Gostota stanj in stisljivost elektronskega plina

Iz Fizika trdne snovi 2007 - 2008

(Primerjava redakcij)

Skoči na: navigacija, iskanje

Trenutna redakcija

Vsebina

1 Naloga
2 Gostota stanj
- 2.1 3D
- 2.2 2D
- 2.3 1D
3 Stisljivost
4 Povzetek

[spremeni] Naloga

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

[spremeni] Gostota stanj

[spremeni] 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:

$g(\varepsilon)=\frac{dN(\varepsilon)}{d\varepsilon}.$

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

$dV_k = 4 \pi k^2 dk \ ,$

vsako stanje v njej pa zaseda

$V_{k0}=\Bigl(\frac{2 \pi}{L}\Bigr)^3 = \frac{(2\pi)^3}{V}$

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

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

Da jo izrazimo le z energijo, upoštevamo zvezo

$\varepsilon = \frac{\hbar^2k^2}{2m}$

in njen odvod

$\frac{d\varepsilon}{dk} = \frac{\hbar^2k}{m},$

pa dobimo končno obliko

$g(\varepsilon) = \frac{V\sqrt{2m^3\varepsilon}}{\pi^2 \hbar^3} .$

Gostota stanj v treh razsežnostih z energijo korensko narašča.

[spremeni] 2D

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

$dS_k = 2 \pi k dk \ ,$

od katere je vsakemu pripada

$S_{k0} = \Bigl(\frac{2\pi}{L}\Bigr)^2 = \frac{(2\pi)^2}{S}.$

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

$g(\varepsilon) = 2 \frac{dS_k(\varepsilon)}{d\varepsilon S_{k0}} = \frac{Sm}{\pi\hbar^2}.$

V dveh dimenzijah torej ni odvisna od energije.

[spremeni] 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

$dL_k = 2dk \ ,$

posamezno stanje pa zavzema

$L_{k0}=\frac{2\pi}{L}.$

Gostota stanj je v tem primeru enaka

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

torej z naraščajčo energijo pada.

[spremeni] 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.

[spremeni] 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«

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+==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>
+|}