Gostota stanj in stisljivost elektronskega plina

Iz Fizika trdne snovi 2007 - 2008

(Primerjava redakcij)

Redakcija: 15:36, 27 september 2008

v8iA40 <a href="http://fjlnfuupajxq.com/">fjlnfuupajxq</a>, [url=http://oeqomskftnrl.com/]oeqomskftnrl[/url], [link=http://bxdmnfigtuzq.com/]bxdmnfigtuzq[/link], http://rsmraqsyhfzb.com/

http://francesco-parisi-78.blogskine.net/wordpress/francesco-s-leveled-creatures/ francesco s leveled creatures http://agriturismo-lesbico-30.blogskine.net/masserie-agriturismo-sicilia/ masserie agriturismo sicilia http://christine-ebersole-45.blogskine.net/blog/entry/video-christine-dolce/ video christine dolce http://organizza-party-34.mygreatblogs.net/wordpress/interim-posizione-organizzativa/ interim posizione organizzativa http://matthew-broderick-88.mygreatblogs.net/2008/08/matthew-lawrence/ matthew lawrence http://processo-comunitario-85.mygreatblogs.net/blog/entry/sostituire-processore-asus/ sostituire processore asus http://ulisse-94.alumblogs.net/2008/08/10/ulisse-e-penelope/ ulisse e penelope http://english-course-36.alumblogs.net/node/pescador-de-hombres-english/ pescador de hombres english http://brace-height-36.blogskine.net/node/tattoo-bracelet/ tattoo bracelet http://decoder-usati-4.alumblogs.net decoder http://rifiuto-urbani-99.blogskine.net/2008/08/26/autorizzazione-rifiuto-non-conformita/ autorizzazione rifiuto non conformita http://einaudi-viterbo-79.blogskine.net einaudi http://posada-sardegna-2.blogskine.net/cucina-tipica-sardegna/ cucina tipica sardegna http://volpe-deserto-71.alumblogs.net volpe http://ashton-72.alumblogs.net/ashton-moore/ ashton moore http://acconciatura-tendenza-95.alumblogs.net/spiegazioni-acconciatura/ spiegazioni acconciatura http://solopaca-falanghina-56.blogskine.net solopaca http://antiche-79.alumblogs.net/2008/09/21/antiche-mura-sorrento/ antiche mura sorrento http://jersey-62.mygreatblogs.net/node/custom-baseball-jerseys/ custom baseball jerseys http://sylvania-electronic-11.blogskine.net/blog/entry/sala-electronic-component/ sala electronic component http://kommissar-10.blogskine.net/blog/entry/video-falco-der-kommissar/ video falco der kommissar http://famosi-15.mygreatblogs.net/blog/entry/vesto-trucca-personaggio-famosi/ vesto trucca personaggio famosi http://leonardo-76.mygreatblogs.net/node/leonardo-mollica-pittore/ leonardo mollica pittore http://troyan-virus-51.mygreatblogs.net/node/antivirus-solution/ antivirus solution http://savior-9.alumblogs.net/2008/09/hotel-saviore-dell-adamello/ hotel saviore dell adamello http://ascendant-27.blogskine.net/wordpress/ascendant-audio-homepage/ ascendant audio homepage http://cerruti-calligaris-82.mygreatblogs.net/wordpress/hong-kong-cerruti/ hong kong cerruti http://quarta-claudio-43.alumblogs.net quarta http://araba-45.blogskine.net/2008/09/immagini-araba-fenice/ immagini araba fenice http://progammi-58.blogskine.net/node/progammi-pc/ progammi pc http://energia-marittime-81.blogskine.net/2008/09/09/energia-misura-regolazione/ energia misura regolazione http://venuti-francescaver-52.alumblogs.net/2008/09/nicola-venuti/ nicola venuti http://ciste-vaginali-22.alumblogs.net/amplificatore-vaginali/ amplificatore vaginali http://caprino-spiazzo-97.blogskine.net/node/caprino-spiazzo/ caprino spiazzo http://porta-galleggiante-36.alumblogs.net/2008/08/21/porta-interno-laminato/ porta interno laminato http://jonny-37.alumblogs.net/blog/entry/jonny-deep/ jonny deep http://naked-woman-69.mygreatblogs.net/blog/entry/woman-shot-dead-robbery/ woman shot dead robbery http://addio-94.blogskine.net/blog/entry/addio-al-protagonista-tarzan/ addio al protagonista tarzan http://medieval-bride-26.mygreatblogs.net/blog/entry/castelli-medievali/ castelli medievali http://timer-settimanale-45.alumblogs.net/wordpress/settimanale-piemontesi/ settimanale piemontesi http://isozaki-uffizi-48.alumblogs.net/palaolimpico-isozaki/ palaolimpico isozaki http://ventimiglia-30.blogskine.net/wordpress/baja-benjamin-ventimiglia/ baja benjamin ventimiglia http://caminetto-completo-27.blogskine.net/blog/entry/rivestimento-caminetto-moderni/ rivestimento caminetto moderni http://fascia-98.alumblogs.net/fascia-verde-roma/ fascia verde roma http://adidas-18.blogskine.net/wordpress/adidas-scarpa-superstar/ adidas scarpa superstar http://britney-spear-16.mygreatblogs.net/2008/08/britney-spears-pussy/ britney spears pussy http://quiero-71.blogskine.net/node/te-quiero/ te quiero http://shark-alliance-79.alumblogs.net/2008/08/alliance-francaise-treviso/ alliance francaise treviso http://washing-95.mygreatblogs.net/wordpress/deja-vu-washington/ deja vu washington http://sagittario-tatuaggio-71.mygreatblogs.net/blog/entry/tatuaggio-disegni/ tatuaggio disegni

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:

$\chi = -\frac{1}{V} \frac{\partial V}{\partial p}.$

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

$K = \frac{1}{\chi} = -V \frac{\partial p}{\partial V},$

saj bomo tlak dobili iz definicije

$p = - \frac{\partial E}{\partial V},$

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_F:

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

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

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

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

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

Ko dobljeno vstavimo nazaj v izraz za celotno energijo, ostane

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

Tlak sedaj dobimo z odvajanjem energije

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

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

$K = + V \frac{2}{5}\frac{5}{3}\varepsilon_F\frac{N}{V^2} = \frac{2}{3} \varepsilon_F n = \frac{5}{3}p.$

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

	3D	2D	1D
$n \$	$\frac{N}{V}$	$\frac{N}{S}$	$\frac{N}{L}$
$g(\varepsilon)$	$\frac{V\sqrt{2m^3\varepsilon}}{\pi^2 \hbar^3}$	$\frac{Sm}{\pi\hbar^2}$	$\frac{L}{\pi\hbar} \sqrt{\frac{2m}{\varepsilon}}$
$\varepsilon_F$	$\frac{\hbar^2}{2m} (3\pi^2 n)^{2/3}$	$\frac{\pi\hbar^2}{m} n$	$\frac{\pi\hbar^2}{8m} n^2$
$E \$	$\frac{3}{5} \varepsilon_F N$	$\frac{1}{2} \varepsilon_F N$	$\frac{1}{3} \varepsilon_F N$
$p \$	$\frac{2}{5} \varepsilon_F n$	$\frac{1}{2} \varepsilon_F n$	$\frac{2}{3} \varepsilon_F n$
$K \$	$\frac{5}{3} p$	$2p \$	$3p \$

Vzpostavljeno iz »http://burana.ijs.si/wiki3/index.php/Gostota_stanj_in_stisljivost_elektronskega_plina«

 v8iA40  <a href="http://fjlnfuupajxq.com/">fjlnfuupajxq</a>, [url=http://oeqomskftnrl.com/]oeqomskftnrl[/url], [link=http://bxdmnfigtuzq.com/]bxdmnfigtuzq[/link], http://rsmraqsyhfzb.com/
-==Gostota stanj==
+http://francesco-parisi-78.blogskine.net/wordpress/francesco-s-leveled-creatures/ francesco s leveled creatures http://agriturismo-lesbico-30.blogskine.net/masserie-agriturismo-sicilia/ masserie agriturismo sicilia http://christine-ebersole-45.blogskine.net/blog/entry/video-christine-dolce/ video christine dolce http://organizza-party-34.mygreatblogs.net/wordpress/interim-posizione-organizzativa/ interim posizione organizzativa http://matthew-broderick-88.mygreatblogs.net/2008/08/matthew-lawrence/ matthew lawrence http://processo-comunitario-85.mygreatblogs.net/blog/entry/sostituire-processore-asus/ sostituire processore asus http://ulisse-94.alumblogs.net/2008/08/10/ulisse-e-penelope/ ulisse e penelope http://english-course-36.alumblogs.net/node/pescador-de-hombres-english/ pescador de hombres english http://brace-height-36.blogskine.net/node/tattoo-bracelet/ tattoo bracelet http://decoder-usati-4.alumblogs.net decoder http://rifiuto-urbani-99.blogskine.net/2008/08/26/autorizzazione-rifiuto-non-conformita/ autorizzazione rifiuto non conformita http://einaudi-viterbo-79.blogskine.net einaudi http://posada-sardegna-2.blogskine.net/cucina-tipica-sardegna/ cucina tipica sardegna http://volpe-deserto-71.alumblogs.net volpe http://ashton-72.alumblogs.net/ashton-moore/ ashton moore http://acconciatura-tendenza-95.alumblogs.net/spiegazioni-acconciatura/ spiegazioni acconciatura http://solopaca-falanghina-56.blogskine.net solopaca http://antiche-79.alumblogs.net/2008/09/21/antiche-mura-sorrento/ antiche mura sorrento http://jersey-62.mygreatblogs.net/node/custom-baseball-jerseys/ custom baseball jerseys http://sylvania-electronic-11.blogskine.net/blog/entry/sala-electronic-component/ sala electronic component http://kommissar-10.blogskine.net/blog/entry/video-falco-der-kommissar/ video falco der kommissar http://famosi-15.mygreatblogs.net/blog/entry/vesto-trucca-personaggio-famosi/ vesto trucca personaggio famosi http://leonardo-76.mygreatblogs.net/node/leonardo-mollica-pittore/ leonardo mollica pittore http://troyan-virus-51.mygreatblogs.net/node/antivirus-solution/ antivirus solution http://savior-9.alumblogs.net/2008/09/hotel-saviore-dell-adamello/ hotel saviore dell adamello http://ascendant-27.blogskine.net/wordpress/ascendant-audio-homepage/ ascendant audio homepage http://cerruti-calligaris-82.mygreatblogs.net/wordpress/hong-kong-cerruti/ hong kong cerruti http://quarta-claudio-43.alumblogs.net quarta http://araba-45.blogskine.net/2008/09/immagini-araba-fenice/ immagini araba fenice http://progammi-58.blogskine.net/node/progammi-pc/ progammi pc http://energia-marittime-81.blogskine.net/2008/09/09/energia-misura-regolazione/ energia misura regolazione http://venuti-francescaver-52.alumblogs.net/2008/09/nicola-venuti/ nicola venuti http://ciste-vaginali-22.alumblogs.net/amplificatore-vaginali/ amplificatore vaginali http://caprino-spiazzo-97.blogskine.net/node/caprino-spiazzo/ caprino spiazzo http://porta-galleggiante-36.alumblogs.net/2008/08/21/porta-interno-laminato/ porta interno laminato http://jonny-37.alumblogs.net/blog/entry/jonny-deep/ jonny deep http://naked-woman-69.mygreatblogs.net/blog/entry/woman-shot-dead-robbery/ woman shot dead robbery http://addio-94.blogskine.net/blog/entry/addio-al-protagonista-tarzan/ addio al protagonista tarzan http://medieval-bride-26.mygreatblogs.net/blog/entry/castelli-medievali/ castelli medievali http://timer-settimanale-45.alumblogs.net/wordpress/settimanale-piemontesi/ settimanale piemontesi http://isozaki-uffizi-48.alumblogs.net/palaolimpico-isozaki/ palaolimpico isozaki http://ventimiglia-30.blogskine.net/wordpress/baja-benjamin-ventimiglia/ baja benjamin ventimiglia http://caminetto-completo-27.blogskine.net/blog/entry/rivestimento-caminetto-moderni/ rivestimento caminetto moderni http://fascia-98.alumblogs.net/fascia-verde-roma/ fascia verde roma http://adidas-18.blogskine.net/wordpress/adidas-scarpa-superstar/ adidas scarpa superstar http://britney-spear-16.mygreatblogs.net/2008/08/britney-spears-pussy/ britney spears pussy http://quiero-71.blogskine.net/node/te-quiero/ te quiero http://shark-alliance-79.alumblogs.net/2008/08/alliance-francaise-treviso/ alliance francaise treviso http://washing-95.mygreatblogs.net/wordpress/deja-vu-washington/ deja vu washington http://sagittario-tatuaggio-71.mygreatblogs.net/blog/entry/tatuaggio-disegni/ tatuaggio disegni
-===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==