In physics and electronic engineering, an electron hole is the lack of an electron at a position where one could exist in an atom or atomic lattice. Since in a normal atom or crystal lattice the negative charge of the electrons is balanced by the positive charge of the atomic nuclei, the absence of an electron leaves a net positive charge at the hole's location. Holes are not particles, but rather quasiparticles. Holes in a metal or semiconductor crystal lattice can move through the lattice as electrons can, act to positively-charged particles, they play an important role in the operation of semiconductor devices such as transistors and integrated circuits. If an electron is excited into a higher state it leaves a hole in its old state; this meaning is used in Auger electron spectroscopy, in computational chemistry, to explain the low electron-electron scattering-rate in crystals. In crystals, electronic band structure calculations lead to an effective mass for the electrons, negative at the top of a band.
The negative mass is an unintuitive concept, in these situations a more familiar picture is found by considering a positive charge with a positive mass. In solid-state physics, an electron hole is the absence of an electron from a full valence band. A hole is a way to conceptualize the interactions of the electrons within a nearly full valence band of a crystal lattice, missing a small fraction of its electrons. In some ways, the behavior of a hole within a semiconductor crystal lattice is comparable to that of the bubble in a full bottle of water. Hole conduction in a valence band can be explained by the following analogy. Imagine a row of people seated in an auditorium, where there are no spare chairs. Someone in the middle of the row wants to leave, so he jumps over the back of the seat into another row, walks out; the empty row is analogous to the conduction band, the person walking out is analogous to a conduction electron. Now imagine someone else comes along and wants to sit down; the empty row has a poor view.
Instead, a person in the crowded row moves into the empty seat the first person left behind. The empty seat moves one spot closer to the person waiting to sit down; the next person follows, the next, et cetera. One could say. Once the empty seat reaches the edge, the new person can sit down. In the process everyone in the row has moved along. If those people were negatively charged, this movement would constitute conduction. If the seats themselves were positively charged only the vacant seat would be positive; this is a simple model of how hole conduction works. Instead of analyzing the movement of an empty state in the valence band as the movement of many separate electrons, a single equivalent imaginary particle called a "hole" is considered. In an applied electric field, the electrons move in one direction, corresponding to the hole moving in the other. If a hole associates itself with a neutral atom, that atom becomes positive. Therefore, the hole is taken to have positive charge of +e the opposite of the electron charge.
In reality, due to the uncertainty principle of quantum mechanics, combined with the energy levels available in the crystal, the hole is not localizable to a single position as described in the previous example. Rather, the positive charge which represents the hole spans an area in the crystal lattice covering many hundreds of unit cells; this is equivalent to being unable to tell. Conduction band electrons are delocalized; the analogy above is quite simplified, cannot explain why holes create an opposite effect to electrons in the Hall effect and Seebeck effect. A more precise and detailed explanation follows; the dispersion relation determines. A dispersion relation is the relationship between wavevector and energy in a band, part of the electronic band structure. In quantum mechanics, the electrons are waves, energy is the wave frequency. A localized electron is a wavepacket, the motion of an electron is given by the formula for the group velocity of a wave. An electric field affects an electron by shifting all the wavevectors in the wavepacket, the electron accelerates when its wave group velocity changes.
Therefore, the way an electron responds to forces is determined by its dispersion relation. An electron floating in space has the dispersion relation E=ℏ2k2/, where m is the electron mass and ℏ is reduced Planck constant. Near the bottom of the conduction band of a semiconductor, the dispersion relation is instead E=ℏ2k2/, so a conduction-band electron responds to forces as if it had the mass m*. Electrons near the top of the valence band behave; the dispersion relation near the top of the valence band is E=ℏ2k2/ with negative effective mass. So electrons near the top of the valence band behave; when a force pulls the electrons to the right, these electrons move left. This is due to the shape of the valence band, is unrelated to whether the band is full or empty. If you could somehow empty out the valence band and just put one electron near the valence band maximum, this electron would move the "wrong way" in response to forces. Po
Fluorescence is the emission of light by a substance that has absorbed light or other electromagnetic radiation. It is a form of luminescence. In most cases, the emitted light has a longer wavelength, therefore lower energy, than the absorbed radiation; the most striking example of fluorescence occurs when the absorbed radiation is in the ultraviolet region of the spectrum, thus invisible to the human eye, while the emitted light is in the visible region, which gives the fluorescent substance a distinct color that can be seen only when exposed to UV light. Fluorescent materials cease to glow nearly when the radiation source stops, unlike phosphorescent materials, which continue to emit light for some time after. Fluorescence has many practical applications, including mineralogy, medicine, chemical sensors, fluorescent labelling, biological detectors, cosmic-ray detection, most fluorescent lamps. Fluorescence occurs in nature in some minerals and in various biological states in many branches of the animal kingdom.
An early observation of fluorescence was described in 1560 by Bernardino de Sahagún and in 1565 by Nicolás Monardes in the infusion known as lignum nephriticum. It was derived from the wood of Pterocarpus indicus and Eysenhardtia polystachya; the chemical compound responsible for this fluorescence is matlaline, the oxidation product of one of the flavonoids found in this wood. In 1819, Edward D. Clarke and in 1822 René Just Haüy described fluorescence in fluorites, Sir David Brewster described the phenomenon for chlorophyll in 1833 and Sir John Herschel did the same for quinine in 1845. In his 1852 paper on the "Refrangibility" of light, George Gabriel Stokes described the ability of fluorspar and uranium glass to change invisible light beyond the violet end of the visible spectrum into blue light, he named this phenomenon fluorescence: "I am inclined to coin a word, call the appearance fluorescence, from fluor-spar, as the analogous term opalescence is derived from the name of a mineral." The name was derived from the mineral fluorite, some examples of which contain traces of divalent europium, which serves as the fluorescent activator to emit blue light.
In a key experiment he used a prism to isolate ultraviolet radiation from sunlight and observed blue light emitted by an ethanol solution of quinine exposed by it. Fluorescence occurs when an orbital electron of a molecule, atom, or nanostructure, relaxes to its ground state by emitting a photon from an excited singlet state: Excitation: S 0 + h ν e x → S 1 Fluorescence: S 1 → S 0 + h ν e m + h e a t Here h ν is a generic term for photon energy with h = Planck's constant and ν = frequency of light; the specific frequencies of exciting and emitted lights are depended on the particular system. S0 is called the ground state of the fluorophore, S1 is its first excited singlet state. A molecule in S1 can relax by various competing pathways, it can undergo non-radiative relaxation in which the excitation energy is dissipated as heat to the solvent. Excited organic molecules can relax via conversion to a triplet state, which may subsequently relax via phosphorescence, or by a secondary non-radiative relaxation step.
Relaxation from S1 can occur through interaction with a second molecule through fluorescence quenching. Molecular oxygen is an efficient quencher of fluorescence just because of its unusual triplet ground state. In most cases, the emitted light has a longer wavelength, therefore lower energy, than the absorbed radiation. However, when the absorbed electromagnetic radiation is intense, it is possible for one electron to absorb two photons; the emitted radiation may be of the same wavelength as the absorbed radiation, termed "resonance fluorescence". Molecules that are excited through light absorption or via a different process can transfer energy to a second'sensitized' molecule, converted to its excited state and can fluoresce; the fluorescence quantum yield gives the efficiency of the fluorescence process. It is defined as the ratio of the number of photons emitted to the number of photons absorbed. Φ = Number of photons emitted Number of photons absorbed The maximum possible fluorescence quantum yield is 1.0.
Compounds with quantum yields of 0.10 are still considered quite fluorescent. Another way to define the quantum yield of fluorescence is by the rate of excited state decay: Φ = k f ∑ i k i where k f is the rate constant of spontaneous emission of radiation and ∑ i k i is the sum of all rates of
Resonant inelastic X-ray scattering
Resonant inelastic X-ray scattering is an X-ray spectroscopy technique used to investigate the electronic structure of molecules and materials. Inelastic X-ray scattering is a fast developing experimental technique in which one scatters high energy, X-ray photons inelastically off matter, it is a photon-in/photon-out spectroscopy where one measures both the energy and momentum change of the scattered photon. The energy and momentum lost by the photon are transferred to intrinsic excitations of the material under study and thus RIXS provides information about those excitations; the RIXS process can be described as a resonant X-ray Raman or resonant X-ray emission process. RIXS is a resonant technique because the energy of the incident photon is chosen such that it coincides with, hence resonates with, one of the atomic X-ray absorption edges of the system; the resonance can enhance the inelastic scattering cross section, sometimes by many orders of magnitudeThe RIXS event can be thought of as a two-step process.
Starting from the initial state, absorption of an incident photon leads to creation of an excited intermediate state, that has a core hole. From this state, emission of a photon leads to the final state. In a simplified picture the absorption process gives information of the empty electronic states, while the emission gives information about the occupied states. In the RIXS experiment these two pieces of information come together in a convolved manner perturbed by the core-hole potential in the intermediate state. RIXS studies can be performed using both hard X-rays. Compared to other scattering techniques, RIXS has a number of unique features: it covers a large scattering phase-space, is polarization dependent and orbital specific, bulk sensitive and requires only small sample volumes. In RIXS one measures both the energy and momentum change of the scattered photon. Comparing the energy of a neutron, electron or photon with a wavelength of the order of the relevant length scale in a solid— as given by the de Broglie equation considering the interatomic lattice spacing is in the order of Ångströms—it derives from the relativistic energy–momentum relation that an X-ray photon has more energy than a neutron or electron.
The scattering phase space of X-rays is therefore without equal. In particular, high-energy X-rays carry a momentum, comparable to the inverse lattice spacing of typical condensed matter systems so that, unlike Raman scattering experiments with visible or infrared light, RIXS can probe the full dispersion of low energy excitations in solids. RIXS can utilize the polarization of the photon: the nature of the excitations created in the material can be disentangled by a polarization analysis of the incident and scattered photons, which allow one, through the use of various selection rules, to characterize the symmetry and nature of the excitations. RIXS is element and orbital specific: chemical sensitivity arises by tuning to the absorption edges of the different types of atoms in a material. RIXS can differentiate between the same chemical element at sites with inequivalent chemical bondings, with different valencies or at inequivalent crystallographic positions as long as the X-ray absorption edges in these cases are distinguishable.
In addition, the type of information on the electronic excitations of a system being probed can be varied by tuning to different X-ray edges of the same chemical element, where the photon excites core-electrons into different valence orbitals. RIXS is bulk sensitive: the penetration depth of resonant X-ray photons is material and scattering geometry- specific, but is on the order of a few micrometre in the hard X-ray regime and on the order of 0.1 micrometre in the soft X-ray regime. RIXS needs only small sample volumes: the photon-matter interaction is strong, compared to for instance the neutron-matter interaction strength; this makes RIXS possible on small volume samples, thin films and nano-objects, in addition to bulk single crystal or powder samples. In principle RIXS can probe a broad class of intrinsic excitations of the system under study—as long as the excitations are overall charge neutral; this constraint arises from the fact that in RIXS the scattered photons do not add or remove charge from the system under study.
This implies that, in principle RIXS has a finite cross section for probing the energy and polarization dependence of any type of electron-hole excitation: for instance the electron-hole continuum and excitons in band metals and semiconductors, charge transfer and crystal field excitations in correlated materials, lattice excitations, orbital excitations, so on. In addition magnetic excitations are symmetry-allowed in RIXS, because the angular momentum that the photons carry can in principle be transferred to the electron's spin moment. Moreover, it has been theoretically shown that RIXS can probe Bogoliubov quasiparticles in high-temperature superconductors, shed light on the nature and symmetry of the electron-electron pairing of the superconducting state; the energy and momentum resolution of RIXS do not depend on the core-hole, present in the intermediate state. In general the natural linewidth of a spectral feature is determined by the life-times of initial and final states. In X-ray absorption and non-resonant emission spectroscopy, the resolution is limited by the short life-time of the final state core-hole.
As in RIXS a high energy core-hole is absent in final state, this leads to intrinsically sharp spectra with energy and momentum resolution determined by the in
In optics, a diffraction grating is an optical component with a periodic structure that splits and diffracts light into several beams travelling in different directions. The emerging coloration is a form of structural coloration; the directions of these beams depend on the spacing of the grating and the wavelength of the light so that the grating acts as the dispersive element. Because of this, gratings are used in monochromators and spectrometers. For practical applications, gratings have ridges or rulings on their surface rather than dark lines; such gratings can be either reflective. Gratings that modulate the phase rather than the amplitude of the incident light are produced using holography; the principles of diffraction gratings were discovered by James Gregory, about a year after Newton's prism experiments with items such as bird feathers. The first man-made diffraction grating was made around 1785 by Philadelphia inventor David Rittenhouse, who strung hairs between two finely threaded screws.
This was similar to notable German physicist Joseph von Fraunhofer's wire diffraction grating in 1821. In the 1860s, gratings with the lowest line-distance d were created by Friedrich Adolph Nobert in Greifswald the two Americans Lewis Morris Rutherfurd and William B. Rogers took over the lead, by the end of the 19th century, the concave gratings of Henry Augustus Rowland were the best gratings available. Diffraction can create "rainbow" colors; the sparkling effects from the spaced narrow tracks on optical storage disks such as CDs or DVDs are an example, while the similar rainbow effects caused by thin layers of oil on water are not caused by a grating, but rather by interference effects in reflections from the spaced transmissive layers. A grating has parallel lines. Diffraction colors appear when one looks at a bright point source through a translucent fine-pitch umbrella-fabric covering. Decorative patterned plastic films based on reflective grating patches are inexpensive, are commonplace.
The relationship between the grating spacing and the angles of the incident and diffracted beams of light is known as the grating equation. According to the Huygens–Fresnel principle, each point on the wavefront of a propagating wave can be considered to act as a point source, the wavefront at any subsequent point can be found by adding together the contributions from each of these individual point sources. Gratings may be of the'reflective' or'transmissive' type, analogous to a mirror or lens, respectively. A grating has a'zero-order mode', in which there is no diffraction and a ray of light behaves according to the laws of reflection and refraction the same as with a mirror or lens, respectively. An idealised grating is made up of a set of slits of spacing d, that must be wider than the wavelength of interest to cause diffraction. Assuming a plane wave of monochromatic light of wavelength λ with normal incidence, each slit in the grating acts as a quasi point-source from which light propagates in all directions.
After light interacts with the grating, the diffracted light is composed of the sum of interfering wave components emanating from each slit in the grating. At any given point in space through which diffracted light may pass, the path length to each slit in the grating varies. Since path length varies so do the phases of the waves at that point from each of the slits. Thus, they add or subtract from each other to create peaks and valleys through additive and destructive interference; when the path difference between the light from adjacent slits is equal to half the wavelength, λ/2, the waves are out of phase, thus cancel each other to create points of minimum intensity. When the path difference is λ, the phases add together and maxima occur; the maxima occur at angles θm, which satisfy the relationship d sinθm/λ = | m |, where θm is the angle between the diffracted ray and the grating's normal vector, d is the distance from the center of one slit to the center of the adjacent slit, m is an integer representing the propagation-mode of interest.
Thus, when light is incident on the grating, the diffracted light has maxima at angles θm given by: d sin θ m = m λ. It is straightforward to show that if a plane wave is incident at any arbitrary angle θi, the grating equation becomes: d = m λ; when solved for the diffracted angle maxima, the equation is: θ m = arcsin. Please note that these equations assume that both sides of the grating are in contact with the same medium; the light that corresponds to direct transmission is called the zero order, is denoted m = 0. The other maxima occur at angles represented by non-zero integers m. Note that m can be positive or negative, resulting in diffracted orders on b
A chemical element is a species of atom having the same number of protons in their atomic nuclei. For example, the atomic number of oxygen is 8, so the element oxygen consists of all atoms which have 8 protons. 118 elements have been identified, of which the first 94 occur on Earth with the remaining 24 being synthetic elements. There are 80 elements that have at least one stable isotope and 38 that have radionuclides, which decay over time into other elements. Iron is the most abundant element making up Earth, while oxygen is the most common element in the Earth's crust. Chemical elements constitute all of the ordinary matter of the universe; however astronomical observations suggest that ordinary observable matter makes up only about 15% of the matter in the universe: the remainder is dark matter. The two lightest elements and helium, were formed in the Big Bang and are the most common elements in the universe; the next three elements were formed by cosmic ray spallation, are thus rarer than heavier elements.
Formation of elements with from 6 to 26 protons occurred and continues to occur in main sequence stars via stellar nucleosynthesis. The high abundance of oxygen and iron on Earth reflects their common production in such stars. Elements with greater than 26 protons are formed by supernova nucleosynthesis in supernovae, when they explode, blast these elements as supernova remnants far into space, where they may become incorporated into planets when they are formed; the term "element" is used for atoms with a given number of protons as well as for a pure chemical substance consisting of a single element. For the second meaning, the terms "elementary substance" and "simple substance" have been suggested, but they have not gained much acceptance in English chemical literature, whereas in some other languages their equivalent is used. A single element can form multiple substances differing in their structure; when different elements are chemically combined, with the atoms held together by chemical bonds, they form chemical compounds.
Only a minority of elements are found uncombined as pure minerals. Among the more common of such native elements are copper, gold and sulfur. All but a few of the most inert elements, such as noble gases and noble metals, are found on Earth in chemically combined form, as chemical compounds. While about 32 of the chemical elements occur on Earth in native uncombined forms, most of these occur as mixtures. For example, atmospheric air is a mixture of nitrogen and argon, native solid elements occur in alloys, such as that of iron and nickel; the history of the discovery and use of the elements began with primitive human societies that found native elements like carbon, sulfur and gold. Civilizations extracted elemental copper, tin and iron from their ores by smelting, using charcoal. Alchemists and chemists subsequently identified many more; the properties of the chemical elements are summarized in the periodic table, which organizes the elements by increasing atomic number into rows in which the columns share recurring physical and chemical properties.
Save for unstable radioactive elements with short half-lives, all of the elements are available industrially, most of them in low degrees of impurities. The lightest chemical elements are hydrogen and helium, both created by Big Bang nucleosynthesis during the first 20 minutes of the universe in a ratio of around 3:1 by mass, along with tiny traces of the next two elements and beryllium. All other elements found in nature were made by various natural methods of nucleosynthesis. On Earth, small amounts of new atoms are produced in nucleogenic reactions, or in cosmogenic processes, such as cosmic ray spallation. New atoms are naturally produced on Earth as radiogenic daughter isotopes of ongoing radioactive decay processes such as alpha decay, beta decay, spontaneous fission, cluster decay, other rarer modes of decay. Of the 94 occurring elements, those with atomic numbers 1 through 82 each have at least one stable isotope. Isotopes considered stable are those. Elements with atomic numbers 83 through 94 are unstable to the point that radioactive decay of all isotopes can be detected.
Some of these elements, notably bismuth and uranium, have one or more isotopes with half-lives long enough to survive as remnants of the explosive stellar nucleosynthesis that produced the heavy metals before the formation of our Solar System. At over 1.9×1019 years, over a billion times longer than the current estimated age of the universe, bismuth-209 has the longest known alpha decay half-life of any occurring element, is always considered on par with the 80 stable elements. The heaviest elements undergo radioactive decay with half-lives so short that they are not found in nature and must be synthesized; as of 2010, there are 118 known elements (in this context, "known" means observed well enough from just a few de
X-ray spectroscopy is a general term for several spectroscopic techniques for characterization of materials by using x-ray excitation. When an electron from the inner shell of an atom is excited by the energy of a photon, it moves to a higher energy level; when it returns to the low energy level, the energy which it gained by the excitation is emitted as a photon which has a wavelength, characteristic for the element. Analysis of the X-ray emission spectrum produces qualitative results about the elemental composition of the specimen. Comparison of the specimen's spectrum with the spectra of samples of known composition produces quantitative results. Atoms can be excited by a high-energy beam of charged particles such as electrons, protons or a beam of X-rays; these methods enable elements from the entire periodic table to be analysed, with the exception of H, He and Li. In electron microscopy an electron beam excites X-rays. In X-Ray Transmission, the equivalent atomic composition is captured based on photo electric and Compton effects.
In an energy-dispersive X-ray spectrometer, a semiconductor detector measures energy of incoming photons. To maintain detector integrity and resolution it should be cooled with liquid nitrogen or by Peltier cooling. EDS is employed in electron microscopes and in cheaper and/or portable XRF units. In a wavelength dispersive X-ray spectrometer the single crystal diffracts the photons which are collected by a detector. Without any motion there will be just one wavelength detected. By moving crystal and detector, a wide region of spectrum is observed. In contrast to EDS, WDS method is a method of sequential spectrum acquisition. While WDS is slower than EDS and more sensitive to positioning specimen in the spectrometer, it has superior spectral resolution and sensitivity. WDS is used in microprobes and in XRF; the father-and-son scientific team of William Lawrence Bragg and William Henry Bragg, who were 1915 Nobel Prize Winners, were the original pioneers in developing X-ray emission spectroscopy. Jointly they measured the X-ray wavelengths of many elements to high precision, using high-energy electrons as excitation source.
The cathode ray tube or an x-ray tube was the method used to pass electrons through a crystal of numerous elements. They painstakingly produced numerous diamond-ruled glass diffraction gratings for their spectrometers; the law of diffraction of a crystal is called Bragg's law in their honor. Intense and wavelength-tunable X-rays are now generated with synchrotrons. In a material, the X-rays may suffer an energy loss compared to the incoming beam; this energy loss of the re-emerging beam reflects an internal excitation of the atomic system, an X-ray analogue to the well-known Raman spectroscopy, used in the optical region. In the X-ray region there is sufficient energy to probe changes in the electronic state. For instance, in the ultra soft X-ray region, crystal field excitations give rise to the energy loss; the photon-in-photon-out process may be thought of as a scattering event. When the x-ray energy corresponds to the binding energy of a core-level electron, this scattering process is resonantly enhanced by many orders of magnitude.
This type of X-ray emission spectroscopy is referred to as resonant inelastic X-ray scattering. Due to the wide separation of orbital energies of the core levels, it is possible to select a certain atom of interest; the small spatial extent of core level orbitals forces the RIXS process to reflect the electronic structure in close vicinity of the chosen atom. Thus, RIXS experiments give valuable information about the local electronic structure of complex systems, theoretical calculations are simple to perform. There exist several efficient designs for analyzing an X-ray emission spectrum in the ultra soft X-ray region; the figure of merit for such instruments is the spectral throughput, i.e. the product of detected intensity and spectral resolving power. It is possible to change these parameters within a certain range while keeping their product constant... X-ray diffraction in spectrometers is achieved on crystals, but in Grating spectrometers, the X-rays emerging from a sample must pass a source-defining slit optical elements disperse them by diffraction according to their wavelength and a detector is placed at their focal points.
Henry Augustus Rowland devised an instrument that allowed the use of a single optical element that combines diffraction and focusing: a spherical grating. Reflectivity of X-rays is low, regardless of the used material and therefore, grazing incidence upon the grating is necessary. X-ray beams impinging on a smooth surface at a few degrees glancing angle of incidence undergo external total reflection w
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is thus the inverse of the spatial frequency. Wavelength is determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is designated by the Greek letter lambda; the term wavelength is sometimes applied to modulated waves, to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids. Assuming a sinusoidal wave moving at a fixed wave speed, wavelength is inversely proportional to frequency of the wave: waves with higher frequencies have shorter wavelengths, lower frequencies have longer wavelengths. Wavelength depends on the medium. Examples of wave-like phenomena are sound waves, water waves and periodic electrical signals in a conductor.
A sound wave is a variation in air pressure, while in light and other electromagnetic radiation the strength of the electric and the magnetic field vary. Water waves are variations in the height of a body of water. In a crystal lattice vibration, atomic positions vary. Wavelength is a measure of the distance between repetitions of a shape feature such as peaks, valleys, or zero-crossings, not a measure of how far any given particle moves. For example, in sinusoidal waves over deep water a particle near the water's surface moves in a circle of the same diameter as the wave height, unrelated to wavelength; the range of wavelengths or frequencies for wave phenomena is called a spectrum. The name originated with the visible light spectrum but now can be applied to the entire electromagnetic spectrum as well as to a sound spectrum or vibration spectrum. In linear media, any wave pattern can be described in terms of the independent propagation of sinusoidal components; the wavelength λ of a sinusoidal waveform traveling at constant speed v is given by λ = v f, where v is called the phase speed of the wave and f is the wave's frequency.
In a dispersive medium, the phase speed itself depends upon the frequency of the wave, making the relationship between wavelength and frequency nonlinear. In the case of electromagnetic radiation—such as light—in free space, the phase speed is the speed of light, about 3×108 m/s, thus the wavelength of a 100 MHz electromagnetic wave is about: 3×108 m/s divided by 108 Hz = 3 metres. The wavelength of visible light ranges from deep red 700 nm, to violet 400 nm. For sound waves in air, the speed of sound is 343 m/s; the wavelengths of sound frequencies audible to the human ear are thus between 17 m and 17 mm, respectively. Note that the wavelengths in audible sound are much longer than those in visible light. A standing wave is an undulatory motion. A sinusoidal standing wave includes stationary points of no motion, called nodes, the wavelength is twice the distance between nodes; the upper figure shows three standing waves in a box. The walls of the box are considered to require the wave to have nodes at the walls of the box determining which wavelengths are allowed.
For example, for an electromagnetic wave, if the box has ideal metal walls, the condition for nodes at the walls results because the metal walls cannot support a tangential electric field, forcing the wave to have zero amplitude at the wall. The stationary wave can be viewed as the sum of two traveling sinusoidal waves of oppositely directed velocities. Wavelength and wave velocity are related just as for a traveling wave. For example, the speed of light can be determined from observation of standing waves in a metal box containing an ideal vacuum. Traveling sinusoidal waves are represented mathematically in terms of their velocity v, frequency f and wavelength λ as: y = A cos = A cos where y is the value of the wave at any position x and time t, A is the amplitude of the wave, they are commonly expressed in terms of wavenumber k and angular frequency ω as: y = A cos = A cos in which wavelength and wavenumber are related to velocity and frequency as: k = 2 π λ = 2 π f v = ω