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In atomic physics, the **Rydberg formula** calculates the wavelengths of a spectral line in many chemical elements. The formula was primarily presented as a generalization of the Balmer series for all atomic electron transitions of hydrogen. It was first empirically stated in 1888 by the Swedish physicist Johannes Rydberg,^{ [1] } then theoretically by Niels Bohr in 1913, who used a primitive form of quantum mechanics. The formula directly generalizes the equations used to calculate the wavelengths of the hydrogen spectral series.

In 1880, Rydberg worked on a formula describing the relation between the wavelengths in spectral lines of alkali metals. He noticed that lines came in series and he found that he could simplify his calculations using the wavenumber (the number of waves occupying the unit length, equal to 1/*λ*, the inverse of the wavelength) as his unit of measurement. He plotted the wavenumbers (*n*) of successive lines in each series against consecutive integers which represented the order of the lines in that particular series. Finding that the resulting curves were similarly shaped, he sought a single function which could generate all of them, when appropriate constants were inserted.

First he tried the formula: , where *n* is the line's wavenumber, *n*_{0} is the series limit, *m* is the line's ordinal number in the series, *m'* is a constant different for different series and *C*_{0} is a universal constant. This did not work very well.

Rydberg was trying: when he became aware of Balmer's formula for the hydrogen spectrum In this equation, *m* is an integer and *h* is a constant (not to be confused with the later Planck constant).

Rydberg therefore rewrote Balmer's formula in terms of wavenumbers, as .

This suggested that the Balmer formula for hydrogen might be a special case with and , where , the reciprocal of Balmer's constant (this constant **h** is written **B** in the Balmer equation article, again to avoid confusion with Planck's constant).

The term was found to be a universal constant common to all elements, equal to 4/*h*. This constant is now known as the Rydberg constant, and *m*′ is known as the quantum defect.

As stressed by Niels Bohr,^{ [2] } expressing results in terms of wavenumber, not wavelength, was the key to Rydberg's discovery. The fundamental role of wavenumbers was also emphasized by the Rydberg-Ritz combination principle of 1908. The fundamental reason for this lies in quantum mechanics. Light's wavenumber is proportional to frequency , and therefore also proportional to light's quantum energy *E*. Thus, . Modern understanding is that Rydberg's findings were a reflection of the underlying simplicity of the behavior of spectral lines, in terms of fixed (quantized) *energy* differences between electron orbitals in atoms. Rydberg's 1888 classical expression for the form of the spectral series was not accompanied by a physical explanation. Walther Ritz's *pre-quantum* 1908 explanation for the *mechanism* underlying the spectral series was that atomic electrons behaved like magnets and that the magnets could vibrate with respect to the atomic nucleus (at least temporarily) to produce electromagnetic radiation,^{ [3] } but this theory was superseded in 1913 by Niels Bohr's model of the atom.

In Bohr's conception of the atom, the integer Rydberg (and Balmer) *n* numbers represent electron orbitals at different integral distances from the atom. A frequency (or spectral energy) emitted in a transition from *n*_{1} to *n*_{2} therefore represents the photon energy emitted or absorbed when an electron makes a jump from orbital 1 to orbital 2.

Later models found that the values for *n*_{1} and *n*_{2} corresponded to the principal quantum numbers of the two orbitals.

where

- is the wavelength of electromagnetic radiation emitted in vacuum,
- is the Rydberg constant for hydrogen, approximately 1.09677583×10
^{7}m^{−1}, - is the principal quantum number of an energy level, and
- is the principal quantum number of an energy level for the atomic electron transition.

Note:- Here, >

By setting to 1 and letting run from 2 to infinity, the spectral lines known as the Lyman series converging to 91 nm are obtained, in the same manner:

n_{1} | n_{2} | Name | Converge toward |
---|---|---|---|

1 | 2 – ∞ | Lyman series | 91.13 nm (UV) |

2 | 3 – ∞ | Balmer series | 364.51 nm (Visible) |

3 | 4 – ∞ | Paschen series | 820.14 nm (IR) |

4 | 5 – ∞ | Brackett series | 1458.03 nm (Far IR) |

5 | 6 – ∞ | Pfund series | 2278.17 nm (Far IR) |

6 | 7 – ∞ | Humphreys series | 3280.56 nm (Far IR) |

The formula above can be extended for use with any hydrogen-like chemical elements with

where

- is the wavelength (in vacuum) of the light emitted,
- is the Rydberg constant for this element,
- is the atomic number, i.e. the number of protons in the atomic nucleus of this element,
- is the principal quantum number of the lower energy level, and
- is the principal quantum number of the higher energy level for the atomic electron transition.

This formula can be directly applied only to hydrogen-like, also called *hydrogenic* atoms of chemical elements, i.e. atoms with only one electron being affected by an effective nuclear charge (which is easily estimated). Examples would include He^{+}, Li^{2+}, Be^{3+} etc., where no other electrons exist in the atom.

But the Rydberg formula also provides correct wavelengths for distant electrons, where the effective nuclear charge can be estimated as the same as that for hydrogen, since all but one of the nuclear charges have been screened by other electrons, and the core of the atom has an effective positive charge of +1.

Finally, with certain modifications (replacement of *Z* by *Z* − 1, and use of the integers 1 and 2 for the *n*s to give a numerical value of 3⁄4 for the difference of their inverse squares), the Rydberg formula provides correct values in the special case of K-alpha lines, since the transition in question is the K-alpha transition of the electron from the 1s orbital to the 2p orbital. This is analogous to the Lyman-alpha line transition for hydrogen, and has the same frequency factor. Because the 2p electron is not screened by any other electrons in the atom from the nucleus, the nuclear charge is diminished only by the single remaining 1s electron, causing the system to be effectively a hydrogenic atom, but with a diminished nuclear charge *Z* − 1. Its frequency is thus the Lyman-alpha hydrogen frequency, increased by a factor of (*Z* − 1)^{2}. This formula of *f* = *c*/*λ* = (Lyman-alpha frequency)⋅(*Z* − 1)^{2} is historically known as Moseley's law (having added a factor *c* to convert wavelength to frequency), and can be used to predict wavelengths of the K_{α} (K-alpha) X-ray spectral emission lines of chemical elements from aluminum to gold. See the biography of Henry Moseley for the historical importance of this law, which was derived empirically at about the same time it was explained by the Bohr model of the atom.

For other spectral transitions in multi-electron atoms, the Rydberg formula generally provides *incorrect* results, since the magnitude of the screening of inner electrons for outer-electron transitions is variable and not possible to compensate for in the simple manner above. The correction to the Rydberg formula for these atoms is known as the quantum defect.

In atomic physics, the **Bohr model** or **Rutherford–Bohr model**, presented by Niels Bohr and Ernest Rutherford in 1913, is a system consisting of a small, dense nucleus surrounded by orbiting electrons—similar to the structure of the Solar System, but with attraction provided by electrostatic forces in place of gravity. After the cubical model (1902), the plum pudding model (1904), the Saturnian model (1904), and the Rutherford model (1911) came the *Rutherford–Bohr model* or just *Bohr model* for short (1913). The improvement over the 1911 Rutherford model mainly concerned the new quantum physical interpretation.

**Diatomic molecules** are molecules composed of only two atoms, of the same or different chemical elements. The prefix *di-* is of Greek origin, meaning "two". If a diatomic molecule consists of two atoms of the same element, such as hydrogen (H_{2}) or oxygen (O_{2}), then it is said to be homonuclear. Otherwise, if a diatomic molecule consists of two different atoms, such as carbon monoxide (CO) or nitric oxide (NO), the molecule is said to be heteronuclear. The bond in a homonuclear diatomic molecule is non-polar.

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. **Atomic hydrogen** constitutes about 75% of the baryonic mass of the universe.

A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called **energy levels**. This contrasts with classical particles, which can have any amount of energy. The term is commonly used for the energy levels of the electrons in atoms, ions, or molecules, which are bound by the electric field of the nucleus, but can also refer to energy levels of nuclei or vibrational or rotational energy levels in molecules. The energy spectrum of a system with such discrete energy levels is said to be quantized.

In the physical sciences, the **wavenumber** is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. Whereas temporal frequency can be thought of as the number of waves per unit time, wavenumber is the number of waves per unit distance.

The **Bohr radius** (*a*_{0}) is a physical constant, equal to the most probable distance between the nucleus and the 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. Its value is 5.29177210903(80)×10^{−11} m.

**Planck's law** describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature *T*, when there is no net flow of matter or energy between the body and its environment.

In spectroscopy, the **Rydberg constant**, symbol for heavy atoms or for hydrogen, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to the electromagnetic spectra of an atom. The constant first arose as an empirical fitting parameter in the Rydberg formula for the hydrogen spectral series, but Niels Bohr later showed that its value could be calculated from more fundamental constants via his Bohr model. As of 2018, and electron spin *g*-factor are the most accurately measured physical constants.

In physics, the **Lyman-alpha line**, sometimes written as **Ly-α line**, is a spectral line of hydrogen, or more generally of one-electron ions, in the Lyman series, emitted when the electron falls from the *n* = 2 orbital to the *n* = 1 orbital, where *n* is the principal quantum number. In hydrogen, its wavelength of 1215.67 angstroms (121.567 nm or 1.21567×10^{−7} m), corresponding to a frequency of 2.47×10^{15} hertz, places the Lyman-alpha line in the vacuum ultraviolet part of the electromagnetic spectrum, which is absorbed by air. Lyman-alpha astronomy must therefore ordinarily be carried out by satellite-borne instruments, except for extremely distant sources whose redshifts allow the hydrogen line to penetrate the atmosphere.

**H-alpha** (**Hα**) is a specific deep-red visible spectral line in the Balmer series with a wavelength of 656.28 nm in air; it occurs when a hydrogen electron falls from its third to second lowest energy level. H-alpha light is the brightest hydrogen line in the visible spectral range. It is important to astronomers as it is emitted by many emission nebulae and can be used to observe features in the Sun's atmosphere, including solar prominences and the chromosphere.

The **Balmer series**, or **Balmer lines** in atomic physics, is one of a set of six named series describing the spectral line emissions of the hydrogen atom. The Balmer series is calculated using the Balmer formula, an empirical equation discovered by Johann Balmer in 1885.

In physics and chemistry, the **Lyman series** is a hydrogen spectral series of transitions and resulting ultraviolet emission lines of the hydrogen atom as an electron goes from *n* ≥ 2 to *n* = 1, the lowest energy level of the electron. The transitions are named sequentially by Greek letters: from *n* = 2 to *n* = 1 is called Lyman-alpha, 3 to 1 is Lyman-beta, 4 to 1 is Lyman-gamma, and so on. The series is named after its discoverer, Theodore Lyman. The greater the difference in the principal quantum numbers, the higher the energy of the electromagnetic emission.

**Johann Jakob Balmer** was a Swiss mathematician best known for his work in physics, the Balmer series of Hydrogen atom.

The **Saha ionisation equation** is an expression that relates the ionisation state of a gas in thermal equilibrium to the temperature and pressure. The equation is a result of combining ideas of quantum mechanics and statistical mechanics and is used to explain the spectral classification of stars. The expression was developed by Indian physicist Meghnad Saha in 1920.

The **Compton wavelength** is a quantum mechanical property of a particle. The Compton wavelength of a particle is equal to the wavelength of a photon whose energy is the same as the mass of that particle. It was introduced by Arthur Compton in his explanation of the scattering of photons by electrons.

**Quantum mechanics** is the study of very small things. It explains the behavior of matter and its interactions with energy on the scale of atomic and subatomic particles. By contrast, classical physics explains matter and energy only on a scale familiar to human experience, including the behavior of astronomical bodies such as the Moon. Classical physics is still used in much of modern science and technology. However, towards the end of the 19th century, scientists discovered phenomena in both the large (macro) and the small (micro) worlds that classical physics could not explain. The desire to resolve inconsistencies between observed phenomena and classical theory led to two major revolutions in physics that created a shift in the original scientific paradigm: the *theory of relativity* and the development of *quantum mechanics*. This article describes how physicists discovered the limitations of classical physics and developed the main concepts of the quantum theory that replaced it in the early decades of the 20th century. It describes these concepts in roughly the order in which they were first discovered. For a more complete history of the subject, see *History of quantum mechanics*.

The emission spectrum of atomic hydrogen has been divided into a number of **spectral series**, with wavelengths given by the Rydberg formula. These observed spectral lines are due to the electron making transitions between two energy levels in an atom. The classification of the series by the Rydberg formula was important in the development of quantum mechanics. The spectral series are important in astronomical spectroscopy for detecting the presence of hydrogen and calculating red shifts.

The **Rydberg–Ritz combination principle** is an empirical generalization proposed by Walther Ritz in 1908 to describe the relationship of the spectral lines for all atoms. The principle states that the spectral lines of any element include frequencies that are either the sum or the difference of the frequencies of two other lines. Lines of the spectra of elements could be predicted from existing lines. Since the frequency of light is proportional to the wavenumber or reciprocal wavelength, the principle can also be expressed in terms of wavenumbers which are the sum or difference of wavenumbers of two other lines.

The **Planck constant**, or **Planck's constant**, is a fundamental physical constant denoted , and is of fundamental importance in quantum mechanics. A photon's energy is equal to its frequency multiplied by the Planck constant. Due to mass–energy equivalence, the Planck constant also relates mass to frequency.

The **Planck relation** is a fundamental equation in quantum mechanics which states that the energy of a photon, *E*, known as photon energy, is proportional to its frequency, *ν*:

- ↑ See:
- Rydberg, J.R. (1889). "Researches sur la constitution des spectres d'émission des éléments chimiques" [Investigations of the composition of the emission spectra of chemical elements].
*Kongliga Svenska Vetenskaps-Akademiens Handlingar [Proceedings of the Royal Swedish Academy of Science]*. 2nd series (in French).**23**(11): 1–177. - English summary: Rydberg, J.R. (1890). "On the structure of the line-spectra of the chemical elements".
*Philosophical Magazine*. 5th series.**29**: 331–337.

- Rydberg, J.R. (1889). "Researches sur la constitution des spectres d'émission des éléments chimiques" [Investigations of the composition of the emission spectra of chemical elements].
- ↑ Bohr, N. (1985). "Rydberg's discovery of the spectral laws". In Kalckar, J. (ed.).
*Collected works*.**10**. Amsterdam: North-Holland Publ. Cy. pp. 373–379. - ↑ Ritz, W. (1908). "Magnetische Atomfelder und Serienspektren" [The magnetic fields of atoms and spectral series].
*Annalen der Physik*(in German).**330**(4): 660–696. Bibcode:1908AnP...330..660R. doi:10.1002/andp.19083300403.

- Sutton, Mike (July 2004). "Getting the numbers right: The lonely struggle of the 19th century physicist/chemist Johannes Rydberg".
*Chemistry World*.**1**(7): 38–41. ISSN 1473-7604. - Martinson, I.; Curtis, L.J. (2005). "Janne Rydberg – his life and work".
*Nuclear Instruments and Methods in Physics Research Section B*.**235**(1–4): 17–22. Bibcode:2005NIMPB.235...17M. CiteSeerX 10.1.1.602.6210 . doi:10.1016/j.nimb.2005.03.137.

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