{"id": "Wpo8QDSVEeawUzuNC0Vlhw", "created": "2016-06-17T14:11:15.619049+00:00", "updated": "2016-06-17T14:11:15.619056+00:00", "user": "acct:bolverk@hypothes.is", "uri": "urn:x-pdf:f4383218052dc2af68de32271945e9e2", "text": "The constraint on the number of the mode in each direction is\n$$n_x^2 + n_y^2 +n_z^2 < n_{max}^2 = \\frac{m_e E}{h^2} V^{2/3}$$\nThe naive number of states is\n$$N_0 = \\int_0^{n_{\\max}} d n_x \\int_0^{{\\sqrt{n_{\\max}^2-n_x^2}}} dn_y \\int_0^{\\sqrt{n_{\\max}^2-n_x^2-n_y^2}} d n_z$$\n$$\\approx n_{max}^3 \\approx \\frac{m_e^{3/2} E^{3/2}}{h^3} V$$\nThe number of modes where one of the modes disappears is\n$$N_1 = 3 \\int_0^{n_{\\max}} d n_x \\int_0^{{\\sqrt{n_{\\max}^2-n_x^2}}} dn_y \\approx $$\n$$\\approx n_{max}^2 \\approx \\frac{m_e E}{h^2} V^{2/3}$$\nThe second term is proportional to the area of the cell.", "tags": [], "group": "__world__", "moderation_status": "APPROVED", "permissions": {"read": ["group:__world__"], "admin": ["acct:bolverk@hypothes.is"], "update": ["acct:bolverk@hypothes.is"], "delete": ["acct:bolverk@hypothes.is"]}, "target": [{"source": "urn:x-pdf:f4383218052dc2af68de32271945e9e2", "selector": [{"end": 96826, "type": "TextPositionSelector", "start": 96749}, {"type": "TextQuoteSelector", "exact": "Exercise The number density in phase space, excluding  spin  effccts. is 2.25", "prefix": "simple Pauli Principle picture. ", "suffix": " Calculate the  nuclear surface "}]}], "document": {"title": ["Black Holes, White Dwarfs and Neutron Stars: The Physics of Compact Objects "]}, "links": {"html": "https://hypothes.is/a/Wpo8QDSVEeawUzuNC0Vlhw", "incontext": "https://hyp.is/Wpo8QDSVEeawUzuNC0Vlhw/www.phys.huji.ac.il/~almog.yalin/bhwdns.pdf", "json": "https://hypothes.is/api/annotations/Wpo8QDSVEeawUzuNC0Vlhw"}, "actions": [], "mentions": [], "user_info": {"display_name": null}, "flagged": false, "hidden": false}