# Diffuse series

The diffuse series is a series of spectral lines in the atomic emission spectrum caused when electrons jump between the lowest p orbital and d orbitals of an atom. The total orbital angular momentum changes between 1 and 2. The spectral lines include some in the visible light, and may extend into ultraviolet or near infrared. The lines get closer and closer together as the frequency increases never exceeding the series limit. The diffuse series was important in the development of the understanding of electron shells and subshells in atoms. The diffuse series has given the letter d to the d atomic orbital or subshell.

The diffuse series has values given by

${\displaystyle v={\frac {R}{\left[2+p\right]^{2}}}-{\frac {R}{\left[m+d\right]^{2}}}with\ m=2,3,4,5,6,...}$

The series is caused by transitions from the lowest P state to higher energy D orbitals. One terminology to identify the lines is: 1P-mD[1] But note that 1P just means the lowest P state in the valence shell of an atom and that the modern designation would start at 2P, and is larger for higher atomic numbered atoms.

The terms can have different designations, mD for single line systems, mδ for doublets and md for triplets.[2]

Since the Electron in the D subshell state is not the lowest energy level for the alkali atom (the S is) the diffuse series will not show up as absorption in a cool gas, however it shows up as emission lines. The Rydberg correction is largest for the S term as the electron penetrates the inner core of electrons more.

The limit for the series corresponds to electron emission, where the electron has so much energy it escapes the atom.[3]

In alkali metals the P terms are split ${\displaystyle 2P_{\frac {3}{2}}}$ and ${\displaystyle 2P_{\frac {1}{2}}}$. This causes the spectral lines to be doublets, with a constant spacing between the two parts of the double line.[4]

This splitting is called fine structure. The splitting is larger for atoms with higher atomic number. The splitting decreases towards the series limit. Another splitting occurs on the redder line of the doublet. This is because of splitting in the D level ${\displaystyle nd^{2}D_{\frac {3}{2}}}$ and ${\displaystyle nd^{2}D_{\frac {5}{2}}}$. Splitting in the D level has a lesser amount than the P level, and it reduces as the series limit is approached.[5]

## History

The diffuse series used to be called the first subordinate series, with the sharp series being the second subordinate, both being subordinate to the principal series.[2]

## Laws for alkali metals

The diffuse series limit is the same as the sharp series limit. In the late 1800s these two were termed supplementary series.

Spectral lines of the diffuse series are split into three lines in what is called fine structure. These lines cause the overall line to look diffuse. The reason this happens is that both the P and D levels are split into two closely spaced energies. P is split into ${\displaystyle P_{\frac {1}{2}}\ and\ P_{\frac {3}{2}}}$ . D is split into ${\displaystyle D_{\frac {3}{2}}\ and\ D_{\frac {5}{2}}}$. Only three of the possible four transitions can take place because the angular momentum change cannot have a magnitude greater than one.[6]

In 1896 Arthur Schuster stated his law: "If we subtract the frequency of the fundamental vibration from the convergence frequency of the principal series , we obtain the convergence frequency of the supplementary series".[7] But in the next issue of the journal he realised that Rydberg had published the idea a few months earlier.[8]

Rydberg Schuster Law: Using wave numbers, the difference between the diffuse and sharp series limits and principal series limit is the same as the first transition in the principal series.

This difference is the lowest P level.[9]

Runge's Law: Using wave numbers the difference between the diffuse series limit and fundamental series limit is the same as the first transition in the diffuse series.

This difference is the lowest D level energy.[9]

### Lithium

Lithium has a diffuse series with diffuse lines averaged around 6103.53, 4603.0, 4132.3, 3915.0 and 3794.7 Å.[10]

### Sodium

Graph showing wavelengths of the diffuse series of sodium plotted against N−2 (inverse square) making assumptions of different starting point of n. Blue diamond starts with n=2, red square starts with n=3, green triangle starts with n=4, violet X starts with n=5. Only with starting n of 3 is a straight line achieved[11]

The sodium diffuse series has wave numbers given by: ${\displaystyle \nu _{d}=R\left({\frac {Z_{3p}^{2}}{3^{2}}}-{\frac {Z_{nd}^{2}}{n^{2}}}\right)n=3,4,5,6,...}$

The sharp series has wave numbers given by: ${\displaystyle \nu _{s}=R\left({\frac {Z_{3p}^{2}}{3^{2}}}-{\frac {Z_{ns}^{2}}{n^{2}}}\right)n=4,5,6,...}$

when n tends to infinity the diffuse and sharp series end up with the same limit.[11]

sodium diffuse series[12]
transition wavelength 1 Å wavelength 2 Å wavelength 3 Å
3P-3D 8194.82 8183.26 8194.79
3P-4D 5688.21 5682.63 5688.19
3P-5D 4982.81 4978.54 4982.8
3P-6D 4668.56 4664.81 4668.6
3P-7D 4497.66 4494.18 4497.7
3P-8D 4393.34 4390.03 4393.3
3P-9D 4324.62 4321.40 4324.6
3P-10D 4276.79 4273.64 4276.8
3P-11D 4242.08 4238.99 4242.0
3P-12D 4215
3P-13D 4195

### Potassium

potassium diffuse series[13]
transition wavelength 1 Å wavelength 2 Å wavelength 3 Å
4P-3D 11772.8 11690.2 11769.7
4P-4D 6964.69 6936.27 6964.18
4P-5D 5831.9 5812.2 5831.7
4P-6D 5359.7 5343.1 5359.6
4P-7D 5112.2 5097.2 5112.2
4P-8D 4965.0 4950.8 4965.0
4P-9D 4869.8 4856.1 4869.8
4P-10D 4804.3 4791.0 4804.3
4P-11D 4757.4 4744.4 4757.4

## Alkaline earths

A diffuse series of triplet lines is designated by series letter d and formula 1p-md. The diffuse series of singlet lines has series letter S and formula 1P-mS.[3]

### Helium

Helium is in the same category as alkaline earths with respect to spectroscopy, as it has two electrons in the S subshell as do the other alkaline earths. Helium has a diffuse series of doublet lines with wavelengths 5876, 4472 and 4026 Å. Helium when ionised is termed HeII and has a spectrum very similar to hydrogen but shifted to shorter wavelengths. This has a diffuse series as well with wavelengths at 6678, 4922 and 4388 Å.[14]

### Magnesium

Magnesium has a diffuse series of triplets and a sharp series of singlets.[3]

### Calcium

Calcium has a diffuse series of triplets and a sharp series of singlets.[15]

### Strontium

With strontium vapour, the most prominent lines are from the diffuse series.[16]

### Barium

Barium has a diffuse series running from infrared to ultraviolet with wavelengths at 25515.7, 23255.3, 22313.4; 5818.91, 5800.30, 5777.70; 4493.66, 4489.00; 4087.31, 4084.87; 3898.58, 3894.34; 3789.72, 3788.18; 3721.17, and 3720.85 Å[17]

## History

At Cambridge University George Liveing and James Dewar set out to systematically measure spectra of elements from groups I, II and III in visible light and longer wave ultraviolet that would transmit though air. They noticed that lines for sodium were alternating sharp and diffuse. They were the first to use the term "diffuse" for the lines.[18] They classified alkali metal spectral lines into sharp and diffuse categories. In 1890 the lines that also appeared in the absorption spectrum were termed the principal series. Rydberg continued the use of sharp and diffuse for the other lines,[19] whereas Kayser and Runge preferred to use the term first subordinate series for the diffuse series.[20]

Arno Bergmann found a fourth series in infrared in 1907, and this became known as Bergmann Series or fundamental series.[20]

Heinrich Kayser, Carl Runge and Johannes Rydberg found mathematical relations between the wave numbers of emission lines of the alkali metals.[21]

Friedrich Hund introduced the s, p, d, f notation for subshells in atoms.[21][22] Others followed this use in the 1930s and the terminology has remained to this day.

## References

1. ^ Fowler, A. (1924). "The Origin of Spectra". Journal of the Royal Astronomical Society of Canada. 18: 373–380. Bibcode:1924JRASC..18..373F.
2. ^ a b Saunders, F. A. (1915). "Some Recent Discoveries in Spectrum Series". Astrophysical Journal. 41: 323. Bibcode:1915ApJ....41..323S. doi:10.1086/142175. Retrieved 26 August 2015.
3. ^ a b c Saunders, F. A. (1915). "Some Recent Discoveries in Spectrum Series". ApJ. 41: 323–327. Bibcode:1915ApJ....41..323S. doi:10.1086/142175.
4. ^ Rydberg, J. R. (1897). "The New Series in the Spectrum of Hydrogen". Astrophysical Journal. 6: 233–236. Bibcode:1897ApJ.....6..233R. doi:10.1086/140393.
5. ^ Band, Yehuda B. (14 September 2006). Light and Matter: Electromagnetism, Optics, Spectroscopy and Lasers. John Wiley. ISBN 9780471899310. Retrieved 3 July 2015.
6. ^ Band, Yehuda B. (2006-09-14). Light and Matter: Electromagnetism, Optics, Spectroscopy and Lasers. John Wiley & Sons. pp. 321–322. ISBN 9780471899310. Retrieved 10 January 2014.
7. ^ Schuster, Arthur (31 December 1986). "On a New Law Connecting the Periods of Molecular Vibrations". Nature. 55 (1418): 200–201. Bibcode:1896Natur..55..200S. doi:10.1038/055200a0.
8. ^ Schuster, Arthur (7 January 1987). "On a New Law Connecting the Periods of Molecular Vibrations". Nature. 55 (1419): 223. Bibcode:1897Natur..55..223S. doi:10.1038/055223a0.
9. ^ a b Atomic, Molecular and Laser Physics. Krishna Prakashan Media. p. 2.59.
10. ^ atomic spectra and the vector model. volume 1. series spectra. CUP Archive. p. 19. ISBN 9781001286228.
11. ^ a b Sala, O.; Araki, K.; Noda, L. K. (September 1999). "A Procedure to Obtain the Effective Nuclear Charge from the Atomic Spectrum of Sodium" (PDF). Journal of Chemical Education. 76 (9): 1269. Bibcode:1999JChEd..76.1269S. doi:10.1021/ed076p1269.
12. ^ Wiese, W.; Smith, M. W.; Miles, B. M. (October 1969). Atomic Transition Probabilities Volume II Sodium Through Calcium A Critical Data Compilation. Washington: National Bureau of Standards. pp. 39–41.
13. ^ Wiese, W.; Smith, M. W.; Miles, B. M. (October 1969). Atomic Transition Probabilities Volume II Sodium Through Calcium A Critical Data Compilation (PDF). Washington: National Bureau of Standards. pp. 228–230.
14. ^ Saunders, F. A. (1919). "Review of Recent Work on the Series Spectra of Helium and of Hydrogen". Astrophysical Journal. 50: 151–154. Bibcode:1919ApJ....50..151S. doi:10.1086/142490. Retrieved 3 July 2015.
15. ^ Saunders, F. A. (December 1920). "Revision of the Series in the Spectrum of Calcium". The Astrophysical Journal. 52 (5): 265. Bibcode:1920ApJ....52..265S. doi:10.1086/142578.
16. ^ Saunders, F. A. (1922). "Revision of the Series in the Spectrum of Strontium". Astrophysical Journal. 56: 73–82. Bibcode:1922ApJ....56...73S. doi:10.1086/142690.
17. ^ Saunders, F. A. (1920). "Revision of the Series in the Spectrum of Barium". Astrophysical Journal. 51: 23–36. Bibcode:1920ApJ....51...23S. doi:10.1086/142521.
18. ^ Brand, John Charles Drury (1995-10-01). Lines Of Light: The Sources Of Dispersive Spectroscopy, 1800-1930. CRC Press. pp. 123–. ISBN 9782884491624. Retrieved 30 December 2013.
19. ^ Rydberg, J. R. (April 1890). "XXXIV. On the structure of the line-spectra of the chemical elements". Philosophical Magazine. Series 5. 29 (179): 331–337. doi:10.1080/14786449008619945.
20. ^ a b Mehra, Jagdish; Rechenberg, Helmut (2001-01-01). The Historical Development of Quantum Theory. Springer. pp. 165–166. ISBN 9780387951744. Retrieved 30 December 2013.
21. ^ a b William B. Jensen (2007). "The Origin of the S, p, d, f Orbital Labels". Journal of Chemical Education. 84 (5): 757–758. Bibcode:2007JChEd..84..757J. doi:10.1021/ed084p757.
22. ^ Hund, Friedrich (1927). Linienspektren und Periodisches System der Elemente. Struktur der Materie in Einzeldarstellungen. 4. Springer. pp. 55–56. ISBN 9783709156568.

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