A diffraction pattern of a red laser beam projected onto a plate after passing through a small circular aperture in another plate. Credit: Wisky.
Interference fringes are produced in overlapping plane waves. Credit: Fffred~commonswiki.

Def. the "breaking up of an electromagnetic wave as it passes a geometric structure (e.g. a slit), followed by reconstruction of the wave by interference"[1] is called diffraction.

In classical physics, the diffraction phenomenon is described by the Huygens–Fresnel principle that treats each point in a propagating wavefront as a collection of individual spherical wavelets.[2]

From wave theory, specifically, the interaction of identical waves from at least two sources produces interference fringes, where maxima are separated by the fringe spacing

${\displaystyle d_{f}={\frac {\lambda }{\sin \theta }}}$

and df increases with increase in wavelength or decreasing angle θ.

Def. a "light or dark band formed by the diffraction of light",[3] is called a fringe.

## Quasicrystals

Ho-Mg-Zn dodecahedral quasicrystal, grown by using the self-flux method (excess Mg), and slowly cooling from 700 C to 480 C. Credit: Ames Laboratory, US Department of Energy.

A crystalline substance that falls into a periodic pattern, or space group, in three dimensions, can be a mineral, or crystalline solid.

A quasicrystal consists of an ordered array of atoms or molecules without periodicity. They display a discrete pattern in X-ray diffraction but do not fall into any space group.

## Continua

X-ray continuum emission "can arise both from a jet and from the hot corona of the accretion disc via a scattering process: in both cases it shows a power-law spectrum. In some radio-quiet active galactic nuclei (AGN) there is an excess of soft X-ray emission in addition to the power-law component.

X-ray line emission is a result of illumination of cold heavy elements by the X-ray continuum that causes fluorescence of X-ray emission lines.

Using X-rays to determine a crystal structure results in diffraction intensities that are represented in reciprocal space as peaks. These have a finite width due to a variety of defects away from a perfectly periodic lattice. There may be significant diffuse scattering, a continuum of scattered X-rays that fall between the Bragg peaks.

The X-ray continuum can arise from bremsstrahlung, black-body radiation, synchrotron radiation, or what is called inverse Compton scattering of lower-energy photons by relativistic electrons, knock-on collisions of fast protons with atomic electrons, and atomic recombination, with or without additional electron transitions.[4]

## A-fringes

Dark-field image shows two families of Brazil twin lamellae at reflection (1,-1,0,0). Credit: L. van Goethem, J. van Landuyt, and S. Amelinckx.{{fairuse}}

A-fringes are often called α-fringes.

"Since the crystal axes in both parts of a Brazil twin are parallel, it is expected that if the boundary produces diffraction contrast fringes, then these will be a-fringes."[5]

"The fringes from such a boundary have been termed a-fringes by Amelinckx".[5]

In the image on the right, the a-fringes occur at or immediately next to the twin boundaries. Some are dark fringes. Others are smooth and lighter due to the overlap of two boundaries.

## Neutrons

By bombarding a silicon single crystal wedge with thickness ranging from ~0.3 cm to 1 cm in a slow neutron beam, a series of Pendellösung interference fringes parallel to the thinner edge of the wedge appear within Bragg reflections formed by the diffraction of the neutrons.[6]

## Electrons

Map of Kikuchi line pairs is down to 1/1Å for 300 keV electrons in hexagonal sapphire (Al2O3), with some intersections labeled. Credit: P. Fraundorf.{{free media}}
HRTEM lattice images and electron diffraction patterns (top-right insets) are taken along the [0001] direction. Credit: Materialscientist.

Kikuchi lines pair up to form bands in electron diffraction from single crystal specimens. They serve as "roads in orientation-space" for microscopists not certain at what they are looking. In transmission electron microscopes, they are easily seen in diffraction from regions of the specimen thick enough for multiple scattering.[7] Unlike diffraction spots, which blink on and off as one tilts the crystal, Kikuchi bands mark orientation space with well-defined intersections (called zones or poles) as well as paths connecting one intersection to the next.

Experimental and theoretical maps of Kikuchi band geometry, as well as their direct-space analogs e.g. bend contours, electron channeling patterns, and fringe visibility maps are increasingly useful tools in electron microscopy of crystalline and nanocrystalline materials.[8] Because each Kikuchi line is associated with Bragg diffraction from one side of a single set of lattice planes, these lines can be labeled with the same Miller or reciprocal-lattice indices that are used to identify individual diffraction spots. Kikuchi band intersections, or zones, on the other hand are indexed with direct-lattice indices i.e. indices which represent integer multiples of the lattice basis vectors a, b and c.

Kikuchi lines are formed in diffraction patterns by diffusely scattered electrons, e.g. as a result of thermal atom vibrations.[9] The main features of their geometry can be deduced from a simple elastic mechanism proposed in 1928 by Seishi Kikuchi,[10] although the dynamical theory of diffuse inelastic scattering is needed to understand them quantitatively.[11]

In x-ray scattering these lines are referred to as Kossel lines[12] (named after Walther Kossel).

In the well-known experiment with diffraction of light at two close slits on the screen, we can see interference pattern appear on the observation plane of the screen. The similar pattern is observed for electrons. Thus, experiments were conducted to prove the existence of diffraction of electrons flying almost one by one. [13] If an electron is regarded as a point object, it is very difficult to understand its diffraction, because the theoretical analysis of the diffraction experiment predicts simultaneous passage of the electron through both slits. [14]

"From the point of view of the substantial model, electrons in atoms represent the objects similar to charged clouds. At the level of stars, the similar clouds consist of the matter, which contains negative charges − electrons. The external electromagnetic field interacts with these charges, creating electrical currents as well as charge-density fluctuations in the clouds’ matter. Therefore, a moving cloud in an excited state is accompanied by its intrinsic electromagnetic oscillations, which are converted for an external observer to the corresponding de Broglie wavelength. When the cloud is detached from star attracting it (as in case of electron’s detachment from an atom or a piece of matter), the cloud may break into pieces, which then will move in different directions. This is due to the fact that the gravitational forces of the cloud are unable to keep its matter from the electrical forces of repulsion. Therefore, it can be expected that some electrons during the diffraction experiment will fall into pieces, which in this case can be coherent with respect to de Broglie waves. If such pieces of electrons fall into different slits of the screen, then after diffraction, at an appropriate path difference, there should be places on the observation plane, where these pieces come to either in phase or in antiphase of oscillations. This leads to either addition of oscillation energies or to their subtraction. As a result, on the observation plane we periodically see the recurrent places, where the incident electrons are strongly excited and have increased energy, and the places, where the electrons have almost no excitation energy. This gives the possibility of emergence of the interference pattern."[15]

The second image down on the right is a high-resolution, transmission electron micrograph (HRTEM) lattice image with an electron diffraction pattern (top-right inset) taken along the [0001] direction; i.e., looking down the [0001] axis. An image simulation is added in the bottom-left inset, and a model fragment of the crystal structure is on the right of the bottom-left inset. On the lower right of the composite image is a 1 nanometre (nm) marker.

## Synchrotron x-ray diffraction

These black crystals of pyroxferroite are from the famous Eifel quarry field, Bellerberg, Ettringen, Eifel Mountains, Germany. Credit: Giuseppe Siccardi.{{fairuse|permission:Giuseppe Siccardi}}
Photograph shows a stilbite specimen from Kiui Island, Alaska, USA. Credit: Dlloyd.{{fairuse}}

Def. the "oxyanion of silicon SiO32- or any salt or mineral containing this ion"[16] is called a metasilicate.

"This hydrated sodium-manganese silicate [raite] extends the already wide range of manganese crystal chemistry (3), which includes various complex oxides in ore deposits and nodules from the sea floor and certain farming areas, the pyroxmangite analog of the lunar volcanic metasilicate pyroxferroite, the Mn analog yofortierite of the clay mineral palygorskite, and the unnamed Mn analog of sepiolite."[17]

The pyroxferroite crystals in the image on the right are 0.6 x 1.1 x 0.7 cm in dimensions.

Stilbite-Ca can have the chemical formula NaCa4Al8Si28O72•30(H2O).[18]

## X-rays

This Hubble Space Telescope image shows Sirius A, the brightest star in our nighttime sky, along with its faint, tiny stellar companion, Sirius B. Credit: H. Bond (STScI) and M. Barstow (University of Leicester), NASA, ESA.
An X-ray image of the Sirius star system located 8.6 light years from Earth is shown. Credit: NASA/SAO/CXC.

An example of the differences between visual stellar classification and a possible X-ray classification is the disparity between the image of Sirius A [at above centre in the overexposed Hubble image] with the dim Sirius B [tiny dot at lower left]. [The cross-shaped diffraction spikes and concentric rings around Sirius A, and the small ring around Sirius B, are artifacts produced within the telescope's imaging system.] And, the lower image of the same two stars in X-rays.

This image shows two sources and a spike-like pattern due to the support structure for the transmission grating. The bright source is Sirius B, a white dwarf star that has a surface temperature of about 25,000 degrees Celsius which produces very low energy X-rays. The dim source at the position of Sirius A – a normal star more than twice as massive as the Sun – may be due to ultraviolet radiation from Sirius A leaking through the filter on the detector. In contrast, Sirius A is the brightest star in the northern sky when viewed with an optical telescope, while Sirius B is 10,000 times dimmer.

In the bottom image, Sirius B clearly outshines Sirius A. However, in the visual range the reverse is the case as shown in the top image. The surface effective temperature of Sirius A (spectral type A1V) is only 9,940±210 K,[19] while that of Sirius B (a white dwarf, DA2) is 25,200 K.[20] On the surface temperature of the photosphere alone, Sirius B would be a Class B star.

## Airy diffraction

An Airy diffraction pattern as shown is generated by a plane wave falling on a circular aperture, such as the pupil of the eye. Credit: Inductiveload.

A telescope's imaging system's resolution can be limited either by aberration or by diffraction causing blurring of the image. These two phenomena have different origins and are unrelated. Aberrations can be explained by geometrical optics and can in principle be solved by increasing the optical quality — and cost — of the system. On the other hand, diffraction comes from the wave nature of light and is determined by the finite aperture of the optical elements. The lens' circular aperture is analogous to a two-dimensional version of the single-slit experiment. Light passing through the lens interferes with itself creating a ring-shape diffraction pattern, known as the Airy pattern, if the wavefront of the transmitted light is taken to be spherical or plane over the exit aperture.

The interplay between diffraction and aberration can be characterised by the point spread function (PSF). The narrower the aperture of a lens the more likely the PSF is dominated by diffraction.

Two point sources are regarded as just resolved when the principal diffraction maximum of one image coincides with the first minimum of the other.[21]

## Diffraction limits

This is a log-log plot of aperture diameter vs angular resolution at the diffraction limit for various light wavelengths compared with various astronomical instruments. Credit: Cmglee.

The log-log plot at right is of aperture diameter vs angular resolution at the diffraction limit for various light wavelengths compared with various astronomical instruments. For example, the blue star shows that the Hubble Space Telescope is almost diffraction-limited in the visible spectrum at 0.1 arcsecs, whereas the red circle shows that the human eye should have a resolving power of 20 arcsecs in theory, though normally only 60 arcsecs.

## Reds

The Hubble Space Telescope Advanced Camera for Surveys (ACS) image has H-alpha emission of the Red Rectangle shown in blue. Credit: .
The Red Rectangle is a proto-planetary nebula. Here is the Hubble Space Telescope Advanced Camera for Surveys (ACS) image. Broadband red light is shown in red. Credit: .

"The ERE was first recognized clearly in the peculiar reflection nebula called the Red Rectangle by Schmidt, Cohen, & Margon (1980)."[22]

The Red Rectangle Nebula, so called because of its red color and unique rectangular shape, is a protoplanetary nebula in the Monoceros constellation. Also known as HD 44179, the nebula was discovered in 1973 during a rocket flight associated with the AFCRL Infrared Sky Survey called Hi Star.

In the Red Rectangle Nebula, diffraction-limited speckle images of it in visible and near infrared light reveal a highly symmetric, compact bipolar nebula with X-shaped spikes which imply toroidal dispersion of the circumstellar material. The central binary system is completely obscured, providing no direct light.[23]

## Infrareds

An Atlas Image Mosaic is of the carbon star IRAS 06088+1909. Credit: S. Van Dyk (IPAC).
ESO PR Photo 34g/04 shows the stellar cluster NGC 2093 Credit: ESO.
Note also the distinctly red star to the East (left) of the center of this cluster. Credit: ESO.

"Carbon stars are evolved stars, similar to the Sun, which are nearing the ends of their lives, in the so-called asymptotic giant branch phase. Many form dusty, carbon-rich envelopes, due to mass loss, which makes the very red in color, especially in the near-infrared. IRAS 06088+1909 is a very dusty carbon star toward the Galactic anticenter (Jura & Kleinmann 1990, ApJ 364, 663). 2MASS is particularly sensitive to carbon stars. Liebert et al. (2000, PASP, 112, 1315) report on several very cool carbon stars in or beyond the Galactic halo, some of which are heavily dust enshrouded. They conclude that 2MASS can be used to define a useful sample of carbon stars at high Galactic latitude as tracers of the halo out to distances comparable to the Magellanic Clouds. In the 2MASS image, the fainter reddish "stars" immediately east of due north and west of due south of both IRAS 06088+1909 and the bluer bright star to its northeast are known 2MASS "filter glint" artifacts; known diffraction spike and persistence artifacts are also seen associated with these two bright stars."[24]

"ESO PR Photo 34g/04 [second down on the right] shows the stellar cluster NGC 2093, a comparatively rich aggregate of young stars, a few tens of millions of years old. The hot temperature of the most massive of such young stars is responsible for its predominantly blue colour. The sky field measures 5.6 x 5.1 arcmin. North is up and East is left. ESO PR Photo 34h/04 shows the stellar cluster NGC 2108, a rich "mid-aged" cluster, about 600 million years old. A careful comparison with its neighbour NGC 2093 (PR Photo 34g/04) shows that the brightest stars of NGC 2108 are fainter and whiter than the brightest members of NGC 2093; this indicates that NGC 2108 is older. Note also the distinctly red star to the East (left) of the center of this cluster. This is a member of a stellar class referred to as "Carbon stars", cool giant stars that are characterized by the presence of carbon molecules (C2) in their atmospheres and having extremely red colours. This sky field measures 2.8 x 2.6 arcsec; the orientation is the same."[25]

"NGC 2108 is a rich "mid-aged" cluster, about 600 million years old. A careful comparison with its neighbour NGC 2093 shows that the brightest stars of NGC 2108 are fainter and whiter than the brightest members of NGC 2093; this indicates that NGC 2108 is older. Note also the distinctly red star to the East (left) of the center of this cluster. This is a member of a stellar class referred to as "Carbon stars", cool giant stars that are characterized by the presence of carbon molecules (C2) in their atmospheres and having extremely red colours. This sky field measures 2.8 x 2.6 arcsec. North is up and East is left."[26]

## Submillimeters

Stars "believed to have circumstellar disks similar to the primitive solar nebula [are] based on the criteria [...]:

1. high far-infrared optical depths around visible stars,
2. shallow spectral energy densities longward of 5 µm, and
3. large millimeter-wave flux densities indicative of ≳ 0.01 M of H2."[27]

"Evidence for changes in particle composition, size, or shape, reflected in the emissivity index, could therefore be relevant to theories of cosmogony."[27]

"The observations were carried out at the Caltech Submillimeter Observatory (CSO) in Hawaii during 1989 November through December 4, and 1990 December 4 through 9. The detector was a silicon composite bolometer fed by a Winston cone and cooled to a few tenths of a degree with a 3He refrigerator. The filtering employed standard techniques: a scattering filter of black polyethylene fused to fluorogold at 77 K blocked wavelengths in the far-infrared; a crystal quartz filter coated with black polyethylene at 4 K eliminated all near-infrared radiation; and bandpass filters made of metal mesh on nylon or polyethylene, defined the actual wavebands (e.g., Whitcomb & Keene 1980; Cunningham 1982). Different Winston cones were used with each filter to match the diffraction limit of the 10 m telescope, giving different beam sizes on the sky."[27]

## Sun

The image shows solar coronal loops observed by the Transition Region And Coronal Explorer (TRACE), 171 Å filter. Credit: TRACE/NASA.
The image shows the cooling post-flare arcade (rotated by -90 degrees so that north is to the right) 6h after the flare (at 00:11 UT on September 8. Credit: TRACE/NASA.
Evolution of the solar radius since 1567, before the Maunder Minimum. Credit: A. Kilcik, C. Sigismondi, J.P. Rozelot, K. Guhl.{{fairuse}}

"Almost as soon as Active Region 10808 rotated onto the solar disk, it spawned a major X17 flare. TRACE was pointed at the other edge of the Sun at the time, but was repointed 6 hours after the flare started. The image on the left shows the cooling post-flare arcade (rotated by -90 degrees so that north is to the right) 6h after the flare (at 00:11 UT on September 8); the loop tops still glow so brightly that the diffraction pattern repeats them on diagonals away from the brightest spots. Some 18h after the flare, the arcade is still glowing, as seen in the image on the right (at 11:42 UT on September 8). In such big flares, magnetic loops generally light up successively higher in the corona, as can be seen here too: the second image shows loops that are significantly higher than those seen in the first. Note also that the image on the right also contains a much smaller version of the cooling arcade in a small, very bright loop low over the polarity inversion line of the region."[28]

Nearly all of the TRACE images of coronal loops and the transition region indicate that material in these loops and loop-like structures returns to the chromosphere.

"Normally, solar energetic particle (SEP) events associated with disturbances in the eastern hemisphere are characterized by slow onset and lack of high-energy particles. The SEP event associated with the first major flare (X17) [...] is among very few such events over several decades in that although the source region was on the east limb, the particle flux started to rise only a few hours from the flare onset, while the flux of protons with energies in excess of 100 MeV went up by more than a factor of one hundred. We do not understand how these energetic particles can reach the Earth from that side of the Sun, because there should be no magnetic connectivity."[28]

The image on the second right graphs the evolution "of the solar radius since 1567, before the Maunder Minimum. The T+WD data are after Toulmonde (1997) and Wittmann and Débarbat (1990), whose error bars were computed by one of us from the records kept at the Paris Observatory. The triangles are the data of Toulmonde (T corr series) with his correction for diffraction and refraction. Historical eclipses represented by circles are after Fiala (1994) and Sigismondi (2008). Their error bars are smaller than the dimension of the circle. Toulmonde’s data are averaged over many observations made by respective authors, each lasting for several years; therefore they seemingly distribute around an average value of 960 arc sec."[29]

## Earth

Circular waves are generated by diffraction from the narrow entrance of a flooded coastal quarry. Credit: Verbcatcher.
A solar glory is shown on steam from hot springs. Credit: Brocken Inaglory.
Diffraction of sea waves is at breakwater, Ashkelon, Israel. Credit: Dmitris1.
Water wave is diffracted around the rocky outcrop of Point Reyes, forming circular ripples. Reflection of the Sun is seen on the left of the photo. The photo was taken along the staircase leading down to the Point Reyes lighthouse, Point Reyes National Seashore, California, USA. Credit: Wing-Chi Poon.

At right is the Blue Lagoon, Abereiddy, Pembrokeshire, Wales, a disused coastal slate quarry with a narrow entrance to the sea. Single-slit wave diffraction and reflection patterns can be seen on the water in the lagoon.

At left is a solar glory through the steam from hot springs at Yellowstone National Park. A glory is an optical phenomenon produced by light backscattered (a combination of diffraction, reflection and refraction) towards its source by a cloud of uniformly sized water droplets.

"Radioactive potassium [...] appears also to be a substantial source of heat in the Earth's core"[30]

"Radioactive potassium, uranium and thorium are thought to be the three main sources of heat in the Earth's interior, aside from that generated by the formation of the planet. Together, the heat keeps the mantle actively churning and the core generating a protective magnetic field."[30]

Much "less potassium [occurs] in the Earth's crust and mantle than [is] expected based on the composition of rocky meteors that supposedly formed the Earth. If, as some have proposed, the missing potassium resides in the Earth's iron core, how did an element as light as potassium get there, especially since iron and potassium don't mix?"[30]

At "the high pressures and temperatures in the Earth's interior, potassium can form an alloy with iron never before observed. During the planet's formation, this potassium-iron alloy could have sunk to the core, depleting potassium in the overlying mantle and crust and providing a radioactive potassium heat source in addition to that supplied by uranium and thorium in the core."[30]

The "new alloy [is created] by squeezing iron and potassium between the tips of two diamonds [a diamond anvil] to temperatures and pressures characteristic of 600-700 kilometers below the surface - 2,500 degrees Celsius and nearly 4 million pounds per square inch, or a quarter of a million times atmospheric pressure."[30]

"Our new findings indicate that the core may contain as much as 1,200 parts per million potassium -just over one tenth of one percent."[31]

"This amount may seem small, and is comparable to the concentration of radioactive potassium naturally present in bananas. Combined over the entire mass of the Earth's core, however, it can be enough to provide one-fifth of the heat given off by the Earth."[31]

"With one experiment, Lee and Jeanloz demonstrated that potassium may be an important heat source for the geodynamo, provided a way out of some troublesome aspects of the core's thermal evolution, and further demonstrated that modern computational mineral physics not only complements experimental work, but that it can provide guidance to fruitful experimental explorations,"[32]

"More experiments need to be done to show that iron can actually pull potassium away from the silicate rocks that dominate in the Earth's mantle."[33]

"They proved it would be possible to dissolve potassium into liquid iron."[33]

"Modelers need heat, so this is one source, because the radiogenic isotope of potassium can produce heat and that can help power convection in the core and drive the magnetic field. They proved it could go in. What's important is how much is pulled out of the silicate. There's still work to be done."[33]

"If a significant amount of potassium does reside in the Earth's core, this would clear up a lingering question - why the ratio of potassium to uranium in stony meteorites (chondrites), which presumably coalesced to form the Earth, is eight times greater than the observed ratio in the Earth's crust. Though some geologists have asserted that the missing potassium resides in the core, there was no mechanism by which it could have reached the core. Other elements like oxygen and carbon form compounds or alloys with iron and presumably were dragged down by iron as it sank to the core. But at normal temperature and pressure, potassium does not associate with iron."[30]

"Early in Earth's history, the interior temperature and pressure would not have been high enough to make this alloy."[31]

"But as more and more meteorites piled on, the pressure and temperature would have increased to the point where this alloy could form."[31]

"The Earth is thought to have formed from the collision of many rocky asteroids, perhaps hundreds of kilometers in diameter, in the early solar system. As the proto-Earth gradually bulked up, continuing asteroid collisions and gravitational collapse kept the planet molten. Heavier elements - in particular iron - would have sunk to the core in 10 to 100 million years' time, carrying with it other elements that bind to iron."[30]

"Gradually, however, the Earth would have cooled off and become a dead rocky globe with a cold iron ball at the core if not for the continued release of heat by the decay of radioactive elements like potassium-40, uranium-238 and thorium-232, which have half-lives of 1.25 billion, 4 billion and 14 billion years, respectively. About one in every thousand potassium atoms is radioactive."[30]

"The heat generated in the core turns the iron into a convecting dynamo that maintains a magnetic field strong enough to shield the planet from the solar wind. This heat leaks out into the mantle, causing convection in the rock that moves crustal plates and fuels volcanoes."[30]

Pure "iron and pure potassium [combined] in a diamond anvil cell [that] squeezed the small sample to 26 gigapascals of pressure while heating the sample with a laser above 2,500 Kelvin (4,000 degrees Fahrenheit), which is above the melting points of both potassium and iron. [Repeat] six times in the high-intensity X-ray beams of two different accelerators - Lawrence Berkeley National Laboratory's Advanced Light Source and the Stanford Synchrotron Radiation Laboratory - to obtain X-ray diffraction images of the samples' internal structure. The images confirmed that potassium and iron had mixed evenly to form an alloy, much as iron and carbon mix to form steel alloy."[30]

"In the theoretical magma ocean of a proto-Earth, the pressure at a depth of 400-1,000 kilometers (270-670 miles) would be between 15 and 35 gigapascals and the temperature would be 2,200-3,000 Kelvin."[34]

"At these temperatures and pressures, the underlying physics changes and the electron density shifts, making potassium look more like iron."[34]

"At high pressure, the periodic table looks totally different."[34]

"The work by Lee and Jeanloz provides the first proof that potassium is indeed miscible in iron at high pressures and, perhaps as significantly, it further vindicates the computational physics that underlies the original prediction."[32]

"If it can be further demonstrated that potassium would enter iron in significant amounts in the presence of silicate minerals, conditions representative of likely core formation processes, then potassium could provide the extra heat needed to explain why the Earth's inner core hasn't frozen to as large a size as the thermal history of the core suggests it should."[32]

## Mars

This is the first ever view of Martian soil in X-rays. Credit: NASA/JPL-Caltech/Ames.
This self-portrait of NASA's Curiosity Mars rover shows the vehicle at the "Big Sky" site, where its drill collected the mission's fifth taste of Mount Sharp. Credit: NASA.

The image at right is an X-ray diffraction pattern from Martian soil. The image is from "the Chemistry and Mineralogy (CheMin) experiment on NASA's Curiosity rover. The image reveals the presence of crystalline feldspar, pyroxenes and olivine mixed with some amorphous (non-crystalline) material. The soil sample, taken from a wind-blown deposit within Gale Crater, where the rover landed, is similar to volcanic soils in Hawaii."[35]

"Curiosity scooped the soil on Oct. 15, 2012, the 69th sol, or Martian day, of operations. It was delivered to CheMin for X-ray diffraction analysis on October 17, 2012, the 71st sol. By directing an X-ray beam at a sample and recording how X-rays are scattered by the sample at an atomic level, the instrument can definitively identify and quantify minerals on Mars for the first time. Each mineral has a unique pattern of rings, or "fingerprint," revealing its presence. The colors in the graphic represent the intensity of the X-rays, with red being the most intense."[35]

## Fizeau fringes

A Fizeau scheme spatially filters light from an exoplanet relative to its star. Credit: P. Baudoz, A. Boccaletti, J. Baudrand, & D. Rouan.{{fairuse}}
The Fizeau image in the focal plane shows fringes pinning the intensity distribution of the stellar flux. Credit: P. Baudoz, A. Boccaletti, J. Baudrand, & D. Rouan.{{fairuse}}

The technique [relies] on a simple Fizeau recombination of the science beam [from a beam splitter in the diagram on the right] and the reference beam. It enables to discriminate speckles from a planet without using fast [optical path difference] OPD modulation or deformable mirrors."[36]

The science beam "is spatially filtered using a pinhole or optical fiber. The two beams are recombined in the focal plane in a Fizeau scheme. [...] The Fizeau image in the focal plane will show fringes [see the images on the left, third image] pinning the intensity distribution of the stellar flux. Since the flux of the companion is fully removed from the reference beam by spatial filtering, the intensity of the companion will be unaltered by the reference beam. This is what can be seen in [the images on the left] where the point spread function of the stellar flux interfere with the reference beam creating a fringe pattern over all the diffraction rings and speckles. A companion located at a distance of 2 λ/D and in the upper right quadrant does not show fringes because it is not coherent with the reference beam."[36]

## Pendellösung fringes

Def. interference fringes produced by radiation passing through thick wedge plates at near-normal incidence are called Pendellösung fringes.

Pendellösung fringes, the name, as coined by the German physicist Paul Peter Ewald, means “spherical solution”.[37]

The "dark and light fringes (Pendellösung fringes) will have a narrower spacing at the edges of the diffraction region and a wider spacing in the centre[.] [The] central part of the fringe turns out to be bent towards the thick part of the wedge. The degree of bending of the fringes depends on L [the source-film distance as in a camera or length to the plane of focus]. In the limit L ⟶ ∞ the fringes are straight, which corresponds to the plane-wave incidence at the Bragg angle [...]. In practice, the fringes are always bent due to L finiteness, and the first fringe broadens out in the centre."[38]

## Seyfert galaxy

These images both depict the same area - the region of ionized gas around the nucleus of the bright Seyfert galaxy NGC 4151. Credit: Hutchings et al. in Astrophysical Journal Letters 492, L115 (1998).

"These images [at right] both depict the same area - the region of ionized gas around the nucleus of the bright Seyfert galaxy NGC 4151. In one case, the observations used a narrow-band filter to isolate the bright gaseous emission of doubly ionized oxygen - [O III] - at 5007 Angstroms, while the other used a diffraction grating to spread the light at each point into a spectrum. The color coding was used to indicate this and to show the direction of spectral dispersion, though the wavelength range spanned by the observation would not produce a vivid color range visually. The brilliant nucleus is spread into a horizontal line, producing radiation at all wavelengths in the form of a continuum. In the dispersed image, a single cloud of gas with small internal motions will have a single Doppler shift, and will have the same appearance as in the filtered image, perhaps with a position shift due to its overall Doppler shift with respect to the average. However, a parcel of gas with a large velocity dispersion, such as one might see in a turbulent situation or near a shock front, will be preferentially smeared along the wavelength direction (left-right in this depiction). Careful comparison of these two images shows that there are many such regions, mostly located in the locations close to where the galaxy's small radio jets emerge from the nucleus. These radio jets lie approximately along the axes of the twin emission-line cones. This connection between rapid local gas motions and the emerging jets has been interpreted as evidence that much of the "turbulent" motion in the outer regions of Seyfert nuclei is powered by the radio jets, as they transfer energy to their surroundings."[39]

"The emission-line image shows clearly the biconical region where most of the ionized gas appears, in support of a beaming picture" for the various kinds of Seyfert nuclei. It poses an interesting puzzle, however, because to get this plan view we must be outside the cones, but we see the broad-line region and thus classify NGC 4151 as a type 1 Seyfert - in fact, it was the prototype of the class. Thus, there are directions outside these cones where we can have a fairly clear view of the core, so the simple mental picture of a solid torus and clear views along its axis cannot be taken quite literally."[39]

## Talbot fringes

Schematic of the Talbot fringes experimental setup, where the pulsed metastable helium beam is produced in a switched dc gas discharge, and time-of-flight detection is done by the time-resolving detection scheme. Credit: S. Nowak, Ch. Kurtsiefer, and T. Pfau and C. David.{{fairuse}}
Experimental results are for two different settings of L1: (a) L1 = 0.24 m, (b) L1 = 0.79 m. Credit: S. Nowak, Ch. Kurtsiefer, and T. Pfau and C. David.{{fairuse}}

"The Talbot effect is the self-imaging of a grating il-luminated with plane coherent waves, showing up at a characteristic distance LT = 2d2/λ from the diffraction grating. LT is called the Talbot length. More-general self-images with subharmonic periods also occur at distances LT/(2m) (integer m) but with reduced periodicity d/m. In the paraxial approximation, when an ideal line grating with an infinite slit number is used, a small open fraction and perfect coherent illumination lead to patterns with 100% contrast; self-images then occur at all integer multiples of the Talbot length, nLT, and the subharmonics at all distances nLT/2m, where m can be infinitely large. In practice the achievable contrast is limited by the finite size and slit width of the grating and by the finite transverse coherence length and monochromaticity of the incident beam. In the case of finite geometry (distance L0 from source to grating and L1 from grating to detection plane; [...]), one has to deal with curved wave fronts instead of plane waves. This results in a scaling of the Talbot length15 to an effective Talbot length LTE = 2(1 + L1/L0)d2/λ. In this case the respective diffraction patterns are geometrically magnified, and the periodicity of the fringes takes values deff/m = (1 + L1/L0)d/m."[40]

Plotted are "the spatial atom distribution versus the de Broglie wavelength λdB. The intensity is encoded in the gray scale. The arrows indicate the locations at which Talbot fringes of the mth order are observed. [...] Atomic distributions are shown for λdB = 1–3 pm and λdB = 30–60 pm, where the fast and slow atoms are produced by the source. [Also plotted are] the spatial atom distribution versus the de Broglie wavelength.﻿﻿﻿﻿"[40]

## Young's fringes

Measured vertical intensity distribution is taken through the cross section of a set of fringes. Credit: Graham H. Cross and Yitao Ren, Neville J. Freeman.{{fairuse}}

"On the conventional Huygens model each waveguide will act as a source of cylindrical waves diffracting into the far-field where they will overlap to give Young’s interference fringes."[41]

"The full spatial intensity pattern is described by a set of fringes modulated in the x direction and contained within a Gaussian envelope which results from diffraction. According to the simple theory of interference fringes maxima and minima of intensity should be spaced with spacing, a, according to

${\displaystyle a={\frac {\lambda s}{d}},}$

where λ is the wavelength (632.8 nm in this case) and d, the separation of the sources."[41]

"To confirm the Young’s fringe condition [for the above equation] an intensity scan was taken through the cross section of a set of fringes. At a distance to the screen, s, of 9.6 cm we observed a separation between adjacent fringe maxima, a, of 1.76 cm. At a wavelength of 632.8 nm this gives a waveguide center separation for this sample, d, of 3.45 µm. This compares extremely favorably with the expected separation of between 3.4 and 3.8 µm and is thus within experimental error."[41]

## Laser diode arrays (LDAs)

"Using diffraction gratings as beam splitters in LDA systems the resulting fringe spacing becomes achromatic. Thus employing unstabilized laser diodes of different wavelengths, distinguishable fringe systems with identical fringe spacings in the measuring volume can be generated. The evaluation of at least two LDA burst signals and their signal phase corresponding to the different fringe systems locally parallel shifted and/or tilted against each other allows the directional measurement of lateral and axial velocity components without using frequency shift techniques."[42]

## Modulation collimators

The diagram shows the principles of operation of the four-grid modulation collimator. Credit: H. Bradt, G. Garmire, M. Oda, G. Spada, and B.V. Sreekantan, P. Gorenstein and H. Gursky.

A modulation collimator consists of “two or more wire grids [diffraction gratings] placed in front of an X-ray sensitive Geiger tube or proportional counter.”[43] Relative to the path of incident X-rays (incoming X-rays) the wire grids are placed one beneath the other with a slight offset that produces a shadow of the upper grid over part of the lower grid.[44]

### Use of wire grids

Each grid consists of only parallel wires (like a diffraction grating, not a network of crossing wires) of diameter d and a center-to-center spacing of 2d.[44] Let D be the distance between the grids for a bigrid, or the distance between the uppermost grid and the lower most grid (the grid immediately in front of the detector) in a multigrid system.

Incident parallel radiation from a distant point source "falls upon the first grid" so that "depending upon the angle of incidence, the portions of the beam ... transmitted by the first grid fall

1. solely on the wires of the second grid,
2. "solely [through] the open spaces, or
3. upon both wires and spaces of the second grid."[43]

The transmission of the grids in the first two cases is 0% and 50%, respectively.[44] In the third case, it varies linearly with incident angle.[44]

The planes of 50% transmittance, planes of maximum transmittance (PMT), through the bigrid or multiple grid system, intersect "the celestial sphere [to] form [two or] multiple great circles ('lines-of-position') upon one of which the [astronomical] X-ray source must lie."[43]

### Net angular responses

"[T]he net angular response of [a] two-grid or bigrid modulation collimator to a parallel X-ray beam is cyclic and trangular in shape with a peak transmission of 50%".[43]

Def. the full width at half maximum (FWHM):

θr = d/D.[44]

is called "[t]he response angle" (θr).[43]

"The two-grid system unambiguously determines the angular size of an X-ray source with size between about θr/4 and 2θr, and clearly distinguishes sizes above and below this range."[43]

### Grid enclosure

The collimating effects of the grid enclosure or external metal slats determine the envelope for the triangular transmission peaks.[43] The enclosure or slats, in general, slowly modulate the peak heights.[43]

### Multigrid collimators

The multigrid collimator has the additional grid (third grid or more) inserted

1. at a specified intermediate position between the two grids,
2. aligned approximately parallel to them, and
3. "positioned and rotated so that each [third] wire lies in a plane defined by a wire in [the] outer grid and a wire in the [inner] grid."[43]

This positioning is such that every other triangular peak of the bigrid system is removed.[43] An additional grid would be placed midway between one of the initial grids "and the adjacent intermediate grid."[43]

### Modulation

In electronics and telecommunications, modulation is the process of varying one or more properties of a high frequency periodic waveform, called the carrier signal, with a modulating signal. This is done in a similar fashion as a musician modulating a tone (a periodic waveform) from a musical instrument by varying its volume, timing and pitch. The three key parameters of a periodic waveform are its amplitude ("volume"), its phase ("timing") and its frequency ("pitch"). Any of these properties can be modified in accordance with a low frequency signal to obtain the modulated signal. Typically a high-frequency sinusoid waveform is used as carrier signal, but a square wave pulse train may also be used.

Here with the 'modulation collimator' the amplitude (intensity) of the incoming X-rays is reduced by the presence of two or more 'diffraction gratings' of parallel wires that block or greatly reduce that portion of the signal incident upon the wires.

### Collimators

This diagram shows how a lead (Söller) collimator filters a stream of rays. Credit: Pete Verdon.

A collimator is a device that narrows a beam of particles or waves. To "narrow" can mean either to cause the directions of motion to become more aligned in a specific direction (i.e. collimated or parallel) or to cause the spatial cross section of the beam to become smaller.

The figure to the right illustrates how a Söller collimator is used in neutron and X-ray machines. The upper panel shows a situation where a collimator is not used, while the lower panel introduces a collimator. In both panels the source of radiation is to the right, and the image is recorded on the gray plate at the left of the panels.

For the modulation collimator, the collimating slats as represented in the diagram are replaced by wires (end on, ⊗←D→⊗, rather than a slat ▬).

## Scanning Modulation Collimator, HEAO A-3

The Scanning Modulation Collimator is onboard HEAO as experiment A-3. Credit: Wallace H. Tucker, Scientific and Technical Information Branch, NASA, Washington, D.C.
This is a schematic of the scanning modulation collimator in experiment A-3. Credit: Wallace H. Tucker.

The A-3 experiment consisted of two scanning modulation collimators with proportional counters, aspect sensors, and electronics. The major purpose of this experiment was to determine accurately the position of selected cosmic X-ray sources and to investigate their size and structure. Principal Investigators were Herbert Gursky and Daniel Schwartz of the Smithsonian Astrophysical Observatory and Hale Brad t of MI T. Hard ware was provided by American Science and Engineering, Inc.

## Small Astronomy Satellite 3

This is a diagram of NASA's SAS 3 Small Astronomy Satellite. Credit: NASA.

The SAS 3 diagram shows an overall view of the spacecraft as it appeared deployed, with major features labeled. The S/C z-axis points to the upper right. The scientific instrument package is above, and the spacecraft section (power conditioning and batteries, command and data electronics, communications equipment, momentum wheel for attitude control) are in the lower section. One star sensor views along the +z axis, and one looks out along +y, which is pointed out of the picture. The Rotating Modulation Collimator system looks up along the +z axis, while the slat and tube collimators, and the Soft X-ray experiment (LEDs), look out along the +y axis. The four solar power panels, which look like the sails of a windmill, could be rotated by command to catch the maximum solar flux.

## Satelite de Aplicaciones Cientificas - 1 (SAC - 1)

Argentinean Satellite SAC-A is deployed from Space Shuttle STS-88 on 14 December 1998. Credit: NASA.

The Fourier Imaging X-ray spectrometer (FIXS) is one of four instruments on SAC-1. The FIXS instrument, consisting of six modulation collimators mounted on a rotating spacecraft, is designed to provide images of solar flares at X-ray energies between 5 keV and 35 keV.

## Rossi X-ray Timing Explorer

Artist impression shows the Rossi X-ray Timing Explorer (RXTE). Credit: GDK.{{free media}}

The Rossi X-ray Timing Explorer uses a 'rotation modulation collimator'.

## Far Ultraviolet Spectroscopic Explorer

The Far Ultraviolet Spectroscopic Explorer is shown in a pre-launch clean room. Credit: NASA.

The Far Ultraviolet Spectroscopic Explorer (FUSE) detected light in the far ultraviolet portion of the electromagnetic spectrum, between 90.5-119.5 nanometres, which is mostly unobservable by other telescopes. Its primary mission was to characterize universal deuterium in an effort to learn about the stellar processing times of deuterium left over from the Big Bang. The telescope comprises four individual mirrors. Each of the four mirrors is a 39-by-35 cm (15.4-by-13.8 in) off-axis parabola. Two mirror segments are coated with silicon carbide for reflectivity at the shortest ultraviolet wavelengths, and two mirror segments are coated with lithium fluoride over aluminum that reflects better at longer wavelengths. Each mirror has a corresponding astigmatism-corrected, holographically-ruled diffraction grating, each one on a curved substrate so as to produce four 1.65 m (5.4 ft) Rowland circle spectrographs. The dispersed ultraviolet light is detected by two microchannel plate intensified double delay-line detectors, whose surfaces are curved to match the curvature of the focal plane.[45]

## Gemini South observatory

Gemini South is on Cerro Pachón in Chile. Credit: Denys.{{free media}}`
Gemini Planet Imager (GPI) image is of a planet orbiting a distant star known as 51 Eridani. Credits: J. Rameau (Univ. of Montreal) and C. Marois (NRC Herzberg, Canada).

In the image on the right, the bright central star 51 Eridani has been mostly removed by a hardware and software mask to enable the detection of the exoplanet (labelled "b") that is one millionth as bright.

At Gemini-S the Gemini Multi-Conjugate Adaptive Optics System (GeMS) may be used with the FLAMINGOS-2 near-infrared imager and spectrometry, or the Gemini South Adaptive Optics Imager (GSAOI), which provides uniform, diffraction-limited image quality to arcminute-scale fields of view, GeMS achieved first light on December 16, 2011.[46] Using a constellation of five laser guide stars, it achieved Full width at half maximum (FWHM) of 0.08 arc-seconds in the H band over a field of 87 arc-seconds square.

The detectors in each instrument have recently been upgraded with Hamamatsu Photonics devices, which significantly improve performance in the far red part of the optical spectrum (700–1,000 nm).[47]

Near-infrared imaging and spectroscopy are provided by the NIRI, NIFS, GNIRS, FLAMINGOS-2, and GSAOI instruments.[48]

The Gemini Planet Imager (GPI) is a high contrast imaging instrument that was built for the Gemini South Telescope that achieves high contrast at small angular separations, allowing for the direct imaging and integral field spectroscopy of extrasolar planets around nearby stars.[49][50]

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