View source for Hydrogen spectral series

{{Short description|Important atomic emission spectra}}
{{Use American English|date = February 2019}}
[[File:Hydrogen spectrum.svg|frame|right|The spectral series of hydrogen, on a [[logarithm]]ic scale.]]

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 [[atomic electron transition|transitions]] between two [[energy level]]s 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 shift]]s.

==Physics==
{{Further|Hydrogen atom }}
[[File:Hydrogen transitions.svg|thumb|right|400px|Electron transitions and their resulting wavelengths for hydrogen. Energy levels are not to scale.]]
A hydrogen atom consists of an electron orbiting its [[atomic nucleus|nucleus]]. The [[electromagnetic force]] between the electron and the nuclear [[proton]] leads to a set of [[quantum state]]s for the electron, each with its own energy. These states were visualized by the [[Bohr model]] of the hydrogen atom as being distinct orbits around the nucleus. Each energy state, or orbit, is designated by an integer, {{mvar|n}} as shown in the figure. The Bohr model was later replaced by quantum mechanics in which the electron occupies an [[atomic orbital]] rather than an orbit, but the allowed energy levels of the hydrogen atom remained the same as in the earlier theory.

Spectral emission occurs when an electron transitions, or jumps, from a higher energy state to a lower energy state. To distinguish the two states, the lower energy state is commonly designated as {{mvar|n′}}, and the higher energy state is designated as {{mvar|n}}. The energy of an emitted [[photon]] corresponds to the energy difference between the two states. Because the energy of each state is fixed, the energy difference between them is fixed, and the transition will always produce a photon with the same energy.

The spectral lines are grouped into series according to {{mvar|n′}}. Lines are named sequentially starting from the longest wavelength/lowest frequency of the series, using Greek letters within each series. For example, the {{nowrap|2 → 1}} line is called "Lyman-alpha" (Ly-α), while the {{nowrap|7 → 3}} line is called "Paschen-delta” (Pa-δ).
[[File:05-07-2015 Problem 7.64.jpg|thumb|Energy level diagram of electrons in hydrogen atom]]
There are emission lines from hydrogen that fall outside of these series, such as the [[Hydrogen line|21 cm line]]. These emission lines correspond to much rarer atomic events such as [[hyperfine structure|hyperfine]] transitions.<ref name="Hyperphysics21">{{cite web |url=http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/h21.html |title=The Hydrogen 21-cm Line |accessdate=2009-03-18 |work=[[Hyperphysics]] |publisher=[[Georgia State University]] |date=2005-10-30 }}</ref> The [[fine structure]] also results in single spectral lines appearing as two or more closely grouped thinner lines, due to relativistic corrections.<ref name="Liboff">{{cite book |last=Liboff |first=Richard L. |author-link=Richard Liboff |title=Introductory Quantum Mechanics |publisher=[[Addison-Wesley]] |year=2002 |isbn=978-0-8053-8714-8}}</ref>

In quantum mechanical theory, the discrete spectrum of atomic emission was based on the [[Schrödinger equation]], which is mainly devoted to the study of energy spectra of hydrogenlike atoms, whereas the time-dependent [[Heisenberg picture#Equivalence of Heisenberg's equation to the Schrödinger equation|equivalent Heisenberg]] equation is convenient when studying an atom driven by an external [[electromagnetic waves|electromagnetic wave]]<ref name ="Andrew, chapter 2" >{{cite book|title= Atomic spectroscopy. Introduction of theory to Hyperfine Structure|language= en|first1= A. V.|last1= Andrew|date= 2006|p=274|isbn= 978-0-387-25573-6|chapter= 2. [[Schrödinger equation]]}}</ref>.

In the processes of absorption or emission of photons by an atom, the [[conservation law|conservation laws]] hold for the whole [[isolated system]], such as an atom plus a photon. Therefore the motion of the electron in the process of photon absorption or emission is always accompanied by motion of the nucleus, and, because the mass of the nucleus is always finite, the energy spectra of [[hydrogen-like atom]]s must [[isotopic shift|depend on the nuclear mass]]. And since hydrogen atoms have a nucleus of only one proton, the spectrum energy of an hydrogen atom depends only by the nucleus (e.g. in the Coulomb field): in fact, the mass of one proton is ca <math>\frac {10^4}{5}</math> times the mass of an electron, which gives only the zero [[order of approximation]] and thus may be not taken into account.<ref name ="Andrew, chapter 2" />{{clarify |date=March 2018 |reason=This sentence is quite incomprehensible. Please rewrite it more clearly, or else explain more fully on the talk page so someone else can rewrite it.}}

==Rydberg formula==
{{Main|Rydberg formula}}
The energy differences between levels in the Bohr model, and hence the wavelengths of emitted/absorbed photons, is given by the Rydberg formula:<ref>{{citation |last=Bohr |first=Niels |author-link=Niels Bohr |chapter=Rydberg's discovery of the spectral laws |editor-last=Kalckar |editor-first=J. |title=N. Bohr: Collected Works |publisher=North-Holland Publ. |location=Amsterdam |date=1985 |volume=10 |pages=373–9}}</ref>
:<math> {1 \over \lambda} = Z^2 R_\infty \left( {1 \over {n_1}^2} - {1 \over {n_2}^2} \right)</math>

where 

:{{mvar|Z}} is the [[atomic number]],
: <math>n_1</math> is the [[principal quantum number]] of the lower energy level,
: <math>n_2</math> is the principal quantum number of the upper energy level, and
: <math>R_\infty</math> is the [[Rydberg constant]]. ({{val|1.09737|e=7|u=m{{sup|−1}}}}).<ref name="CODATA">{{cite journal |url=http://physics.nist.gov/cuu/Constants/codata.pdf |title=CODATA Recommended Values of the Fundamental Physical Constants: 2006 |volume=80 |issue=2 |pages=633–730 |journal=Committee on Data for Science and Technology (CODATA) |bibcode=2008RvMP...80..633M |last1=Mohr |first1=Peter J. |last2=Taylor |first2=Barry N. |last3=Newell |first3=David B. |year=2008 |arxiv=0801.0028 |doi=10.1103/RevModPhys.80.633 |citeseerx=10.1.1.150.3858 }}</ref>

Meaningful values are returned only when <math>n_1</math> is less than <math>n_2</math>. Note that this equation is valid for all hydrogen-like species, i.e. atoms having only a single electron, and the particular case of hydrogen spectral lines is given by Z=1.

==Series==

===Lyman series ({{mvar|n′}}&nbsp;=&nbsp;1)===
[[File:LymanSeries.svg|thumb|right|400px|[[Lyman series]] of [[hydrogen atom]] spectral lines in the [[ultraviolet]]]]
{{main|Lyman series}}
In the Bohr model, the Lyman series includes the lines emitted by transitions of the electron from an outer orbit of quantum number n > 1 to the 1st orbit of quantum number n' = 1.

The series is named after its discoverer, [[Theodore Lyman]], who discovered the spectral lines from 1906–1914. All the wavelengths in the Lyman series are in the [[ultraviolet]] band.<ref>{{citation |last=Lyman |first=Theodore |author-link=Theodore Lyman |title=The Spectrum of Hydrogen in the Region of Extremely Short Wave-Length |journal=Memoirs of the American Academy of Arts and Sciences |volume=13 |issue=3 |series=New Series |pages=125–146 |jstor=25058084 |issn=0096-6134 |year=1906|bibcode=1906ApJ....23..181L |doi=10.1086/141330 }}</ref><ref>{{citation |last=Lyman |first=Theodore |author-link=Theodore Lyman |title=An Extension of the Spectrum in the Extreme Ultra-Violet |year=1914 |journal=Nature |volume=93 |issue=2323 |pages=241 |doi=10.1038/093241a0|bibcode = 1914Natur..93..241L |url=https://zenodo.org/record/1429587 }}</ref>

{| class="wikitable" border="1" style="text-align: center"
|-
! {{mvar|n}}
! {{mvar|λ}}, vacuum
(nm)
|-
| 2 || 121.57
|-
| 3 || 102.57
|-
| 4 || 97.254
|-
| 5 || 94.974
|-
| 6 || 93.780
|-
| ∞ || 91.175
|- 
| colspan="2" style="text-align: center;" | '''Source:<ref name="Wiese et al. 2009">{{cite journal|last1=Wiese|first1=W. L.|last2=Fuhr|first2=J. R.|title=Accurate Atomic Transition Probabilities for Hydrogen, Helium, and Lithium|journal=Journal of Physical and Chemical Reference Data|date=2009|volume=38|issue=3|pages=565|doi=10.1063/1.3077727|bibcode = 2009JPCRD..38..565W |url=https://zenodo.org/record/1232309}}</ref>'''
|}
{{-}}

===Balmer series ({{mvar|n′}}&nbsp;=&nbsp;2)===
{{main|Balmer series}}
[[File:Emission spectrum-H.svg|757px|thumb|center|The four visible hydrogen emission spectrum lines in the Balmer series. H-alpha is the red line at the right.]]

The Balmer series includes the lines due to transitions from an outer orbit n > 2 to the orbit n' = 2.

Named after [[Johann Balmer]], who discovered the '''Balmer formula''', an [[empirical]] equation to predict the Balmer series, in 1885. Balmer lines are historically referred to as "[[H-alpha]]", "H-beta", "H-gamma" and so on, where H is the element hydrogen.<ref>{{citation |last=Balmer |first=J. J. |author-link=Johann Jakob Balmer |title=Notiz uber die Spectrallinien des Wasserstoffs |journal=Annalen der Physik |volume=261 |issue=5 |pages=80–87 |year=1885 |url=http://www3.interscience.wiley.com/journal/112487600/abstract |doi=10.1002/andp.18852610506|bibcode = 1885AnP...261...80B }}{{dead link|date=February 2019|bot=medic}}{{cbignore|bot=medic}}</ref> Four of the Balmer lines are in the technically "visible" part of the spectrum, with wavelengths longer than 400&nbsp;nm and shorter than 700&nbsp;nm. Parts of the Balmer series can be seen in the [[Fraunhofer lines|solar spectrum]]. H-alpha is an important line used in astronomy to detect the presence of hydrogen.

{| class="wikitable" border="1" style="text-align: center"
|-
! {{mvar|n}} 
! {{mvar|λ}}, air 
(nm)
|-
| 3 || 656.3
|-
| 4 || 486.1
|-
| 5 || 434.0
|-
| 6 || 410.2
|-
| 7 || 397.0
|-
| ∞ || 364.6
|- 
| colspan="2" style="text-align: center;" | '''Source:'''<ref name="Wiese et al. 2009"/>
|}
{{-}}

=== {{anchor|Paschen series}} Paschen series (Bohr series, {{mvar|n′}}&nbsp;=&nbsp;3)===
Named after the [[Germany|German]] physicist [[Friedrich Paschen]] who first observed them in 1908. The Paschen lines all lie in the [[infrared]] band.<ref>{{citation |last=Paschen |first=Friedrich |author-link=Friedrich Paschen |year=1908 |title=Zur Kenntnis ultraroter Linienspektra. I. (Normalwellenlängen bis 27000 Å.-E.) |journal=Annalen der Physik |volume=332 |issue=13 |pages=537–570 |url=http://www3.interscience.wiley.com/journal/112500956/abstract |archive-url=https://archive.today/20121217200521/http://www3.interscience.wiley.com/journal/112500956/abstract |url-status=dead |archive-date=2012-12-17 |doi=10.1002/andp.19083321303|bibcode = 1908AnP...332..537P }}</ref> This series overlaps with the next (Brackett) series, i.e. the shortest line in the Brackett series has a wavelength that falls among the Paschen series. All subsequent series overlap.

{| class="wikitable" border="1" style="text-align: center"
|-
! {{mvar|n}}
! {{mvar|λ}}, air
(nm)
|-
| 4 || 1875
|-
| 5 || 1282
|-
| 6 || 1094
|-
| 7 || 1005
|-
| 8 || 954.6
|-
| ∞ || 820.4
|- 
| colspan="2" style="text-align: center;" | '''Source:'''<ref name="Wiese et al. 2009"/>
|}
{{-}}

===Brackett series ({{mvar|n′}}&nbsp;=&nbsp;4)===
Named after the American physicist [[Frederick Sumner Brackett]] who first observed the spectral lines in 1922.<ref>{{citation |last=Brackett |first=Frederick Sumner |author-link=Frederick Sumner Brackett |year=1922 |title=Visible and Infra-Red Radiation of Hydrogen
|journal=Astrophysical Journal |volume=56 |pages=154 |doi=10.1086/142697|bibcode = 1922ApJ....56..154B |hdl=2027/uc1.$b315747 |url=http://hdl.handle.net/2027/uc2.ark:/13960/t2h70975p }}</ref>The spectral lines of Brackett series lie in far infrared band. 

{| class="wikitable" border="1" style="text-align: center"
|-
! {{mvar|n}}
! {{mvar|λ}}, air
(nm)
|-
| 5 || 4051
|-
| 6 || 2625
|-
| 7 || 2166
|-
| 8 || 1944
|-
| 9 || 1817
|-
| ∞ || 1458
|- 
| colspan="2" style="text-align: center;" | '''Source:'''<ref name="Wiese et al. 2009"/>
|}
{{-}}

===Pfund series ({{mvar|n′}}&nbsp;=&nbsp;5)===
Experimentally discovered in 1924 by [[August Herman Pfund]].<ref>{{citation |last=Pfund |first=A. H. |author-link=August Herman Pfund |title=The emission of nitrogen and hydrogen in infrared |year=1924 |journal=J. Opt. Soc. Am. |volume=9 |issue=3 |pages=193–196 |doi=10.1364/JOSA.9.000193}}</ref>

{| class="wikitable" border="1" style="text-align: center"
|-
! {{mvar|n}}
! {{mvar|λ}}, vacuum
(nm)
|-
|  6 || 7460
|-
|  7 || 4654
|-
|  8 || 3741
|-
|  9 || 3297
|-
| 10 || 3039
|-
|  ∞ || 2279
|- 
| colspan="2" style="text-align: center;" | '''Source:'''<ref name="Kramida et al. 2010">{{cite journal|last1=Kramida|first1=A. E.|display-authors=etal|title=A critical compilation of experimental data on spectral lines and energy levels of hydrogen, deuterium, and tritium|journal=Atomic Data and Nuclear Data Tables|date=November 2010|volume=96|issue=6|pages=586–644|doi=10.1016/j.adt.2010.05.001|bibcode = 2010ADNDT..96..586K }}</ref>
|}
{{-}}

===Humphreys series ({{mvar|n′}}&nbsp;=&nbsp;6)===
Discovered in 1953 by American physicist [[Curtis J. Humphreys]].<ref>{{citation |last=Humphreys |first=C.J. |author-link=Curtis J. Humphreys |title=The Sixth Series in the Spectrum of Atomic Hydrogen|journal=J. Research Natl. Bur. Standards|year=1953|volume=50|pages=1 |doi=10.6028/jres.050.001 }}</ref>

{| class="wikitable" border="1" style="text-align: center"
|-
! {{mvar|n}}
! {{mvar|λ}}, vacuum
(μm)
|-
|  7 || 12.37
|-
|  8 || 7.503
|-
|  9 || 5.908
|-
| 10 || 5.129
|-
| 11 || 4.673
|-
|  ∞ || 3.282
|- 
| colspan="2" style="text-align: center;" | '''Source:'''<ref name="Kramida et al. 2010"/>
|}
{{-}}

===Further ({{mvar|n′}}&nbsp;>&nbsp;6)===
Further series are unnamed, but follow the same pattern as dictated by the Rydberg equation. Series are increasingly spread out and occur in increasing wavelengths. The lines are also increasingly faint, corresponding to increasingly rare atomic events. The seventh series of atomic hydrogen was first demonstrated experimentally at infrared wavelengths in 1972 by John Strong and Peter Hansen at the University of Massachusetts Amherst.<ref>{{cite journal |last1=Hansen |first1=Peter |last2=Strong |first2=John |title=Seventh Series of Atomic Hydrogen |journal=Applied Optics |date=1973 |volume=12 |issue=2 |pages=429–430 |doi=10.1364/AO.12.000429 }}</ref>

== Extension to other systems ==
The concepts of the Rydberg formula can be applied to any system with a single particle orbiting a nucleus, for example a [[Helium|He]]<sup>+</sup> ion or a [[muonium]] exotic atom. The equation must be modified based on the system's [[Bohr radius]]; emissions will be of a similar character but at a different range of energies.  The [[Pickering series|Pickering–Fowler series]] was originally attributed to an unknown form of hydrogen with half-integer transition levels by both [[Edward Charles Pickering|Pickering]]<ref>{{cite journal|last = Pickering|first = E. C.|authorlink = Edward Charles Pickering|journal = [[Harvard College Observatory Circular]]|volume = 12|title = Stars having peculiar spectra. New variable stars in Crux and Cygnus.|pages = 1–2|year = 1896|bibcode = 1896HarCi..12....1P}}  Also published as: {{cite journal|title = Stars having peculiar spectra. New variable stars in Crux and Cygnus.|last1 = Pickering|first1 = E. C.|authorlink = Edward Charles Pickering|last2 = Fleming|first2 = W. P.|authorlink2 = Williamina Fleming|journal = [[Astrophysical Journal]]|volume = 4|pages = 369–370|year = 1896|doi = 10.1086/140291|bibcode = 1896ApJ.....4..369P}}</ref><ref>{{cite journal|title = Stars having peculiar spectra. New variable Stars in Crux and Cygnus.|first = E. C.|last = Pickering|authorlink = Edward Charles Pickering|year = 1897|journal = [[Astronomische Nachrichten]]|volume = 142|issue = 6|pages = 87–90|doi = 10.1002/asna.18971420605|bibcode = 1896AN....142...87P|url = https://zenodo.org/record/1424755/files/article.pdf}}</ref><ref>{{cite journal|title = The spectrum of zeta Puppis|last = Pickering|first = E. C.|authorlink = Edward Charles Pickering|year = 1897|journal = [[Astrophysical Journal]]|volume = 5|pages = 92–94|doi = 10.1086/140312|bibcode = 1897ApJ.....5...92P}}</ref> and [[Alfred Fowler|Fowler]],<ref>{{cite journal|title = Observations of the Principal and other Series of Lines in the Spectrum of Hydrogen.|first = A.|last = Fowler|authorlink = Alfred Fowler|journal = [[Monthly Notices of the Royal Astronomical Society]]|volume = 73|issue = 2|year = 1912|pages = 62–63|doi = 10.1093/mnras/73.2.62|bibcode = 1912MNRAS..73...62F}}</ref> but Bohr correctly recognised them as spectral lines arising from the He<sup>+</sup> nucleus.<ref>{{cite journal|title = The Spectra of Helium and Hydrogen|first = N.|last = Bohr|authorlink = Niels Bohr|journal = [[Nature (journal)|Nature]]|volume = 92|issue = 2295|year = 1913|pages = 231–232|doi = 10.1038/092231d0|bibcode = 1913Natur..92..231B|url = https://zenodo.org/record/1429570}}</ref><ref>{{cite book|first = Ulrich|last = Hoyer|chapter = Constitution of Atoms and Molecules|pages = 103–316 (esp. pp. 116–122)|title = Niels Bohr &ndash; Collected Works: Volume 2 &ndash; Work on Atomic Physics (1912–1917)|chapter-url = https://books.google.com/books?id=zGczmJjSO6kC&pg=PA117|editor-first = Ulrich|editor-last = Hoyer|publisher = [[North Holland Publishing Company]]|location = Amsterdam|year = 1981|isbn = 978-0720418002}}</ref><ref>{{cite journal|title = The Spectrum of &zeta; Puppis and the Historical Evolution of Empirical Data|first = Nadia|last = Robotti|journal = [[Historical Studies in the Physical Sciences]]|volume = 14|issue = 1|year = 1983|pages =  123–145|doi = 10.2307/27757527|jstor = 27757527}}</ref>

All other atoms possess at least two electrons in their [[ionization|neutral]] form and the interactions between these electrons makes analysis of the spectrum by such simple methods as described here impractical.  The deduction of the Rydberg formula was a major step in physics, but it was long before an extension to the spectra of other elements could be accomplished.

==See also==
* [[Astronomical spectroscopy]]
* The [[hydrogen line]] (21&nbsp;cm)
* [[Lamb shift]]
* [[Moseley's law]]
* [[Quantum optics]]
* [[Theoretical and experimental justification for the Schrödinger equation]]

==References==
{{reflist|colwidth=30em}}

== External links ==
{{Commonscat|Hydrogen spectral series}}
* [http://www.bigs.de/BLH/en/index.php?option=com_content&view=category&layout=blog&id=50&Itemid=221 Spectral series of hydrogen animation]

{{Hydrogen spectral series-footer}}

[[Category:Hydrogen physics]]
[[Category:Emission spectroscopy]]
[[Category:Hydrogen]]