{"product_id":"sponara-pixel-hoodie","title":"Sponara – Hertha Sponer and the Birge–Sponer Method Pixel Hoodie","description":"\u003cp\u003eSponara is a respectful pixel portrait of Hertha Sponer, after a 1913 public-domain photograph, beside a glowing ladder whose rungs close up toward a bright limit line.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eHertha Sponer (1895–1968) took her doctorate at Göttingen in 1920 under Peter Debye and worked with James Franck on molecular spectroscopy. On a fellowship at Berkeley she and Raymond T. Birge showed in 1926 how to estimate a molecule’s dissociation energy from its band spectrum. The vibrational levels of a real bond crowd together as they climb. Adding up the gaps until they shrink to zero gives the energy at which the molecule falls apart. Dismissed from Göttingen under Nazi law in 1934, she went to Oslo and in 1936 joined Duke University, the first woman on its physics faculty.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eG(v) = ω\u003csub\u003ee\u003c\/sub\u003e(v + ½) − ω\u003csub\u003ee\u003c\/sub\u003ex\u003csub\u003ee\u003c\/sub\u003e(v + ½)²\u003c\/strong\u003e\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eD₀ ≈ Σ\u003csub\u003ev\u003c\/sub\u003e ΔG\u003csub\u003ev+½\u003c\/sub\u003e   (for this two-term form, D\u003csub\u003ee\u003c\/sub\u003e ≈ ω\u003csub\u003ee\u003c\/sub\u003e² \/ 4ω\u003csub\u003ee\u003c\/sub\u003ex\u003csub\u003ee\u003c\/sub\u003e)\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eG(v): vibrational term value of level v (cm⁻¹)\u003c\/li\u003e\n\u003cli\u003ev: vibrational quantum number, 0, 1, 2, …\u003c\/li\u003e\n\u003cli\u003eω\u003csub\u003ee\u003c\/sub\u003e: harmonic vibrational constant (cm⁻¹)\u003c\/li\u003e\n\u003cli\u003eω\u003csub\u003ee\u003c\/sub\u003ex\u003csub\u003ee\u003c\/sub\u003e: anharmonicity constant (cm⁻¹)\u003c\/li\u003e\n\u003cli\u003eΔG\u003csub\u003ev+½\u003c\/sub\u003e: spacing G(v+1) − G(v) between neighbouring levels (cm⁻¹)\u003c\/li\u003e\n\u003cli\u003eD₀: dissociation energy from the lowest level v = 0 (cm⁻¹ or eV)\u003c\/li\u003e\n\u003cli\u003eD\u003csub\u003ee\u003c\/sub\u003e: dissociation energy measured from the bottom of the potential well\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003ePlot ΔG against v + ½ and extrapolate to ΔG = 0; the area under the line is D₀. Source: R. T. Birge \u0026amp; H. Sponer, “The heat of dissociation of non-polar molecules”, \u003cem\u003ePhys. Rev.\u003c\/em\u003e 28, 259 (1926), doi:10.1103\/PhysRev.28.259; Duke Physics, “Hertha Sponer”.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e Birge–Sponer is an extrapolation and only an estimate: real level spacings near the dissociation limit bend away from a straight line, so the method can overestimate or underestimate D₀.\u003c\/p\u003e\u003ch3\u003eIn character\u003c\/h3\u003e\u003cp\u003e\u003cem\u003eCount the steps, watch them shrink, and follow them to where there are no more. That is where the bond lets go, and that is the number.\u003c\/em\u003e\u003c\/p\u003e\u003ch3\u003eOn the hoodie\u003c\/h3\u003e\u003cp\u003eFront: a tiny ladder with shrinking gaps under a pixel crown. Back: a pixel portrait of Hertha Sponer with “CROWNED IN HER OWN LIGHT” and “HERTHA SPONER - 1926”. Gildan 18500 Heavy Blend hoodie, Black, S to 5XL.\u003c\/p\u003e\u003cp\u003ePart of the \u003cstrong\u003eUnsung Hands\u003c\/strong\u003e set with \u003ca href=\"\/it\/products\/recoilwyn-pixel-hoodie\"\u003eRecoilwyn\u003c\/a\u003e, \u003ca href=\"\/it\/products\/ayrtha-pixel-hoodie\"\u003eAyrtha\u003c\/a\u003e. Browse all \u003ca href=\"\/it\/collections\/hoodies\"\u003ephysics hoodies\u003c\/a\u003e, more \u003ca href=\"\/it\/collections\/quantum-physics-t-shirts\"\u003equantum designs\u003c\/a\u003e, or the \u003ca href=\"\/it\/collections\/deep-cuts\"\u003eDeep Cuts\u003c\/a\u003e.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"Black \/ S","offer_id":50964469383271,"sku":"21994265210478376952","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ M","offer_id":50964469416039,"sku":"32205395793648449551","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ L","offer_id":50964469448807,"sku":"59886375318620799164","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ XL","offer_id":50964469481575,"sku":"32306011724677858891","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 2XL","offer_id":50964469514343,"sku":"19658560549868276941","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 3XL","offer_id":50964469547111,"sku":"24150544771332124072","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 4XL","offer_id":50964469579879,"sku":"22576485165543150713","price":64.99,"currency_code":"USD","in_stock":true},{"title":"Black \/ 5XL","offer_id":50964469612647,"sku":"26212679212273883683","price":64.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/03_H05_72846dbf-4aef-4ee6-b837-e32ca67a9537.png?v=1793678478","url":"https:\/\/pixelatedphysics.com\/it\/products\/sponara-pixel-hoodie","provider":"Pixelated Physics","version":"1.0","type":"link"}