Secondary Bow Research Articles

Dynamical phase transitions, caustics, and quantum dark bands

Abstract We provide a new perspective on quantum dynamical phase transitions (DPTs) by explaining their origin in terms of caustics that form in the Fock space representation of the many-body state over time, using the fully connected transverse field Ising model as an example. In this way we establish a connection between DPTs in a quantum spin system and an everyday natural phenomenon: The dark band between the primary and secondary bows (caustics) in rainbows known as Alexander’s dark band. The DPT occurs when the Loschmidt echo crosses the switching line between the evanescent tails of two back-to-back Airy functions that dress neighbouring fold caustics in Fock space and is the time-dependent analogue of what is seen as a function of angle in the sky. The structural stability and universal properties of caustics, as described mathematically by catastrophe theory, explains the generic occurrence of DPTs in the model and suggests that our analysis has wide applicability. Based on our thorough analytical understanding we propose a protocol which can be used to verify the existence of a DPT in a finite system experiment.

Icebows.

Theoretical and experimental studies show that water ice spheres can produce a rainbow in which the primary and secondary bows overlap. To our knowledge, no such natural "icebow" has ever been reported.

Further evidence for a dynamically generated secondary bow inC13+C12rainbow scattering

The existence of a secondary bow is confirmed for 13C+12C nuclear rainbow scattering in addition to the 16O+12C system. This is found by studying the experimental angular distribution of 13C+12C scattering at the incident 13C energy $E_L$=250 MeV with an extended double folding (EDF) model that describes all the diagonal and off-diagonal coupling potentials derived from the microscopic wave functions for 12C using a density-dependent nucleon-nucleon force. The Airy minimum at \theta$ $\approx$70$^\circ$, which is not reproduced by a conventional folding potential, is revealed to be a secondary bow generated dynamically by a coupling to the excited state 2+ (4.44 MeV) of 12C. The essential importance of the quadruple {\it Y2} term (reorientation term) of potential of the excited state 2+ of 12C for the emergence of a secondary bow is found. The mechanism of the secondary bow is intuitively explained by showing how the trajectories are refracted dynamically into the classically forbidden angular region beyond the rainbow angle of the primary rainbow.

27aSP-4 原子核の副虹の存在と動的核間相互作用

27aSP-4 原子核の副虹の存在と動的核間相互作用

Three-dimensional graphic physically based simulator of rainbows together with the background scene.

This paper presents a single scattering 3D graphics simulator of rainbows that includes the thickness of the rain shaft and the background scenery. The simulator is devised so that we can find a good configuration of the sun, the viewers, and the volume of water drops in a complicated geometric setting. The background-scene geometry and light-reflecting properties are modelled using 3D graphics tools. The simulator allows both the light reflected from the background surface and the light scattered by water drops to contribute to the final image by taking the depth to the background surface into account. The simulator generates an image of the rainbow by using the radiative transfer equation (RTE). We use ray optics to compute the average scattering cross section and the average phase function of particles that are the main parameters of the RTE. Depending on the density distribution of the water drops, the rainbow is perceived to be translucent, and the background scene is visible through the rainbow. We simulate other effects of the variation of the water-dropdensity and the location of the viewer, e.g., the visibility of the secondary rainbow, the brightness of the sky around the rainbow, the close-up view of the rainbow, and the full-circle rainbow. We explain these effects partly by computing the luminance contrasts of the primary and secondary bows against their local backgrounds.

Evidence for a secondary bow in Newton's zero-order nuclear rainbow

Rainbows are generally considered to be caused by static refraction and reflection. A primary and a secondary rainbow appear due to refraction and internal reflection in a raindrop as explained by Newton. The quantum nuclear rainbow, which is generated by refraction in the nucleus droplet, only has a "primary" rainbow. Here we show for the first time evidence for the existence of a secondary nuclear rainbow generated dynamically by coupling to an excited state without internal reflection. This has been demonstrated for experimental $^{16}$O+$^{12}$C scattering using the coupled channel method with an extended double folding potential derived from microscopic realistic wave functions for $^{12}$C and $^{16}$O.

Fountain rainbows

We present the first measurements of radiance spectra of rainbows. The bows on two sunny days (3 and 6 June 2008) were produced by the fountain in the Parque de las Ciencias, Granada, Spain, that consists of a rectangular perimeter of 40 spray nozzles. Optical thickness of the spray from each nozzle was approximately 0.5. Spectral purity of the primary bow was highest for orange and blue, reaching values of 23% and 7%, respectively, while skylight 90 degrees from the Sun had a color purity of 34% (on 6 June). The secondary bow had much lower color purity with red absent because the regions around the bows and in Alexander's dark band were pale blue. The narrow sickle shape of the chromaticity curves for the primary bows and the absence of supernumerary bows indicated that the drop radius was between 0.2 and 0.4 mm.

Simulating rainbows in their atmospheric environment

Light and color of geometric optics rainbows are simulated in their atmospheric environment. Sunlight passes through a molecular atmosphere with ozone and an aerosol layer near the ground to strike a cuboidal rain shaft below an overhanging cuboidal cloud. The rainbows are treated as singly scattered sunbeams that are depleted as they pass through the atmosphere and rain shaft. They appear in a setting illuminated by scattered light from behind the observer, from the background beyond the rain shaft, and from the rain shaft. In dark backgrounds the primary and secondary bows first become visible when the optical thickness of rain shafts tau(R) congruent with 0.0003 and tau(R) congruent with 0.003, respectively. The bows are brightest and most colorful for 0.1<or= tau(R) <or= 3, a range that is typical for most showers. The peaks of the scattering phase function for raindrops that correspond to the geometric optics rainbow are so pronounced that rainbows remain bright and colorful for optically thick rain shafts seen against dark backgrounds, but the bows appear washed out or vanish as the background brightens or where the rain shaft is shaded by an overhanging cloud. Rainbows also redden as the Sun approaches the horizon.

A Triple Rainbow?

It's a good question — is there such a thing as a triple rainbow? No doubt we've all admired spectacular rainbows, including some double rainbows. I must also admit that most times when I'm out watering the yard on a sunny day, I cannot help myself from trying to trace out the primary and secondary circular bows in the spray. But a triple rainbow?

Supernumerary rainbows

Late in the afternoon of July 19, 1999, we were able to photograph some very unusual rainbows in the sky over Boulder, Colorado. The picture here shows a primary rainbow, a fainter secondary bow above it, and several pastel-shaded rainbows inside the primary one. While we think we counted as many as four concurrent rainbows in that afternoon, only three of them show up in developed prints.

From Alexander of Aphrodisias to Young and Airy

A didactic discussion of the physics of rainbows is presented, with some emphasis on the history, especially the contributions of Thomas Young nearly 200 years ago. We begin with the simple geometrical optics of Descartes and Newton, including the reasons for Alexander's dark band between the main and secondary bows. We then show how dispersion produces the familiar colorful spectacle. Interference between waves emerging at the same angle, but traveling different optical paths within the water drops, accounts for the existence of distinct supernumerary rainbows under the right conditions (small drops, uniform in size). Young's and Airy's contributions are given their due.

Rainbows

The formation of a rainbow is a subtle and often misunderstood process, and yet it can be explained mostly by simple optics and geometry. The theory of primary and secondary bow formation is presented along with details of a suitable classroom experiment.

Infrared Rainbow

Radition in the near-infrared spectral region should produce a rainbow that is not visible to the human eye. An infrared photograph is shown which displays the primary bow, the secondary bow, and two supernumerary bows in side the primary bow.

Robert Grosseteste on the Rainbow

Previous articleNext article No AccessRobert Grosseteste on the RainbowCarl B. BoyerCarl B. Boyer Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by Osiris Volume 111954 Published for the History of Science Society Article DOIhttps://doi.org/10.1086/368581 Views: 8Total views on this site Citations: 7Citations are reported from Crossref PDF download Crossref reports the following articles citing this article: Bibliographie, (May 2019): 377–402.https://doi.org/10.3917/puf.mehl.2019.01.0377Joshua S. Harvey, Hannah E. Smithson, Clive R. Siviour, Giles E. M. Gasper, Sigbj?rn O. S?nnesyn, Brian K. Tanner, Tom C. B. McLeish Bow-shaped caustics from conical prisms: a 13th-century account of rainbow formation from Robert Grosseteste?s De iride, Applied Optics 56, no.1919 (Jun 2017): G197.https://doi.org/10.1364/AO.56.00G197Massimo Corradi A short history of the rainbow, Lettera Matematica 4, no.11 (May 2016): 49–57.https://doi.org/10.1007/s40329-016-0127-3Nader El-Bizri Grosseteste’s Meteorological Optics: Explications of the Phenomenon of the Rainbow After Ibn al-Haytham, (Aug 2016): 21–39.https://doi.org/10.1007/978-3-319-33468-4_2Cecilia Panti The Theological Use of Science in Robert Grosseteste and Adam Marsh According to Roger Bacon: The Case Study of the Rainbow, (Aug 2016): 143–163.https://doi.org/10.1007/978-3-319-33468-4_8Hannah E. Smithson, Philip S. Anderson, Greti Dinkova-Bruun, Robert A. E. Fosbury, Giles E. M. Gasper, Philip Laven, Tom C. B. McLeish, Cecilia Panti, Brian K. Tanner Color-coordinate system from a 13th-century account of rainbows, Journal of the Optical Society of America A 31, no.44 (Mar 2014): A341.https://doi.org/10.1364/JOSAA.31.00A341Ivan R. Schwab, W. Barry Lee, David Bisno The Rainbow: From Ancient Greece to Modern Optics, Survey of Ophthalmology 54, no.66 (Nov 2009): 714–720.https://doi.org/10.1016/j.survophthal.2009.03.003

A Triple Rainbow

MR. R. C. T. EVANS, of 10 Eddington Lane, Herne Bay, has forwarded a description of an unusual rainbow seen by him at Herne Bay that differs from any of those described in recent numbers of NATURE, in that it made a considerable angle with the ordinary primary and secondary bows, implying that the sun was not at the centre of curvature of the unusual bow. The phenomenon was seen at 18h. 20m. G.M.T., presumably on a date in the third week of August it was visible for about five minutes, the secondary bow for only about three minutes, while the primary bow lasted much longer. The tide was low at the time and the sun was clearly reflected on a distant wet mud flat, no doubt forming an extension of the northward sloping beach, so as to give an image below the level of the observer's eye and to the left of the sun, that is to say, lying in a direction somewhat farther round towards the south. Mr. Evans attributes the third bow to this additional source of light. It appeared as a short arc with colours arranged in the same order as those in the primary bow, with an apparent centre of curvature above the horizon and north of the centre of curvature of the two ordinary bows. He estimated that about two-thirds of a complete circle would have been above the horizon had the arc continued far enough, while the two normal bows would have shown to only half that extent above the horizon. The explanation offered seems to accord well with the observed effect.

Unusual Rainbow Phenomena

A NUMBER of correspondents have added further descriptions of unusual rainbow phenomena to the account by Mr. J. L. Horton of the display of June 26 (NATURE, July 8, p. 57). The month of June was unprecedented during at least the past sixty years for the number of thunderstorms, and was at the same time a generally sunny month, so that opportunities for seeing rainbows were unusually frequent. An account from J. O. Ewing, of bows seen from Brandon, Suffolk, on the evening of June 17, described three closely adjacent bows with the red farthest from the sun, of which only the outermost showed the complete range from violet to red, this being the brightest, while another bow much nearer to the sun is said to have shown colours in the reverse order. It appears possible, seeing that the effect was described from memory, that errors have been made over the colour sequences, as the bows corresponding presumably with the ordinary primary and secondary bows have the usual order of colour reversed, but even so the phenomenon was evidently very different from that described by Mr. Horton.

Letters to Editor

THERE was observed over the Medway estuary on August 18 (7.15 p.m., B.S.T.) an extremely brilliant rainbow. In addition to secondary bows concentric with the primary (all less than a semicircle), there was seen a bow of considerable brightness having an arc greater than a semicircle. This “anomalous bow” appeared to be of the same radius as the primary bow, had its colours in the same order (i.e. red outermost), and cut the horizon at the same point. It in fact, the remainder of the circle of which the primary arc formed a part.

An Unusual Rainbow

AN unusual rainbow display was visible at sea between 6.30 and 7 p.m. on the evening of August 16. The primary and secondary bows were complete and of exceptional brilliancy. Between these two lay two arcs of a third bow, cutting the primary bow near the horizon and ending in the secondary bow about 20 above the horizon in the manner shown in the accompanying diagram.

Remarkable Rainbow Phenomena

ON Monday, September 3, a very heavy thunder-shower passed from west to east over the parish of Deerness, Orkney, from 5.30 p.m. to 6.25 p.m. When the dark nimbus cloud to the west had lifted its pall, the sun came out in great brilliancy. A rainbow now began to form to the north-east, but instead of the ordinary bow there was one of a bifurcated nature. Two stumps which coalesced on the horizon gradually developed into two magnificent bows, which met on both horizons, viz. north-east and south-west, but were about five or six degrees apart at the apex. All the colours of a radiant bow were present in both, and both had the colours arranged in the order of the primary bow. The secondary bow also appeared with the colours reversed and the same bifurcation, but in this case it extended only to about thirty or thirty-five degrees above the horizon, as secondaries generally do. As I had never seen or heard of anything like this, my first impulse was to find a cause. When the double rainbows were at their best, there was a bar of stratus cloud extending across the middle of the sun, and in breadth about one-sixth of its diameter. The two primary bows remained complete from 6.30 p.m. to 6.35 p.m., and without the arc of the apex for about another five minutes.

A Phenomenal Rainbow

A VERY beautiful rainbow was observed here on the evening of May 26 last, just before sunset. A light easterly air prevailed at the time; but the thin bank of stratus cloud upon which the bow was projected had drifted slowly across from the south-west, and now hung in the eastern sky. The sun was quite low at the time, and during the last two or three minutes before setting was shining through a thin layer of stratus which lay just above the horizon; but there was no apparent diminution in the startling vividness of colour exhibited in the arch. This extraordinary brightness, however, was not the only noticeable feature; immediately below the great arch, and contiguous to it and to each other, were four narrow arches, in which the vivid colours were repeated; these did not reach the horizon, but faded when about three parts of the way down. There was also, some distance above the main arch, a secondary bow, with the four narrow arches appearing again; but here, instead of being below, they were directly above the arch, and, of course, not so bright as the primary set. The whole appearance was curiously like some of the solar phenomena observed in the Arctic sky, and was so beautiful as to attract the attention of several working bushmen, who are not prone to fall into ecstacies over any natural wonders.