Bohr model

Bohr model

Overview


In atomic physics
Atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and...

, the Bohr model, introduced by Niels Bohr
Niels Bohr
Niels Henrik David Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. Bohr mentored and collaborated with many of the top physicists of the century at his institute in...

 in 1913, depicts the atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

 as a small, positively charged nucleus
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...

 surrounded by electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s that travel in circular orbits around the nucleus—similar in structure to the solar system
Solar System
The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...

, but with electrostatic forces providing attraction, rather than gravity. This was an improvement on the earlier cubic model (1902), the plum-pudding model (1904), the Saturnian model (1904), and the Rutherford model
Rutherford model
The Rutherford model or planetary model is a model of the atom devised by Ernest Rutherford. Rutherford directed the famous Geiger-Marsden experiment in 1909, which suggested on Rutherford's 1911 analysis that the so-called "plum pudding model" of J. J. Thomson of the atom was incorrect...

 (1911).
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In atomic physics
Atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and...

, the Bohr model, introduced by Niels Bohr
Niels Bohr
Niels Henrik David Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. Bohr mentored and collaborated with many of the top physicists of the century at his institute in...

 in 1913, depicts the atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

 as a small, positively charged nucleus
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...

 surrounded by electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s that travel in circular orbits around the nucleus—similar in structure to the solar system
Solar System
The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...

, but with electrostatic forces providing attraction, rather than gravity. This was an improvement on the earlier cubic model (1902), the plum-pudding model (1904), the Saturnian model (1904), and the Rutherford model
Rutherford model
The Rutherford model or planetary model is a model of the atom devised by Ernest Rutherford. Rutherford directed the famous Geiger-Marsden experiment in 1909, which suggested on Rutherford's 1911 analysis that the so-called "plum pudding model" of J. J. Thomson of the atom was incorrect...

 (1911). Since the Bohr model is a quantum-physics–based modification of the Rutherford model, many sources combine the two, referring to the Rutherford–Bohr model.

The model's key success lay in explaining the Rydberg formula
Rydberg formula
The Rydberg formula is used in atomic physics to describe the wavelengths of spectral lines of many chemical elements. It was formulated by the Swedish physicist Johannes Rydberg, and presented on November 5, 1888.-History:...

 for the spectral emission lines
Hydrogen spectral series
The emission spectrum of atomic hydrogen is divided into a number of spectral series, with wavelengths given by the Rydberg formula. These observed spectral lines are due to electrons moving between energy levels in the atom. The spectral series are important in astronomy for detecting the presence...

 of atomic hydrogen
Hydrogen
Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...

. 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 primitive model of the hydrogen atom
Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...

. 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
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, 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
Atomic orbital
An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...

. A related model was originally proposed by Arthur Erich Haas
Arthur Erich Haas
Arthur Erich Haas was an Austrian physicist, noted for a 1910 paper he submitted in support of this habilitation as Privatdocent at the University of Vienna that outlined a treatment of the hydrogen atom involving quantization of electronic orbitals, thus anticipating the Bohr model by three...

 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
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

 (1925) is often referred to as the old quantum theory
Old quantum theory
The old quantum theory was a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was a collection of heuristic prescriptions which are now understood to be the first quantum corrections to classical mechanics...

.

Origin


In the early 20th century, experiments by Ernest Rutherford
Ernest Rutherford
Ernest Rutherford, 1st Baron Rutherford of Nelson OM, FRS was a New Zealand-born British chemist and physicist who became known as the father of nuclear physics...

 established that atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

s consisted of a diffuse cloud of negatively charged electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s surrounding a small, dense, positively charged nucleus. Given this experimental data, Rutherford naturally considered a planetary-model atom, the Rutherford model
Rutherford model
The Rutherford model or planetary model is a model of the atom devised by Ernest Rutherford. Rutherford directed the famous Geiger-Marsden experiment in 1909, which suggested on Rutherford's 1911 analysis that the so-called "plum pudding model" of J. J. Thomson of the atom was incorrect...

 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
Larmor formula
In physics, in the area of electrodynamics, the Larmor formula is used to calculate the total power radiated by a nonrelativistic point charge as it accelerates. It was first derived by J. J...

), predict that the electron will release electromagnetic radiation
Electromagnetic radiation
Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...

 while orbiting a nucleus. Because the electron would lose energy, it would gradually spiral inwards, collapsing into the nucleus. This atom model is disastrous, because it predicts that all atoms are unstable.

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 discharge
Electric discharge
Electric discharge describes any flow of electric charge through a gas, liquid or solid. Electric discharges include:*Electric glow discharge*Electric arc*Electrostatic discharge*Electric discharge in gases*Leader *Partial discharge...

s through various low-pressure gas
Gas
Gas is one of the three classical states of matter . Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point , boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons...

es in evacuated glass tubes had shown that atoms will only emit light (that is, electromagnetic radiation) at certain discrete frequencies.

To overcome this difficulty, Niels Bohr
Niels Bohr
Niels Henrik David Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. Bohr mentored and collaborated with many of the top physicists of the century at his institute in...

 proposed, in 1913, what is now called the Bohr model of the atom. He suggested that electrons could only have certain classical motions:
  1. The electrons can only travel in certain orbits: at a certain discrete set of distances from the nucleus with specific energies.
  2. The electrons of an atom revolve around the nucleus in orbits. These orbits are associated with definite energies and are also called energy shells or energy levels. Thus, the electrons do not continuously lose energy as they travel in a particular orbit. They 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.
  3. 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
Principal quantum number
In atomic physics, the principal quantum symbolized as n is the firstof a set of quantum numbers of an atomic orbital. The principal quantum number can only have positive integer values...

, 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
Bohr radius
The Bohr radius is a physical constant, approximately equal to the most probable distance between the proton and electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an atom...

. 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:
  1. Like Einstein's theory of the Photoelectric effect
    Photoelectric effect
    In the photoelectric effect, electrons are emitted from matter as a consequence of their absorption of energy from electromagnetic radiation of very short wavelength, such as visible or ultraviolet light. Electrons emitted in this manner may be referred to as photoelectrons...

    , 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
    Maxwell's equations
    Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...

     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 photon
    Photon
    In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

    s.
  2. According to the Maxwell theory the frequency ν of classical radiation is equal to the rotation frequency νrot 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 En and Enk 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
    Correspondence principle
    In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers....

    , requiring quantum theory to agree with the classical theory only in the limit of large quantum numbers.
  3. The Bohr-Kramers-Slater theory
    BKS theory
    The Bohr-Kramers-Slater theory was perhaps the final attempt at understanding the interaction of matter and electromagnetic radiation on the basis of the so-called Old quantum theory, in which quantum phenomena are treated by imposing quantum restrictions on classically describable behaviour...

     (BKS theory) is a failed attempt to extend the Bohr model which violates the conservation of energy
    Conservation of energy
    The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...

     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 ini 1924 by de Broglie as a standing wave
Standing wave
In physics, a standing wave – also known as a stationary wave – is a wave that remains in a constant position.This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling...

 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 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
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, in which Bohr's model of electrons traveling in quantized orbits was extended into a more accurate model
Matrix mechanics
Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.Matrix mechanics was the first conceptually autonomous and logically consistent formulation of quantum mechanics. It extended the Bohr Model by describing how the quantum jumps...

 of electron motion. The new theory was proposed by Werner Heisenberg
Werner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...

. Another form
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

 of the same theory, wave mechanics, was discovered by the Austrian physicist Erwin Schrödinger
Erwin Schrödinger
Erwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...

 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
Hydrogen-like atom
A hydrogen-like ion is any atomic nucleus with one electron and thus is isoelectronic with hydrogen. Except for the hydrogen atom itself , these ions carry the positive charge e, where Z is the atomic number of the atom. Examples of hydrogen-like ions are He+, Li2+, Be3+ and B4+...

, 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
Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...

, singly ionized helium
Helium
Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table...

, doubly ionized lithium
Lithium
Lithium is a soft, silver-white metal that belongs to the alkali metal group of chemical elements. It is represented by the symbol Li, and it has the atomic number 3. Under standard conditions it is the lightest metal and the least dense solid element. Like all alkali metals, lithium is highly...

, but it includes positronium
Positronium
Positronium is a system consisting of an electron and its anti-particle, a positron, bound together into an "exotic atom". Being unstable, the two particles annihilate each other to produce two gamma ray photons after an average lifetime of 125 ps or three gamma ray photons after 142 ns in...

 and Rydberg states
Rydberg states
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. Although the Rydberg formula was developed to describe atomic energy levels, it has been used to describe many other systems...

 of any atom where one electron is far away from everything else. It can be used for K-line
K-line (spectrometry)
The K-line is a spectral peak in astronomical spectrometry used, along with the L-line, to observe and describe the light spectrum stars.The K-line is associated with iron , and is described as being from emissions at ~6.14keV .On 5 October 2006 NASA announced the results of research using the...

 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
Quark
A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...

 meson
Meson
In particle physics, mesons are subatomic particles composed of one quark and one antiquark, bound together by the strong interaction. Because mesons are composed of sub-particles, they have a physical size, with a radius roughly one femtometer: 10−15 m, which is about the size of a proton...

s.

To calculate the orbits requires two assumptions:
  • Classical mechanics
The electron is held in a circular orbit by electrostatic attraction. The centripetal force
Centripetal force
Centripetal force is a force that makes a body follow a curved path: it is always directed orthogonal to the velocity of the body, toward the instantaneous center of curvature of the path. The mathematical description was derived in 1659 by Dutch physicist Christiaan Huygens...

 is equal to the Coulomb force.
where me is the mass, e is the charge of the electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

 and ke is Coulomb's constant. This determines the speed at any radius:

It also determines the 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
Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

, which is true for non circular orbits too by the virial theorem.
For positronium
Positronium
Positronium is a system consisting of an electron and its anti-particle, a positron, bound together into an "exotic atom". Being unstable, the two particles annihilate each other to produce two gamma ray photons after an average lifetime of 125 ps or three gamma ray photons after 142 ns in...

, me is replaced by its reduced mass
Reduced mass
Reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. This is a quantity with the unit of mass, which allows the two-body problem to be solved as if it were a one-body problem. Note however that the mass determining the gravitational force is not...

 .

  • Quantum rule

The angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

  is an integer multiple of ħ:

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 is called the Bohr radius
Bohr radius
The Bohr radius is a physical constant, approximately equal to the most probable distance between the proton and electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an atom...

 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
Electronvolt
In physics, the electron volt is a unit of energy equal to approximately joule . By definition, it is equal to the amount of kinetic energy gained by a single unbound electron when it accelerates through an electric potential difference of one volt...

 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 (RE):

This expression is clarified by interpreting it in combinations which form more natural units
Natural units
In physics, natural units are physical units of measurement based only on universal physical constants. For example the elementary charge e is a natural unit of electric charge, or the speed of light c is a natural unit of speed...

:
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
Atomic number
In chemistry and physics, the atomic number is the number of protons found in the nucleus of an atom and therefore identical to the charge number of the nucleus. It is conventionally represented by the symbol Z. The atomic number uniquely identifies a chemical element...

. 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
N-body problem
The n-body problem is the problem of predicting the motion of a group of celestial objects that interact with each other gravitationally. Solving this problem has been motivated by the need to understand the motion of the Sun, planets and the visible stars...

) because the electrons are not only affected by the nucleus
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...

 but also interact with each other via the Coulomb Force.

When Z = 1/α (Z ≈ 137), the motion becomes highly relativistic, and Z2 cancels the α2 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
Reduced mass
Reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. This is a quantity with the unit of mass, which allows the two-body problem to be solved as if it were a one-body problem. Note however that the mass determining the gravitational force is not...

 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 4.0.

For positronium, the formula uses the reduced mass
Reduced mass
Reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. This is a quantity with the unit of mass, which allows the two-body problem to be solved as if it were a one-body problem. Note however that the mass determining the gravitational force is not...

 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)

Rydberg formula


The Rydberg formula
Rydberg formula
The Rydberg formula is used in atomic physics to describe the wavelengths of spectral lines of many chemical elements. It was formulated by the Swedish physicist Johannes Rydberg, and presented on November 5, 1888.-History:...

, 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.

When the electron gets moved from its original energy level to a higher one, it then jumps back each level till it comes to the original position, which results in a photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

 being 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.

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

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

Since the energy of a photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

 is


the wavelength of the photon given off is given by


This is known as the Rydberg formula
Rydberg formula
The Rydberg formula is used in atomic physics to describe the wavelengths of spectral lines of many chemical elements. It was formulated by the Swedish physicist Johannes Rydberg, and presented on November 5, 1888.-History:...

, and the Rydberg constant R is , or in natural units
Natural units
In physics, natural units are physical units of measurement based only on universal physical constants. For example the elementary charge e is a natural unit of electric charge, or the speed of light c is a natural unit of speed...

. This formula was known in the nineteenth century to scientists studying spectroscopy
Spectroscopy
Spectroscopy is the study of the interaction between matter and radiated energy. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, e.g., by a prism. Later the concept was expanded greatly to comprise any interaction with radiative...

, 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
Lyman series
In physics and chemistry, the Lyman series is the series of transitions and resulting ultraviolet emission lines of the hydrogen atom as an electron goes from n ≥ 2 to n = 1...

 (), Balmer
Balmer series
The Balmer series or Balmer lines in atomic physics, is the designation of one of a set of six different named series describing the spectral line emissions of the hydrogen atom....

 (), and Paschen () series, and successful theoretical prediction of other lines not yet observed, was one reason that his model was immediately accepted.

To apply to atoms with more than one electron, the Rydberg formula can be modified by replacing "Z" with "Z - b" or "n" with "n - b" where b is constant representing a screening effect due to the inner-shell and other electrons (see Electron shell
Electron shell
An electron shell may be thought of as an orbit followed by electrons around an atom's nucleus. The closest shell to the nucleus is called the "1 shell" , followed by the "2 shell" , then the "3 shell" , and so on further and further from the nucleus. The shell letters K,L,M,.....

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

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
Electron configuration
In atomic physics and quantum chemistry, electron configuration is the arrangement of electrons of an atom, a molecule, or other physical structure...

, 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 of screening
Shielding effect
The shielding effect describes the decrease in attraction between an electron and the nucleus in any atom with more than one electron shell. It is also referred to as the screening effect or atomic shielding.-Cause:...

. 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.

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 1/4 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
Moseley's law
Moseley's law is an empirical law concerning the characteristic x-rays that are emitted by atoms. The law was discovered and published by the English physicist Henry Moseley in 1913...

 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.

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
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

 of gases and density of pure crystalline solids. Atoms tend to get smaller toward the right in the periodic table, and become much larger at the next line of the table. Atoms to the right of the table tend to gain electrons, while atoms to the left tend to lose them. Every element on the last column of the table is chemically inert (noble gas
Noble gas
The noble gases are a group of chemical elements with very similar properties: under standard conditions, they are all odorless, colorless, monatomic gases, with very low chemical reactivity...

).

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, 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.

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
Henry Moseley
Henry Gwyn Jeffreys Moseley was an English physicist. Moseley's outstanding contribution to the science of physics was the justification from physical laws of the previous empirical and chemical concept of the atomic number. This stemmed from his development of Moseley's law in X-ray spectra...

 found an empirical relationship between the strongest X-ray line emitted by atoms under electron bombardment (then known as the K-alpha
K-alpha
In X-ray spectroscopy, K-alpha emission lines result when an electron transitions to the innermost "K" shell from a 2p orbital of the second or "L" shell...

 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
Ernest Rutherford
Ernest Rutherford, 1st Baron Rutherford of Nelson OM, FRS was a New Zealand-born British chemist and physicist who became known as the father of nuclear physics...

 and Antonius Van den Broek
Antonius Van den Broek
Antonius Johannes van den Broek was a Dutch amateur physicist notable for being the first who realized that the number of an element in the periodic table corresponds to the charge of its atomic nucleus....

 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.

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 an electron from 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
Moseley's law
Moseley's law is an empirical law concerning the characteristic x-rays that are emitted by atoms. The law was discovered and published by the English physicist Henry Moseley in 1913...

 for K-alpha lines:


or


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, Moseley's law not only established the objective meaning of atomic number (see Henry Moseley
Henry Moseley
Henry Gwyn Jeffreys Moseley was an English physicist. Moseley's outstanding contribution to the science of physics was the justification from physical laws of the previous empirical and chemical concept of the atomic number. This stemmed from his development of Moseley's law in X-ray spectra...

 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 (place on the periodic table) standing for whole units of nuclear charge.

The K-alpha
K-alpha
In X-ray spectroscopy, K-alpha emission lines result when an electron transitions to the innermost "K" shell from a 2p orbital of the second or "L" shell...

 line of Moseley's time is now known to be a pair of close lines, written as (Kα1 and Kα2) in Siegbahn notation
Siegbahn notation
The Siegbahn notation is used in X-ray spectroscopy to name the spectral lines that are characteristic to elements. It was created by Manne Siegbahn....

.

Shortcomings


The Bohr model gives an incorrect value 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. 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 a spherical cloud of probability which 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." (Though many such coincidental agreements are found between the semi-classical 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 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
    K-alpha
    In X-ray spectroscopy, K-alpha emission lines result when an electron transitions to the innermost "K" shell from a 2p orbital of the second or "L" shell...

     and some L-alpha X-ray emission spectra for larger atoms, if two additional ad hoc assumptions are made (see Moseley's law
    Moseley's law
    Moseley's law is an empirical law concerning the characteristic x-rays that are emitted by atoms. The law was discovered and published by the English physicist Henry Moseley in 1913...

     above). Emission spectra for atoms with a single outer-shell electron (atoms in the lithium
    Lithium
    Lithium is a soft, silver-white metal that belongs to the alkali metal group of chemical elements. It is represented by the symbol Li, and it has the atomic number 3. Under standard conditions it is the lightest metal and the least dense solid element. Like all alkali metals, lithium is highly...

     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
    Rydberg formula
    The Rydberg formula is used in atomic physics to describe the wavelengths of spectral lines of many chemical elements. It was formulated by the Swedish physicist Johannes Rydberg, and presented on November 5, 1888.-History:...

    ). 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
    Stark effect
    The Stark effect is the shifting and splitting of spectral lines of atoms and molecules due to presence of an external static electric field. The amount of splitting and or shifting is called the Stark splitting or Stark shift. In general one distinguishes first- and second-order Stark effects...

    ).
  • The existence of fine structure
    Fine structure
    In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to first order relativistic corrections.The gross structure of line spectra is the line spectra predicted by non-relativistic electrons with no spin. For a hydrogenic atom, the gross structure energy...

     and hyperfine structure
    Hyperfine structure
    The term hyperfine structure refers to a collection of different effects leading to small shifts and splittings in the energy levels of atoms, molecules and ions. The name is a reference to the fine structure which results from the interaction between the magnetic moments associated with electron...

     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
    Zeeman effect
    The Zeeman effect is the splitting of a spectral line into several components in the presence of a static magnetic field. It is analogous to the Stark effect, the splitting of a spectral line into several components in the presence of an electric field...

     - changes in spectral lines due to external magnetic field
    Magnetic field
    A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

    s; these are also due to more complicated quantum principles interacting with electron spin and orbital magnetic fields.
  • The model also violates the uncertainty principle
    Uncertainty principle
    In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...

     in that it considers electrons to have known orbits and definite radius, two things which can not be directly known at once.
  • Doublets and Triplets: Appear in the spectra of some atoms: Very close pairs of lines. Bohr’s model cannot say why some energy levels should be very close together.
  • Multi-electron Atoms: don’t have energy levels predicted by the model. It doesn’t work for (neutral) helium.

Refinements




Several enhancements to the Bohr model were proposed; most notably the Sommerfeld model
Old quantum theory
The old quantum theory was a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was a collection of heuristic prescriptions which are now understood to be the first quantum corrections to classical mechanics...

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


where pr 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
Action (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...

 of action-angle coordinates
Action-angle coordinates
In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequencies of oscillatory or rotational motion without solving the equations of motion. Action-angle coordinates...

. This condition, suggested by the correspondence principle
Correspondence principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers....

, is the only one possible, since the quantum numbers are adiabatic invariant
Adiabatic invariant
An adiabatic invariant is a property of a physical system that stays constant when changes occur slowly.In thermodynamics, an adiabatic process is a change that occurs without heat flow, and slowly compared to the time to reach equilibrium. In an adiabatic process, the system is in equilibrium at...

s.

The Bohr-Sommerfeld model was fundamentally inconsistent and led to many paradoxes. The magnetic quantum number
Magnetic quantum number
In atomic physics, the magnetic quantum number is the third of a set of quantum numbers which describe the unique quantum state of an electron and is designated by the letter m...

 measured the tilt of the orbital plane relative to the xy-plane, and it could only take a few discrete values. This contradicted the obvious fact that an atom could be turned this way and that relative to the coordinates without restriction. The Sommerfeld quantization can be performed in different canonical coordinates, and sometimes gives answers which are different. 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
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

 treatment of the hydrogen atom
Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...

, which was first given by Wolfgang Pauli
Wolfgang Pauli
Wolfgang Ernst Pauli was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after being nominated by Albert Einstein, he received the Nobel Prize in Physics for his "decisive contribution through his discovery of a new law of Nature, the exclusion principle or...

 in 1925, using Heisenberg
Werner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...

's matrix mechanics
Matrix mechanics
Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.Matrix mechanics was the first conceptually autonomous and logically consistent formulation of quantum mechanics. It extended the Bohr Model by describing how the quantum jumps...

. The current picture of the hydrogen atom is based on the atomic orbitals of wave mechanics
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

 which Erwin Schrödinger
Erwin Schrödinger
Erwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...

 developed in 1926.

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
Perturbation theory
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 Bohr model and quantum mechanics make the same predictions for the spectral line splitting in the Stark effect
Stark effect
The Stark effect is the shifting and splitting of spectral lines of atoms and molecules due to presence of an external static electric field. The amount of splitting and or shifting is called the Stark splitting or Stark shift. In general one distinguishes first- and second-order Stark effects...

. 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 the curvature form
Curvature form
In differential geometry, the curvature form describes curvature of a connection on a principal bundle. It can be considered as an alternative to or generalization of curvature tensor in Riemannian geometry.-Definition:...

 of a connection
Connection (mathematics)
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. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...

 of a Hermitian
Charles Hermite
Charles Hermite was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra....

 line bundle
Line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these...

, which is called a prequantization
Geometric quantization
In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory...

.

Primary sources

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.)

Further reading

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