{"product_id":"penninga-penning-trap","title":"Penninga – The Penning Trap \/ Invariance Theorem Physics Sticker","description":"\u003cp\u003ePenninga, an armoured warden, holds a single charged particle captive in a glowing cage of orbits.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eEarnshaw's theorem says no arrangement of static electric charges alone can hold a charge in stable equilibrium. A Penning trap gets around it by adding a strong, uniform magnetic field to an electric quadrupole field. A trapped particle then has three motions: a fast modified cyclotron circle (ω\u003csub\u003e+\u003c\/sub\u003e), an axial bounce along the field (ω\u003csub\u003ez\u003c\/sub\u003e), and a slow magnetron drift (ω\u003csub\u003e−\u003c\/sub\u003e). L. S. Brown and G. Gabrielse showed that, even in an imperfect trap, the three combine to give the true cyclotron frequency (“Geonium theory”, Rev. Mod. Phys. 58, 233, 1986).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eω\u003csub\u003ec\u003c\/sub\u003e² = ω\u003csub\u003e+\u003c\/sub\u003e² + ω\u003csub\u003ez\u003c\/sub\u003e² + ω\u003csub\u003e−\u003c\/sub\u003e²\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eω\u003csub\u003ec\u003c\/sub\u003e: free-space cyclotron frequency, qB\/m (rad\/s)\u003c\/li\u003e\n\u003cli\u003eω\u003csub\u003e+\u003c\/sub\u003e: modified cyclotron frequency (rad\/s)\u003c\/li\u003e\n\u003cli\u003eω\u003csub\u003ez\u003c\/sub\u003e: axial frequency (rad\/s)\u003c\/li\u003e\n\u003cli\u003eω\u003csub\u003e−\u003c\/sub\u003e: magnetron frequency (rad\/s)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eBecause ω\u003csub\u003ec\u003c\/sub\u003e = qB\/m, measuring it gives a particle's charge-to-mass ratio, and Penning traps make some of the most precise mass and magnetic-moment measurements in physics.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e The invariance theorem holds for small misalignment and ellipticity of the trap. Large imperfections and field anharmonicities still need corrections.\u003c\/p\u003e\u003ch3\u003eOn the sticker\u003c\/h3\u003e\u003cp\u003ePenninga, in armour, holds glowing atom-style orbits, a stylised picture rather than the trap's three motions. Printed text: \u003cstrong\u003ePENNINGA\u003c\/strong\u003e with the line \u003cstrong\u003eωc² = ω+² + ωz² + ω−²\u003c\/strong\u003e. Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"2\" × 2\" \/ White","offer_id":50964021706855,"sku":"18816880546893225528","price":4.99,"currency_code":"USD","in_stock":true},{"title":"3\" × 3\" \/ White","offer_id":50964021739623,"sku":"14208629226980313300","price":6.99,"currency_code":"USD","in_stock":true},{"title":"4\" × 4\" \/ White","offer_id":50964021772391,"sku":"20451783407035206904","price":8.99,"currency_code":"USD","in_stock":true},{"title":"6\" × 6\" \/ White","offer_id":50964021805159,"sku":"14483893476484592420","price":11.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/14948270397598841019_1200.jpg?v=1793658719","url":"https:\/\/pixelatedphysics.com\/zh-hant\/products\/penninga-penning-trap","provider":"Pixelated Physics","version":"1.0","type":"link"}