).The Life of Stars: The Controversial Inception and Emergence of the Theory of Stellar Structure. Springer. p. 203.ISBN 978-3642020872.
  45. ^Бор Н. (1970).Избранные научные труды (статьи 1909–1925). Vol. 1. М.: «Наука». p. 133.
  46. ^Svidzinsky, Anatoly A.; Scully, Marlan O.; Herschbach, Dudley R. (23 August 2005)."Bohr's 1913 molecular model revisited".Proceedings of the National Academy of Sciences of the United States of America.102 (34):11985–11988.arXiv:physics/0508161.Bibcode:2005PNAS..10211985S.doi:10.1073/pnas.0505778102.PMC 1186029.PMID 16103360.
  47. ^Schirrmacher, Arne (2009). "Bohr's Atomic Model". In Greenberger, Daniel M.; Hentschel, Klaus; Weinert, Friedel (eds.).Compendium of quantum physics: concepts, experiments, history, and philosophy. Heidelberg New York: Springer.ISBN 978-3-540-70626-7.
  48. ^"Logo Usage Guidelines".www.iaea.org. 2021-11-23. Retrieved2024-09-04.

Primary sources

[edit]

Further reading

[edit]
  • Linus Carl Pauling (1970). "Chapter 5-1".General Chemistry (3rd ed.). San Francisco: W.H. Freeman & Co.
  • George Gamow (1985). "Chapter 2".Thirty Years That Shook Physics. Dover Publications.
  • Walter J. Lehmann (1972). "Chapter 18".Atomic and Molecular Structure: the development of our concepts. John Wiley and Sons.ISBN 0-471-52440-9.
  • Paul Tipler and Ralph Llewellyn (2002).Modern Physics (4th ed.). W. H. Freeman.ISBN 0-7167-4345-0.
  • Klaus Hentschel: Elektronenbahnen, Quantensprünge und Spektren, in: Charlotte Bigg & Jochen Hennig (eds.) Atombilder. Ikonografien des Atoms in Wissenschaft und Öffentlichkeit des 20. Jahrhunderts, Göttingen: Wallstein-Verlag 2009, pp. 51–61
  • Steven and Susan Zumdahl (2010). "Chapter 7.4".Chemistry (8th ed.). Brooks/Cole.ISBN 978-0-495-82992-8.
  • Kragh, Helge (November 2011). "Conceptual objections to the Bohr atomic theory — do electrons have a 'free will'?".The European Physical Journal H.36 (3):327–352.Bibcode:2011EPJH...36..327K.doi:10.1140/epjh/e2011-20031-x.S2CID 120859582.

External links

[edit]
Wikimedia Commons has media related toBohr model.
Historic models
Current models
Retrieved from "https://en.wikipedia.org/w/index.php?title=Bohr_model&oldid=1283764177"
Categories:
Hidden categories:
[8]ページ先頭
©2009-2025 Movatter.jp