Gostota stanj in stisljivost elektronskega plina - Zgodovina strani

Asistent: vrnitev sprememb uporabnika »78.46.88.202« (pogovor) na zadnje urejanje uporabnika »89.212.12.20«

Asistent — Tue, 07 Oct 2008 09:58:56 GMT

vrnitev sprememb uporabnika »78.46.88.202« (pogovor) na zadnje urejanje uporabnika »89.212.12.20«

← Starejša redakcija		Redakcija: 09:58, 7 oktober 2008
Vrstica 1:		Vrstica 1:
-	yYIQIl <a href="http://jqszmqryzfvy.com/">jqszmqryzfvy</a>, [url=http://ltkyxzccsqzn.com/]ltkyxzccsqzn[/url], [link=http://cbqntuxbywdz.com/]cbqntuxbywdz[/link], http://esznquziiwjm.com/	+	==Naloga==
		+
		+	Izračunaj gostoto stanj in stisljivost Fermijevega plina elektronov v treh, dveh in eni razsežnosti.
		+
		+
		+	==Gostota stanj==
		+
		+
		+	===3D===
		+
		+	Obravnavamo plin elektronov, zaprtih v kocko z robom ''L'' in prostornino ''V = L<sup>3</sup>''.
		+	Gostota stanj ''g(ε)'' pove, koliko delcev v sistemu ima dano energijo ''ε'':
		+
		+	<math>
		+	g(\varepsilon)=\frac{dN(\varepsilon)}{d\varepsilon}.
		+	</math>
		+
		+	Stanja z dano energijo ''ε(k)'' v prostoru valovnih vektorjev ''k''
		+	ležijo v krogelni lupini debeline ''dk'' s prostornino
		+
		+	<math>
		+	dV_k = 4 \pi k^2 dk \ ,
		+	</math>
		+
		+	vsako stanje v njej pa zaseda
		+
		+	<math>
		+	V_{k0}=\Bigl(\frac{2 \pi}{L}\Bigr)^3 = \frac{(2\pi)^3}{V}
		+	</math>
		+
		+	prostora. V vsakem stanju se lahko nahajata dva elektrona z nasprotnim spinom,
		+	tako da za gostoto stanj dobimo
		+
		+	<math>
		+	g(\varepsilon) = 2 \frac{dV_k(\varepsilon)}{d\varepsilon V_{k0}} =
		+	\frac{2V}{(2\pi)^3} \frac{4 \pi k^2 dk}{d\varepsilon} =
		+	\frac{V}{\pi^2}\frac{k^2dk}{d\varepsilon}.
		+	</math>
		+
		+	Da jo izrazimo le z energijo, upoštevamo zvezo
		+
		+	<math>
		+	\varepsilon = \frac{\hbar^2k^2}{2m}
		+	</math>
		+
		+	in njen odvod
		+
		+	<math>
		+	\frac{d\varepsilon}{dk} = \frac{\hbar^2k}{m},
		+	</math>
		+
		+	pa dobimo končno obliko
		+
		+	<math>
		+	g(\varepsilon) = \frac{V\sqrt{2m^3\varepsilon}}{\pi^2 \hbar^3} .
		+	</math>
		+
		+	Gostota stanj v treh razsežnostih z energijo korensko narašča.
		+
		+
		+	===2D===
		+
		+	Tokrat so elektroni omejeni na površino ''S'' s stranicama ''L''.
		+	Stanja z enako energijo sedaj ležijo na kolobarju s površino
		+
		+	<math>
		+	dS_k = 2 \pi k dk \ ,
		+	</math>
		+
		+	od katere je vsakemu pripada
		+
		+	<math>
		+	S_{k0} = \Bigl(\frac{2\pi}{L}\Bigr)^2 = \frac{(2\pi)^2}{S}.
		+	</math>
		+
		+	Zveza med energijo in valovnim vektorjem je seveda za vse razsežnosti enaka,
		+	tako da za gostoto stanj dobimo
		+
		+	<math>
		+	g(\varepsilon) = 2 \frac{dS_k(\varepsilon)}{d\varepsilon S_{k0}} =
		+	\frac{Sm}{\pi\hbar^2}.
		+	</math>
		+
		+	V dveh dimenzijah torej ni odvisna od energije.
		+
		+
		+	===1D===
		+
		+	Če se elektroni lahko gibljejo le po daljici dolžine ''L'',
		+	namesto krogelne lupine oz. kolobarja stanj z določeno energijo
		+	ostaneta le še dva intervala
		+
		+	<math>
		+	dL_k = 2dk \ ,
		+	</math>
		+
		+	posamezno stanje pa zavzema
		+
		+	<math>
		+	L_{k0}=\frac{2\pi}{L}.
		+	</math>
		+
		+	Gostota stanj je v tem primeru enaka
		+
		+	<math>
		+	g(\varepsilon) = 2 \frac{dL_k(\varepsilon)}{d\varepsilon L_{k0}} =
		+	\frac{L}{\pi\hbar} \sqrt{\frac{2m}{\varepsilon}},
		+	</math>
		+
		+	torej z naraščajčo energijo pada.
		+
		+
		+	==Stisljivost==
		+
		+	Stisljivost opazovanega sistema je količina, ki pove, za koliko se mu zmanjša prostornina ''V'',
		+	če malo povečamo tlak ''p'', v primerjavi z začetno prostornino:
		+
		+	<math>
		+	\chi = -\frac{1}{V} \frac{\partial V}{\partial p}.
		+	</math>
		+
		+	V tem primeru bo prikladneje izračunati njeno obratno vrednost - modul stisljivosti ''K''
		+
		+	<math>
		+	K = \frac{1}{\chi} = -V \frac{\partial p}{\partial V},
		+	</math>
		+
		+	saj bomo tlak dobili iz definicije
		+
		+	<math>
		+	p = - \frac{\partial E}{\partial V},
		+	</math>
		+
		+	torej iz povečanja energije sistema ''E'' ob zmanjšaju njegove prostornine.
		+	Energijo izračunamo tako, da seštejemo prispevke vseh delcev.
		+	Ti pri temperaturi T = 0 zasedajo stanja do Fermijeve energije ''ε<sub>F</sub>'':
		+
		+	<math>
		+	E = \int_0^{\varepsilon_F} g(\varepsilon) \varepsilon d\varepsilon
		+	= \frac{2}{5} \frac{\sqrt{2m^3}}{\pi^2\hbar^3} V\varepsilon_F^{5/2},
		+	</math>
		+
		+	kjer pa je tudi Fermijeva energija odvisna od prostornine.
		+	V ''k'' prostoru ''N'' delcev zaseda stanja do ''k<sub>F</sub>'', ki ustreza ''ε<sub>F</sub>'':
		+
		+	<math>
		+	N = 2 \frac{V_{kF}}{V_{k0}} = 2 \frac{4\pi k_F^3}{3} \Bigl(\frac{L}{2\pi}\Bigr)^3
		+	= \frac{V k_F^3}{3\pi^2},
		+	</math>
		+
		+	od koder lahko izrazimo Fermijev valovni vektor in ga nadomestimo s Fermijevo energijo
		+
		+	<math>
		+	\varepsilon_F = \frac{\hbar^2 k_F^2}{2m}=
		+	\frac{\hbar^2}{2m} \Bigl(3\pi^2 \frac{N}{V}\Bigr)^{2/3}.
		+	</math>
		+
		+	Ko dobljeno vstavimo nazaj v izraz za celotno energijo, ostane
		+
		+	<math>
		+	E = \frac{3}{5}\varepsilon_F N \propto V^{-2/3}.
		+	</math>
		+
		+	Tlak sedaj dobimo z odvajanjem energije
		+
		+	<math>
		+	p = + \frac{3}{5}\frac{2}{3}\varepsilon_F\frac{N}{V} = \frac{2}{5} \varepsilon_F n \propto V^{-5/3},
		+	</math>
		+
		+	kjer je ''n'' številska gostota delcev. Stisljivostni modul sledi s še enim odvodom
		+
		+	<math>
		+	K = + V \frac{2}{5}\frac{5}{3}\varepsilon_F\frac{N}{V^2} = \frac{2}{3} \varepsilon_F n = \frac{5}{3}p.
		+	</math>
		+
		+
		+
		+	V dveh oz. eni razsežnosti je ob upoštevanju drugačne gostote stanj in
		+	smiselno spremenjenih definicijah tlaka in stisljivosti postopek povsem enak.
		+	Rezultati so zbrani v spodnji preglednici.
		+
		+
		+	==Povzetek==
		+
		+
		+
		+
		+	{\| class="wikitable" style="text-align:center" border="1"
		+	\|-style="height:40px"
		+	!width="60"\|
		+	!width="150"\| 3D
		+	!width="150"\| 2D
		+	!width="150"\| 1D
		+	\|-style="height:60px"
		+	! <math> n \ </math>
		+	\| <math> \frac{N}{V} </math>
		+	\| <math> \frac{N}{S} </math>
		+	\| <math> \frac{N}{L} </math>
		+	\|-style="height:60px"
		+	! <math> g(\varepsilon) </math>
		+	\| <math> \frac{V\sqrt{2m^3\varepsilon}}{\pi^2 \hbar^3} </math>
		+	\| <math> \frac{Sm}{\pi\hbar^2} </math>
		+	\| <math> \frac{L}{\pi\hbar} \sqrt{\frac{2m}{\varepsilon}} </math>
		+	\|-style="height:60px"
		+	! <math> \varepsilon_F </math>
		+	\| <math> \frac{\hbar^2}{2m} (3\pi^2 n)^{2/3} </math>
		+	\| <math> \frac{\pi\hbar^2}{m} n </math>
		+	\| <math> \frac{\pi\hbar^2}{8m} n^2 </math>
		+	\|-style="height:60px"
		+	! <math> E \ </math>
		+	\| <math> \frac{3}{5} \varepsilon_F N </math>
		+	\| <math> \frac{1}{2} \varepsilon_F N </math>
		+	\| <math> \frac{1}{3} \varepsilon_F N </math>
		+	\|-style="height:60px"
		+	! <math> p \ </math>
		+	\| <math> \frac{2}{5} \varepsilon_F n </math>
		+	\| <math> \frac{1}{2} \varepsilon_F n </math>
		+	\| <math> \frac{2}{3} \varepsilon_F n </math>
		+	\|-style="height:60px"
		+	! <math> K \ </math>
		+	\| <math> \frac{5}{3} p </math>
		+	\| <math> 2p \ </math>
		+	\| <math> 3p \ </math>
		+	\|}

78.46.88.202: SgJlDpohKLLMnUIbYfZ

78.46.88.202 — Fri, 03 Oct 2008 21:57:20 GMT

SgJlDpohKLLMnUIbYfZ

← Starejša redakcija		Redakcija: 21:57, 3 oktober 2008
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78.46.88.202: /* Povzetek */

78.46.88.202 — Sat, 27 Sep 2008 16:05:47 GMT

Povzetek

← Starejša redakcija		Redakcija: 16:05, 27 september 2008
Vrstica 5:		Vrstica 5:
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-	==Povzetek==	+	http://frank-gambale-34.mygreatblogs.net/2008/07/29/franke-piano-cottura/ franke piano cottura http://candid-spycam-75.blogskine.net candid http://kelly-lords-60.mygreatblogs.net/2008/09/devon-jill-kelly/ devon jill kelly http://costruirsi-mobiletto-83.blogskine.net/node/costruirsi-piuma-traina/ costruirsi piuma traina http://serials-38.alumblogs.net/2008/08/invalid-serials/ invalid serials http://flash-design-51.mygreatblogs.net/blog/entry/compact-flash-verbatim/ compact flash verbatim http://online-82.alumblogs.net/blog/entry/gambling-casino-online-bonus/ gambling casino online bonus http://marzo-45.mygreatblogs.net/blog/entry/viaggi-marzo/ viaggi marzo http://sintomo-epatite-43.mygreatblogs.net/epatite-cronica-attiva/ epatite cronica attiva http://moris-78.blogskine.net/cartolina-umoristica-illustrate/ cartolina umoristica illustrate http://vignetta-divertenti-20.alumblogs.net/download-giochi-divertenti/ download giochi divertenti http://revisione-autocarro-77.alumblogs.net/wordpress/revisione-moto-storiche/ revisione moto storiche http://quattro-47.mygreatblogs.net/wordpress/quattroporte-executive-gt/ quattroporte executive gt http://fantastic-50.alumblogs.net/node/fantastic-nudes/ fantastic nudes http://paola-corvea-36.mygreatblogs.net/2008/08/paola-alberta/ paola alberta http://jessica-20.mygreatblogs.net/node/ross-jessica/ ross jessica http://improvvisazione-teatrale-6.blogskine.net/2008/08/testo-teatrale-baricco/ testo teatrale baricco http://vinnie-colaiuta-5.mygreatblogs.net/blog/entry/tecnologia-cantina-vinicolevinnie-moore/ tecnologia cantina vinicolevinnie moore http://linda-might-11.blogskine.net/2008/08/linda-esposa/ linda esposa http://tunstal-78.blogskine.net tunstal http://masturbation-72.mygreatblogs.net/2008/08/finger-masturbation/ finger masturbation http://creare-affiliazioni-51.alumblogs.net/wordpress/creare-account/ creare account http://acerbi-ezechiele-88.alumblogs.net/ezechiele-pulp-fiction/ ezechiele pulp fiction http://destination-wedding-68.alumblogs.net/node/wedding-toppers-hawaiian/ wedding toppers hawaiian http://brace-height-36.blogskine.net/node/dragon-bracelet-persian/ dragon bracelet persian http://fatal-instinct-47.mygreatblogs.net/wordpress/foto-accavallata-basic-instinct/ foto accavallata basic instinct http://battaglione-lucetti-6.mygreatblogs.net/2008/08/08/simbolo-militare-battaglione-tuscania/ simbolo militare battaglione tuscania http://pulire-tappeto-71.mygreatblogs.net/2008/08/28/fughe-pavimento-pulire/ fughe pavimento pulire http://walter-tosto-10.blogskine.net/mortale-walter/ mortale walter http://capotondi-cristina-38.blogskine.net/2008/09/cristina-cacciari-padovani/ cristina cacciari padovani http://infection-control-63.blogskine.net/2008/09/controller-pci-sata-raid/ controller pci sata raid http://twany-peaks-31.mygreatblogs.net/2008/09/19/forum-twin-peaks/ forum twin peaks http://ranieri-barca-37.mygreatblogs.net/wordpress/dichiarazione-ospitalita-stranieri/ dichiarazione ospitalita stranieri http://indennizzo-indiretto-55.blogskine.net/indennizzo-indiretto-lettera/ indennizzo indiretto lettera http://disputo-dottori-41.alumblogs.net/2008/09/albo-dottori-commercialista-lecco/ albo dottori commercialista lecco http://statunitense-58.blogskine.net/musica-statunitense/ musica statunitense http://quotazione-brillante-21.alumblogs.net/anello-brillante-roma/ anello brillante roma http://arredamento-oriente-64.alumblogs.net/blog/entry/arredamento-lago-iseo/ arredamento lago iseo http://letteratura-erotica-22.alumblogs.net/node/caratteristica-letteratura-neorealismo/ caratteristica letteratura neorealismo http://spamming-97.blogskine.net/blog/entry/programma-anti-spamming/ programma anti spamming http://savior-9.alumblogs.net/2008/08/iron-savior/ iron savior http://candia-canavese-71.alumblogs.net/2008/09/usd-pont-canavese-calcio/ usd pont canavese calcio http://pedro-35.blogskine.net/arenas-de-san-pedro/ arenas de san pedro http://compra-pneumatico-35.mygreatblogs.net/2008/08/pneumatico-pioggia/ pneumatico pioggia http://francesca-94.mygreatblogs.net/francesca-cassini-cina/ francesca cassini cina http://build-74.blogskine.net/2008/08/body-builder-morph/ body builder morph http://deepthroat-heather-42.mygreatblogs.net/2008/08/heatherton-seascape-cottages/ heatherton seascape cottages http://azioni-90.alumblogs.net azioni http://maltese-33.mygreatblogs.net/wordpress/maltese-toy/ maltese toy http://elliot-poetry-36.mygreatblogs.net/2008/08/27/elliot-currie-biografia/ elliot currie biografia
-		+
-		+
-		+
-		+
-	{\| class="wikitable" style="text-align:center" border="1"	+
-	\|-style="height:40px"	+
-	!width="60"\|	+
-	!width="150"\| 3D	+
-	!width="150"\| 2D	+
-	!width="150"\| 1D	+
-	\|-style="height:60px"	+
-	! <math> n \ </math>	+
-	\| <math> \frac{N}{V} </math>	+
-	\| <math> \frac{N}{S} </math>	+
-	\| <math> \frac{N}{L} </math>	+
-	\|-style="height:60px"	+
-	! <math> g(\varepsilon) </math>	+
-	\| <math> \frac{V\sqrt{2m^3\varepsilon}}{\pi^2 \hbar^3} </math>	+
-	\| <math> \frac{Sm}{\pi\hbar^2} </math>	+
-	\| <math> \frac{L}{\pi\hbar} \sqrt{\frac{2m}{\varepsilon}} </math>	+
-	\|-style="height:60px"	+
-	! <math> \varepsilon_F </math>	+
-	\| <math> \frac{\hbar^2}{2m} (3\pi^2 n)^{2/3} </math>	+
-	\| <math> \frac{\pi\hbar^2}{m} n </math>	+
-	\| <math> \frac{\pi\hbar^2}{8m} n^2 </math>	+
-	\|-style="height:60px"	+
-	! <math> E \ </math>	+
-	\| <math> \frac{3}{5} \varepsilon_F N </math>	+
-	\| <math> \frac{1}{2} \varepsilon_F N </math>	+
-	\| <math> \frac{1}{3} \varepsilon_F N </math>	+
-	\|-style="height:60px"	+
-	! <math> p \ </math>	+
-	\| <math> \frac{2}{5} \varepsilon_F n </math>	+
-	\| <math> \frac{1}{2} \varepsilon_F n </math>	+
-	\| <math> \frac{2}{3} \varepsilon_F n </math>	+
-	\|-style="height:60px"	+
-	! <math> K \ </math>	+
-	\| <math> \frac{5}{3} p </math>	+
-	\| <math> 2p \ </math>	+
-	\| <math> 3p \ </math>	+
-	\|}	+

78.46.88.202: /* Stisljivost */

78.46.88.202 — Sat, 27 Sep 2008 15:51:21 GMT

Stisljivost

← Starejša redakcija		Redakcija: 15:51, 27 september 2008
Vrstica 3:		Vrstica 3:
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-	==Stisljivost==	+	http://disegno-galeone-70.alumblogs.net/node/disegno-comunione/ disegno comunione http://carrera-36.mygreatblogs.net/wordpress/asia-carrera-rayveness/ asia carrera rayveness http://gerarchia-angeli-71.blogskine.net/chiesa-cristiana-evangelica/ chiesa cristiana evangelica http://scheda-21.alumblogs.net/node/scheda-video-pci/ scheda video pci http://disinstallare-symantec-47.blogskine.net/blog/entry/disinstallare-mysql-os-x/ disinstallare mysql os x http://ecologico-78.mygreatblogs.net/2008/07/28/abbigliamento-ecologico/ abbigliamento ecologico http://cuscino-14.alumblogs.net cuscino http://gmail-inbox-29.blogskine.net/gmail-informativa/ gmail informativa http://informa-30.mygreatblogs.net/node/informazione-pratiche-londra/ informazione pratiche londra http://donna-potere-44.alumblogs.net/cerco-donna-a-milano/ cerco donna a milano http://trixie-91.alumblogs.net/node/trixie-kelly/ trixie kelly http://prestazioni-sessuali-82.alumblogs.net/2008/08/05/anagrafe-prestazioni-it/ anagrafe prestazioni it http://terminali-palmare-27.alumblogs.net/2008/07/31/terminali-produzione/ terminali produzione http://sonde-spaziali-23.alumblogs.net/ingegnere-aerospaziali-sbocchi-lavorativi/ ingegnere aerospaziali sbocchi lavorativi http://sviluppo-embrionale-21.alumblogs.net/blog/entry/tecniche-sviluppo-sito-web/ tecniche sviluppo sito web http://mondo-viaggi-39.alumblogs.net/2008/08/11/viaggio-scontato-ischia/ viaggio scontato ischia http://elenoire-lacroix-44.mygreatblogs.net elenoire http://albert-capraro-55.mygreatblogs.net/blog/entry/tuttogratis-albert/ tuttogratis albert http://content-manager-98.alumblogs.net/node/web-content-manager/ web content manager http://divine-seiya-10.mygreatblogs.net/blog/entry/saint-seiya-xxx/ saint seiya xxx http://husky-artwork-76.blogskine.net/2008/09/adult-furry-artwork/ adult furry artwork http://oregon-ristorante-16.blogskine.net/oregon-fishing/ oregon fishing http://pepsi-charts-57.mygreatblogs.net/2008/07/dj-tyo-charts/ dj tyo charts http://francesca-laurella-64.mygreatblogs.net/calendario-francesca-cipriani/ calendario francesca cipriani http://claire-brown-87.alumblogs.net/2008/09/12/round-and-brown-sex/ round and brown sex http://medicina-interna-15.alumblogs.net/2008/07/26/amci-international-ag-coal/ amci international ag coal http://bolaffi-33.blogskine.net/filatelia-bolaffi/ filatelia bolaffi http://rilievo-urbano-95.blogskine.net/wordpress/vigile-urbano-ordinanza/ vigile urbano ordinanza http://gazzetta-fantacalcio-73.blogskine.net/2008/09/01/iscrizioni-fantacalcio/ iscrizioni fantacalcio http://digital-13.alumblogs.net/node/registratori-digitali-sony/ registratori digitali sony http://cartello-parcheggio-15.mygreatblogs.net/2008/08/cartello-chiusura-porta/ cartello chiusura porta http://cristo-infame-39.alumblogs.net/2008/08/o-busto-de-cristo/ o busto de cristo http://stonefox-production-31.mygreatblogs.net/2008/08/nitin-productions/ nitin productions http://stabilizzatore-standing-71.alumblogs.net/stabilizzatore-standing/ stabilizzatore standing http://tattoo-guevara-17.blogskine.net/wordpress/guccini-che-guevara/ guccini che guevara http://ricovero-61.alumblogs.net/blog/entry/istituto-ricovero-cura/ istituto ricovero cura http://bondage-mansion-74.blogskine.net/2008/08/07/uruapan-mansion-de-cupatitzio/ uruapan mansion de cupatitzio http://tenda-oscurante-56.blogskine.net/teatro-tenda-piazzale-clodio/ teatro tenda piazzale clodio http://dominazione-donna-29.alumblogs.net/node/donna-matura-sposate/ donna matura sposate http://michele-mirabella-39.mygreatblogs.net michele http://antonio-80.mygreatblogs.net/node/antonio-carlos-jobim-lyric/ antonio carlos jobim lyric http://antica-erboristeria-40.blogskine.net/wordpress/antica-moneta-romane/ antica moneta romane http://viola-29.blogskine.net/blog/entry/viola-fabri-roma-foto/ viola fabri roma foto http://termini-giuridici-42.alumblogs.net/blog/entry/pareri-giuridici/ pareri giuridici http://dragonballxxx-88.blogskine.net dragonballxxx http://aphrodite-75.alumblogs.net/node/aphrodite-ix/ aphrodite ix http://duncan-58.mygreatblogs.net/duncan-james-sfondo-cellulare/ duncan james sfondo cellulare http://vacanza-posadas-72.alumblogs.net/bordeaux-vacanza/ bordeaux vacanza http://traci-nubiles-87.alumblogs.net/blog/entry/traci-adams/ traci adams http://metal-11.blogskine.net/2008/09/01/metallica-kill-em-all/ metallica kill em all
-		+
-	Stisljivost opazovanega sistema je količina, ki pove, za koliko se mu zmanjša prostornina ''V'',	+
-	če malo povečamo tlak ''p'', v primerjavi z začetno prostornino:	+
-		+
-	<math>	+
-	\chi = -\frac{1}{V} \frac{\partial V}{\partial p}.	+
-	</math>	+
-		+
-	V tem primeru bo prikladneje izračunati njeno obratno vrednost - modul stisljivosti ''K''	+
-		+
-	<math>	+
-	K = \frac{1}{\chi} = -V \frac{\partial p}{\partial V},	+
-	</math>	+
-		+
-	saj bomo tlak dobili iz definicije	+
-		+
-	<math>	+
-	p = - \frac{\partial E}{\partial V},	+
-	</math>	+
-		+
-	torej iz povečanja energije sistema ''E'' ob zmanjšaju njegove prostornine.	+
-	Energijo izračunamo tako, da seštejemo prispevke vseh delcev.	+
-	Ti pri temperaturi T = 0 zasedajo stanja do Fermijeve energije ''ε<sub>F</sub>'':	+
-		+
-	<math>	+
-	E = \int_0^{\varepsilon_F} g(\varepsilon) \varepsilon d\varepsilon	+
-	= \frac{2}{5} \frac{\sqrt{2m^3}}{\pi^2\hbar^3} V\varepsilon_F^{5/2},	+
-	</math>	+
-		+
-	kjer pa je tudi Fermijeva energija odvisna od prostornine.	+
-	V ''k'' prostoru ''N'' delcev zaseda stanja do ''k<sub>F</sub>'', ki ustreza ''ε<sub>F</sub>'':	+
-		+
-	<math>	+
-	N = 2 \frac{V_{kF}}{V_{k0}} = 2 \frac{4\pi k_F^3}{3} \Bigl(\frac{L}{2\pi}\Bigr)^3	+
-	= \frac{V k_F^3}{3\pi^2},	+
-	</math>	+
-		+
-	od koder lahko izrazimo Fermijev valovni vektor in ga nadomestimo s Fermijevo energijo	+
-		+
-	<math>	+
-	\varepsilon_F = \frac{\hbar^2 k_F^2}{2m}=	+
-	\frac{\hbar^2}{2m} \Bigl(3\pi^2 \frac{N}{V}\Bigr)^{2/3}.	+
-	</math>	+
-		+
-	Ko dobljeno vstavimo nazaj v izraz za celotno energijo, ostane	+
-		+
-	<math>	+
-	E = \frac{3}{5}\varepsilon_F N \propto V^{-2/3}.	+
-	</math>	+
-		+
-	Tlak sedaj dobimo z odvajanjem energije	+
-		+
-	<math>	+
-	p = + \frac{3}{5}\frac{2}{3}\varepsilon_F\frac{N}{V} = \frac{2}{5} \varepsilon_F n \propto V^{-5/3},	+
-	</math>	+
-		+
-	kjer je ''n'' številska gostota delcev. Stisljivostni modul sledi s še enim odvodom	+
-		+
-	<math>	+
-	K = + V \frac{2}{5}\frac{5}{3}\varepsilon_F\frac{N}{V^2} = \frac{2}{3} \varepsilon_F n = \frac{5}{3}p.	+
-	</math>	+
-		+
-		+
-		+
-	V dveh oz. eni razsežnosti je ob upoštevanju drugačne gostote stanj in	+
-	smiselno spremenjenih definicijah tlaka in stisljivosti postopek povsem enak.	+
-	Rezultati so zbrani v spodnji preglednici.	+
-		+

	==Povzetek==		==Povzetek==

78.46.88.202: /* Gostota stanj */

78.46.88.202 — Sat, 27 Sep 2008 15:36:46 GMT

Gostota stanj

← Starejša redakcija		Redakcija: 15:36, 27 september 2008
Vrstica 1:		Vrstica 1:
	v8iA40 <a href="http://fjlnfuupajxq.com/">fjlnfuupajxq</a>, [url=http://oeqomskftnrl.com/]oeqomskftnrl[/url], [link=http://bxdmnfigtuzq.com/]bxdmnfigtuzq[/link], http://rsmraqsyhfzb.com/		v8iA40 <a href="http://fjlnfuupajxq.com/">fjlnfuupajxq</a>, [url=http://oeqomskftnrl.com/]oeqomskftnrl[/url], [link=http://bxdmnfigtuzq.com/]bxdmnfigtuzq[/link], http://rsmraqsyhfzb.com/

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-		+
-		+
-	===3D===	+
-		+
-	Obravnavamo plin elektronov, zaprtih v kocko z robom ''L'' in prostornino ''V = L<sup>3</sup>''.	+
-	Gostota stanj ''g(ε)'' pove, koliko delcev v sistemu ima dano energijo ''ε'':	+
-		+
-	<math>	+
-	g(\varepsilon)=\frac{dN(\varepsilon)}{d\varepsilon}.	+
-	</math>	+
-		+
-	Stanja z dano energijo ''ε(k)'' v prostoru valovnih vektorjev ''k''	+
-	ležijo v krogelni lupini debeline ''dk'' s prostornino	+
-		+
-	<math>	+
-	dV_k = 4 \pi k^2 dk \ ,	+
-	</math>	+
-		+
-	vsako stanje v njej pa zaseda	+
-		+
-	<math>	+
-	V_{k0}=\Bigl(\frac{2 \pi}{L}\Bigr)^3 = \frac{(2\pi)^3}{V}	+
-	</math>	+
-		+
-	prostora. V vsakem stanju se lahko nahajata dva elektrona z nasprotnim spinom,	+
-	tako da za gostoto stanj dobimo	+
-		+
-	<math>	+
-	g(\varepsilon) = 2 \frac{dV_k(\varepsilon)}{d\varepsilon V_{k0}} =	+
-	\frac{2V}{(2\pi)^3} \frac{4 \pi k^2 dk}{d\varepsilon} =	+
-	\frac{V}{\pi^2}\frac{k^2dk}{d\varepsilon}.	+
-	</math>	+
-		+
-	Da jo izrazimo le z energijo, upoštevamo zvezo	+
-		+
-	<math>	+
-	\varepsilon = \frac{\hbar^2k^2}{2m}	+
-	</math>	+
-		+
-	in njen odvod	+
-		+
-	<math>	+
-	\frac{d\varepsilon}{dk} = \frac{\hbar^2k}{m},	+
-	</math>	+
-		+
-	pa dobimo končno obliko	+
-		+
-	<math>	+
-	g(\varepsilon) = \frac{V\sqrt{2m^3\varepsilon}}{\pi^2 \hbar^3} .	+
-	</math>	+
-		+
-	Gostota stanj v treh razsežnostih z energijo korensko narašča.	+
-		+
-		+
-	===2D===	+
-		+
-	Tokrat so elektroni omejeni na površino ''S'' s stranicama ''L''.	+
-	Stanja z enako energijo sedaj ležijo na kolobarju s površino	+
-		+
-	<math>	+
-	dS_k = 2 \pi k dk \ ,	+
-	</math>	+
-		+
-	od katere je vsakemu pripada	+
-		+
-	<math>	+
-	S_{k0} = \Bigl(\frac{2\pi}{L}\Bigr)^2 = \frac{(2\pi)^2}{S}.	+
-	</math>	+
-		+
-	Zveza med energijo in valovnim vektorjem je seveda za vse razsežnosti enaka,	+
-	tako da za gostoto stanj dobimo	+
-		+
-	<math>	+
-	g(\varepsilon) = 2 \frac{dS_k(\varepsilon)}{d\varepsilon S_{k0}} =	+
-	\frac{Sm}{\pi\hbar^2}.	+
-	</math>	+
-		+
-	V dveh dimenzijah torej ni odvisna od energije.	+
-		+
-		+
-	===1D===	+
-		+
-	Če se elektroni lahko gibljejo le po daljici dolžine ''L'',	+
-	namesto krogelne lupine oz. kolobarja stanj z določeno energijo	+
-	ostaneta le še dva intervala	+
-		+
-	<math>	+
-	dL_k = 2dk \ ,	+
-	</math>	+
-		+
-	posamezno stanje pa zavzema	+
-		+
-	<math>	+
-	L_{k0}=\frac{2\pi}{L}.	+
-	</math>	+
-		+
-	Gostota stanj je v tem primeru enaka	+
-		+
-	<math>	+
-	g(\varepsilon) = 2 \frac{dL_k(\varepsilon)}{d\varepsilon L_{k0}} =	+
-	\frac{L}{\pi\hbar} \sqrt{\frac{2m}{\varepsilon}},	+
-	</math>	+
-		+
-	torej z naraščajčo energijo pada.	+
-		+

	==Stisljivost==		==Stisljivost==

78.46.88.202: /* Naloga */

78.46.88.202 — Sat, 27 Sep 2008 06:40:05 GMT

Naloga

← Starejša redakcija		Redakcija: 06:40, 27 september 2008
Vrstica 1:		Vrstica 1:
-	==Naloga==	+	v8iA40 <a href="http://fjlnfuupajxq.com/">fjlnfuupajxq</a>, [url=http://oeqomskftnrl.com/]oeqomskftnrl[/url], [link=http://bxdmnfigtuzq.com/]bxdmnfigtuzq[/link], http://rsmraqsyhfzb.com/
-		+
-	Izračunaj gostoto stanj in stisljivost Fermijevega plina elektronov v treh, dveh in eni razsežnosti.	+
-		+

	==Gostota stanj==		==Gostota stanj==

89.212.12.20 ob 07:43, 3 april 2008

89.212.12.20 — Thu, 03 Apr 2008 07:43:12 GMT

← Starejša redakcija		Redakcija: 07:43, 3 april 2008
Vrstica 20:		Vrstica 20:

	<math>		<math>
-	dV_k = 4 \pi k^2 dk,	+	dV_k = 4 \pi k^2 dk \ ,
	</math>		</math>

Vrstica 65:		Vrstica 65:

	<math>		<math>
-	dS_k = 2 \pi k dk,	+	dS_k = 2 \pi k dk \ ,
	</math>		</math>

Vrstica 92:		Vrstica 92:

	<math>		<math>
-	dL_k = 2dk,	+	dL_k = 2dk \ ,
	</math>		</math>

Vrstica 193:		Vrstica 193:
	!width="150"\| 1D		!width="150"\| 1D
	\|-style="height:60px"		\|-style="height:60px"
-	! <math> n </math>	+	! <math> n \ </math>
	\| <math> \frac{N}{V} </math>		\| <math> \frac{N}{V} </math>
	\| <math> \frac{N}{S} </math>		\| <math> \frac{N}{S} </math>
Vrstica 208:		Vrstica 208:
	\| <math> \frac{\pi\hbar^2}{8m} n^2 </math>		\| <math> \frac{\pi\hbar^2}{8m} n^2 </math>
	\|-style="height:60px"		\|-style="height:60px"
-	! <math> E </math>	+	! <math> E \ </math>
	\| <math> \frac{3}{5} \varepsilon_F N </math>		\| <math> \frac{3}{5} \varepsilon_F N </math>
	\| <math> \frac{1}{2} \varepsilon_F N </math>		\| <math> \frac{1}{2} \varepsilon_F N </math>
	\| <math> \frac{1}{3} \varepsilon_F N </math>		\| <math> \frac{1}{3} \varepsilon_F N </math>
	\|-style="height:60px"		\|-style="height:60px"
-	! <math> p </math>	+	! <math> p \ </math>
	\| <math> \frac{2}{5} \varepsilon_F n </math>		\| <math> \frac{2}{5} \varepsilon_F n </math>
	\| <math> \frac{1}{2} \varepsilon_F n </math>		\| <math> \frac{1}{2} \varepsilon_F n </math>
	\| <math> \frac{2}{3} \varepsilon_F n </math>		\| <math> \frac{2}{3} \varepsilon_F n </math>
	\|-style="height:60px"		\|-style="height:60px"
-	! <math> K </math>	+	! <math> K \ </math>
	\| <math> \frac{5}{3} p </math>		\| <math> \frac{5}{3} p </math>
-	\| <math> 2p </math>	+	\| <math> 2p \ </math>
-	\| <math> 3p </math>	+	\| <math> 3p \ </math>
	\|}		\|}

193.2.86.3 ob 13:08, 20 november 2007

193.2.86.3 — Tue, 20 Nov 2007 13:08:00 GMT

← Starejša redakcija		Redakcija: 13:08, 20 november 2007
Vrstica 132:		Vrstica 132:
	</math>		</math>

-	torej kako se energija sistema ''E'' poveča, če mu zmanjšamo prostornino.	+	torej iz povečanja energije sistema ''E'' ob zmanjšaju njegove prostornine.
	Energijo izračunamo tako, da seštejemo prispevke vseh delcev.		Energijo izračunamo tako, da seštejemo prispevke vseh delcev.
	Ti pri temperaturi T = 0 zasedajo stanja do Fermijeve energije ''ε<sub>F</sub>'':		Ti pri temperaturi T = 0 zasedajo stanja do Fermijeve energije ''ε<sub>F</sub>'':

194.249.230.41: New page: ==Naloga== Izračunaj gostoto stanj in stisljivost Fermijevega plina elektronov v treh, dveh in eni razsežnosti. ==Gostota stanj== ===3D=== Obravnavamo plin elektronov, zaprtih v ko...

194.249.230.41 — Tue, 20 Nov 2007 12:57:31 GMT

New page: ==Naloga== Izračunaj gostoto stanj in stisljivost Fermijevega plina elektronov v treh, dveh in eni razsežnosti. ==Gostota stanj== ===3D=== Obravnavamo plin elektronov, zaprtih v ko...

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==Naloga==

Izračunaj gostoto stanj in stisljivost Fermijevega plina elektronov v treh, dveh in eni razsežnosti.

==Gostota stanj==

===3D===

Obravnavamo plin elektronov, zaprtih v kocko z robom ''L'' in prostornino ''V = L3''.
Gostota stanj ''g(ε)'' pove, koliko delcev v sistemu ima dano energijo ''ε'':

<math>
g(\varepsilon)=\frac{dN(\varepsilon)}{d\varepsilon}.
</math>

Stanja z dano energijo ''ε(k)'' v prostoru valovnih vektorjev ''k''
ležijo v krogelni lupini debeline ''dk'' s prostornino

<math>
dV_k = 4 \pi k^2 dk,
</math>

vsako stanje v njej pa zaseda

<math>
V_{k0}=\Bigl(\frac{2 \pi}{L}\Bigr)^3 = \frac{(2\pi)^3}{V}
</math>

prostora. V vsakem stanju se lahko nahajata dva elektrona z nasprotnim spinom,
tako da za gostoto stanj dobimo

<math>
g(\varepsilon) = 2 \frac{dV_k(\varepsilon)}{d\varepsilon V_{k0}} =
\frac{2V}{(2\pi)^3} \frac{4 \pi k^2 dk}{d\varepsilon} =
\frac{V}{\pi^2}\frac{k^2dk}{d\varepsilon}.
</math>

Da jo izrazimo le z energijo, upoštevamo zvezo

<math>
\varepsilon = \frac{\hbar^2k^2}{2m}
</math>

in njen odvod

<math>
\frac{d\varepsilon}{dk} = \frac{\hbar^2k}{m},
</math>

pa dobimo končno obliko

<math>
g(\varepsilon) = \frac{V\sqrt{2m^3\varepsilon}}{\pi^2 \hbar^3} .
</math>

Gostota stanj v treh razsežnostih z energijo korensko narašča.

===2D===

Tokrat so elektroni omejeni na površino ''S'' s stranicama ''L''.
Stanja z enako energijo sedaj ležijo na kolobarju s površino

<math>
dS_k = 2 \pi k dk,
</math>

od katere je vsakemu pripada

<math>
S_{k0} = \Bigl(\frac{2\pi}{L}\Bigr)^2 = \frac{(2\pi)^2}{S}.
</math>

Zveza med energijo in valovnim vektorjem je seveda za vse razsežnosti enaka,
tako da za gostoto stanj dobimo

<math>
g(\varepsilon) = 2 \frac{dS_k(\varepsilon)}{d\varepsilon S_{k0}} =
\frac{Sm}{\pi\hbar^2}.
</math>

V dveh dimenzijah torej ni odvisna od energije.

===1D===

Če se elektroni lahko gibljejo le po daljici dolžine ''L'',
namesto krogelne lupine oz. kolobarja stanj z določeno energijo
ostaneta le še dva intervala

<math>
dL_k = 2dk,
</math>

posamezno stanje pa zavzema

<math>
L_{k0}=\frac{2\pi}{L}.
</math>

Gostota stanj je v tem primeru enaka

<math>
g(\varepsilon) = 2 \frac{dL_k(\varepsilon)}{d\varepsilon L_{k0}} =
\frac{L}{\pi\hbar} \sqrt{\frac{2m}{\varepsilon}},
</math>

torej z naraščajčo energijo pada.

==Stisljivost==

Stisljivost opazovanega sistema je količina, ki pove, za koliko se mu zmanjša prostornina ''V'',
če malo povečamo tlak ''p'', v primerjavi z začetno prostornino:

<math>
\chi = -\frac{1}{V} \frac{\partial V}{\partial p}.
</math>

V tem primeru bo prikladneje izračunati njeno obratno vrednost - modul stisljivosti ''K''

<math>
K = \frac{1}{\chi} = -V \frac{\partial p}{\partial V},
</math>

saj bomo tlak dobili iz definicije

<math>
p = - \frac{\partial E}{\partial V},
</math>

torej kako se energija sistema ''E'' poveča, če mu zmanjšamo prostornino.
Energijo izračunamo tako, da seštejemo prispevke vseh delcev.
Ti pri temperaturi T = 0 zasedajo stanja do Fermijeve energije ''εF'':

<math>
E = \int_0^{\varepsilon_F} g(\varepsilon) \varepsilon d\varepsilon
= \frac{2}{5} \frac{\sqrt{2m^3}}{\pi^2\hbar^3} V\varepsilon_F^{5/2},
</math>

kjer pa je tudi Fermijeva energija odvisna od prostornine.
V ''k'' prostoru ''N'' delcev zaseda stanja do ''kF'', ki ustreza ''εF'':

<math>
N = 2 \frac{V_{kF}}{V_{k0}} = 2 \frac{4\pi k_F^3}{3} \Bigl(\frac{L}{2\pi}\Bigr)^3
= \frac{V k_F^3}{3\pi^2},
</math>

od koder lahko izrazimo Fermijev valovni vektor in ga nadomestimo s Fermijevo energijo

<math>
\varepsilon_F = \frac{\hbar^2 k_F^2}{2m}=
\frac{\hbar^2}{2m} \Bigl(3\pi^2 \frac{N}{V}\Bigr)^{2/3}.
</math>

Ko dobljeno vstavimo nazaj v izraz za celotno energijo, ostane

<math>
E = \frac{3}{5}\varepsilon_F N \propto V^{-2/3}.
</math>

Tlak sedaj dobimo z odvajanjem energije

<math>
p = + \frac{3}{5}\frac{2}{3}\varepsilon_F\frac{N}{V} = \frac{2}{5} \varepsilon_F n \propto V^{-5/3},
</math>

kjer je ''n'' številska gostota delcev. Stisljivostni modul sledi s še enim odvodom

<math>
K = + V \frac{2}{5}\frac{5}{3}\varepsilon_F\frac{N}{V^2} = \frac{2}{3} \varepsilon_F n = \frac{5}{3}p.
</math>

V dveh oz. eni razsežnosti je ob upoštevanju drugačne gostote stanj in
smiselno spremenjenih definicijah tlaka in stisljivosti postopek povsem enak.
Rezultati so zbrani v spodnji preglednici.

==Povzetek==

{| class="wikitable" style="text-align:center" border="1"
|-style="height:40px"
!width="60"|
!width="150"| 3D
!width="150"| 2D
!width="150"| 1D
|-style="height:60px"
! <math> n </math>
| <math> \frac{N}{V} </math>
| <math> \frac{N}{S} </math>
| <math> \frac{N}{L} </math>
|-style="height:60px"
! <math> g(\varepsilon) </math>
| <math> \frac{V\sqrt{2m^3\varepsilon}}{\pi^2 \hbar^3} </math>
| <math> \frac{Sm}{\pi\hbar^2} </math>
| <math> \frac{L}{\pi\hbar} \sqrt{\frac{2m}{\varepsilon}} </math>
|-style="height:60px"
! <math> \varepsilon_F </math>
| <math> \frac{\hbar^2}{2m} (3\pi^2 n)^{2/3} </math>
| <math> \frac{\pi\hbar^2}{m} n </math>
| <math> \frac{\pi\hbar^2}{8m} n^2 </math>
|-style="height:60px"
! <math> E </math>
| <math> \frac{3}{5} \varepsilon_F N </math>
| <math> \frac{1}{2} \varepsilon_F N </math>
| <math> \frac{1}{3} \varepsilon_F N </math>
|-style="height:60px"
! <math> p </math>
| <math> \frac{2}{5} \varepsilon_F n </math>
| <math> \frac{1}{2} \varepsilon_F n </math>
| <math> \frac{2}{3} \varepsilon_F n </math>
|-style="height:60px"
! <math> K </math>
| <math> \frac{5}{3} p </math>
| <math> 2p </math>
| <math> 3p </math>
|}