# Balmer series

The Balmer series or Balmer lines in atomic physics, is the designation of 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.

The visible spectrum of light from hydrogen displays four wavelengths, 410 nm, 434 nm, 486 nm, and 656 nm, that correspond to emissions of photons by electrons in excited states transitioning to the quantum level described by the principal quantum number n equals 2.[1] There are several prominent ultraviolet Balmer lines with wavelengths shorter than 400 nm. The number of these lines is an infinite continuum as it approaches a limit of 364.6 nm in the ultraviolet.

After Balmer's discovery, five other hydrogen spectral series were discovered, corresponding to electrons transitioning to values of n other than 2.

The "visible" hydrogen emission spectrum lines in the Balmer series. H-alpha is the red line at the right. Four lines (counting from the right) are formally in the visible range. Lines five and six can be seen with the naked eye, but are considered to be ultraviolet as they have wavelengths less than 400 nm.

## Overview

In the simplified Rutherford Bohr model of the hydrogen atom, the Balmer lines result from an electron jump between the second energy level closest to the nucleus, and those levels more distant. Shown here is a photon emission. The 3→2 transition depicted here produces H-alpha, the first line of the Balmer series. For hydrogen (Z = 1) this transition results in a photon of wavelength 656 nm (red).

The Balmer series is characterized by the electron transitioning from n ≥ 3 to n = 2, where n refers to the radial quantum number or principal quantum number of the electron. The transitions are named sequentially by Greek letter: n = 3 to n = 2 is called H-α, 4 to 2 is H-β, 5 to 2 is H-γ, and 6 to 2 is H-δ. As the first spectral lines associated with this series are located in the visible part of the electromagnetic spectrum, these lines are historically referred to as "H-alpha", "H-beta", "H-gamma" and so on, where H is the element hydrogen.

 Transition of n Name Wavelength (nm) Energy difference (eV) Color 3→2 4→2 5→2 6→2 7→2 8→2 9→2 ∞→2 H-α / Ba-α H-β / Ba-β H-γ / Ba-γ H-δ / Ba-δ H-ε / Ba-ε H-ζ / Ba-ζ H-η / Ba-η Balmer break 656.45377[2] 486.13615[3] 434.0462[3] 410.174[4] 397.0072[4] 388.9049[4] 383.5384[4] 364.6 1.89 2.55 2.86 3.03 3.13 3.19 3.23 3.40 Red Aqua Blue Violet (Ultraviolet) (Ultraviolet) (Ultraviolet) (Ultraviolet)

Although physicists were aware of atomic emissions before 1885, they lacked a tool to accurately predict where the spectral lines should appear. The Balmer equation predicts the four visible absorption/emission lines of hydrogen with high accuracy. Balmer's equation inspired the Rydberg equation as a generalization of it, and this in turn led physicists to find the Lyman, Paschen, and Brackett series which predicted other absorption/emission lines of hydrogen found outside the visible spectrum.

The familiar red H-alpha spectral line of the Balmer series of atomic hydrogen, which is the transition from the shell n = 3 to the shell n = 2, is one of the conspicuous colours of the universe. It contributes a bright red line to the spectra of emission or ionisation nebula, like the Orion Nebula, which are often H II regions found in star forming regions. In true-colour pictures, these nebula have a distinctly pink colour from the combination of visible Balmer lines that hydrogen emits.

Later, it was discovered that when the Balmer series lines of the hydrogen spectrum were examined at very high resolution, they were closely spaced doublets. This splitting is called fine structure. It was also found that excited electrons from shells with n greater than 6 could jump to the n = 2 shell, emitting shades of ultraviolet when doing so.

Two of the Balmer lines (α and β) are clearly visible in this emission spectrum of a deuterium lamp.

## Balmer's formula

Balmer noticed that a single wavelength had a relation to every line in the hydrogen spectrum that was in the visible light region. That wavelength was 364.50682 nm. When any integer higher than 2 was squared and then divided by itself squared minus 4, then that number multiplied by 364.50682 nm (see equation below) gave the wavelength of another line in the hydrogen spectrum. By this formula, he was able to show that some measurements of lines made in his time by spectroscopy were slightly inaccurate and his formula predicted lines that were later found although had not yet been observed. His number also proved to be the limit of the series.

The Balmer equation could be used to find the wavelength of the absorption/emission lines and was originally presented as follows (save for a notation change to give Balmer's constant as B):

${\displaystyle \lambda \ =B\left({\frac {n^{2}}{n^{2}-m^{2}}}\right)=B\left({\frac {n^{2}}{n^{2}-2^{2}}}\right)}$

Where

λ is the wavelength.
B is a constant with the value of 3.6450682×10−7 m or 364.50682 nm.
m is equal to 2
n is an integer such that n > m.

In 1888 the physicist Johannes Rydberg generalized the Balmer equation for all transitions of hydrogen. The equation commonly used to calculate the Balmer series is a specific example of the Rydberg formula and follows as a simple reciprocal mathematical rearrangement of the formula above (conventionally using a notation of m for n as the single integral constant needed):

${\displaystyle {\frac {1}{\lambda }}={\frac {4}{B}}\left({\frac {1}{2^{2}}}-{\frac {1}{n^{2}}}\right)=R_{\mathrm {H} }\left({\frac {1}{2^{2}}}-{\frac {1}{n^{2}}}\right)\quad \mathrm {for~} n=3,4,5,...}$

where λ is the wavelength of the absorbed/emitted light and RH is the Rydberg constant for hydrogen. The Rydberg constant is seen to be equal to 4/B in Balmer's formula, and this value, for an infinitely heavy nucleus, is 4/3.6450682×10−7 m = 10973731.57 m−1.[5]

## Role in astronomy

The Balmer series is particularly useful in astronomy because the Balmer lines appear in numerous stellar objects due to the abundance of hydrogen in the universe, and therefore are commonly seen and relatively strong compared to lines from other elements.

The spectral classification of stars, which is primarily a determination of surface temperature, is based on the relative strength of spectral lines, and the Balmer series in particular is very important. Other characteristics of a star that can be determined by close analysis of its spectrum include surface gravity (related to physical size) and composition.

Because the Balmer lines are commonly seen in the spectra of various objects, they are often used to determine radial velocities due to doppler shifting of the Balmer lines. This has important uses all over astronomy, from detecting binary stars, exoplanets, compact objects such as neutron stars and black holes (by the motion of hydrogen in accretion disks around them), identifying groups of objects with similar motions and presumably origins (moving groups, star clusters, galaxy clusters, and debris from collisions), determining distances (actually redshifts) of galaxies or quasars, and identifying unfamiliar objects by analysis of their spectrum.

Balmer lines can appear as absorption or emission lines in a spectrum, depending on the nature of the object observed. In stars, the Balmer lines are usually seen in absorption, and they are "strongest" in stars with a surface temperature of about 10,000 kelvins (spectral type A). In the spectra of most spiral and irregular galaxies, AGNs, H II regions and planetary nebulae, the Balmer lines are emission lines.

In stellar spectra, the H-epsilon line (transition 7→2) is often mixed in with another absorption line caused by ionized calcium known by astronomers as "H" (the original designation given by Fraunhofer). That is, H-epsilon's wavelength is quite close to Ca H at 396.847 nm, and cannot be resolved in low resolution spectra. The H-zeta line (transition 8→2) is similarly mixed in with a neutral helium line seen in hot stars.

## Notes

1. ^ Nave, C. R. (2006). "Hydrogen Spectrum". HyperPhysics. Georgia State University. Retrieved 2008-03-01.
2. ^ "NIST ASD Output: Lines". physics.nist.gov. Retrieved 2018-07-08.
3. ^ a b P. Mohr and S. Kotochigova, unpublished calculations (2000). The wavelengths for the Balmer-alpha and Balmer-beta transitions at 6563 and 4861 Å include only the stronger components of more extensive fine structures.
4. ^ a b c d J. Reader, C. H. Corliss, W. L. Wiese, and G. A. Martin, Natl. Stand. Ref. Data Ser., Natl. Bur. Stand. (U.S.) 68 (1980).
5. ^ "CODATA Recommended Values of the Fundamental Physical Constants: 2006" (PDF). Committee on Data for Science and Technology (CODATA). NIST.
AR Andromedae

AR Andromedae (AR And) is a dwarf nova of the SS Cygni type in the constellation Andromeda. Its typical apparent visual magnitude is 17.6, but increases up to 11.0 magnitude during outbursts. The outbursts occur approximately every 23 days.

B-type main-sequence star

A B-type main-sequence star (B V) is a main-sequence (hydrogen-burning) star of spectral type B and luminosity class V. These stars have from 2 to 16 times the mass of the Sun and surface temperatures between 10,000 and 30,000 K. B-type stars are extremely luminous and blue. Their spectra have neutral helium, which are most prominent at the B2 subclass, and moderate hydrogen lines. Examples include Regulus and Algol A.This class of stars was introduced with the Harvard sequence of stellar spectra and published in the Revised Harvard photometry catalogue. The definition of type B-type stars was the presence of non-ionized helium lines with the absence of singly ionized helium in the blue-violet portion of the spectrum. All of the spectral classes, including the B type, were subdivided with a numerical suffix that indicated the degree to which they approached the next classification. Thus B2 is 1/5 of the way from type B (or B0) to type A.Later, however, more refined spectra showed lines of ionized helium for stars of type B0. Likewise, A0 stars also show weak lines of non-ionized helium. Subsequent catalogues of stellar spectra classified the stars based on the strengths of absorption lines at specific frequencies, or by comparing the strengths of different lines. Thus, in the MK Classification system, the spectral class B0 has the line at wavelength 439 nm being stronger than the line at 420 nm. The Balmer series of hydrogen lines grows stronger through the B class, then peak at type A2. The lines of ionized silicon are used to determine the sub-class of the B-type stars, while magnesium lines are used to distinguish between the temperature classes.Type-B stars don't have a corona and lack a convection zone in their outer atmosphere. They have a higher mass loss rate than smaller stars such as the Sun, and their stellar wind has velocities of about 3,000 km/s. The energy generation in main-sequence B-type stars comes from the CNO cycle of thermonuclear fusion. Because the CNO cycle is very temperature sensitive, the energy generation is heavily concentrated at the center of the star, which results in a convection zone about the core. This results in a steady mixing of the hydrogen fuel with the helium byproduct of the nuclear fusion. Many B-type stars have a rapid rate of rotation, with an equatorial rotation velocity of about 200 km/s.

Balmer

Balmer is a surname. Notable people with the surname include:

Earl Balmer (born 1935), former NASCAR Cup Series driver

Thomas Balmer (born 1952), Oregon Chief Justice

Edwin Balmer (1883–1959), American science fiction writer

Florian Balmer (born 1979), independent software developer living in Switzerland

Jack Balmer (1916–1984), English football player

Jacqueline Balmer (died 2011), British stunt woman, known professionally as Jacquie de Creed

John Balmer (1910–1944), Royal Australian Air Force officer

Johann Jakob Balmer (1825–1898), Swiss mathematician and physicist

Balmer series, is the designation of one of a set of six different named series describing the spectral line emissions of the hydrogen atom

Lori Balmer, Australian pop singer

Randall Balmer (born 1954), American author

Robert Balmer (1787–1844), Scottish theologian

Balmer jump

The Balmer jump or Balmer discontinuity is the difference of intensity of the stellar continuum spectrum on both sides of the limit of the Balmer series of hydrogen at 364.6 nm. It is caused by electrons being completely ionized directly from the second energy level of a hydrogen atom (bound-free absorption), which creates a continuum absorption at wavelengths shorter than 364.6 nm.In some cases the Balmer discontinuity can show continuum emission, usually when the Balmer lines themselves are strongly in emission. Other hydrogen spectral series also show bound-free absorption and hence a continuum discontinuity, but the Balmer jump in the near UV has been the most observed.The strength of the continuum absorption, and hence the size of the Balmer jump, depends on temperature and density in the region responsible for the absorption. At cooler stellar temperatures, the density most strongly affects the strength of the discontinuity and this can be used to classify stars on the basis of their surface gravity and hence luminosity. This effect is strongest in A class stars, but in hotter stars temperature has a much larger effect on the Balmer jump than surface gravity.

Be star

Be Stars are a heterogeneous set of stars with B spectral types and emission lines. A narrower definition, sometimes referred to as Classical Be Stars, is a non-supergiant B star whose spectrum has, or had at some time, one or more Balmer emission lines.

Emission nebula

An emission nebula is a nebula formed of ionized gases that emit light of various wavelengths. The most common source of ionization is high-energy photons emitted from a nearby hot star. Among the several different types of emission nebulae are H II regions, in which star formation is taking place and young, massive stars are the source of the ionizing photons; and planetary nebulae, in which a dying star has thrown off its outer layers, with the exposed hot core then ionizing them.

H-alpha

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 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.

HR 4049

HR 4049, also known as HD 89353 and AG Antliae, is a binary post-asymptotic-giant-branch (post-AGB) star in the constellation Antlia. A very metal-poor star, it is surrounded by a thick unique circumbinary disk enriched in several molecules. With an apparent magnitude of about 5.5, the star can readily be seen under ideal conditions. It is located approximately 1,700 parsecs (5,500 ly) distant.

HR 4049 has a peculiar spectrum. The star appears, based on its spectrum in the Balmer series, to be a blue supergiant, although in reality it is an old low-mass star on the post-AGB phase of its life. Its atmosphere is extremely deficient in heavy elements, over with a metallicity over 30,000 lower than the Sun. It also shows a strong infrared excess, corresponding closely to a 1,200 K blackbody produced by a disk of material surrounding the star. The star is also undergoing intense mass-lossHR 4049 has an unseen companion, detected from variations in the doppler shift of its spectral lines. The properties of the companion can only be estimated by making certain assumptions about the inclination of the orbit and the mass function. Given those assumptions, it is thought to be a low luminosity main sequence star.HR 4049 is an unusual variable star, ranging between magnitudes 5.29 and 5.83 with a period of 429 days. It has been given the variable star designation AG Antliae, but is still more commonly referred to as HR 4049. It has been described as pulsating in a similar fashion to an RV Tauri variable, although the preferred interpretation is that the variations are produced by variable extinction produced by the material around the star and that the period is the same as the orbital period.Although HR 4049 apparently has the spectrum of a blue supergiant, it is an old low-mass star which has exhausted nuclear fusion and is losing its outer layers as it transitions towards a white dwarf and possibly a planetary nebula. During this phase it has a luminosity several thousand times that of the Sun, although a mass around half that of the sun. The mass can only be guessed from the expected mass of the white dwarf that it is becoming.

Hydrogen spectral series

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.

Inglis–Teller equation

The Inglis–Teller equation represents an approximate relationship between an energy level and the electron number density. This equation is related to the Stark effect in which spectral lines are split explicitly due to the presence of an electric field. The equation was derived by David R. Inglis and Edward Teller in 1939.

The equation is directly related to astrophysics as the electron densities of stars are determined using this equation. Since the Stark effect shifts spectral lines, for some energy level n, the splitting is equal to the difference between the adjoining energy level. Beyond this level, spectral lines merge.

Electron density and the merging energy level relationship for hydrogen atom is given by;

${\displaystyle \log(N_{i}+N_{e})=23.491-7.5\log(n_{m})\,,}$

where, Ni= no: of positive ions per cm3, Ne= no: of negative electrons per cm3, nm= last observable energy level of the Balmer series.

Kappa1 Lupi

Kappa1 Lupi is a solitary star in the southern constellation of Lupus. It is visible to the naked eye with an apparent visual magnitude of 3.86. Based upon an annual parallax shift of 18.12 mas as seen from Earth, it is located about 180 light years from the Sun. Both Kappa1 Lupi and its neighbor Kappa2 Lupi are members of the Hyades Stream, which is a moving group that is coincident with the proper motions of the Hyades cluster.This is a B-type main sequence star with a stellar classification of B9.5 Vne. The 'n' suffix indicates the spectrum shows "nebulous" absorption lines due to rapid rotation, while the 'e' means this is a Be star that displays Balmer series emission lines. With an estimated age of 195 million years, it is about 75% of the way through its life span on the main sequence. The star is rotating with a projected rotational velocity of 191 km/s. This rate of spin is giving the star an oblate shape with an equatorial bulge that is an estimated 9% larger than the polar radius.In Chinese astronomy, Kappa1 Lupi is called 騎陣將軍, Pinyin: Qízhènjiāngjūn, meaning Chariots and Cavalry General, because this star is marking itself and stand alone in Chariots and Cavalry General asterism, Root mansion (see : Chinese constellation).

Lyman series

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 (where n is the principal quantum number), 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.

QR Andromedae

QR Andromedae (often abbreviated to QR And) is an eclipsing binary star in the constellation Andromeda. Its maximum apparent visual magnitude is 12.16, but its light curve shows clearly eclipsing events where its brightness can drop to a magnitude of 13.07. This leads to its classification as an Algol variable star.

SN 2011dh

SN 2011dh is a supernova in the Whirlpool Galaxy (M51). On 31 May 2011 an apparent magnitude 13.5 type II supernova (the explosion of a single massive star) was detected in M51 at coordinates 13:30:05.08 +47:10:11.2. It was discovered by Tom Reiland; Thomas Griga; Amédée Riou; and Stephane Lamotte Bailey and confirmed by several sources, including the Palomar Transient Factory. A candidate progenitor has been detected in Hubble Space Telescope images at coordinates 13:30:05.119 +47:10:11.55. The progenitor may have been a highly luminous yellow supergiant with an initial mass of 18-24 solar masses. The supernova appears to have peaked near apparent magnitude 12.1 on 19 June 2011.Emission spectra from W. M. Keck Observatory, obtained by Palomar Transient Factory indicate that this is a type II supernova with a relatively blue continuum with P Cygni profiles in the Balmer series. This is a unique event, because it occurs in a galaxy that is imaged almost constantly. It is expected to be observable for northern hemisphere observers for several months.This is the third supernova to be recorded in the Whirlpool galaxy in 17 years (following SN 1994I and SN 2005cs) which is a lot for a single galaxy. The galactic supernova frequency is estimated to be around one event every 40 years.

Shell star

A shell star is a star having a spectrum that shows extremely broad absorption lines, plus some very narrow absorption lines. They typically also show some emission lines, usually from the Balmer series but occasionally of other lines. The broad absorption lines are due to rapid rotation of the photosphere, the emission lines from an equatorial disk, and the narrow absorption lines are produced when the disc is seen nearly edge-on.

Shell stars have spectral types O7.5 to F5, with rotation velocities of 200–300 km/s, not far from the point when the rotational acceleration would disrupt the star.

Theoretical and experimental justification for the Schrödinger equation

The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics determined by Maxwell's equations, as an analogue for all types of particles.

V529 Andromedae

V529 Andromedae, also known as HD 8801, is a variable star in the constellation of Andromeda. It has a 13th magnitude visual companion star 15" away, which is just a distant star on the same line of sight.

It is also an Am star with a spectral classification Am(kA5/hF1/mF2), meaning that it has the calcium K line of a star with spectral type A5, the Balmer series of a F1 star, and metallic lines of an F2 star.

WR 148

WR 148 is a spectroscopic binary in the constellation Cygnus. The primary star is a Wolf-Rayet star and one of the most luminous stars known. The secondary has been suspected of being a stellar-mass black hole but may be a class O main sequence star.

WR 148 shows a classic WN8h spectrum, but with the addition of weak central absorption on some of the emission lines. NIII and NIV emission lines are stronger than NV, and HeI lines are stronger than HeII, The Balmer series hydrogen lines and some other lines have P Cygni profiles.WR 148 is erratically variable on timescales ranging from seconds to years, but it shows consistent brightness and radial velocity variations with a period of 4.32 days. There is little doubt that it is a binary system, due to the regular variations and the presence of hard x-ray radiation from colliding winds, but the secondary is not clearly detectable in the spectrum. One proposal for a companion that would match the faint absorption features would be a B3 subgiant, but that is not compatible with the orbit. An early calculated orbit based on faint absorption features gave a relatively large mass ratio which imply either a very high companion mass, meaning a black hole, or an unreasonably low primary mass for a luminous WR star. Another analysis of the spectrum finds absorption features consistent with an O5 star, similar masses for the two components, and only a small orbital inclination.Because of its erratic changes in apparent magnitude at so many frequencies WR 148 is classified in the General Catalogue of Variable Stars as a unique type of variable, not a member of any of the defined classes. The shape of the light curve is unusual and has been modelled as being produced by an extended secondary object which may be an ionised cavity in the dense wind of the primary star, produced as the secondary orbits at a distance comparable to the radius of the primary star.WR 148 is found unusually far from the galactic plane for a Wolf-Rayet star, at 500 - 800 pc. Young massive stars such as WN8h WR stars are members of the thin disc population, on average only 60 pc from the galactic plane. It is suggested that WR 148 is a runaway from a supernova explosion. Calculations based on its large peculiar velocity of 197 km/s, current binary orbit, and likely lifetime since any supernova, are consistent with expulsion from a very massive triple system.

WR 42e

WR 42e (2MASS J11144550-115001) is a Wolf-Rayet star in the massive H II region NGC 3603 in the constellation of the Carina. It is around 25,000 light years or 7,600 parsec from the Sun. WR 42e is one of the most massive and most luminous stars known.

WR 42e was first catalogued in 2004 as a member of NGC 3603, numbered 954. It was noted as having x-ray and Hα emission. A detailed study published in 2012 showed that the faint red star was actually a highly obscured (6.4 magnitudes in the visual) hot blue Wolf Rayet star and gave it the name WR 42e. Subsequent changes to the naming conventions for new galactic Wolf Rayet stars mean it is also called WR 42-1.WR 42e is located 2.7 arcmin west-northwest of the massive open cluster HD 97950 at the heart of NGC 3603, corresponding to 6 parsecs at the distance of NGC 3603. This is outside the compact core of the cluster where similar massive luminous stars are found. It is speculated that WR 42e was ejected in an unusual three-body encounter possibly involving the merger of two of the stars and the ejection of both the resulting objects.The spectrum of WR 42e shows many characteristics of an OIf* star, such as hydrogen Balmer series absorption lines and emission lines of ionised nitrogen and helium. The relative strengths of the nitrogen emission lines and the lack of absorption in the 468.4 m, helium line indicate a spectral class of O3 If*. However, the Hβ line shows a distinct emission wing. A P Cygni profile for this line is a defining character of the OIf*/WN class and so WR 42e is assigned the type O3If*/WN6.

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