Rydberg constant

Rydberg constant is a fundamental physical constant used in formulas to calculate energy levels and atomic frequencies . Introduced by Swedish scientist Johannes Robert Rydberg in 1890 when studying the emission spectra of atoms. Designated as $R$ ${\ displaystyle R}$ $R$ ^[1] .

This constant initially appeared as an empirical fitting parameter in Rydberg's formula describing the spectral series of hydrogen . Niels Bohr later showed that its value can be calculated from more fundamental constants , explaining their relationship using his atom model ( Bohr model ). The Rydberg constant is the limit value of the highest wave number of any photon that can be emitted by a hydrogen atom; on the other hand, this is the wave number of the lowest-energy photon capable of ionizing a hydrogen atom in its ground state.

It also uses a unit of energy closely related to the Rydberg constant, simply called Rydberg and denoted by $\mathrm {Ry}$ ${\ displaystyle \ mathrm {Ry}}$ $\ mathrm {Ry}$ . It corresponds to the energy of a photon, the wave number of which is equal to the Rydberg constant, that is, the ionization energy of a hydrogen atom.

As of 2012, the Rydberg constant and the electron g-factor are the most accurately measured fundamental physical constants. ^[2]

Content

1 Numerical value
2 Properties
3 See also
4 notes
5 Literature

Numerical value

The numerical value of the Rydberg constant recommended by CODATA in 2014 is ^[3] :

R

{\ displaystyle R}

= 10973731.568508 (65) m ⁻¹ .

For light atoms, the Rydberg constant has the following meanings:

Hydrogen : $R_{H}$ ${\ displaystyle R_ {H}}$ = 109677.583407 cm ⁻¹ ;
Deuterium : $R_{D}$ ${\ displaystyle R_ {D}}$ = 109707.417 cm ⁻¹ ;
Helium : $R_{He}$ ${\ displaystyle R_ {He}}$ = 109722.267 cm ⁻¹ .

In atomic physics, a constant is often used as an energy unit (Rydberg):

\mathrm {Ry} =R\cdot h\cdot c=2\pi \hbar cR=me^{4}/2\hbar ^{2}=e^{2}/2a_{0}

{\ displaystyle \ mathrm {Ry} = R \ cdot h \ cdot c = 2 \ pi \ hbar cR = me ^ {4} / 2 \ hbar ^ {2} = e ^ {2} / 2a_ {0}}

where

a_{0}

{\ displaystyle a_ {0}}

- Bohr radius .

Numerical value ^[4] ^[5] :

\mathrm {Ry} =13{,}605693009(84)

{\ displaystyle \ mathrm {Ry} = 13 {,} 605693009 (84)}

eV =

2{,}179872325(27)\times 10^{-18}

{\ displaystyle 2 {,} 179872325 (27) \ times 10 ^ {- 18}}

J.

Properties

Rydberg constant is included in the general law for spectral frequencies as follows:

\nu =R{Z^{2}}\left({\frac {1}{n^{2}}}-{\frac {1}{m^{2}}}\right)

{\ displaystyle \ nu = R {Z ^ {2}} \ left ({\ frac {1} {n ^ {2}}} - {\ frac {1} {m ^ {2}}} \ right)}

Where $\nu$ ${\ displaystyle \ nu}$ - wave number (by definition, this is the reciprocal wavelength or the number of wavelengths that fit 1 cm), Z is the atomic number.

\nu ={\frac {1}{\lambda }}

{\ displaystyle \ nu = {\ frac {1} {\ lambda}}}

cm ⁻¹

Accordingly,

{\frac {1}{\lambda }}=R{Z^{2}}\left({\frac {1}{n^{2}}}-{\frac {1}{m^{2}}}\right)

{\ displaystyle {\ frac {1} {\ lambda}} = R {Z ^ {2}} \ left ({\ frac {1} {n ^ {2}}} - {\ frac {1} {m ^ {2}}} \ right)}

If we consider the mass of the nucleus of an atom to be infinitely large compared to the mass of the electron (that is, assume that the nucleus is motionless), then the Rydberg constant for the frequency in Hz will be defined as

R={\frac {me^{4}}{4\pi c\hbar ^{3}}}

{\ displaystyle R = {\ frac {me ^ {4}} {4 \ pi c \ hbar ^ {3}}}}

in the GHS system, where $m$ ${\ displaystyle m}$ and $e$ ${\ displaystyle e}$ - mass and charge of an electron , $c$ ${\ displaystyle c}$ Is the speed of light , and $\hbar$ ${\ displaystyle \ hbar}$ Is the Dirac constant or the reduced Planck constant .

In the International System of Units (SI) for frequency in Hz:

R_{c}={\frac {mk^{2}e^{4}}{4\pi \hbar ^{3}}}

{\ displaystyle R_ {c} = {\ frac {mk ^ {2} e ^ {4}} {4 \ pi \ hbar ^ {3}}}}

R_{c}={\frac {2m\pi ^{2}k^{2}e^{4}}{h^{3}}}

{\ displaystyle R_ {c} = {\ frac {2m \ pi ^ {2} k ^ {2} e ^ {4}} {h ^ {3}}}}

Where $k=c^{2}\times 10^{-7}$ ${\ displaystyle k = c ^ {2} \ times 10 ^ {- 7}}$ - coefficient from Coulomb's law . Numerical value ^[6] :

R_{c}=3{,}289841960355(19)\times 10^{15}

{\ displaystyle R_ {c} = 3 {,} 289841960355 (19) \ times 10 ^ {15}}

s ⁻¹

Usually, when they talk about Rydberg's constant, they mean the constant calculated with a motionless core. When taking into account the motion of the nucleus, the electron mass is replaced by the reduced mass of the electron and the nucleus, and then

R_{i}={\frac {R}{1+m/M_{i}}}

{\ displaystyle R_ {i} = {\ frac {R} {1 + m / M_ {i}}}}

where

M_{i}

{\ displaystyle M_ {i}}

Is the mass of the nucleus of an atom.

Notes

↑ Rydberg constant // Physical Encyclopedia / Ch. ed. A.M. Prokhorov . - M .: Big Russian Encyclopedia , 1994. - T. 4. - S. 391. - 704 p. - 40,000 copies. - ISBN 5-85270-087-8 .
↑ Pohl, Randolf; Antognini, Aldo; Nez, François; Amaro, Fernando D .; Biraben, François; Cardoso, João MR; Covita, Daniel S .; Dax, Andreas; Dhawan, Satish; Fernandes, Luis MP; Giesen, Adolf; Graf, Thomas; Hänsch, Theodor W .; Indelicato, Paul; Julien, Lucile; Kao, Cheng-Yang; Knowles, Paul; Le Bigot, Eric-Olivier; Liu, Yi-Wei; Lopes, José AM; Ludhova, Livia; Monteiro, Cristina MB; Mulhauser, Françoise; Nebel, Tobias; Rabinowitz, Paul; Dos Santos, Joaquim MF; Schaller, Lukas A .; Schuhmann, Karsten; Schwob, Catherine; Taqqu, David. The size of the proton (Eng.) // Nature. - 2010 .-- Vol. 466 , no. 7303 . - P. 213-216 . - DOI : 10.1038 / nature09250 . - . - PMID 20613837 .
↑ Rydberg constant // 2014 CODATA recommended values
↑ Rydberg constant times hc in eV // 2014 CODATA recommended values
↑ Rydberg constant times hc in J // 2014 CODATA recommended values
↑ Rydberg constant times c in Hz // 2014 CODATA recommended values

Literature

Shpolsky E.V. Atomic Physics. Volume 1 - M.: Science, 1974.
Born M. Atomic Physics. - M.: Mir, 1970.
Saveliev I.V. Course in General Physics. Book 5. Quantum optics. Atomic physics. Solid State Physics. Physics of the atomic nucleus and elementary particles. - M .: AST, Astrel, 2003.

[1] Rydberg constant // Physical Encyclopedia / Ch. ed. A.M. Prokhorov . - M .: Big Russian Encyclopedia , 1994. - T. 4. - S. 391. - 704 p. - 40,000 copies. - ISBN 5-85270-087-8 .

[pohl-2] Pohl, Randolf; Antognini, Aldo; Nez, François; Amaro, Fernando D .; Biraben, François; Cardoso, João MR; Covita, Daniel S .; Dax, Andreas; Dhawan, Satish; Fernandes, Luis MP; Giesen, Adolf; Graf, Thomas; Hänsch, Theodor W .; Indelicato, Paul; Julien, Lucile; Kao, Cheng-Yang; Knowles, Paul; Le Bigot, Eric-Olivier; Liu, Yi-Wei; Lopes, José AM; Ludhova, Livia; Monteiro, Cristina MB; Mulhauser, Françoise; Nebel, Tobias; Rabinowitz, Paul; Dos Santos, Joaquim MF; Schaller, Lukas A .; Schuhmann, Karsten; Schwob, Catherine; Taqqu, David. The size of the proton (Eng.) // Nature. - 2010 .-- Vol. 466 , no. 7303 . - P. 213-216 . - DOI : 10.1038 / nature09250 . - . - PMID 20613837 .

[3] Rydberg constant // 2014 CODATA recommended values

[4] Rydberg constant times hc in eV // 2014 CODATA recommended values

[5] Rydberg constant times hc in J // 2014 CODATA recommended values

[6] Rydberg constant times c in Hz // 2014 CODATA recommended values