{"title":"Forbidden Transitions","description":"Slow \"forbidden\" spectral lines from long-lived states, seen best where collisions are rare.","products":[{"product_id":"nebulia-pixel-hoodie","title":"Nebulia – The Forbidden [O III] 500.7 nm Line Pixel Hoodie","description":"\u003cp\u003eNebulia, a veiled oracle in deep-navy robes, holds a ring of doubly ionised oxygen whose green light fills a planetary-nebula halo.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eThe bright green nebular lines at 5006.84 Å and 4958.91 Å (air wavelengths) were long blamed on a hypothetical element, “nebulium”. In 1927 I. S. Bowen identified them as forbidden transitions of doubly ionised oxygen, O²⁺ (spectroscopic notation O III), from the metastable ¹D₂ level of the 2p² ground configuration down to the ³P₂ and ³P₁ levels. Because both levels share the same configuration and parity, an electric-dipole (E1) transition is forbidden; the lines go mainly by the much slower magnetic-dipole (M1) channel. In a lab gas, collisions empty ¹D₂ long before it can radiate. In a nebula they do not, because the density sits far below the level’s critical density.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003en\u003csub\u003ecrit\u003c\/sub\u003e = A\u003csub\u003eul\u003c\/sub\u003e \/ q\u003csub\u003eul\u003c\/sub\u003e\u003c\/strong\u003e\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eλ = hc \/ ΔE\u003csub\u003eul\u003c\/sub\u003e\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003en\u003csub\u003ecrit\u003c\/sub\u003e: critical electron density, where collisional de-excitation equals radiative decay (m⁻³)\u003c\/li\u003e\n\u003cli\u003eA\u003csub\u003eul\u003c\/sub\u003e: Einstein spontaneous-emission coefficient from upper level u to lower level l (s⁻¹)\u003c\/li\u003e\n\u003cli\u003eq\u003csub\u003eul\u003c\/sub\u003e: collisional de-excitation rate coefficient (m³ s⁻¹)\u003c\/li\u003e\n\u003cli\u003eλ: emitted wavelength (m)\u003c\/li\u003e\n\u003cli\u003eh: Planck constant, 6.626 070 15 × 10⁻³⁴ J s (exact)\u003c\/li\u003e\n\u003cli\u003ec: speed of light in vacuum, 299 792 458 m s⁻¹ (exact)\u003c\/li\u003e\n\u003cli\u003eΔE\u003csub\u003eul\u003c\/sub\u003e: energy gap between the two levels (J)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eBelow n\u003csub\u003ecrit\u003c\/sub\u003e nearly every excitation ends in a photon, so a weak transition can still be one of the brightest lines in the spectrum. William Huggins first saw these lines in 1864 in NGC 6543. Sources: I. S. Bowen, \u003cem\u003eNature\u003c\/em\u003e 120, 473 (1927), doi:10.1038\/120473a0; A. Fowler, \u003cem\u003eNature\u003c\/em\u003e 120, 582 (1927); NIST Atomic Spectra Database (O III).\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e “Forbidden” means strongly suppressed, not impossible. Quoted wavelengths are in air (500.68 nm); the vacuum value is slightly longer.\u003c\/p\u003e\u003cp\u003eThe glow is the approximate line colour: a pure spectral colour lies outside the sRGB range a print can show.\u003c\/p\u003e\u003ch3\u003eIn character\u003c\/h3\u003e\u003cp\u003e\u003cem\u003eI was called a new element for sixty years. I was oxygen all along, speaking only where the dark is quiet enough to hear me.\u003c\/em\u003e\u003c\/p\u003e\u003ch3\u003eOn the hoodie\u003c\/h3\u003e\u003cp\u003eFront: a three-level ladder emblem with a dashed green ¹D→³P arrow. Back: Nebulia in full pixel art with “FORBIDDEN. OBSERVED ANYWAY.” and “NEBULIUM WAS OXYGEN.” Gildan 18500 Heavy Blend hoodie, Black, S to 5XL.\u003c\/p\u003e\u003cp\u003ePart of the \u003cstrong\u003eForbidden Transitions\u003c\/strong\u003e set with \u003ca href=\"\/pl\/products\/coronix-pixel-hoodie\"\u003eCoronix\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/flipwarden-pixel-hoodie\"\u003eFlipwarden\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/viridael-pixel-hoodie\"\u003eViridael\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/sanguora-pixel-hoodie\"\u003eSanguora\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/sulphra-pixel-hoodie\"\u003eSulphra\u003c\/a\u003e. Browse all \u003ca href=\"\/pl\/collections\/hoodies\"\u003ephysics hoodies\u003c\/a\u003e, more \u003ca href=\"\/pl\/collections\/astrophysics\"\u003eastrophysics \u0026amp; cosmology designs\u003c\/a\u003e, or the \u003ca href=\"\/pl\/collections\/deep-cuts\"\u003eDeep Cuts\u003c\/a\u003e.\u003c\/p\u003e","brand":"Pixelated Physics","offers":[{"title":"Black \/ S","offer_id":50964464762983,"sku":"23364393242333113942","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ M","offer_id":50964464795751,"sku":"86037443440874778334","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ L","offer_id":50964464828519,"sku":"18292876638205299572","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ XL","offer_id":50964464861287,"sku":"24641802712931612922","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 2XL","offer_id":50964464894055,"sku":"48639394253285931488","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 3XL","offer_id":50964464926823,"sku":"67718269315531573106","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 4XL","offer_id":50964464959591,"sku":"12022848020915244183","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 5XL","offer_id":50964464992359,"sku":"24807988142212132362","price":64.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/03_H02.png?v=1793678357"},{"product_id":"coronix-pixel-hoodie","title":"Coronix – The Forbidden Fe XIV Coronal Line Pixel Hoodie","description":"\u003cp\u003eCoronix, a knight in blackened iron, wears a crown of 13 broken spikes against an eclipsed Sun, each tip burning coronal green.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eThe green coronal line at 530.3 nm (5302.86 Å in air), seen at total eclipses, was blamed on an unknown element, “coronium”. Walter Grotrian (1939) matched the red 637.4 nm line to Fe X, and Bengt Edlén (announced 1941, full paper in \u003cem\u003eZ. Astrophys.\u003c\/em\u003e 22, 30, 1942) identified the green line as Fe XIV: iron with 13 of its 26 electrons stripped away, Fe¹³⁺. The transition is the magnetic-dipole decay ²P°\u003csub\u003e3\/2\u003c\/sub\u003e → ²P°\u003csub\u003e1\/2\u003c\/sub\u003e inside the 3s²3p ground term, a fine-structure flip that an electric dipole cannot drive.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eFe + 13 ionisations → Fe¹³⁺ (Fe XIV)\u003c\/strong\u003e\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eχ(Fe¹²⁺ → Fe¹³⁺) ≈ 361 eV ≫ k\u003csub\u003eB\u003c\/sub\u003eT\u003csub\u003ephotosphere\u003c\/sub\u003e ≈ 0.5 eV\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eFe XIV: spectroscopic notation for the ion with 13 electrons removed (Roman numeral = charge + 1)\u003c\/li\u003e\n\u003cli\u003eχ: ionisation energy needed to remove the 13th electron (eV)\u003c\/li\u003e\n\u003cli\u003ek\u003csub\u003eB\u003c\/sub\u003e: Boltzmann constant, 1.380 649 × 10⁻²³ J K⁻¹ (exact)\u003c\/li\u003e\n\u003cli\u003eT\u003csub\u003ephotosphere\u003c\/sub\u003e: temperature of the visible solar surface, about 5800 K\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eMaking Fe¹³⁺ in collisional equilibrium needs electron temperatures of about a million kelvin or more, so the identification proved the corona is far hotter than the surface below it. Sources: B. Edlén, \u003cem\u003eMNRAS\u003c\/em\u003e 105, 323 (1945), doi:10.1093\/mnras\/105.6.323; W. Grotrian, \u003cem\u003eNaturwiss.\u003c\/em\u003e 27, 214 (1939); P. Swings, \u003cem\u003eApJ\u003c\/em\u003e 98, 116 (1943); NIST Atomic Spectra Database (Fe XIV).\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e Why the corona is so hot (the coronal heating problem) is still unsolved; wave heating and many small reconnection events (“nanoflares”) are both studied. The ionisation-energy figure is from the NIST database and rounded.\u003c\/p\u003e\u003cp\u003eThe glow is the approximate line colour: a pure spectral colour lies outside the sRGB range a print can show.\u003c\/p\u003e\u003ch3\u003eIn character\u003c\/h3\u003e\u003cp\u003e\u003cem\u003eThirteen spikes, thirteen electrons surrendered to a million-kelvin crown. They named me coronium. I am iron that refused to stay cool.\u003c\/em\u003e\u003c\/p\u003e\u003ch3\u003eOn the hoodie\u003c\/h3\u003e\u003cp\u003eFront: an eclipsed disc emblem with a green ring and pixel “XIV”. Back: Coronix in full pixel art with “FORBIDDEN. OBSERVED ANYWAY.” and “CORONIUM WAS IRON.” Gildan 18500 Heavy Blend hoodie, Black, S to 5XL.\u003c\/p\u003e\u003cp\u003ePart of the \u003cstrong\u003eForbidden Transitions\u003c\/strong\u003e set with \u003ca href=\"\/pl\/products\/nebulia-pixel-hoodie\"\u003eNebulia\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/flipwarden-pixel-hoodie\"\u003eFlipwarden\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/viridael-pixel-hoodie\"\u003eViridael\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/sanguora-pixel-hoodie\"\u003eSanguora\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/sulphra-pixel-hoodie\"\u003eSulphra\u003c\/a\u003e. Browse all \u003ca href=\"\/pl\/collections\/hoodies\"\u003ephysics hoodies\u003c\/a\u003e, more \u003ca href=\"\/pl\/collections\/astrophysics\"\u003eastrophysics \u0026amp; cosmology designs\u003c\/a\u003e, or the \u003ca href=\"\/pl\/collections\/deep-cuts\"\u003eDeep Cuts\u003c\/a\u003e.\u003c\/p\u003e","brand":"Pixelated Physics","offers":[{"title":"Black \/ S","offer_id":50964465156199,"sku":"11321626474428457629","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ M","offer_id":50964465188967,"sku":"16887944514829888669","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ L","offer_id":50964465221735,"sku":"30752224223844355610","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ XL","offer_id":50964465254503,"sku":"13968668967395919465","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 2XL","offer_id":50964465287271,"sku":"27594334369793468256","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 3XL","offer_id":50964465320039,"sku":"28271541142016493123","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 4XL","offer_id":50964465352807,"sku":"15837063366898941815","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 5XL","offer_id":50964465385575,"sku":"28835126505246358574","price":64.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/03_H13.png?v=1793678366"},{"product_id":"flipwarden-pixel-hoodie","title":"Flipwarden – The 21 cm Hydrogen Spin-Flip Line Pixel Hoodie","description":"\u003cp\u003eFlipwarden, a cowled monk-warden, waits beside an almost-full hourglass while one hydrogen atom above his palm flips its spin.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eIn ground-state hydrogen the electron and proton spins couple through their magnetic moments. The level with total spin F = 1 (spins parallel) lies just above F = 0 (antiparallel), and the magnetic-dipole spin flip between them emits a photon at 1420.405751768 MHz, a wavelength of 21.106 cm. The spontaneous rate is tiny, so each atom waits about 11 million years on average.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΔE = hν = 5.874 µeV,   λ = c\/ν = 21.106 cm\u003c\/strong\u003e\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eA\u003csub\u003e10\u003c\/sub\u003e ≈ 2.9 × 10⁻¹⁵ s⁻¹  ⇒  τ = 1\/A\u003csub\u003e10\u003c\/sub\u003e ≈ 3.5 × 10¹⁴ s ≈ 11 Myr\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eΔE: hyperfine splitting of the hydrogen 1s level (eV)\u003c\/li\u003e\n\u003cli\u003eh: Planck constant, 6.626 070 15 × 10⁻³⁴ J s\u003c\/li\u003e\n\u003cli\u003eν: transition frequency, 1420.405751768 MHz\u003c\/li\u003e\n\u003cli\u003eλ: wavelength (m)\u003c\/li\u003e\n\u003cli\u003ec: speed of light, 299 792 458 m s⁻¹\u003c\/li\u003e\n\u003cli\u003eA\u003csub\u003e10\u003c\/sub\u003e: Einstein coefficient for spontaneous F = 1 → F = 0 decay (s⁻¹)\u003c\/li\u003e\n\u003cli\u003eτ: mean radiative lifetime of the upper level, per atom (s)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eRare per atom, the line is still radio astronomy’s principal tracer of neutral hydrogen, because the Galaxy holds so much of it. The frequency is known to 13 significant figures thanks to hydrogen masers, which run on this very transition by stimulated emission. Sources: Kulkarni, ASTR501 lecture notes (A = 2.8843 × 10⁻¹⁵ s⁻¹); Zaroubi, \u003cem\u003eThe Epoch of Reionization\u003c\/em\u003e (A₁₀ = 2.85 × 10⁻¹⁵ s⁻¹).\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e The 11 Myr is a per-atom average lifetime, not a waiting time anyone has watched. The violet glow is symbolic: 21 cm is a radio wave and has no visible colour.\u003c\/p\u003e\u003ch3\u003eIn character\u003c\/h3\u003e\u003cp\u003e\u003cem\u003eOne grain falls. One spin turns. I have kept this watch eleven million years, and the sky still hears every flip I make.\u003c\/em\u003e\u003c\/p\u003e\u003ch3\u003eOn the hoodie\u003c\/h3\u003e\u003cp\u003eFront: two violet spin-arrow pairs, ↑↑ to ↑↓. Back: Flipwarden in full pixel art with “FORBIDDEN. OBSERVED ANYWAY.” and “ONE FLIP PER 11 MILLION YEARS.” Gildan 18500 Heavy Blend hoodie, Black, S to 5XL.\u003c\/p\u003e\u003cp\u003ePart of the \u003cstrong\u003eForbidden Transitions\u003c\/strong\u003e set with \u003ca href=\"\/pl\/products\/nebulia-pixel-hoodie\"\u003eNebulia\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/coronix-pixel-hoodie\"\u003eCoronix\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/viridael-pixel-hoodie\"\u003eViridael\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/sanguora-pixel-hoodie\"\u003eSanguora\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/sulphra-pixel-hoodie\"\u003eSulphra\u003c\/a\u003e. Browse all \u003ca href=\"\/pl\/collections\/hoodies\"\u003ephysics hoodies\u003c\/a\u003e, more \u003ca href=\"\/pl\/collections\/astrophysics\"\u003eastrophysics \u0026amp; cosmology designs\u003c\/a\u003e, or the \u003ca href=\"\/pl\/collections\/deep-cuts\"\u003eDeep Cuts\u003c\/a\u003e.\u003c\/p\u003e","brand":"Pixelated Physics","offers":[{"title":"Black \/ S","offer_id":50964465713255,"sku":"13078080911991187289","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ M","offer_id":50964465746023,"sku":"15526425071000076403","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ L","offer_id":50964465778791,"sku":"52243355899520614626","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ XL","offer_id":50964465811559,"sku":"10625248146536432879","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 2XL","offer_id":50964465844327,"sku":"14351088639385296846","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 3XL","offer_id":50964465877095,"sku":"15495369749119146940","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 4XL","offer_id":50964465909863,"sku":"21869795945365652919","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 5XL","offer_id":50964465942631,"sku":"18837958601923582490","price":64.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/03_H05.png?v=1793678376"},{"product_id":"viridael-pixel-hoodie","title":"Viridael – The Forbidden [O I] 557.7 nm Green Aurora Line Pixel Hoodie","description":"\u003cp\u003eViridael, a hooded keeper with closed eyes, holds a single droplet of green light beneath auroral curtains that stop at a haze line.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eThe aurora’s dominant green line at 557.7 nm, first examined spectroscopically by Anders Ångström in 1867, was identified in 1925 by J. C. McLennan and G. M. Shrum as the forbidden ¹S₀ → ¹D₂ transition of atomic oxygen, both levels inside the 2p⁴ ground configuration (an electric-quadrupole transition). The ¹S₀ level is metastable, lifetime about 0.74 s. Below roughly 100 to 120 km, collisions de-excite it before it can radiate, so the green curtain has a lower edge.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eτ(¹S₀) ≈ 0.74 s\u003c\/strong\u003e\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eLine emitted when  A ≳ k\u003csub\u003eq\u003c\/sub\u003e[M]   (quenching rate per atom below radiative rate)\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eτ: radiative lifetime of the upper level (s)\u003c\/li\u003e\n\u003cli\u003eA: Einstein spontaneous-emission coefficient, = 1\/τ for a single decay channel (s⁻¹)\u003c\/li\u003e\n\u003cli\u003ek\u003csub\u003eq\u003c\/sub\u003e: collisional quenching rate coefficient (m³ s⁻¹)\u003c\/li\u003e\n\u003cli\u003e[M]: number density of colliding air molecules (m⁻³), which rises steeply with falling altitude\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eStatistically the green emission peaks near 114 km. Sources: McLennan \u0026amp; Shrum, \u003cem\u003eProc. R. Soc. A\u003c\/em\u003e 108, 501 (1925), doi:10.1098\/rspa.1925.0088; \u003cem\u003eAstron. \u0026amp; Geophys.\u003c\/em\u003e 47, 1.11 (2006); Whiter et al., \u003cem\u003eAnn. Geophys.\u003c\/em\u003e 41, 1 (2023).\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e Published lifetimes for ¹S₀ range from about 0.7 to 0.8 s; we use 0.74 s. The quench altitude is a range, not a sharp line.\u003c\/p\u003e\u003cp\u003eThe glow is the approximate line colour: a pure spectral colour lies outside the sRGB range a print can show.\u003c\/p\u003e\u003ch3\u003eIn character\u003c\/h3\u003e\u003cp\u003e\u003cem\u003eHold still. I can keep this light for less than a second, and only above the noise of the crowded air. That is enough to paint the sky.\u003c\/em\u003e\u003c\/p\u003e\u003ch3\u003eOn the hoodie\u003c\/h3\u003e\u003cp\u003eFront: a green droplet in a ¹S→¹D ladder glyph. Back: Viridael in full pixel art with “FORBIDDEN. OBSERVED ANYWAY.” and “HOLD IT 0.74 SECONDS.” Gildan 18500 Heavy Blend hoodie, Black, S to 5XL.\u003c\/p\u003e\u003cp\u003ePart of the \u003cstrong\u003eForbidden Transitions\u003c\/strong\u003e set with \u003ca href=\"\/pl\/products\/nebulia-pixel-hoodie\"\u003eNebulia\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/coronix-pixel-hoodie\"\u003eCoronix\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/flipwarden-pixel-hoodie\"\u003eFlipwarden\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/sanguora-pixel-hoodie\"\u003eSanguora\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/sulphra-pixel-hoodie\"\u003eSulphra\u003c\/a\u003e. Browse all \u003ca href=\"\/pl\/collections\/hoodies\"\u003ephysics hoodies\u003c\/a\u003e, more \u003ca href=\"\/pl\/collections\/astrophysics\"\u003eastrophysics \u0026amp; cosmology designs\u003c\/a\u003e, or the \u003ca href=\"\/pl\/collections\/deep-cuts\"\u003eDeep Cuts\u003c\/a\u003e.\u003c\/p\u003e","brand":"Pixelated Physics","offers":[{"title":"Black \/ S","offer_id":50964466106471,"sku":"19414848035613386043","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ M","offer_id":50964466139239,"sku":"17016648611276429798","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ L","offer_id":50964466172007,"sku":"87744045058325625784","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ XL","offer_id":50964466204775,"sku":"11275585377535559326","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 2XL","offer_id":50964466237543,"sku":"21643665374794089763","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 3XL","offer_id":50964466270311,"sku":"29664513403508579823","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 4XL","offer_id":50964466303079,"sku":"20794520443654310472","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 5XL","offer_id":50964466335847,"sku":"30785834400081188521","price":64.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/03_H14.png?v=1793678386"},{"product_id":"sanguora-pixel-hoodie","title":"Sanguora – The Forbidden [O I] 630.0 nm Red Aurora Line Pixel Hoodie","description":"\u003cp\u003eSanguora, a tall oracle on a high spire, trails a slow red veil of light far above a thin band of green aurora.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eAtomic oxygen’s red auroral doublet at 630.0 nm and 636.4 nm is the forbidden ¹D₂ → ³P transition (to ³P₂ for 630.0 nm and ³P₁ for 636.4 nm), mainly magnetic dipole. The ¹D₂ level lives about 110 s, around 150 times longer than the green line’s upper level. Collisions therefore quench it everywhere below roughly 200 km, and the red aurora shines only from the thinnest high air. ¹D₂ is also the lower level of the 557.7 nm green line, so each green photon leaves an atom ready to emit red.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eτ(¹D₂) ≈ 110 s\u003c\/strong\u003e\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eI\u003csub\u003e630\u003c\/sub\u003e ∝ P(¹D₂) · A\u003csub\u003e630\u003c\/sub\u003e \/ (A\u003csub\u003etot\u003c\/sub\u003e + k\u003csub\u003eq\u003c\/sub\u003e[M])\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eτ: radiative lifetime of ¹D₂ (s)\u003c\/li\u003e\n\u003cli\u003eI\u003csub\u003e630\u003c\/sub\u003e: 630.0 nm emission rate per unit volume (photons m⁻³ s⁻¹)\u003c\/li\u003e\n\u003cli\u003eP(¹D₂): production rate of ¹D₂ atoms (m⁻³ s⁻¹)\u003c\/li\u003e\n\u003cli\u003eA\u003csub\u003e630\u003c\/sub\u003e: Einstein coefficient of the 630.0 nm branch (s⁻¹)\u003c\/li\u003e\n\u003cli\u003eA\u003csub\u003etot\u003c\/sub\u003e: total radiative decay rate of ¹D₂, = 1\/τ (s⁻¹)\u003c\/li\u003e\n\u003cli\u003ek\u003csub\u003eq\u003c\/sub\u003e[M]: collisional quenching rate per atom (s⁻¹)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eSources: \u003cem\u003eAstron. \u0026amp; Geophys.\u003c\/em\u003e 47, 1.11 (2006), doi:10.1111\/j.1468-4004.2006.47111.x; Stoffregen \u0026amp; Derblom, “Life-time of the atomic oxygen 6300 Å line”, \u003cem\u003eNature\u003c\/em\u003e 185, 28 (1960), doi:10.1038\/185028a0.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e The green cascade is not the only source of ¹D₂: slow electrons and chemical reactions populate it directly. “Above about 200 km” is approximate and varies with solar activity.\u003c\/p\u003e\u003cp\u003eThe glow is the approximate line colour: a pure spectral colour lies outside the sRGB range a print can show.\u003c\/p\u003e\u003ch3\u003eIn character\u003c\/h3\u003e\u003cp\u003e\u003cem\u003eDown there the air is loud. Up here I have a hundred and ten seconds of silence, and I spend every one of them on red.\u003c\/em\u003e\u003c\/p\u003e\u003ch3\u003eOn the hoodie\u003c\/h3\u003e\u003cp\u003eFront: a red veil curl over a ¹D→³P ladder. Back: Sanguora in full pixel art with “FORBIDDEN. OBSERVED ANYWAY.” and “110 SECONDS, OR NOTHING.” Gildan 18500 Heavy Blend hoodie, Black, S to 5XL.\u003c\/p\u003e\u003cp\u003ePart of the \u003cstrong\u003eForbidden Transitions\u003c\/strong\u003e set with \u003ca href=\"\/pl\/products\/nebulia-pixel-hoodie\"\u003eNebulia\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/coronix-pixel-hoodie\"\u003eCoronix\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/flipwarden-pixel-hoodie\"\u003eFlipwarden\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/viridael-pixel-hoodie\"\u003eViridael\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/sulphra-pixel-hoodie\"\u003eSulphra\u003c\/a\u003e. Browse all \u003ca href=\"\/pl\/collections\/hoodies\"\u003ephysics hoodies\u003c\/a\u003e, more \u003ca href=\"\/pl\/collections\/astrophysics\"\u003eastrophysics \u0026amp; cosmology designs\u003c\/a\u003e, or the \u003ca href=\"\/pl\/collections\/deep-cuts\"\u003eDeep Cuts\u003c\/a\u003e.\u003c\/p\u003e","brand":"Pixelated Physics","offers":[{"title":"Black \/ S","offer_id":50964466434151,"sku":"31589461466487302190","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ M","offer_id":50964466466919,"sku":"13976056328637914017","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ L","offer_id":50964466499687,"sku":"26675432088594547446","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ XL","offer_id":50964466532455,"sku":"11314529587195495811","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 2XL","offer_id":50964466565223,"sku":"29213941198584587947","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 3XL","offer_id":50964466597991,"sku":"33598749605627957709","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 4XL","offer_id":50964466630759,"sku":"13981346293377287018","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 5XL","offer_id":50964466663527,"sku":"94709181752167870614","price":64.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/03_H15.png?v=1793678396"},{"product_id":"sulphra-pixel-hoodie","title":"Sulphra – The [S II] 671.6\/673.1 nm Density Doublet Pixel Hoodie","description":"\u003cp\u003eSulphra, a two-faced porcelain oracle, breathes two unequal ribbons of red light into a balance that tips toward the brighter one.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eSingly ionised sulphur, S⁺ (S II), emits a forbidden red doublet at 671.6 nm and 673.1 nm (air 6716.44 \/ 6730.82 Å) from the ²D°\u003csub\u003e5\/2\u003c\/sub\u003e and ²D°\u003csub\u003e3\/2\u003c\/sub\u003e levels down to the ⁴S°\u003csub\u003e3\/2\u003c\/sub\u003e ground level. The two upper levels lie at almost the same energy, so temperature hardly matters, but they have different radiative rates. That makes the line ratio a gauge of electron density.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eR = I(671.6) \/ I(673.1)\u003c\/strong\u003e\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003en\u003csub\u003ee\u003c\/sub\u003e → 0:  R → g(²D\u003csub\u003e5\/2\u003c\/sub\u003e)\/g(²D\u003csub\u003e3\/2\u003c\/sub\u003e) = 6\/4 ≈ 1.5      n\u003csub\u003ee\u003c\/sub\u003e → ∞:  R → (g A)\u003csub\u003e5\/2\u003c\/sub\u003e\/(g A)\u003csub\u003e3\/2\u003c\/sub\u003e ≈ 0.44\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eR: intensity ratio of the doublet (dimensionless)\u003c\/li\u003e\n\u003cli\u003eI: line intensity (W m⁻² sr⁻¹)\u003c\/li\u003e\n\u003cli\u003en\u003csub\u003ee\u003c\/sub\u003e: electron number density (cm⁻³)\u003c\/li\u003e\n\u003cli\u003eg: statistical weight 2J + 1 of the upper level\u003c\/li\u003e\n\u003cli\u003eA: Einstein spontaneous-emission coefficient of each line (s⁻¹)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eAt low density every collisional excitation radiates, so R follows the collision strengths (proportional to g). At high density the levels reach Boltzmann populations and R follows gA. In between, roughly 50 to 2000 cm⁻³, R measures n\u003csub\u003ee\u003c\/sub\u003e: gas more than ten million billion (10¹⁶) times thinner than air. Sources: Osterbrock \u0026amp; Ferland, \u003cem\u003eAstrophysics of Gaseous Nebulae and Active Galactic Nuclei\u003c\/em\u003e (2006); Sanders et al., \u003cem\u003eApJ\u003c\/em\u003e 816, 23 (2016); Kakkad et al., \u003cem\u003eA\u0026amp;A\u003c\/em\u003e 618, A6 (2018).\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e The limiting ratios (about 1.45 to 1.5 and 0.44) depend on the atomic data adopted and weakly on temperature, so density values carry real systematic uncertainty.\u003c\/p\u003e\u003cp\u003eThe glow is the approximate line colour: a pure spectral colour lies outside the sRGB range a print can show.\u003c\/p\u003e\u003ch3\u003eIn character\u003c\/h3\u003e\u003cp\u003e\u003cem\u003eAsk one tongue and you hear a colour. Ask both and compare, and I will tell you how crowded the dark is.\u003c\/em\u003e\u003c\/p\u003e\u003ch3\u003eOn the hoodie\u003c\/h3\u003e\u003cp\u003eFront: two red bars of unequal height under a tiny scale. Back: Sulphra in full pixel art with “FORBIDDEN. OBSERVED ANYWAY.” and “TWO LINES. ONE DENSITY.” Gildan 18500 Heavy Blend hoodie, Black, S to 5XL.\u003c\/p\u003e\u003cp\u003ePart of the \u003cstrong\u003eForbidden Transitions\u003c\/strong\u003e set with \u003ca href=\"\/pl\/products\/nebulia-pixel-hoodie\"\u003eNebulia\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/coronix-pixel-hoodie\"\u003eCoronix\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/flipwarden-pixel-hoodie\"\u003eFlipwarden\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/viridael-pixel-hoodie\"\u003eViridael\u003c\/a\u003e, \u003ca href=\"\/pl\/products\/sanguora-pixel-hoodie\"\u003eSanguora\u003c\/a\u003e. Browse all \u003ca href=\"\/pl\/collections\/hoodies\"\u003ephysics hoodies\u003c\/a\u003e, more \u003ca href=\"\/pl\/collections\/astrophysics\"\u003eastrophysics \u0026amp; cosmology designs\u003c\/a\u003e, or the \u003ca href=\"\/pl\/collections\/deep-cuts\"\u003eDeep Cuts\u003c\/a\u003e.\u003c\/p\u003e","brand":"Pixelated Physics","offers":[{"title":"Black \/ S","offer_id":50964466827367,"sku":"32611904393364746608","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ M","offer_id":50964466860135,"sku":"35020292875147891017","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ L","offer_id":50964466892903,"sku":"13135463546790499486","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ XL","offer_id":50964466925671,"sku":"13373872655449288521","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 2XL","offer_id":50964466958439,"sku":"64132118080329792959","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 3XL","offer_id":50964466991207,"sku":"88439314632111916075","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 4XL","offer_id":50964467023975,"sku":"77328200518484374728","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 5XL","offer_id":50964467056743,"sku":"76089834310947899755","price":64.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/03_H02_aabed8d2-2644-45a7-be4a-1bbbfceeb38e.png?v=1793678406"}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/collections\/forbidden-transitions.png?v=1793678005","url":"https:\/\/pixelatedphysics.com\/pl\/collections\/forbidden-transitions.oembed","provider":"Pixelated Physics","version":"1.0","type":"link"}