Nikiya Anton Bettey 2021: 09-26-2021-2241 - Hydrogen Spectral Series (hydrogen emission spectra ; emission spectrum of atomic hydrogen ; spectral lines)

Monday, September 27, 2021

09-26-2021-2241 - Hydrogen Spectral Series (hydrogen emission spectra ; emission spectrum of atomic hydrogen ; spectral lines)

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 spectral series of hydrogen, on a logarithmic scale.

Physics[edit]

Electron transitions and their resulting wavelengths for hydrogen. Energy levels are not to scale.

A hydrogen atom consists of an electron orbiting its nucleus. The electromagnetic force between the electron and the nuclear proton leads to a set of quantum states 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 level, or electron shell , or orbit, is designated by an integer, $n$ 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 $n'$ , and the higher energy state is designated as $n$ . 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 $n'$ . Lines are named sequentially starting from the longest wavelength/lowest frequency of the series, using Greek letters within each series. For example, the 2 → 1 line is called "Lyman-alpha" (Ly-α), while the 7 → 3 line is called "Paschen-delta” (Pa-δ).

Energy level diagram of electrons in hydrogen atom

There are emission lines from hydrogen that fall outside of these series, such as the 21 cm line. These emission lines correspond to much rarer atomic events such as hyperfine transitions.^[1] The fine structurealso results in single spectral lines appearing as two or more closely grouped thinner lines, due to relativistic corrections.^[2]

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 equivalent Heisenberg equation is convenient when studying an atom driven by an external electromagnetic wave.^[3]

In the processes of absorption or emission of photons by an atom, the 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 atoms must 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 ${\frac {10^{4}}{5}}$ times the mass of an electron, which gives only the zero order of approximation and thus may be not taken into account.^[3]^{[clarification needed]}

Rydberg formula[edit]

The energy differences between levels in the Bohr model, and hence the wavelengths of emitted or absorbed photons, is given by the Rydberg formula:^[4]

{1 \over \lambda }=Z^{2}R_{\infty }\left({1 \over {n'}^{2}}-{1 \over n^{2}}\right)

where

Z

is the atomic number,

n'

(often written

n_{1}

) is the principal quantum number of the lower energy level,

n

(or

n_{2}

) is the principal quantum number of the upper energy level, and

R_{\infty }

is the Rydberg constant. (1.09677×10⁷ m⁻¹ for hydrogen and 1.09737×10⁷ m⁻¹ for heavy metals).^[5]^[6]

The wavelength will always be positive because $n'$ is defined as the lower level and so is less than $n$ . 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[edit]

Lyman series ( $n'$ = 1)[edit]

Lyman series of hydrogen atom spectral lines in the ultraviolet

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.^[7]^[8]

$n$	$λ$ , vacuum (nm)
2	121.57
3	102.57
4	97.254
5	94.974
6	93.780
∞	91.175
Source:^[9]

Balmer series ( $n'$ = 2)[edit]

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.^[10] Four of the Balmer lines are in the technically "visible" part of the spectrum, with wavelengths longer than 400 nm and shorter than 700 nm. Parts of the Balmer series can be seen in the solar spectrum. H-alpha is an important line used in astronomy to detect the presence of hydrogen.

$n$	$λ$ , air (nm)
3	656.3
4	486.1
5	434.0
6	410.2
7	397.0
∞	364.6
Source:^[9]

Paschen series (Bohr series, $n'$ = 3)[edit]

Named after the German physicist Friedrich Paschen who first observed them in 1908. The Paschen lines all lie in the infrared band.^[11] 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.

$n$	$λ$ , air (nm)
4	1875
5	1282
6	1094
7	1005
8	954.6
∞	820.4
Source:^[9]

Brackett series ( $n'$ = 4)[edit]

Named after the American physicist Frederick Sumner Brackett who first observed the spectral lines in 1922.^[12]The spectral lines of Brackett series lie in far infrared band.

$n$	$λ$ , air (nm)
5	4051
6	2625
7	2166
8	1944
9	1817
∞	1458
Source:^[9]

Pfund series ( $n'$ = 5)[edit]

Experimentally discovered in 1924 by August Herman Pfund.^[13]

$n$	$λ$ , vacuum (nm)
6	7460
7	4654
8	3741
9	3297
10	3039
∞	2279
Source:^[14]

Humphreys series ( $n'$ = 6)[edit]

Discovered in 1953 by American physicist Curtis J. Humphreys.^[15]

$n$	$λ$ , vacuum (μm)
7	12.37
8	7.503
9	5.908
10	5.129
11	4.673
∞	3.282
Source:^[14]

Further series ( $n'$ > 6)[edit]

Further series are unnamed, but follow the same pattern and equation as dictated by the Rydberg equation. Series are increasingly spread out and occur at 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 Peter Hansen and John Strong at the University of Massachusetts Amherst.^[16]

Extension to other systems[edit]

The concepts of the Rydberg formula can be applied to any system with a single particle orbiting a nucleus, for example a He⁺ 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–Fowler series was originally attributed to an unknown form of hydrogen with half-integer transition levels by both Pickering^[17]^[18]^[19] and Fowler,^[20] but Bohr correctly recognised them as spectral lines arising from the He⁺ nucleus.^[21]^[22]^[23]

All other atoms have at least two electrons in their 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.

Emission spectroscopy[edit]

Light consists of electromagnetic radiation of different wavelengths. Therefore, when the elements or their compounds are heated either on a flame or by an electric arc they emit energy in the form of light. Analysis of this light, with the help of a spectroscope gives us a discontinuous spectrum. A spectroscope or a spectrometer is an instrument which is used for separating the components of light, which have different wavelengths. The spectrum appears in a series of lines called the line spectrum. This line spectrum is called an atomic spectrum when it originates from an atom in elemental form. Each element has a different atomic spectrum. The production of line spectra by the atoms of an element indicate that an atom can radiate only a certain amount of energy. This leads to the conclusion that bound electrons cannot have just any amount of energy but only a certain amount of energy.

The emission spectrum can be used to determine the composition of a material, since it is different for each element of the periodic table. One example is astronomical spectroscopy: identifying the composition of stars by analysing the received light. The emission spectrum characteristics of some elements are plainly visible to the naked eye when these elements are heated. For example, when platinum wire is dipped into a sodium nitrate solution and then inserted into a flame, the sodium atoms emit an amber yellow color. Similarly, when indium is inserted into a flame, the flame becomes blue. These definite characteristics allow elements to be identified by their atomic emission spectrum. Not all emitted lights are perceptible to the naked eye, as the spectrum also includes ultraviolet rays and infrared radiation. An emission spectrum is formed when an excited gas is viewed directly through a spectroscope.

Schematic diagram of spontaneous emission

Emission spectroscopy is a spectroscopic technique which examines the wavelengths of photons emitted by atoms or molecules during their transition from an excited state to a lower energy state. Each element emits a characteristic set of discrete wavelengths according to its electronic structure, and by observing these wavelengths the elemental composition of the sample can be determined. Emission spectroscopy developed in the late 19th century and efforts in theoretical explanation of atomic emission spectra eventually led to quantum mechanics.

There are many ways in which atoms can be brought to an excited state. Interaction with electromagnetic radiation is used in fluorescence spectroscopy, protons or other heavier particles in Particle-Induced X-ray Emission and electrons or X-ray photons in Energy-dispersive X-ray spectroscopy or X-ray fluorescence. The simplest method is to heat the sample to a high temperature, after which the excitations are produced by collisions between the sample atoms. This method is used in flame emission spectroscopy, and it was also the method used by Anders Jonas Ångström when he discovered the phenomenon of discrete emission lines in the 1850s.^[1]

Although the emission lines are caused by a transition between quantized energy states and may at first look very sharp, they do have a finite width, i.e. they are composed of more than one wavelength of light. This spectral line broadening has many different causes.

Emission spectroscopy is often referred to as optical emission spectroscopy because of the light nature of what is being emitted.

History[edit]

In 1756 Thomas Melvill observed the emission of distinct patterns of colour when salts were added to alcohol flames.^[2] By 1785 James Gregory discovered the principles of diffraction grating and American astronomer David Rittenhouse made the first engineered diffraction grating.^[3]^[4] In 1821 Joseph von Fraunhofer solidified this significant experimental leap of replacing a prism as the source of wavelength dispersion improving the spectral resolution and allowing for the dispersed wavelengths to be quantified.^[5]

In 1835, Charles Wheatstone reported that different metals could be distinguished by bright lines in the emission spectra of their sparks, thereby introducing an alternative to flame spectroscopy.^[6]^[7] In 1849, J. B. L. Foucault experimentally demonstrated that absorption and emission lines at the same wavelength are both due to the same material, with the difference between the two originating from the temperature of the light source.^[8]^[9] In 1853, the Swedish physicist Anders Jonas Ångström presented observations and theories about gas spectra.^[10] Ångström postulated that an incandescent gas emits luminous rays of the same wavelength as those it can absorb. At the same time George Stokes and William Thomson (Kelvin) were discussing similar postulates.^[8] Ångström also measured the emission spectrum from hydrogen later labeled the Balmer lines.^[11]^[12] In 1854 and 1855, David Alter published observations on the spectra of metals and gases, including an independent observation of the Balmer lines of hydrogen.^[13]^[14]

By 1859, Gustav Kirchhoff and Robert Bunsen noticed that several Fraunhofer lines (lines in the solar spectrum) coincide with characteristic emission lines identified in the spectra of heated elements.^[15]^[16] It was correctly deduced that dark lines in the solar spectrum are caused by absorption by chemical elements in the solar atmosphere.^[17]

Experimental technique in flame emission spectroscopy[edit]

The solution containing the relevant substance to be analysed is drawn into the burner and dispersed into the flame as a fine spray. The solvent evaporates first, leaving finely divided solid particles which move to the hottest region of the flame where gaseous atoms and ions are produced. Here electrons are excited as described above. It is common for a monochromator to be used to allow for easy detection.

On a simple level, flame emission spectroscopy can be observed using just a flame and samples of metal salts. This method of qualitative analysis is called a flame test. For example, sodium salts placed in the flame will glow yellow from sodium ions, while strontium (used in road flares) ions color it red. Copper wire will create a blue colored flame, however in the presence of chloride gives green (molecular contribution by CuCl).

Emission coefficient[edit]

Emission coefficient is a coefficient in the power output per unit time of an electromagnetic source, a calculated value in physics. The emission coefficient of a gas varies with the wavelength of the light. It has units of ms⁻³sr⁻¹.^[18] It is also used as a measure of environmental emissions (by mass) per MWh of electricity generated, see: Emission factor.

Scattering of light[edit]

In Thomson scattering a charged particle emits radiation under incident light. The particle may be an ordinary atomic electron, so emission coefficients have practical applications.

If X dV dΩ dλ is the energy scattered by a volume element dV into solid angle dΩ between wavelengths λ and λ+dλ per unit time then the Emission coefficient is X.

The values of X in Thomson scattering can be predicted from incident flux, the density of the charged particles and their Thomson differential cross section (area/solid angle).

Spontaneous emission[edit]

A warm body emitting photons has a monochromatic emission coefficient relating to its temperature and total power radiation. This is sometimes called the second Einstein coefficient, and can be deduced from quantum mechanical theory.

See also[edit]

Absorption spectroscopy
Absorption spectrum
Atomic spectral line
Electromagnetic spectroscopy
Gas-discharge lamp, Table of emission spectra of gas discharge lamps
Isomeric shift
Isotopic shift
Luminous coefficient
Plasma physics
Rydberg formula
Spectral theory
The Diode equation includes the emission coefficient (which is not related to the one discussed here)
Thermionic emission

https://en.wikipedia.org/wiki/Emission_spectrum

Absorption spectroscopy refers to spectroscopic techniques that measure the absorption of radiation, as a function of frequency or wavelength, due to its interaction with a sample. The sample absorbs energy, i.e., photons, from the radiating field. The intensity of the absorption varies as a function of frequency, and this variation is the absorption spectrum. Absorption spectroscopy is performed across the electromagnetic spectrum.

https://en.wikipedia.org/wiki/Absorption_spectroscopy#Absorption_spectrum

Atomic spectroscopy is the study of the electromagnetic radiation absorbed and emitted by atoms. Since unique elements have characteristic (signature) spectra, atomic spectroscopy, specifically the electromagnetic spectrum or mass spectrum, is applied for determination of elemental compositions. It can be divided by atomization source or by the type of spectroscopy used. In the latter case, the main division is between optical and mass spectrometry. Mass spectrometry generally gives significantly better analytical performance, but is also significantly more complex. This complexity translates into higher purchase costs, higher operational costs, more operator training, and a greater number of components that can potentially fail. Because optical spectroscopy is often less expensive and has performance adequate for many tasks, it is far more common^{[citation needed]} Atomic absorption spectrometers are one of the most commonly sold and used analytical devices.

https://en.wikipedia.org/wiki/Atomic_spectroscopy

Spectroscopy is the study of the interaction between matter and electromagnetic radiation as a function of the wavelength or frequency of the radiation.^[1]^[2]^[3]^[4]^[5]^[6] In simpler terms, spectroscopy is the precise study of color as generalized from visible light to all bands of the electromagnetic spectrum; indeed, historically, spectroscopy originated as the study of the wavelength dependence of the absorption by gas phase matter of visible light dispersed by a prism. Matter waves and acoustic waves can also be considered forms of radiative energy, and recently gravitational waves have been associated with a spectral signature in the context of the Laser Interferometer Gravitational-Wave Observatory (LIGO).

https://en.wikipedia.org/wiki/Spectroscopy

Spectroscopy and spectrography are terms used to refer to the measurement of radiation intensity as a function of wavelength and are often used to describe experimentalspectroscopic methods. Spectral measurement devices are referred to as spectrometers, spectrophotometers, spectrographs or spectral analyzers.

One of the central concepts in spectroscopy is a resonance and its corresponding resonant frequency. Resonances were first characterized in mechanical systems such as pendulums. Mechanical systems that vibrate or oscillate will experience large amplitude oscillations when they are driven at their resonant frequency. A plot of amplitude vs. excitation frequency will have a peak centered at the resonance frequency. This plot is one type of spectrum, with the peak often referred to as a spectral line, and most spectral lines have a similar appearance.

https://en.wikipedia.org/wiki/Spectroscopy

Electron paramagnetic resonance (EPR) or electron spin resonance (ESR) spectroscopy is a method for studying materials with unpaired electrons. The basic concepts of EPR are analogous to those of nuclear magnetic resonance (NMR), but the spins excited are those of the electrons instead of the atomic nuclei. EPR spectroscopy is particularly useful for studying metal complexes and organic radicals. EPR was first observed in Kazan State University by Soviet physicist Yevgeny Zavoisky in 1944,^[1]^[2] and was developed independently at the same time by Brebis Bleaney at the University of Oxford.

https://en.wikipedia.org/wiki/Electron_paramagnetic_resonance

Ferromagnetic resonance, or FMR, is coupling between an electromagnetic wave and the magnetization of a medium through which it passes. This coupling induces a significant loss of power of the wave. The power is absorbed by the precessing magnetization (Larmor precession) of the material and lost as heat. For this coupling to occur, the frequency of the incident wave must be equal to the precession frequency of the magnetization (Larmor frequency) and the polarization of the wave must match the orientation of the magnetization.

This effect can be used for various applications such as spectroscopic techniques or conception of microwave devices.

The FMR spectroscopic technique is used to probe the magnetization of ferromagnetic materials. It is a standard tool for probing spin waves and spin dynamics. FMR is very broadly similar to electron paramagnetic resonance (EPR), and also somewhat similar to nuclear magnetic resonance (NMR), except that FMR probes the sample magnetization resulting from the magnetic moments of dipolar-coupled but unpaired electrons, while NMR probes the magnetic moment of atomic nuclei that are screened by the atomic or molecular orbitals surrounding such nuclei of non-zero nuclear spin.

The FMR resonance is also the basis of various high-frequency electronic devices, such as resonance isolators or circulators.

https://en.wikipedia.org/wiki/Ferromagnetic_resonance

Nikiya Anton Bettey 2021

Blog Archive

Monday, September 27, 2021

09-26-2021-2241 - Hydrogen Spectral Series (hydrogen emission spectra ; emission spectrum of atomic hydrogen ; spectral lines)

Contents

Physics[edit]

Rydberg formula[edit]