ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

tr ♦♠♣♠♥tr♦

❮ Ó

rt♦ rs♦ r♦ ♥ ís ❯

r♦s r♠♦

rr♦

ísst♦ó♦

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❯

❯

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❯

❯

ísst♦ó♦

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❮

♣ít♦ ♥tr♦ó♥

t♦s rs♦

♠t♦♥♥♦ ás♦

♥s str r rst♥

♣ít♦ s rst♥s

rs

♥ ♣r♠t

♥ ♦♥♥♦♥

r rí♣r♦

P♥♦s rst♥♦s

♣ít♦ strtrs rst♥s

strtrs rst♥s

strtrs ús

strtrs ①♦♥s

♠♣qt♠♥t♦s ♦♠♣t♦s

♠♣qt♠♥t♦s s♦♠♣ts

tr ♠♥r sr s strtrs rst♥s

tr♠♥ó♥ ①♣r♠♥t s strtrs rst♥s

♦r♠ó♥ ♦♥ ♣r ró♥ r②♦s ❳ ♣♦r rsts

ó♥ ♥tr ♦r♠ó♥ ② r rí♣r♦

❯♥ ♦r♠ó♥ tr♥t

s♣♦st♦s ①♣r♠♥ts sr♦s ♣♦r ♦♥ó♥

t♦r ♦r♠ ② t♦r strtr

♣ít♦ tr♦♥s ♥ ♥ ♣♦t♥ ♣ró♦ ♦r♠ ♦

t♦r♠ ♦

♦ts s♦r t♦r♠ ♦

s♣r r♠

♥s ♥s ♦ st♦s

♦ts ♦♥s s♦r t♦r♠ ♦

st♦s ♦③♦s ♥ ♥ rst

ísst♦ó♦

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❮ Ó

♣ít♦ ♦♦ r♦♠♠r

♥s ①♣r♠♥ts

♠♦♦

♦♥t étr ♥ ♥ ♠t

♥á♠ ♦s tr♦♥s

t♦

s♣st ♥ ♠♣♦ tr♦♠♥ét♦

Pr♦♣ó♥ ♥

♥s ♥s r

Pr♦♣s tér♠s

♦♥t tér♠ ♥ ♠t t♦

♦s♦♥s tró♥tró♥ ♥ s tr♦♥s

♣ít♦ ♥s tr♦♥s tr♦♥s ♥ ♥ ♣♦t♥ ♣ró♦

é ② ♠ét♦♦ rs rts

♥s tr♦♥s rs

♥s tr♦♥s srs

ó♥ ♥tr strtr ♥s ② strtr rst♥

♠ét♦♦ rs rts ♦ tr♦♥s rt♠♥t ♦s

♥tr♦ó♥ ♦sq♦ ♥ sr♣ó♥ ♥ ♠♦é s♥

♣r♦①♠ó♥ ♦ ♠ét♦♦ rs rts

♦♠♥tr♦s r ♠t♦♦ rs rts

♥♦♥s ❲♥♥r

♣ít♦ ♥á♠ ♦s tr♦♥s

Pr♦♣s ♥rs ♥ó♥ εn(k)

♥ó♠♥♦s tr♥s♣♦rt

♦♦ s♠ás♦ ♥tr ♦s♦♥s

ó♥ ♥tr ♦♥t étr ② s♣r r♠

rít ② ssó♥ ♣r♦①♠ó♥ tr♦♥s ♥♣♥♥ts

♦s

♦♠♥t♦ s♠ás♦ ♥ ♥ ♥♦r♠ ② stát♦

♦♠♥t♦ s♠ás♦ ♥ ♥ ♥♦r♠ ② stát♦

♣ít♦ ♥á♠ r rst♥

♥tr♦ó♥

♦♦ r rst♥ ♥á♠

♦♦s ♥♦r♠s ♥ rst rs ♥♠♥s♦♥ ás♦

♦♦s ♥♦r♠s ♥ r ♥♠♥s♦♥ ♦♥ s

♦s ♠♦♦s ♥♦r♠s s♣rsó♥ ♥tr♦♥s ♣♦r ♥ rst

ísst♦ó♦

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❯

❯

❯

❯

❮

♣ít♦ Pr♦♣s tér♠s ♦s só♦s

♦r s♣í♦ r

í♠t t♠♣rtrs ts

í♠t t♠♣rtrs s

♦♦ ② ♣r t♠♣rtrs ♥tr♠s rst rs

♦♦ ♥st♥

♦♠♣ró♥ ♦s ♦rs s♣í♦s tró♥♦ ② r♦♥ r

t♠♣rtr ② ② ♦s ♦rs ♠♦s ♦s ♦♥♦♥s

t♦s ♥r♠ó♥♦s ♥ ♥ rst

①♣♥só♥ tér♠ ♦s rsts

♠♥s♠♦ ♦só♥ ♥tr ♦♥♦♥s

♦♥t tér♠ r

♦♥t tér♠ ♦s ♠ts ♦♥tró♥ tró♥

sst étr ♦s ♠ts ♣♥♥ ♦♥ T

♣ít♦ t♦s rst♥♦s ♣♥ts

♥tr♦ó♥

t♦s ♣♥ts

♦♠♣♦rt♠♥t♦ tr♠♦♥á♠♦ s ♥ts

♦♥t ó♥

♦♦r ♦s rsts ♥tr♦s ♦♦r

♥tr♦s ♦♦r

♣ít♦ rtrísts ♥♠♥ts ♦s ♠trs

s♠♦♥t♦rs ② ♠♥ét♦s

♥tr♦ó♥

♠♦♥t♦rs

♠♣♦s s♠♦♥t♦rs

♣s ♥rí

strtr ♥s s♠♦♥t♦rs

s♦♥♥ ♦tró♥

♠♣r③s ♥ s♠♦♥t♦rs ♦♥ts s ♠♣r③

ú♠r♦ ♣♦rt♦rs ♥ qr♦ tér♠♦

Pr♦♣s ♠♥éts ♦s só♦s ♥rs

♥♦♠♥♦♦í ♠♥ét ♥ só♦s

ísst♦ó♦

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❯

❯

❯

❯

❯

❯

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

♣ít♦

❯Ó

♥♦s ór♥s ♠♥t ♥ ís st♦ ó♦

á s ♦♥t rtríst ♥ só♦ st♦ s ♦t ♣rá♠tr♦ rrst♥ st♦r srrs ♦s r②♦s ❳ s ♣♥tó ♣♦s str strtr ♦s só♦s ♦♥ ♦s ♠s♠♦s ♣♦r t♥t♦ ♣r♦ r rst♥ sr ♦r♥

á s ♥rí s r♦♥s r P♦r t♦r♠ q♣rtó♥ ás♦ kBTamb ∼140❱

á s ♥rí r tí♣ ♦s tr♦♥s ♥ só♦ ♦♥sr♠♦s q ♥♠t ♣ ♦♥srrs ♥ ♥ ♣r♠r ♣r♦①♠ó♥ ♦♠♦ ♥ ♣♦t♥ rrr ♣♦t♥ q ♦s ♠♥t♥ ♥tr♦ só♦ és r s ♣r♦①♠♠♥t ♥ó♥tr♦ ♠t P♦r ♦ t♥t♦ ♥rí tí♣ r ♦s tr♦♥s só♦ ♣♦♥srrs ♣r♦①♠♠♥t ♥ ♠ts ♦♠♦ ♦r♥ ♠♥t ♥ó♥ tr♦q s ♥ ♥rí ♦r♥ ❱ ♥ ♦s ♠ts

P♦r ♦tr♦ ♦ ♥r s ♥s ♥rí ♠ ♠♥t ♠s r②♦s ❳ t♠é♥ ♥rís ♦r♥ ❱

❯ Ór♥s ♠♥t

é s ís st♦ ó♦

ís st♦ ó♦ t♥ ♥ ♦♥♥t♦ rtrísts áss

sr sst♠s ♠♦s r♣♦s ís tó♠ ② r sr♥ sst♠s ②♠② ♦♠♣♦s ís tr ♦♥♥s ② ♥tr♦ ést sr sst♠s♦♥ ♥ ♥ú♠r♦ ♣rtís tí♣♦ ♦r♥ ♥ú♠r♦ ♦r♦ ♣♦r ♠3

ísst♦ó♦

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❯

❯

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❯

❮ Ó ♥tr♦ó♥

q s ♥r ♥tr♦ ís tr ♦♥♥s s ♥tr ♥ st♦ ♦s só♦s rst♥♦s st♦ ís st♦ ó♦ s ♣s ♥ r♠ ís tr ♦♥♥s st út♠ trt sst♠s ís♦s ② ♥s ♣rtís s ♦r♥ 1023 ♣rtís♠3 P♦r t♥t♦ ♦ só♦s rst♥♦s ♦ ♥♦ ② ♦s íq♦s s ♥②♥ ♥ts sst♠s ② q ♦r♥ ♠♥t ss ♥ss s st♠♥t s ♥ ♠♦ ♥ s♥ ♦♥♦♥s ♥♦r♠s ♣só♥ ② t♠♣rtr t♥ ♥ ♥s ♣r♦①♠ 10−4 10−5 ♠tr ♦♥♥s tí♣

P♦r ♦tr ♣rt ② ♦♠♦ ♦♠♣♥só♥ ♣♦r ♦ ♥tr♦r s ♥tr♦♥s áss q ♣r♥ ♥ s♦♥ ♥ ♦♥♦s s ♥ s♦ó♥ sst♠s ♦♥ s ♥ú♠r♦ ♣rtís stá r s ♣♦ss rs P♦r ♦ s♠♣ó♥ ♥♦r♠ q ♥tr♦ rst♥ q só♦ s ♣ró♦ s ♥♠♥t ♣r st♦ s ♥♠♥t ♣♥trs áss♦♥ ♦s ♦♠♣♦♥♥ts ♥ só♦ ② ás s♦♥ s ♣r♦♣s íss s♦s s♦s ♦♠♣♦♥♥tss ♥s ♥rí ♦s tr♦♥s ♠♥ts♠♦ s♦♦ ♦s s♣♥s sst♠ sí ♦♠♦ ♣r♦♣♦♥r ♠♦♦s q sr♥ s♦s ♦♠♣♦♥♥ts ② ss ♣r♦♣s s ♥ ís ♠♦♦s

♦♦ ♦ q s ♥ ♠③ á♥t stíst ② tó♠

♠♣♦rt♥t ② q ♥t♥r ♦s ór♥s ♠♥t s ♠♥ts q s ♠rá♥ ♥ sí ♦♠♦ t♥r ♥ ó♠♦ s ♠♥ s ♦ss s♦♥s

♠t♦♥♥♦ ♦s ♠♦♦s♥♠♦s ♣s q ♥r ♦ ♥ts ♣♦s qé s ♦ q s ♥t♥ ♣♦r ♥ só♦ r ♥♠t♦♥♥♦ ② ss ♣r♦♣s r qé ♠♦♦s s t③♥ ♣r srr s ♣r♦♣str♠♥r s s ♥sr ♥ sr♣ó♥ ♦♠♣t♠♥t á♥t só♦ ♦ s s ♣ t③r ís s♠ás ♣r srr ♥s ss ♣r♦♣s

rst♥

❨ ♠♦s ♦♠♥t♦ q st ♥ ♣rtr ♦s só♦s rst♥♦s st♦ s ♦s só♦s q♣rs♥t♥ ♥ strtr ♦r♥ s ♣♥t♦ st ♠r♦só♣♦ qí ♣♦♠♦s st♥r ♦r♥ ♦rt♦ ♥ ♦r♥ r♦ ♥ rré♥♦s ♦r♥ ♦rró♥ ♥ s ♣♦s♦♥s ♦s át♦♠♦s ♥♦s

♦r♥ ♦rt♦ ♥ s rr ♠trs ♠♦r♦s ♦ ♦♥stt♦s ♣♦r ♠r♦rstsstrtr rst♥ ♥ s ♥tr♦r ② ♦r♥t♦s ③r ♠♥trs q ♦r♥ r♦ ♥s♥ ♥ r rr q s ①t♥ s♦r ♥ ♦♠♥ ♦♥sr

❯♥ s r③♦♥s ♠♣♦rt♥ts ♣r q ①st♥ ♥ r♥ ♥ú♠r♦ ♥ó♠♥♦s st♥t♦sr♦♥♦s ♦♥ st♦ só♦ str ♥ q ② ♠s strtrs rst♥s q sts♥♥ r ♦♠étr stró♥ s♣ ② ♥ tr♠♥♦ t♣♦ ♥ ③

♠♦s q ♦s sst♠s rst③♦s ♥ ♥r♦ ♥trés s ♥t♦ ♦r♠s ①tr♥srrs ② ♦♠étrs q ♥ ♦ s♦s♣r ♥ rst♥ ás ② s s ❳❱ ♦♥♦♠♥t♦ ♠trs ♣♦rst♥♦s t

♦♥♦ ♣♦r ♦tr ♣rt ♣♦♠♦rs♠♦ q ♣ ♥t♥rs ♦♠♦ ♦s st♥t♦s sts♠tsts ♥ sst♠ ís♦ ♦♥ ♥ tr♠♥ stq♦♠trí

❯♥ s ♦ss ♠ás ♠♣♦rt♥ts q sr♥ ♥ st♦ ♠♣ír♦ ♥s ♣r♦♣síss ♦s só♦s ♣♦r ♠♣♦ q s♥ ♠ts ♦ s♠♦♥t♦rs ♥♦s ♣sts ♣r s♣♦♥r♥ ♥trró♥ ♥tr strtr rst♥ ② s ♣r♦♣s íss ♠tr s s ♠tá♦ ♦s♠♦♥t♦r

♦ ♦♥♥ ♦r s♥ ♠r♦ q trt♠♥t♦ ♦s só♦s ♠♦r♦s ♦ s♦r♥♦s② ♦s íq♦s s♣rt ♥ ♥trés r♥t ♥ ♦s út♠♦s ♥♦s ♥trés q ♥ ♠♦s s♦st♥ ♠♦ q r ♦♥ s ♣♦ss ♣♦♥s t♥♦ós és s♦rs ♠♦r♦s rstsíq♦s t

ísst♦ó♦

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❯

❱ ❯

r③ó♥ q trt ♠②♦r♠♥t ♦s só♦s rst♥♦s s s s♣♠♥t ♥ ss♠♣♦♥s ♥íts q rst♥ ♥tr♦ ♥ ♣r♦♠ srr ♥ú♠r♦♥♥t ♣rtís ♣♦r ♥tí♠tr♦ ú♦ q t♥♥ ♦s sst♠s ♦♥♥s♦s s♠♣♦♥s q♣r♠t♥ ♥ trt♠♥t♦ s♥t♠♥t s♥♦ ♦♠♦ ♣r sr srt♦ qí ♥ t♦ ♣r st♦s s ♣r♦♣s q ♥♦s♦tr♦s st♠♦s ♦ q ♥♦s♦tr♦s sr♠♦s srá s♦ó♥ ó♥ rö♥r ♥ sst♠ ♠s ♣rtís q ♥tr♦♥♥ ♥tr sí ② ♦♥ ú♥♠♣♦ ①tr♥♦ q s ♣q

♥ ♥r s ♥tr♦♥s s♦♥ ♥ ♦♥♦s ♣r♦ té♥t♦ ♣r♦♠ s sr ó♠♦s♦♦♥r ♥ sst♠ ♦♥ ♥ ♥ú♠r♦ r♥ ♣rtís

♥ á♥ ás ♣r♦♠ trs r♣♦s t♥♦ ♦♥ ♥ ♥tró♥ rtt♦r ♥♦t♥ s♦ó♥ ♥ít ①t

♥ á♥ á♥t át♦♠♦ ♦s tr♦♥s ♥trt♥ts ♦ ó♥ ♠♣♦♦♦♠♥♦ ♥ú♦ ♥♦ t♥ t♠♣♦♦ s♦ó♥ ♥ít ①t

sí s ♦ss ♣♦♠♦s s♣rr srr ♥ sst♠ ís♦ ② ♥s s ♦r♥ 1023

♣rtís♠3 ♥t♠♥t ② té♥s ♣r trtr sst♠s ♠s ♣rtís ♣r♦ ①♥♥ sr③♦ ♠t♠át♦ r♥ q ♥♦s♦tr♦s ♥♦ ♥t♥tr♠♦s r③r

♥♥ ♦s ♦♥stt②♥ts

s s♣rr q s ♣r♦♣s rtrísts ♦s át♦♠♦s rs tr♠♥♥ ♥♠♥r ♥tr③ só♦ q s♦♥ ♦♥stt②♥ts sts ♣r♦♣s s♥ ♠r♦ stá♥♠② ♥♥s ♣♦r ♥t♦r♥♦ ♦s át♦♠♦s ♠♦♦ q ♣♥ rr sst♥♠♥t♠ás ② ♣r♦♣s íss q só♦ s ♣♥ ♥r ♣r ♥ ♦t r♥ át♦♠♦s♦♠♦ ♣♦r ♠♣♦ ♦♥t étr rr♦♠♥ts♠♦ ♦r s♣í♦ s tr♥s♦♥s s t

Pró♥ ♦s rsts

② q r ♥♦tr t♠é♥ q ♦s rsts rs s ♣rt♥ strtr ♥♦ s♦♠♥t♣♦r s ts q s♣♦♥ r rr ♥ rst ♣rt♦ ② s♥ ♠♣r③s s♥♦ t♠é♥ ♣♦rr③♦♥s ♣r♠♥t tr♠♦♥á♠s ♦♠♦ r♠♦s ♥ rs♦

♥ t♦ ♥ ♥ rst r ② t♦s ♥ r s♦♦♥s ♠♣r③s t ② s ♥♥♥ s ♣r♦♣s rst s ♥t r♦rr s ♥♦♥s ♣♥ s♠♦♥t♦rs ♦♦r ♦s rsts t

1.1 ❱ ❯

P♦♠♦s ♣♥tr♥♦s ♦s ♣r♥ts áss r♦♥s ♦♥ st♦ só♦

á s st♦ ♥♠♥t ♥ só♦ tr♠♥♦ P♦r qé s st ás s♦♥ sr③s q ♠♥t♥ ♦s át♦♠♦s ♥ ♥ r

ó♠♦ s ♦♠♣♦rt só♦ ♦ ó♥ ♥ ♥♥ ①tr♥

♣r♠r ♣r♥t stá r♦♥ ♦♥ ♦s ♦♥♣t♦s strtr rst♥ ♥ ♥rí ♥ ♦só♥ t Pr♦ ♦♠♦ qr ①♣r♠♥t♦ s♥ ♥tró♥ ② s♣ró♥ st♦♥♠♥t s í ♦♥tstr s ♣r♥ts s ♥♦s ♦♠♦s P♦r ♦♥s♥t ②q str t♠é♥ ♦s ♥ó♠♥♦s ís♦s s♦♦s ♥tró♥ só♦ ♦♥ ♠♣♦s ①tr♥♦s

s ♠♣♦s r ♠ás sts té♥s st ♥ ♣rsó♥ r③♦♥

ísst♦ó♦

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❮ Ó ♥tr♦ó♥

❨ ♠ás ♠♦s t♥r ♦♥♥ q ♦ q ♠♦s s♠♣r s trr ♦♥ ♠♦♦s♣r♦①♠♦s srr♦♥♦ ♥s ♣r♦♣sts ② ♥♦s ♥♦qs ♣r ♠♦rr ♦s ♠♦♦s st♦s♥ q ② q str t♥t♦ tr♦ ①♣r♠♥t q ①st s♦r ♥ó♠♥♦s q st♠♦s♦♥sr♥♦ ♦♠♣r♥♦ ♦♥t♥♠♥t ♦s t♦s ①♣r♠♥ts ♦♥ ♦s á♦s tór♦s

♦t♦ ♥r rs♦ ♦♥sstr ♥ ♥ sr♣ó♥ ás ♥s s ♣r♦♣s♥♠♥ts ♥ só♦ t③♥♦ ♠♦♦s r③♦♥♠♥t s♥♦s

st ♦♥♥t♦ ♣r♦♣s ② ♠♦♦s ♦ ♠♦♦s ② ♣r♦♣s ♦r♠rá♥ ♣r ♥♦s♦tr♦s sss q ♥♦s ♣r♠t♥ ♣tr ♦♠♥t ♠s s ♣♦t♥s ♥ só♦ rst♥♦

♥♠rr♠♦s qí ♥s s q s♦♥ s q ♥♦s ♥ ♣r♠tr ♠♣③r rr♥♦s ís st♦ só♦

♦♥t ♦ rsst étr

♦ ó♥ ♥ ♠♣♦ étr♦ ♦♠♣♦rt♠♥t♦ ♦s st♥t♦s só♦s s r♠♥tst♥t♦ st ♠♥t ♣rs♥t ♥ r♥♦ ró♥ t♥ ♠♣♦ q ♣ ♦♥srrs ♦♠♦ ♠ás ♠♣♦ s ♣r♦♣s íss ♦r♥rs sí rsst ♥ ♠t s t♠♣rtrs♣ sr ♦r♥ 10−10 Ω♠ ♠♥trs q ♥ ♥ s♥t s 1022 Ω♠

② q ♦rr ♣s ♥ t♦rí q ♥♦s ♣r♠t st♥r r♠♥t ♣♦rqé ♦♠♣♦rt♠♥t♦ ♦s st♥t♦s só♦s r♥t ♦♥t étr

♦r s♣í♦

r♦ ♦♥ ís stíst ás ♦r s♣í♦ ♥ só♦ rí sr ♦♥st♥t② ♦♥ ② Ptt ♥ ♠r♦ ♦♠♣♦rt♠♥t♦ ♦r s♣í♦ s t♠♣rtrsrst sr ♣r♦♣♦r♦♥ T 3 ♥ s♥ts ② ♦♠♦ αT +βT 3 ♥ s♦ ♦s ♠ts str♠♦s ♦ r♦ rs♦ ó♠♦ ①♣r st♦s rst♦s

Pr♦♣s ①♣♥só♥ ② ♦♥t tér♠ ♦s rsts

♦♥strr♠♦s ♥ t♦rí r♦♥s ♦s sst♠s rst♥♦s q ①♣q ♠♥r ♥tr♦s rst♦s ①♣r♠♥ts sts ♣r♦♣s

♣r♦♥t

♥♦s ♠trs ♣♦r ♦ ♥ rt t♠♣rtr ♥ ♦rr rsst♥ ♣s♦ ♦rr♥t étr

t ♦s rsts ó♥♦s ♠ts

st♦s ♠trs ♣rs♥t♥ ♠♦s rs♦s ♥ ss rs rt q stá♥ rt♠♥tr♦♥♦s ♦♥ ♦ sr rsts ó♥♦s st♦ s q t♥♥ rs ♥ ♥s ♥ ♣♥t♦str♠♥♦s r rst♥ ♦ ♠ts q stá♥ ♦r♠♦s ♣♦r ♥ ♦♥♥t♦ tr♦♥s q s♠♥ s r♠♥t ♥ t♦♦ rst ♠♥r q s ♥s r t♥ ♥ stró♥s♣ ♠② s♦③

ró♥ r②♦s ❳ ② ♥tr♦♥s ♣♦r strtr rst♥

sts ♦s s♦♥s s♦♥ ♠② st♥ts ② ♣r♠t♥ str ♥ó♠♥♦s ís♦s r♥ts ♣r♠r♥tr♦♥ ♦♥ s ♣rtís rs s♥ ♦♥ s ♣rtís ♠♥éts ② ♦♥ ♦s ♥ú♦s

Pr♦♣s ♠♥éts

s ♣r♦♣s ♠♥éts ♦s só♦s ♣♦r qé s♦♥ ♣r ♦ ♠♥ét♦s ♣♦r qé② rr♦♠♥ts♠♦ ♣♦r qé s ♣r♦♥ ♦r♥♦♥s ♠♥éts ♥ ♥ rst s♦♥ s♠♠♣♦rt♥ t♥t♦ s ♥ ♣♥t♦ st tór♦ ♦♠♦ ss ♣♦♥s

ísst♦ó♦

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t♦s

♦♠♦ ② s ♦♠♥t♦ s ♣rs♥ s ♦♠♣t♠♥t ♥tr ② q ♥ ♣rr ♣♦rr③♦♥s tr♠♦♥á♠s str♠♦s ♥♥ s ♣ró♥ ♥ ♦s só♦s qé ♣r♦♣sstá♥ s ♣rs♥ ♠♣r♦♥s ♦ t♦s ♥ rst

str♠♦s ♣♦r qé ♦♦r ♦s rsts

♦s ♥trsr♠♦s t♠é♥ ♣♦r ♥á♠ ♦s tr♦♥s ♥ rst ♦♥ á♥♠♦ ①trr ♥♦r♠ó♥ ♠s♠ ② sí ♣♦r rs♦r ♦ ♣r♥t q ♥ts s ♣♥t♦r ó♠♦ ♦♥♦r st♦ ♥♠♥t rst ② st♦ ♦s st♦s ♣rtr♦s ♦qr ♥tró♥ q ♥♦s ♠s♠♦

1.2

❯♥ ♦♥ó♥ ♠♣rs♥ ♣r ♦♠♥③r st♦ ♦s só♦s rst♥♦s s q s ♥♣r♥♣♦ s♣♠♦s ♥r ♦♥ t♦t r ♠t♦♥♥♦ q ♣♥s♠♦s sr ♠♥t ♥ só♦ st♦ s q♥t r qé ♥t♥♠♦s ♣♦r ♥ só♦ rst♥♦

♥q ♥ ♥ ♣r♥♣♦ ♣r ♣rr q st♠♦s ♣♥t♥♦ ♥ ♠t♦♥♥♦ ② ♠②s♠♣♦ ② q sr q st ♣r♦①♠ó♥ ♠t♦♥♥♦ ①t♦ sst♠ s ♠② r③♦♥s ♦♠♣r♠♦s ♦♥ s ♣r♦①♠♦♥s q ♥♦s r♠♦s ♦♦s r③r ♣♦str♦r♠♥t

♦s t♣♦s tr♦♥s

♦ ♣r♠r♦ q ♠♦s s st♥r r♠♥t ♦s t♣♦s tr♦♥s ♥ ♦s só♦s rst♥♦s ♠♥r ♣r ♦♠♦ ② s ③♦ srr ♠♦és

♦s tr♦♥s ♥ q s♦♥ q♦s ♦s q s s tr② ♥ ♦♥tró♥♠♣♦rt♥t ♥ qí♠♦

♦s tr♦♥s ♦s ♦♥s ♦ ♦r q stá♥ ♦s rt♠♥t ♦s ♥ú♦s ② q♣♦r ♦♥s♥t ♥♦ r♥ ♥♥ ♣r ♥ s ♣r♦♣s só♦

st r♥ó♥ r ♥tr tr♦♥s stá s ♥ s ♥rás r ♦s tr♦♥s s st♥ts ♣s tó♠s q s ♠str♥ ♥ r

❯ ❮ Ó

Pr r♥♦s ♥ st ♠t♦♥♥♦ s ♥ s♣ ①t♥só♥ ♥tr s ♣r♦①♠♦♥s q s r♦♥ ♥ rs♦s♥tr♦rs ♣♦r ♠♣♦ ♥ s♥tr ís á♥t ♣r ♠♣③r srr ♠♥r s♥ s ♠♦és

ísst♦ó♦

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❮ Ó ♥tr♦ó♥

♥ rá♦ ♠♦s ♥ r♣rs♥tó♥ ♥rí r ♥ ♥ó♥ ♥ú♠r♦ tó♠♦ ♦s ♠♥t♦s s r♦ q ♥rí ♦s ♥s ♠♥t ♥ ♦r s♦t♦ ♦♥ r♣③ ♠♥tr Z ♦ q ♥ q ♦s tr♦♥s ♦s ♥s ♣r♦♥♦s ♦s át♦♠♦s stá♥ ③♠ás ♦s ♥ú♦

❯♥ ♣r♥t q sr ♥♠t♠♥t s s♥t ás st♦s ♥s ♥♥♥ s ♥rí ♣♦r ♣rs♥ ss ♥♦s ♥ ♦r♥ó♥ rst♥ só♦

• Pr rs♣♦♥r ♥ ♠♥r tt ♣♦♠♦s r ♥ st♠ó♥ t③♥♦ ♥♠♦♦ ♠② s♥♦

♥ ♥ ♥ át♦♠♦s s♣r♦s ♥tr sí ♥ st♥ a r r♣rs♥t ♣♦t♥ ♦ r♦ ♥ í♥ r♦ ② ♣♦t♥ s át♦♠♦ str♥♥t♠♥t s♣r♦ ♦s ♥♦s í♥s ♥rs

❯

♥ s♦ át♦♠♦s ♦♠♣t♠♥t s♣r♦s t♦♦s ♦s tr♦♥s ♥ ♥s tó♠♦s♦♥ E < 0 stá♥ ♦s ♥ú♦ ♦r s♥ ♠r♦ ♦s tr♦♥s ♥ ♥s ②♥rí s s♣r♦r −∆E ② ♥♦ stá♥ ♦s ♣♦r ♦ q s ♦♠♣♦rt♠♥t♦ ís♦ sr♥t s♦ át♦♠♦s s♦s

♦♠♦ ♥ sst♠ r ♦sr♠♦s q ∆E = 4q2/a ♦♥ q s r ♥ú♦ t♥♠♦s q ∆E = 11,5 ❱ ♣r a = 5 ② q ∆E = 19,2 ❱ ♣r a = 3 ♦♥♦♥♦ ♦s ♦rs tí♣♦s s st♥s ♥tr♥rs ♥ ♦s só♦s st♦ ♥♦s ♥ r q s♦♠♥t q♦s ♥s tró♥♦s ② ♥rí sté ♥ r♥♦ ♦s10 − 20 ❱ ♣♦r ♦ ♥ ♥ ♥rí r♦ q ♠♦s ♥♦ ♦♠♦ ♦s át♦♠♦s ♥♥t♠♥t s♣r♦s ♥ str ♥♥♦s rt♠♥t ♣♦r ♣rs♥ ss ♥♦s ♥ rst ♦ q rst♦ ♦s tr♦♥s ♥ srr ♥ ♣♦t♥s♠r tó♠♦

• óts q st st♠ó♥ tt t♠é♥ ♦♥ ♦♥ ♦ q ♦♥♦♠♦s r ♦s ♦rs q t♦♠ ♥ó♥ tr♦ ♥ ♠t t♠♥t ♣♦t♥ ♦r♦ ♥ í♥ ♣r♣♥r s♣r ② q ♣s ♣♦r ♥tr♦ ♦s ♥ú♦s s r♣rs♥t ♥ r

ísst♦ó♦

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❯ ❨ ❯Ó

❨ ♦ q ♦s ♦rs tí♣♦s ♥ó♥ tr♦ ♥ ♠t s♦♥ Φ ≃ 2,4 − 4,5 ❱♠♦s ór♥s ♠♥t ♦♥♥ tt♠♥t

st s♣ró♥ ♥tr ♦s tr♦♥s ♥ ♥s ♥rí s♣r♦r ② q♦s q stá♥ ♥ ♥s♠ás rt♠♥t ♦s s s♥ ♣r ♣♥t♠♥t♦ ♠t♦♥♥♦ q qr♠♦s rs♦r♥t♠♥t ♦s tr♦♥s ♥tr♥♦s ♦s ♦♥s ♥♦ ♥ t♥r ♥♥ ♥ ♥ s ♣r♦♣stró♥s ♦s só♦s

♠t♦♥♥♦ ás♦

♦♠♦ s ♦ ♠ás rr ♦t♦ rs♦ s r ♥ sr♣ó♥ ás ♥ só♦ t③♥♦♠♦♦s r③♦♥♠♥t s♥♦s

♥♠♦s ♣s q r ♥ ♣r♦①♠ó♥ ♦ q ♥t♥♠♦s q s ♥ só♦ ♣r ♦s t♦s st s♥tr sí ♣s ♠t♦♥♥♦ q ♣r♦♣♦♥♠♦s ♦♠♦ ♠♦♦ só♦ s

H = Hiones + Helectrones + Helectron−ion

♦♥

Hiones =∑

i

P2i

2Mi+

1

2

∑

i 6=j

V (Ri −Rj)

s ♠t♦♥♥♦ ♥ ♦♥♥t♦ ♦♥s q ♥tr♦♥♥ ♠♥t ♥ ♣♦t♥ V q ♣♥s♦♠♥t s ♣♦s♦♥s rts ♦s ♦♥s ♣rt

Helectrones =∑

i

p2i

2m+

1

2

∑

i 6=j

e2

|ri − rj |

s ♠t♦♥♥♦ q sr ♦s tr♦♥s ♥ ♦ tr♦♥s ♦♠♦ s ♦♥♥tró♥ ♦♦♠♥ ♥tr ♦s ♠♥trs q út♠♦ tér♠♥♦

Helectron−ion =∑

i,j

v(ri −Rj),

r♣rs♥t ♥tró♥ ♥tr ♦s tr♦♥s ② ♦s ♦♥s ♥ s♣st♦ q s ♣ srr♠♥t ♥ ♣♦t♥

s ♠t♦♥♥♦ ♠♦♦ t♥ ② ♥ s ♦r♠ó♥ rs ♣r♦①♠♦♥s ♦ ♠t♦♥s♠♣íts q qr♠♦s r rs s ♥ ♣r♥♣♦ ♥tró♥ ♥tr ♦s ♦♥s ♥♦ s s ♣♦r r♣rs♥tr ♣♦r ♥ ♣♦t♥ st♦ s

ísst♦ó♦

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❮ Ó ♥tr♦ó♥

♣r♥♣♠♥t q ♥tró♥ ♥tr ♦s tr♦♥s s ♣s rrs s ♠② í trtr♥tr ♦trs ♦ss ♣♦r t♦ ♣r♥♣♦ ①só♥ P ♥tró♥ ♥tr ♦♥s ② tr♦♥s t♠♣♦♦ ♣ r♣rs♥trs ♣♦r ♥ ♣♦t♥ s♥♦ ②q ② q ♥r t♦ q ♥tr♦ ♣r♥♣♦ ①só♥ P s♦r ♥tró♥ ♥tr♦s tr♦♥s ♥ tr♦♥s ② ♦s tr♦♥s ♥tr♥♦s ♦r ♦s ♦♥s ♦♥s

♠t♦♥♥♦ sr ♣rtís s♣rs

♣sr ♦ ♠ás ♥t ♠♦s tr♦r ú♥ ♠ás st ♠t♦♥♥♦ ② st♥r♠♦s ♥tr ♣♦só♥ qr♦ ♦s ♦♥s ② s ♣♦só♥ ♥st♥tá♥

Hiones = Heqion−ion +Hvibrac

Helectro−ion = Heqelectron−ion +Helectron−vibrac

st s♣ró♥ ♠t♦♥♥♦ ♥ tr♦③♦s ♥♦s ♣r♠trá str ♠♥r r③♦♥♠♥t♦r♥ st♥ts ♣r♦♣s íss rst s♦s s st♥ts ♣rts ♠t♦♥♥♦

1.3 ❯

♣r♠r ♠♦♦ só♦ q s st♦ st st ♠♦♠♥t♦ s♦ ♠♦♦ tr♦♥srs ♣r ♠ts q ①♣ r③♦♥♠♥t ♥ ♥s s ♣r♦♣s ♠tás ♠ás♦♥♦s ♥ ♠r♦ ② rst♦s ①♣r♠♥ts ♥ ♥t ♦♥tró♥ ♦♥ ♦s á♦s♥ttt♦s ♦ ♠♦♦

♥♦ ♥ r rs♠♥ ♦s q s ♥♥tr ♥ t♦rí ② ♦ q s ♠①♣r♠♥t♠♥t ♥♦♥tr♠♦s

♦s ♥ ♦s ♦♥ts tr♥s♣♦rt

♦♥t ①♣r♠♥t♠♥t ♣♥ ♠♣♦ ♠♥ét♦ ② t♠♣rtr ②♠ás s♥♦ ♦♥t ♥♦ ♦♥ ♥ ♥♦s ♠ts ♦♥ ♣r♦ ♣r tr♦♥srs ♦♠♥t ♦s ♠ts ♥♦s t♥♥ ♥ RH q ♦♥r ♦♥ ♠♦♦

♥t♦rsst♥ ♥ s t♦♦s ♦s s♦s rsst♥ ♥ ♠r ♣r♣♥r ♥ ♥♦r♠ ♣♥ ♥t♥s ♠s♠♦

♠♣♦ tr♠♦étr♦ ♦ s♠♣r s t♥ ♠s♠♦ s♥♦ ♣r Q q q s ♦t♥ ♥ ♠♦♦

② ❲♠♥♥♥③ ①♣r♠♥t♠♥t t♠♣rtrs ♥tr♠s κ/σT ♣♥ t♠♣rtr

♦♥t ♥ ♦rr♥t ♦♥t♥ ♣♥ t♠♣rtr ② ró♥ ♠♣♦étr♦ q ♣q♠♦s

♦♥t ♥ ♦rr♥t tr♥ rt ①♣r♠♥t ♥♦ ♦♥r ♦♥ σ(ω) tr♦♥s rs

♦s s ♣r♦♥s tr♠♦♥á♠s

Pr ♦r s♣í♦ cv ♦s ♠ts t♠♣rtr cv ≃ γT + AT 3 ②tr♠♥♦s ♠ts ②♦ tér♠♥♦ ♥ ♥ T ♥♦ t♥ ♥ q r ♦♥ ♦r γ tr♦♥s rs

P r♦rrs qí s ts q s♦ ♦♥ ♣r trt♠♥t♦ ♦s át♦♠♦s ♠ttró♥♦s ♥ ♥ ♠♦♦t♠é♥ ♠② s♠♣♦ ♣r♦①♠ó♥ rtr♦

s♥♠♥t s r ♥♦ ♣r♦①♠ó♥ ♦r♥♣♣♥♠r q s ♣ó ♥ trt♠♥t♦ s ♠♦ésú♥ s ♠ás s♥s

❱és ♣♦r ♠♣♦ s♥tr ♥♠♥t♦s ís

ísst♦ó♦

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tér♠♥♦ ♥ T 3 s ♦♠♠r t♥ s♥♦ ♦♣st♦ ② s ♠ís♠♦ ♠♥♦r q ①♣r♠♥t

♠ás s♦s ♦s ♥ s ♣r♦♥s ♠♦♦ tr♦♥s rs ② ♦s st♦♥s ássq ♦ ♠♦♦ ♥♦ ①♣ ♣r♠r s á s ♥ú♠r♦ tr♦♥s ♦♥ó♥ ♣♦r át♦♠♦ ♥ ♥ ♠t ②♦ ♠♥t♦♦♥stt②♥t t♥ rs ♥s qí♠s s♥ stó♥ ♥♠♥t s s♥t ♠♦♦ ♥♦ ①♣ ♥ s♦t♦ q ②♦♠♣st♦s q ♥♦ s♦♥ ♠tá♦s á s r③ó♥ ♥t♦♥s q ①st♥ s♠♦♥t♦rs ②s♥ts

ás ♥t s rá q s ♥♦s♦tr♦s t♦♠♠♦s ♥ ♥t ♠♣♦ ♣r♦♦ ♣♦r ♥ ♦♥♥t♦stát♦ ♦♥s s ♣♥ str s ♣r♦♣s tr♥s♣♦rt ♥ só♦ ♥ t♦rís ♥ sq s ♠♦♥ ♥q ♥♦ ♠♥r rást s ♦♥♦♥s ♠♦♦ tr♦♥s rs♦♥ ♠♦ ♠♦♦ r ② ♦♠♠r ② st♦ ♥á♠ ♦s tr♦♥s ♦ ♣♦t♥ r rst♥ ♥♥tr ♣s q ♥♥ ♦s ♦♥s s♦r s ♦s♦♥s ♦s tr♦♥s s♦♠♥t s ♠② ♥♦t ♥♦ ♥ t♦rí s ♥②♥ ♦s ♠♦♠♥t♦s ♦s ♦♥s♦ q ①♣ rsst ♦s ♠ts ② s ró♥ ♦♥ ♠♦♦ ♦s tr♦♥s rs

ísst♦ó♦

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ísst♦ó♦

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♣ít♦

♦♠♦ s ♦ ♥ ♥tr♦ó♥ ♥♦ ♠♦s ♠t♦♥♥♦ ♠♦♦ ♥ só♦♣♦♠♦s str ♥ ♣r♦r♠ ♣r♦①♠♦♥s sr♣ó♥ ③ ♠ás t ♥ só♦ ♥ ♣r♠r r ♦s tr♦♥s q ♥ ♠♦♦ tr♦♥s rs st♦ ♥rs♦s ♥tr♦rs ♥♦ sr♥ ♥♥ú♥ ♣♦t♥ ①tr♥♦ ♠♦s ♦♥srr q s ♠♥ ♦r ♥ ♥♣♦t♥ stát♦ ♦s ♦♥s r rst♥ só♦ ❨ ♣♦str♦r♠♥t ♠♦s str s♦♥s♥s íss ♠♦♠♥t♦ ♦s ♦♥s rs♣t♦ s ♦♥ró♥ stát

Pr♦ ♥ts ♥trr ♥ ts ♠♦♠♥t♦ ♦s tr♦♥s ♥ ♥ ♣♦t♥ ♣ró♦ ♦s ♦♥s qt♦s rá q str ás s♦♥ s ♣♦ss str♦♥s ♣rós q ♣♥♥♦♥trrs ♥ ♥tr③ ❯♥ s r③♦♥s r♥ srr♦♦ s st♠♥t q s②♥ st♦ ♦s só♦s ♦♥ strtr rst♥ ♥ ♦♥tr♣♦só♥ ♦♥ ♦s só♦s s♦r♥♦s♦ ♦♥ ♦ íq♦s ♦ q ♣r♠t ♥s té♥s íss ② ♠t♠áts á♦ ♠♦ ♠ás ♣rss② ♣♦t♥ts

rr s ♦r♠s ①tr♥s ♦s rsts q s ♥♥tr♥ ♥ ♥tr③ ♦ q s♥ rr ♥ ♦s ♦rt♦r♦s ♣rs♣s♦ ② ♥ s♦ ❳❱ r♥ q ♦s rsts st♥♦♥stt♦s ♣♦r ♥ r♣tó♥ ♦qs é♥t♦s ♦② ♥ í s♠♦s q ♦s ♦qs q s r♣t♥♣r ♦r♠r rst ♦♥ át♦♠♦s ♦ r♣♦s át♦♠♦s ♦r♠ q ♦s rsts s♦♥ str♦♥s♣rós tr♠♥s♦♥s r♣♦s át♦♠♦s

♥ ♥tr③ ♦s sst♠s ís♦s ♣rs♥t♥ ♥tr♦♥s ♥tr ss ♦♠♣♦♥♥ts st♦ s ♥tr♦♥s ♥tró♥ tró♥tró♥ ♥tró♥ ♣♦♦♣♦♦ ♥tr ♠♦és t ♥♦srá r ♥ sst♠ ♦♥s q ♣♦r ♦♥♦♥s ♥rí ♠í♥♠ ♥♦s ♥ s st♥s♥tr ♦s át♦♠♦s stró♥ ♦s ♠s♠♦s ♥ s♣♦ st♦ s r rst♥ q ♦r♠♥ s ♥ q sst♠ s ♥♦♥trrá t

♦♦ st♦ ♥♦s qr r q ♥ ♣r♥♣♦ ♠tr t♥♥ ♣rt♠♥t tr♠♥ ás s strtr rst♥ ② ♣♦♠♦rs♠♦ s♦♠♥t s ♥ s♦s qr♦ ♠tst ♥♦s q ♠ st♦ ♠tst ♣ sr ♠② r♥

② q r ♥♦tr qí q strtr rst♥ ② s ♣r♦♣s íss stá♥ str♠♥tr♦♥s P♦r ♠♣♦ ♥ ♦s ♠ts strtr s t q s ♥ s ♣rát♠♥t ♥♦r♦♥ ② st♦ ♣r♠t q ss ♣r♦♣s íss s♥ ♣rát♠♥t sótr♦♣s ② q ♦s ♠ts♥♦ t♥♥ ♥ ró♥ ♣r

1.1 ❱

♦♥♣t♦ q s♣ ♠♥r ♥ q ♦s ♦qs é♥t♦s s r♣♥ ♣r ♦♥sttr rst s r rs st ♦♥♣t♦ s♦♠♥t stá r♦♥♦ ♦♥ ♦♠trí

strtr ♣ró s②♥t rst

❱és ♣♦r ♠♣♦ s♥tr ♥♠♥t♦s ís ♥ ♠r♦ ♣r♥ ①tr♥ ♥ r♣♦ ♥♦ s s♥t ♣r srr ♦ ♥♦ s rst♥ ② q s ♦rr♥t q

♥ ♠str sté ♦♥stt ♣qñ♦s ♠♥t♦s q s ♠r♦só♣ s♦♥ ♠② r♥s ② q ♦♥t♥♥ ♥ r♥ ♥ú♠r♦ ♦♥s ♦♥ ♥ stró♥ ♣ró tr♠♥s♦♥ st♦s st♦s ♣♦rst♥♦s s ♥♥tr♥ ♠ás ♦♠ú♥♠♥t q ♦srsts ♠♦♥♦rst♥♦s ♥ ♦s q ♣r♦ s ♣rt ♥ t♦♦ ♦ ♥♦ ② r♦ ♠str

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❮ Ó s rst♥s

r♠♦s ♦s ♥♦♥s q♥ts

❯♥ r rs s ♥ stró♥ srt ♥♥t ♣♥t♦s s♣st♦s t ♠♥r q stró♥ ♣r é♥t s r ♣♥t♦ stró♥ s q ♠r♠♦s

❯♥ r rs s ♥ ♦♥♥t♦ srt♦ t♦rs ♥♦ ♦♣♥r♦s q ♦r♠♥ ♥♦♥♥t♦ rr♦ ♥ s ♦♣r♦♥s ó♥ ② sstró♥ P♦r ♦♥s♥t ♥ t♦r ♦ ♣♥t♦ r rs ♦ ♣♦♠♦s ♦♥strr ♦♥ ♥ s trs t♦rs ♥♦ ♦♣♥r♦s ♦♥ ♦♥ts♥tr♦s st♦s t♦rs ai ♦s ♠r♠♦s t♦rs ♣r♠t♦s ② ♠♣♥ q

R = n1a1 + n2a2 + n3a3

s♥♦ R ♥ t♦r qr r rs ♦s ni s♦♥ ♥tr♦s

• s ♥trs♥t rr q r rs t♥ ♦♠♦ ♦♥ó♥ q t♥t♦ ♦r♠ ♦♠♦ ♦r♥tó♥ stró♥ ss ♣♥t♦s ♣r♠♥r é♥t ♥♦ ♣s♠♦s ♥ ♣♥t♦ ♦tr♦ r ❱♠♦s st♦ ♥ ♥ ♠♣♦ ♥ r ♠♥s♦♥

❯ ①♦♥ ♥♦ ♥tr

❯♥ ♠♣♦ ♠♥s♦♥ s♥♦s ♥ r ♦r♠ ♣♦r ♥ ♦♥♥t♦ ①á♦♥♦s ②①t♣st♦s st rt♥ s♠trí ①♦♥ ♥ rs ♦sr q ♦s ♣♥t♦s sñ♦st♥♥ ♥ ♥t♦r♥♦ r♥t ② ♣♦r ♦t♥t♦ st r ♥♦ s rs

❯ ①♦♥ ♥tr

♥ ♠r♦ s ñ♠♦s ♥ ♣♥t♦♥ ♥♦ ♦s ♥tr♦s ♦s①á♦♥♦s sí s ♦♥sr t♥t♦ ♦r♠ ♦♠♦ ♦r♥tó♥ rst s qr ♦s ♣♥t♦s ♠s♠ ♣♦r ♦♥s♥t st ♥r sí s ♥ r rsóts q st ♥ r t♥s♠trí tr♥r

♥ ♦s ♣r♦♠s s ♠str q ♦ s♦♥

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❯ ♦s ♦♥s ♣♦ss t♦rs s ♣r r ①♦♥ ♥tr

② q t♥r ♥ ♥t q ♦s t♦rs s q ♥♦s ♥♥ r rs ♥♦ s♦♥ ♥♥♥ ♠♥r ú♥♦s ② s♥ ♠r♦ ♥♦ ♣♦r ♦ ♠♣rs ♣r♠r ♥ó♥ st♦s ♦♥ r ♥ r ♥tr♦r q s s♦ r rs ①♦♥ ♦♥ s ♥♦s ♣♦ss ♦♥s t♦rs s

❯ ♥ ♥ r

• P♥s♠♦s ♥ ♦tr♦ s♦ ♥ trs♠♥s♦♥s ♥ r ú

♥tr ♥ ♥tr♦r q s♦r♠ ñ♥♦ ♥ r ú s♠♣ ♥ ♣♥t♦ ♥ ♠♦ ♦és rrí ♣♥sr q ♦s ♣♥t♦s ♥tr♦ ♦ ♥ r ♠♥r st♥tq ♦s ♣♥t♦s ♦s érts ♦s♦s Pr♦ s♠♦s q ♦s ♣♥t♦s ♦s ♦s q r♦♥ ♦ r♦♥stt②♥ t♠é♥ ♥ ♦ strtr♣ rs ♥t♦♥s ♦♠♦ ①r♦s ♦♥ ♥ ♣♥t♦ ♥ s ♥tr♦ ♦ ①r♦s ♦♥ ♥ ♣♥t♦ ♥ s ♥tr♦r sí ♣s r s ♥ r rs

❯ ♥ st♥ts tr♣ts t♦rs ♣r♠t♦s

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❯ ♥

• st♠♦s ♦r s♦ r

ú ♥tr ♥ s rs q s ♦t♥ r ú ñé♥♦♥ ♣♥t♦ ♥ ♥ s ss rs ♦ Pr♦ ♦s ♣♥t♦s q stá♥ ♥ ♥tr♦r s rs ♦r♠♥ t♠é♥ ♥ r ú♥ ②s rs stá♥ ♦s ♣♥t♦s q ♥ts♦♥sttí♥ r ♦r♥sí ♣s r s ♥ r rst♠é♥

• ①♦♥ ♥ trs ♠♥s♦♥s

❯ ①♦♥ s♠♣

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❯ P❱

• ♠♥ ♥♦s ♠ás ♣ró①♠♦s ♥ ♥ r rs ♦s ♣♥t♦s ♠ás r♥♦s ♥ ♣♥t♦♦ rátr ♣ró♦ ♥♦s ♥ ♥t♦♥s q ♣♥t♦ r t♥ ♠s♠♦♥ú♠r♦ ♠♦s ♠ás ♣ró①♠♦s s ♥ú♠r♦ ♥♦s ♠ás ♣ró①♠♦s q s ♥ ♣r♦♣♥trí♥s r s ♠ ♥ú♠r♦ ♦♦r♥ó♥ r

1.2 ❯ P❱

❱♠♦s ♥r ♦r ♦ q ♥t♥♠♦s ♣♦r ♣r♠t ♦ ♥ ♣r♠t s ♦♠♥ s♣♦ q trs♦ ♣♦r ♦s t♦rs q ♦♥stt②♥ ♥ r rs r♥t♦t♠♥t s♣♦ s♥ s♦♣♠♥t♦s ♥ ♦s

s ♥t q ♥ ♣r♠t ♦♥t♥r ♥ ♣♥t♦ r rs ♦ str ♦♥str ♦r♠ q ♦♠♣rt M ♣♥t♦s ♦♥ M s ♣r♠ts♠é♥ stá r♦ q ♥♦ ② ó♥ ú♥ ♣r ♥ ♣r♠t ♣r♦ qr s t♥r ♠s♠♦ ♦♠♥ s δ s ♥s ♣♥t♦s r ♦♠♥ v ♣r♠t st q ♦♥t♥ ♥ ♣♥t♦ ♣♦r t♥t♦ s ♠♣ q v δ = 1 ② ♣♦r ♦♥s♥t v = 1/δ

❯ st♥ts s ♣r♠ts

❯♥ ♣♦s ó♥ ♣r♠t ♦ ♣♦r ♦tr ♣rt s s♦r ♦♠♥ s♣♦♥♦ ♣♦r ♦s ♣♥t♦s r = x1a1 + x2a2 + x3a3 s♥♦ xi ∈ [0, 1]Pr♦ st♦ ♥♦s ♣r t♥r ♥ ♥ ♣r♠t q t♥ s♠trí sí ♣♦r♠♣♦ ♥ ♦t♥rí♠♦s ♥ ♣r♣í♣♦ ♦♦ q ♥♦ ♣♦s s♠trí ú q s ♠ást r

❯ ♥ ❲♥rt③

tr ♣♦s ó♥ ♣r♠t s s♥t trá♥s ♦ss♠♥t♦s q ♥♥ ♥ ♣♥t♦ r♦♥ t♦♦s ss ♥♦s ♠ás ♣ró①♠♦s♥t♦♥s ♥ ♣r♠t s q♦r♠ ♣♦r ♦♠♥ s♣♦ qq ♥tr♦ ♦s ♣♥♦s st♦rs q♦s s♠♥t♦s srá♣s ♠♥♦r ♣♦r♦ q ♥♥ s♦s♣♥♦s

♦♠♦ st ♦♥stró♥ ♥♦ s rr ♥ s♦t♦ ♦s t♦rs q ♥r♥ r s♥♦ s♦♠♥t r ♠s♠ ♣r♠t sí ♦♥str ♠ ❲♥rt③ t♥ t♦ s♠trí

óts q s ♣ r ♥ r ♦♥ ♥ ♥ú♠r♦ ♦♦r♥ó♥ ♥ ♥♦ ♥q r ♥♦ s ♥ ♦♠♥t s ♥st q t♦♦s ♦s ♣♥t♦s r t♥♥ ♥ ♥ú♠r♦ ♥♦s ♠ás ♣ró①♠♦s ♣r t♦♦s ② s r t♥rá♥ ♥ú♠r♦ ♦♦r♥ó♥ ♥ ♥♦

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♠♣♦

• ①♦♥ s♠♣ ♥ trs ♠♥s♦♥s

❨ s ♦♠♥t♦ q ♥ ♥ r ①♦♥ rs s

❯ ①♦♥ ♥tr

q ♦rrs♣♦♥ ♥ r ①♦♥ ♥tr ② ②♦s t♦rs ♣r♠t♦s stá♥ sñ③♦s ♥ r q ♠str ♦tr ó♥ ♣♦s t♦rs ♣r♠t♦s

ñ♠♦s ♦tr♦ ♣♥♦ ♣r♠♥t ② s♣r♦ ♥ st♥ c ♦t♥♠♦s ♥ r①♦♥ s♠♣ ♥ s ♥

①♦♥ s♠♣

Prs♠ ①♦♥

② ♥ ♦♥♥♦♥ s ♥♣rs♠ ①♦♥ ♦♥ ♥ ♣♥t♦ ♥tr♦♥ s rs s ss ♠♥trs q ♥ ♣r♠t ♥ ♣♦r ♦s t♦rs♣r♠t♦s q ♠♦s s♦♦ s ♥♣r♣í♣♦ ②s ss s♦♥ r♦♠♦s rr ♥tr♦r

1.3 ❯ ❱

♦♠♦ ② s ♦ ♥trr ♥ ♠♦s s♦s s út ♣rsrr s♠trí ♥♥♦s ♥ q ♥♦ s♦♥ ♣r♠ts ♠s s ♥ ♦♥♥♦♥s

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❮P

sts s r♥♥ s♣♦ ♦♠♣t♠♥t ♥♦ s♦♥ trss ♣♦r ♥ s♦♥♥t♦ ♦s t♦rs q ♥♥ P♦r ♦♥s♥t ♥ ♦♥♥♦♥ s t♠ñ♦ ♠②♦rq t♠ñ♦ ♣r♠t ♣r♦ t♥♥ s♠trí r

Pr s♦ s rs rs ② s ♥♥ s s ♦♥♥♦♥s ♦r♠s ♣♦r♦s ♦s q ♥♦s ♣r♠tr♦♥ ♥r ó♠♦ s♦♥ ss rs

❯ s ♦♥♥♦♥s

♥♥ s ♦♥st♥ts r ♦♠♦ ♦s ♥ú♠r♦s q s♣♥ t♠ñ♦ ♥♦♥♥♦♥ ♥ rsts ú♦s ♦♥t a ss rsts

óts q ♥ s♦ r ú s♠♣ ♥ ♦♥♥♦♥ ② ♣r♠t ♦♥♥

1.4 ❮P

❱♠♦s ♥tr♦r ♦r ♦♥♣t♦ r rí♣r♦ ♥ ♠ r rí♣r♦ ♦♥♥t♦ t♦rs G q sts♥ q

eiG·R = 1

♣r t♦♦s ♦s t♦rs R ♥ ♥♦♠r rí♣r♦♦ ♥ s ♠♥s♦♥s s♣♦ ♦♥ stá♥ ♥♦s s♦s t♦rs ♥rs♦ s♣♦ r

st ♦♥♥t♦ t♦rs R rs♣t♦ ♦s q ♥♠♦s s ♠ r rt♥ ró♥ s r rí♣r♦

♥ó♥ s s♦♠♦s ♥ t♦r ♣♦só♥ qr r s ♦♥♥t♦ t♦rs G q ♥ ♦♥s ♣♥s ♦♥ ♣r♦ ♣st♦ q

eiG·(r+R) = eiG·r

② ♣♦r t♥t♦ G ♦r♠ ♦♥s ♣♥s ♦♥ ♣r♦ R ♣r♦

• s t♠é♥ ♥ Pr ♠♦strr♦ ♦♥sr♠♦s ♥ q s ♥ ♦♥ t♦rs♣r♠t♦s a1 a2 a3 ② ♦♥str②♠♦s ♦s t♦rs

b1 = 2π·a2 × a3

a1 · (a2 × a3); b2 = 2π

a3 × a1

a1 · (a2 × a3); b3 = 2π

a1 × a2

a1 · (a2 × a3)

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② rq♠♦s q st♦s t♦rs s♦♥ t♦rs ♣r♠t♦s r rí♣r♦

t♥ qbi · aj = 2πδij =⇒ eibi·R = 1 ∀R ∈ RD

♦♠♣rés

♦s t♦rs ai ♥♦ s♦♥ ♦♣♥r♦s ♣♦r ♦ q ♦s bi t♠♣♦♦ ♦ s♦♥ ② ♦r♠♥ ♥ s s♣♦rí♣r♦♦ P♦r t♥t♦ ♥ t♦r k qr s♣♦ rí♣r♦♦ ♣ ①♣rsrs ♦♠♦

k = k1b1 + k2b2 + k3b3

♦♥ ♦s ♦rs ki ∈ R ♥ ♣r♥♣♦ ♠t♣♠♦s st t♦r k ♣♦r ♥♦ qr R = n1a1 + n2a2 + n3a3 ♦♥ ♦s ni ♥tr♦s

k ·R = (k1n1 + k2n2 + k3n3) · 2π

♦r ♥ k srá ♥ t♦r s s ♠♣ q eik·R = 1 ∀R ∈ st♦ ♥♦s q ♣ré♥tss ♥tr♦r sr ♥ ♥tr♦ ② q ♣♦r ♦ t♥t♦ ♠♣rs q ki ∈ Z ♣r qk = k1b1 + k2b2 + k3b3 s ♥ t♦r

P♦r ♦♥s♥t s ♥ ② ♥♦ ♦s ♣♦ss ♦♥♥t♦s t♦rs ♣r♠t♦ss ♦♥♥t♦ b1,b2,b3

• ♦♠♦ s t♠é♥ ♥ ♣♦♠♦s ♦♥strr s ③ s rí♣r♦ st rí♣r♦ rstsr r rt ♦r♥ q í♠♦s ♣rt♦ Pr ♠♦strr♦ s♠♦s q ♦s t♦rs Pq s♦♥ ♦s rí♣r♦♦s G ♥ sr ts q ♠♣♥ eiP·G = 1 ♣r t♦♦s ♦s t♦rs G P♦r ♠♥s♦♥s ♦s t♦rs P stá♥ ♥ s♣♦ rt♦ ♣r♦ ♠♦s s♣♦♥r q ♥♦♣rt♥♥ s♦ s♥ q ①♣rsó♥ P = x1a1 + x2a2 + x3a3 t♥r ú♥ xi /∈ Z s♦ s rt♦ ② ♦ q ró♥ eiP·G = 1 sr á ♣r qr t♦r G s♦♠♦s t♦r bi ② ♦t♥♠♦s q eiP·bi = e2πixi 6= 1 ♥ ♦♥tró♥ ♦♥ ♦ q s ♦ ♥ts r♠r q P ♣rt♥í

♦t ♦♠♥ ♣r♠t s (2π)3/v s♥♦ v ♦♠♥ ♣r♠t ♥ r rt ♥ t♦ s♠♦s q |a1 · (a2 × a3)| = v ② q s ♦♠♥ ♥rr♦ ♣♦r ♦s t♦rs ai ♠♥r q

b1 · (b2 × b3) =(2π)3

a1 · (a2 × a3).

♦t ♣r♠t ❲♥rt③ r rí♣r♦ s ♠ ♣r♠r ③♦♥

r♦♥ P❩ ♥♦♠r q s♦♠♥t s st ♣r♠t s♣♦ rí♣r♦♦♦♠♦ s ♦ ♥ s♣♦ rt♦ s ♠ ❲♥rt③♠é♥ ①st♥ s♥ trr ③♦♥s r♦♥ q ♦rrs♣♦♥♥ ♦♥♥t♦ r♦♥s s♥ts s q s trs♥♦ ♥♦ ♦s ♣♥♦s rst♥t♦s ♥ ss r♦♥s t♥ ♠s♠♦ ♦♠♥ q P❩ ♦ q ss♥ ♠♦ ♠♥♦s r♥t♠♥t q P❩ ést s ♠ ♠s s ③♦♥ r♦♥

❯♥ ♣♥♦ r s ♥ ♣♥♦ q st ♥ t♦r G rá q st ♣♥♦ ♥rá ♦♥ó♥ ♦♥ ♥ ró♥ ♦s r②♦s ❳ ♥ rst♦rí

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P

♠♣♦s rs rí♣r♦s

♦♠♦ ♠♦s ♣♦r st♦s ♠♣♦s q st♥t trr ② rr ♦s t♦rs s♥tr ② r s r rí♣r♦ ♥ r s rí♣r♦ ② ♥♠♥t s rí♣r♦ ♦♠♦ ② sí♠♦s ♣♦r rst♦ ♥tr♦r

1.5 P

Pr qr r rs trs ♣♥t♦s ♥♦ ♦♥s ♠s♠ ♥♥ ♥ ♣♥♦ rtr

♦ ♣♥♦ r s♠trí tr♥s♦♥ st ♣♥♦ ♦♥t♥ ♥♥t♦s ♣♥t♦s r ♣♥t♦s q ♦♥stt②♥ ♥ ♠♥s♦♥ s♦r ♦ ♣♥♦

♠♠♦s ♠ ♣♥♦s rtrs ♥ ♦♥♥t♦ ♣♥♦s r ♣r♦s ♠♥t s♣r♦s ♦r♠ q qr ♣♥t♦ r stá ♥ ♥♦ ♦s ♣♥♦s sr♦ t♠é♥ q ♥ ♣♥♦ qr r ♣rt♥ ♥ ♠ ts ♣♥♦s rtrs

♦r ♥ ♥ r ♣ rtr③rs ♣♦r ♠ás ♥ ♠ ♣♥♦s rtrs ② ② ♥ró♥ rt ♥tr ♦s t♦rs ② ♦s ♣♥♦s rtrs ①♣rs ♥ s♥t t♦r♠

♥ ♠ ♣♥♦s rtrs Π s♣r♦s ♥ st♥ d ♥t♦♥s ①st♥ t♦rs q s♦♥ ♣r♣♥rs st♦s ♣♥♦s ♠ás ♦rt♦ ♦s t♥ ♥ ♦♥t 2π/d

♠♥r s G s ♥ t♦r ♥t♦♥s ①st ♥ ♠ ♣♥♦s rtrs♣r♣♥rs G q stá♥ s♣r♦s ♥ st♥ d s♥♦ 2π/d ♠ó♦ t♦r q s ♠ás ♦rt♦ ♣r♦ G

♠♦stró♥ ♦♥sr♠♦s ♥ó♥ fK(r) = eiK·r s♥♦ K = 2π

d n n s rs♦r ♣r♣♥r ♠ ♣♥♦s Π st ♥ó♥ s ♣ró ♣r♦♦ 2π/|K| = d ② ♣♦r ♦ t♥t♦t♦♠ ♦s ♠s♠♦s ♦rs s♦r ♦s ♣♥♦s ♠ Π ♣r♣♥rs K

♠♦s ♣♥t♦ r = 0 ♥t♦♥s fK(r) = 1 ♦ q ♥ q s ♣♥t♦♣rt♥ ♥ ♣♥♦ ♠ Π ② q ♦r ♥ó♥ fK(r) s 1 ♥ t♦♦s♦s ♣♥♦s ♦♠♦ s♦s ♣♥♦s ♦♥t♥♥ t♦t ♥t♦♥s s ♠♣ qfK(R) = eiK·R = 1 ∀R ∈ ② ♣♦r ♦ t♥t♦ K s ♥ t♦r

♦r ♥ st K s ♠ás ♦rt♦ ♦s t♦rs q s♦♥ ♣r♣♥rs ♠ ♣♥♦s Π ♣♦♥♠♦s q ①str ♥ G t q |G| < |K| ②q s ♣r♦ K ♥t♦♥s ♦s ♣♥♦s ♥♦s ♣♦r ó♥ eiG·r = 1 t♥♥ ♥

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❮ Ó s rst♥s

s♣ró♥ ♠②♦r q d ② q 2π/|G| > d ② st♦ s♥ q fG(r) ♥♦ ♦s ♠s♠♦s♦rs ♥ t♦ ♠ ♣♥♦s Π q ♣rt♠♦s sí ①strá ú♥ R ∈ tq eiG·R 6= 1 ② ♣♦r t♥t♦ G /∈

K ∈ ♠ás ♦rt♦ ♦s t♦rs ♥ ♥ ró♥ ró♥eiK·r = 1 ♦♥ r r♣rs♥t ♥ ♣♥t♦ qr ♥♦s ♥ ♥ ♦♥♥t♦ ♥♥t♦♣r♦ ② srt♦ ♣♥♦s ② s♣ró♥ ♠r♠♦s d

eiK·r = 1 ♦r K · r =2πn ♦♥ n ∈ Z Pr n ② ♥ ♣♥♦ ♣r♣♥r K q eiK·r = 1 ♣♦♥♠♦s ♦s ♣♥♦s s t♣♦ ♦♥st♦s st♦ s q ♣r r

♥ ♣♥♦ Π1 s ♠♣ K·r =|K|D = 2πn D s st♥ s ♦r♥ ♣♥♦ Π1♠♠trs q ♣r r ♥ ♣♥♦ Π2 s ♠♣ K ·r =|K|(D+d)= 2π(n+1) ♦r D+ds st♥ s ♦r♥ ♣♥♦ Π′ s♣r♦ d Π ♥ rs♠♥ |K| = 2π/d ②♦s ♣♥♦s ♥♦s ♣♦r ró♥ eiK·r = 1 stá♥ s♣r♦s ♥ st♥ d = 2π/|K| ②s♦♥ ♣r♣♥rs K

❯ P♥♦s rst♥♦s ② K

P♦r ♦tr ♣rt eiK·r = 1 ♥♦s ♥ ♠♥♦s ♦♥♥t♦ R ♦s ♣♥t♦s P♦r t♥t♦ ♦s ♣♥♦s Π ♥tr♦rs ♥②♥ ♥ ♠ ♣♥♦s rtrs

♦r ♥ ♦s ♣♥♦s rtrs q st♠♦s ♦♥sr♥♦ stá♥ s♣r♦s ♣♦r♠♣♦ ♥ st♥ md s♥♦ d st♥ ♥tr ♦s ♣♥♦s ♥♦s ♣♦r eiK·r = 1② sú♥ ♣r♠r ♣rt t♦r♠ rá ♥ t♦r ♦♥t 2π/(md) qs ♣r♣♥r st♦s ♣♥♦sst♦ ♦♥tr s♣♦só♥ q ♥str♦R s ♠ás ♦rt♦ ♦s t♦rs ♦♥st ♣r♦♣ ♣♦r ♦♥s♥t st♥ ♥tr ♦s ♣♥♦s rtrs s ♦♠♥td ♦r♠ q ♦♥♥t♦ ♣♥♦s s♣♦ q r♥ q

eiK·r = 1

♣r t♦r R ♠ás ♦rt♦ ♥ s ró♥ ♦♥ ♦♥ ♠ ♣♥♦s

rtrs ② s♣ró♥ s d

ísst♦ó♦

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P

Eq.(1.9)

❯ P♥♦s rtrs ② ♣♥♦s ♣♦r

♦♠♦ ♦♥s♥ t♦r♠ t♥♠♦s q t♦♦ t♦r s ♥♦r♠ ♥ ♠ ♣♥♦s rtrs ♠ás s s ♦♠♣♦♥♥ts ♥ t♦r G ♥♦ t♥♥ ♥ s♦r ♦♠ú♥ ♣♦r ♠♣♦ t♦r ♦♠♣♦♥♥ts ♦ ♦♠♣♦♥♥ts rst q st♥ ♥tr♣♥r ♦s ♣♥♦ss ♥rs♠♥t ♣r♦♣♦r♦♥ |G|

• s r♦ ♥t♦♥s q ♦s ♣♥♦s ♠ás ♣♦♦s ♣♥t♦s r s♦♥ q♦s q stá♥ ♠áss♣r♦s ② ♣♦r ♦ t♥t♦ st♦s ♣♥♦s q s♦♥ ♦s ♠ás ♠♣♦rt♥ts r ♦rrs♣♦♥♥ ♦st♦rs r rí♣r♦ ♠ás ♣qñ♦s

• ♥♠♦s ♦s í♥s r ♥ ♣♥♦ rtr ♦♠♦ s ♦♦r♥s t♦r q s ♥♦r♠ s ♣♥♦ ② q s ♠ás ♦rt♦ s s q ss ♦♠♣♦♥♥ts ♥♦t♥♥ ♥♥ú♥ t♦r ♦♠ú♥♦t s ♦♦r♥s s♦♥ rs♣t♦ ♥ ♦♥♥t♦ tr♠♥♦ t♦rs ♣r♠t♦s ♦r♠ q ♦s í♥s r h k ② l ♣♥♥ ó♥ t♦rs ♣r♠t♦s b1,b2,b3

♦tó♥ ♣r ♦s ♣♥♦s r ♦s í♥s r (h, k, l) ♦ (hkl) s sr♥ ♦♥ s♥♦ ♥t♦ q ♣r♥ t♥r s♦r ♥ú♠r♦ ♥ stó♥ st♦ s s ② q srr −2 s sr 2

s r♦♥s rtrs s s♣♥ ♥tr ♦rts r♦s [n1n2n3] ♦♥n1a1 + n2a2 + n3a3 s ró♥ ♦♥sr

♦♥♥t♦ ♣♥♦s rst♥♦s í♥ (hkl) q s ♥r♥ ♦♥ s ♦♣r♦♥s s♠trí q♥ ♥ sst♠ rst♥♦ ♥r♥t s ♥♦t♥ ♣♦r hkl

♠♥r < n1n2n3 > ♥♦s ♥ ♦♥♥t♦ r♦♥s ♦t♥s ♣r ss♠trís rst [n1n2n3]

❨ q ♥s ♥♦♦s ♥ r s ♦♥st♥t

ísst♦ó♦

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❮ Ó s rst♥s

❯ ♦tó♥ ♣r ♦s ♣♥♦s rtrs

• ♦t s♦r s ③♦♥s r♦♥

♠♦s ♥♦ ♣r♠r ③♦♥ r♦♥ P❩ ♦♠♦ ❲♥rt③ P♦r♦♥s♥t srá ③♦♥ ♦♥t♥ ♣♦r ♦s ♣♥♦s s♦rs ♦s t♦rs ♠ás♣qñ♦s s tr③♥ ♦s t♦rs s ♦r♥ st ♦s ♣♥t♦s ♠ás ♣ró①♠♦s ♠s♠② ♦ s tr③♥ ss ♣♥♦s st♦rs

❯ r♦♥tr ❩

P♦r ♦♥s♥t s G s ♥♦ s♦s t♦rs ♣♥♦ st♦r G stá ♥♦ ♣♦r ó♥

k · 12G =

(12G)2

=(12G)2

⇐⇒ 2k ·G = G2

s♥♦ k ♥ t♦r ②♦ ①tr♠♦ stá ♥ ♦♣♥♦ és r

♦♥sr♠♦s ♦r ♦s t♦rs k ② k′ q t♥♥ ♠s♠♦ ♠ó♦ Pr q k− k′ s♥ t♦r G s t♥rá q

k− k′ = G =⇒ k′2 = k2 +G2 − 2k ·G =⇒ 2k ·G = G2,

♦♥ ♣♦♠♦s ♦♥r q ♣r q ♦s t♦rs ♠s♠♦ ♠ó♦ t♥♥ ♦♠♦ r♥♥ t♦r G ♠♦s t♦rs ♥ t♥r ss ①tr♠♦s s♦r ♣♥♦ st♦r G st♦s ♦s ①tr♠♦s ♠♦s t♦rs ♥ str ♥ r♦♥tr ♥ ③♦♥ r♦♥

❯ r♦♥tr ❩

ísst♦ó♦

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♣ít♦

❯❯

s ♥t q ♥ ♥tr③ ♥♦ s♠♣r s ♥♥tr♥ rs rs ♦♠♦ ♦♥stt②♥ts ♦s rsts ♦ q s♥rá q s♦♠♥t s r♣trí ♥ ♦♠♣♦♥♥t tr♠♥♦ ♥ t♦♦ rst

♦s só♦s s♦♥ strtrs íss ② ♥♦ ♥ ♦♥♥t♦ ♣♥t♦s ♠t♠át♦s P♦r s♦ ♥ rst♣ srrs ♠rs q ♥♦s strtr ♦♠étr s②♥t ♥ rst ♥t♦♦♥ sr♣ó♥ át♦♠♦s ♠♦és ♦♥s t q ② ♥ ♣r♠t q s r♣t

P♦r ♦ t♥t♦

strtr rst♥ r rs s

♦♥ s ♠♦ s ♦♥♥t♦ át♦♠♦s ♠♦és ♦♥s t q s r♣t♥ ♥♥♠♥t♣r ♦r♠r rst

♥ú♠r♦ ♠♥t♦s q t♥♥ s rst ♣ sr ♠② r♥ ♥ ♥♦s rsts ♣r♦tí♥s ♣♥ ♦♥t♥r ♦r♥ 105 át♦♠♦s ② ♦ q ♠♣♦rt st♥r qí s q s q ♥♦s strtr ♣ró rst st♦ ♥♦s ♣r♠trá ♣♦str♦r♠♥t ♥ ①♣r♠♥t♦s♥ ♦s q s ♥t♥t srr strtr ♥ só♦ qr st♥r ♥tr ♦♥tró♥ ♣rt ♣ró r rs ② ♦♥tró♥ s rst

P♦r ♦tr ♣rt st ♥ó♥ strtr rst♥ ♥♦s ♣r♠t sr rts st♥ts♦♠♦ r♥ts ♠s♠ r ♣r♦ ♦♥ st♥t s

♣ ♦♥srrs ♦♠♦ ♥ strtr rst♥ ♦♠♣st ♣♦r ♥ ♠ás ♥s tó♠ st ♥

[0, a2 (x+ y + z)

]

♣ ♦♥srrs ♦♠♦ ♥ strtr rst♥ ♦♠♣st ♣♦r ♥ ♠ás ♥s tr♦ ♣♥t♦s

[0, a2 (x+ y) , a2 (y + z) , a2 (z+ x)

]

2.1 ❯❯

①♣♦♥r♠♦s ♥ st ♣rt♦ ♥s s strtrs rst♥s ♠ás s♥s ♦♥ ss♠♦♥♦tó♠ ② tó♠s

strtrs ♦♥ ♥ s♦♦ t♣♦ át♦♠♦

2.1.1 strtrs ús

• strtr ú s♠♣ ♦rrs♣♦♥ ♥ r rs ♥ q ♣♦só♥ ♦s át♦♠♦s ♦♥ ♦♥ ♦s ♣♥t♦s r

• strtr ú ♥tr ♥ ♥tr♦r r rs ♣ ♦♥srrs ♦♠♦ ♥ strtr rst♥ ♦♠♣st ♣♦r ♥ r rs ♠ás ♥ s tó♠ st ♥

[0, a2 (x+ y + z)

]

ísst♦ó♦

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❮ Ó strtrs rst♥s

• strtr ú ♥tr ♥ s rs r rs ♣ ♦♥srrs ♦♠♦ ♥ strtr rst♥ ♦♠♣st ♣♦r ♥ ♠ás ♥ s tr♦ ♣♥t♦s

[0, a2 (x+ y) , a2 (y + z) , a2 (z+ x)

]

• strtr ♠♥t♦♥sst ♥ ♥ r ♦♥ ♥ s ♦

[(0, 0, 0) , a4 (x+ y + z)

] s ♥ ♣ ♦♥srrs ♦♠♦

♦s rs ♥trr③s ♦r♠ q ♥ s s s♣③♦ ♦ r♦ ♦♥♣r♥♣ ♦tr ♥ ♦♥t 1/4 ♦♥

♦♥♥♦♥ s ♥ ♦♥ s ♦ stró♥ ♦s ♣♥t♦s r s q ♣♥t♦ t♥ tr♦ ♥♦s ♠ás ♣ró①♠♦s st♦s ♥ ♦s érts ♥ ttrr♦

❯ strtr ♠♥t

❯ ② strtr ♠♥t

2.1.2 strtrs ①♦♥s

• strtr ①♦♥ s♠♣♥ ♥ ♣♥♦ ② ♦♠♦ ② s ♦♠♥t♦ s ♣♦♥strr ♥ ①♦♥ ♦♥ s♠trí tr♥r s♦♦♠♦s ♦s ♣♥t♦s r ♥ ♦s érts ♥①á♦♥♦ ② ♥ s ♥tr♦ ①t♥♥♦ s stró♥ t♦♦ ♣♥♦ t♥♠♦s ♥ strtr ①♦♥ s♠♣st♥♦ ♥ s♥ ♣ ♣♥t♦s s♣③ ♥tr♠♥ st♥ rs♣t♦ ♣r♠r ♣

♦♦♠♦s ♥ át♦♠♦ ♥ ♣♥t♦ r rs s♣♦só♥ ♣♦r ♣s s ② strtrs ♥ ①♦♥ s♠♣

ísst♦ó♦

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❯❯

• strtr ①♦♥ ♦♠♣t P strtr ①♦♥ ♥tr ♥ ♣♥♦ ♣ rr ♦tr strtr ①♦♥ st♥t s ♣r♦♠♦s s♥t ♠♥r ♦♦♠♦s ♥ s♥♦ ♣♥♦ ♣♥t♦s ♥ tr♠♥st♥ ♣r♠r ♣♥♦ ♥♦ ♣♥t♦ s♦r ♥tr♦ trá♥♦ q ♦r♠♥ ♦s ♣♥t♦s ♣r♠r♣♥♦ ♥ trr ♣♥♦ ♣♥t♦s s ♦♦ ♠♥t♥♥♦ st♥ ♥tr ♣♥♦s ♥tr♦r s ♦r♠ ♦♦♥♦ ♦s♣♥t♦s ♥ s ♠s♠s ♣♦s♦♥s q t♥♥ ♦s ♣♥t♦s ♥ ♣r♠r ♣♥♦ s♣♦só♥ ♣♦r ♣s s ♥t♦♥s ② strtr s ♥ ①♦♥ s♠♣ ♦♥ ♥ s ♦[(0, 0, 0) ,

(13a,

13b,

12c)]

2.1.3 ♠♣qt♠♥t♦s ♦♠♣t♦s

❯ ♦s rís♦♠♣ts

❱♠♦s ♦r s♣♦♥r q s strtr rst♥s ♥ ♦ rí ♠♥r q strtr stá ♦r♠ ♣♦r ♥♦♥♥t♦ ♦s r q s r♣t♥

• ♥ ♦s ♠♥s♦♥s ♦r♠ ♠ás ♦♠♣t str s ♦s ♥♦s ♣♥t♦s ♥ s q s ♣rs♥t ♥ r

• ♥ trs ♠♥s♦♥s ② ♦s ♠♥r ♣r s ♦s ② s♦ r ♦s strtrs st♥ts

①♦♥ ♦♠♣t P ①♦♥ ♦s♣

❯ ♦s rís ♦♠♣ts P

♦r ♣r♠r ♣ ♦♦♠♦s s♥ ♣ ♦s ♥ q s ♦s s stú♥ s♦r♦s ♦s ♣r♠r ♣ ♦r♠♥♦ ♦tr strtr tr♥r ♦♠♦ r s♣③rs♣t♦ ♣r♠r ♣

ísst♦ó♦

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❮ Ó strtrs rst♥s

trr ♣ s ♦♦ st♦ ♥♠ s ♣♦s♦♥s s ♦s ♣r♠r ♣ rts♦r s s s♥ t

s♣♦só♥ ♣♦r ♣s s ♥t♦♥s strtr rst♥t s ♥ r ①♦♥s♠♣ ♦♥ ♥ s ♦

[(0, 0, 0) ,

(13a,

13b,

12c)]

t♦s s ♦s rís s♦♥ t♥♥ts s ♠♣ q c/a =√8/3 = 1, 6333 ♦♠♦

♥t♠♥t s strtrs rst♥s q s ♥♥tr♥ ♥ ♥tr③ ♥♦ stá♥ ♦♥stts♣♦r ♦s rís ♦s ♦rs ①♣r♠♥ts q s ♦t♥♥ ♣r ró♥ c/a s♦♥ r♥♦s ♦stór♦s c/a =

√8/3 = 1, 6333 ♣r♦ ♥♦ ①t♠♥t s

♦♥s ①♣r♠♥ts c/a só♦s P

❩♥

♦ ❨ ❩r

s♥ ♣ ♦s s stú s♦r ♦s ♦s ♣r♠r ♣r♦ trr ♣ ♥♦ s ♦♦ rt♠♥t♥♠ ♣r♠r s♥♦ ♥ ♦tr ♣♦só♥ ♣♦s ②s r♣t ♣r ♦t♥r ♥ ♣♠♥t♦ ♥t♦♥s strtr q s ♦t♥ s

t♠♥t t♥ ♣♥♦ (111) ♦♠♣t♦♦♠♦ s ♣ ♦srr ♥ r

ísst♦ó♦

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❯❯

strtr

2.1.4 ♠♣qt♠♥t♦s s♦♠♣ts

♥♦ ♦♥ ♦♣r ♣♥t♦ strtr ♦♥ ♥ ♦ rí ♥ stró♥♠♥♦s ♦♠♣t q s ♥tr♦rs P ② ♣r♦ ♠ás ♦♠♣t q ♣♦r ♠♣♦ s ♦r♠ ♣♦r ♥ r ú ② s s ♠♦♥♦tó♠ ♥♦t q s ♥ ♦♥ s s♠♣st♦ s ♥ rst rs

♦♠♦ ② s ♦ ♣♦ró♥ s♣♦ q ♦♣♥ s ♦s ♥ st strtr s ♠♥♦r q q ♦♣♥ ♥ P ②

rst③♥ ♥ s strtr ♦s ♠ts ♥♦s ② ♦tr♦s ♠ts ♣♥t♦ só♥♦ ❲ ♦ sí ♦♠♦ ❱ ② r

strtrs ♦r♠s ♣♦r ♦s t♣♦s át♦♠♦s

Ps♠♦s ♦r ♦♠♥tr ♥s strtrs s♥s ♦♥ ss ♦s át♦♠♦s

• ♣♦ ♦rr♦ s♦♦

s ♥ r ♦♥ s ♦[(0, 0, 0) ,

(12 , 0, 0

)]

r strtr

♦r♦ s ♥♥tr ♥ ♥ ♥t♦r♥♦ ♦tér♦ r♦♦ át♦♠♦s s♦♦ ♦♦r♥ó♥ ♦♥♦s s ss í♣r♦♠♥t s ♦ ♠s♠♦ ♦♥ s♦♦

• ♣♦ ♦rr♦ s♦ s

s ♥ r ♦♥ s ♦[(0, 0, 0) ,

(12 ,

12 ,

12

)]

ísst♦ó♦

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❮ Ó strtrs rst♥s

r strtr s

♦r♦ s ♥♥tr ♥ ♥ ♥t♦r♥♦ ú♦ r♦♦ át♦♠♦s s♦ ♦♦r♥ó♥ ♦♥ ♦ss ♦♦ í♣r♦♠♥t s ♦ ♠s♠♦ ♦♥ s♦

• strtr t♣♦ ③♥♥ sr♦ ③♥

♦♥sst ♥ ♥ r ♦♥ ♥ s ♦ ♦♥ át♦♠♦s ♥ (0, 0, 0) ② át♦♠♦s ♥(1/4, 1/4, 1/4)Prs♥t♥ st strtr ♦s ♦♠♣st♦s ♥r♦s ♦s t♣♦s ❱ ❱ ❱❱ ♦s ♥ú♠r♦s r♦♠♥♦s♥ rr♥ ♦s r♣♦s sst♠ ♣ró♦

❯ ② strtr ③♥♥

2.2 ❯❯

❱♠♦s str ♦r ♥s strtrs rst♥s ♠♣♦rt♥ts q ♣r♥ str ♥s♥♥ú♠r♦ ♦♠♣st♦s ís♦s

strtr ♠♥t

♦♥sst ♥ ♥ r ♦♥ ♥ s ♦[(0, 0, 0) , a4 (x+ y + z)

] s ♥ ♣ ♦♥srrs

♦♠♦ ♦s rs ♥trr③s ♦r♠ q ♥ s s s♣③♦ ♦ r♦ ♦♥♣r♥♣ ♦tr ♥ ♦♥t 1/4 ♦♥

♦♥♥♦♥ s ♥ ♦♥ s ♦ stró♥ ♦s ♣♥t♦s r s q ♣♥t♦ t♥ tr♦ ♥♦s ♠ás ♣ró①♠♦s st♦s ♥ ♦s érts ♥ ttrr♦

ísst♦ó♦

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❯❯

❯ strtr ♠♥t

❯ ② strtr ♠♥t

strtr t♣♦ ③♥♥ sr♦ ③♥

♦♥sst ♥ ♥ r ♦♥ ♥ s ♦ ♦♥ át♦♠♦s ♥ (0, 0, 0) ② át♦♠♦s ♥(1/4, 1/4, 1/4)Prs♥t♥ st strtr ♦s ♦♠♣st♦s ♥r♦s ♦s t♣♦s ❱ ❱ ❱❱ ♦s ♥ú♠r♦s r♦♠♥♦s♥ rr♥ ♦s r♣♦s sst♠ ♣ró♦

❯ ② strtr ③♥♥

ísst♦ó♦

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❮ Ó strtrs rst♥s

strtrs ♦♠♣ts

❯ ♦s rís♦♠♣ts

• ♥ ♦s ♠♥s♦♥s s ♠♥♠♦s ♦s ♣♥t♦s ♥ ♦♠♦♦s rís ♦s r ♦r♠ ♠ás ♦♠♣t strs s q s ♣rs♥t ♥ r

• ♥ trs ♠♥s♦♥s ② ♦s ♠♥r ♣r s ♦s ② s♦ r ♦s strtrs st♥ts

①♦♥ ♦♠♣t P ①♦♥ ♦s♣

❯ ♦s rís ♦♠♣ts P

s♥ ♣ ♦s s stú s♦r ♦s ♦s ♣r♠r ♣ ♦r♠♥♦ ♦tr strtrtr♥r ♦♠♦ r s♣③ rs♣t♦ ♣r♠r ♣

ísst♦ó♦

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❯

❯

❯❯

trr ♣ s ♦♦ st♦ ♥♠ s ♣♦s♦♥s s ♦s ♣r♠r ♣ rts♦r s s s♥ t

s♣♦só♥ ♣♦r ♣s s ② strtr s ♥ ①♦♥ s♠♣ ♦♥ ♥ s♦

[(0, 0, 0) ,

(13a,

13b,

12c)]

t♦s s ♦s s♦♥ t♥♥ts s ♠♣ q c/a =√

8/3 = 1, 6333

♥t♠♥t s strtrs rst♥s q s ♥♥tr♥ ♥ ♥tr③ ♥♦ stá♥ ♦♥stts♣♦r ♦s rís ♠♥r q ①♣r♠♥t♠♥t s ♦t♥♥ r♦♥s c/a ♠② r♥s tór c/a =

√8/3 = 1, 6333 ♣r♦ ♥♦ ①t♠♥t s

♦♥s ①♣r♠♥ts c/a só♦s P

❩♥

♦ ❨ ❩r

s♥ ♣ ♦s s stú s♦r ♦s ♦s ♣r♠r ♣r♦ trr ♣ ♥♦ s ♦♦ rt♠♥t♥♠ ♣r♠r s♥♦ ♥ ♦tr ♣♦só♥ ♣♦s ②s r♣t ♣r ♦t♥r ♥ ♣♠♥t♦ ♥t♦♥s strtr q s ♦t♥ s

t♠♥t t♥ ♣♥♦ (111) ♦♠♣t♦♦♠♦ s ♥ r

strtr

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó strtrs rst♥s

strtrs s♦♠♣ts

stá♥ ♦r♠s ♣♦r ♥ r ú ② s s ♠♦♥♦tó♠ s ♥ ♦♥ s s♠♣st♦ s ♥ rst rs

rst③♥ ♥ s strtr ♦s ♠ts ♥♦s ② ♦tr♦s ♠ts ♣♥t♦ só♥♦ ❲ ♦ sí ♦♠♦ ❱ ② r

strtrs r♦s ♥♦s

♣♦ ♦rr♦ s♦♦

s ♥ r ♦♥ s ♦[(0, 0, 0) ,

(12 , 0, 0

)]

r strtr

♦r♦ s ♥♥tr ♥ ♥ ♥t♦r♥♦ ♦tér♦ r♦♦ át♦♠♦s s♦♦ ♦♦r♥ó♥ ♦♥♦s s ss í♣r♦♠♥t s ♦ ♠s♠♦ ♦♥ s♦♦

♣♦ ♦rr♦ s♦ s

s ♥ r ♦♥ s ♦[(0, 0, 0) ,

(12 ,

12 ,

12

)]

r strtr s

♦r♦ s ♥♥tr ♥ ♥ ♥t♦r♥♦ ú♦ r♦♦ át♦♠♦s s♦ ♦♦r♥ó♥ ♦♥ ♦s s♦♦ í♣r♦♠♥t s ♦ ♠s♠♦ ♦♥ s♦

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

Ó ❳P ❯❯

♥ ♥r s ♣ rs♠r q

♦s ♠ts ② ♦s ss ♥♦s ♣rs♥t♥ strtrs ♦♠♣ts ② s♦♠♣ts

♥ ♦s s♠♦♥t♦rs s♦♥ ♠② ts s strtrs ♠♥t ② ③♥♥

♦s r♦s ♥♦s rst③♥ ♥ s strtrs ♦rr♦ s♦♦ ② ♦rr♦ s♦

♦♠♦ ♠♦s ①st ♥ rt ♥trró♥ ♥tr ♦♥t étr ♦s só♦s ② ssstrtrs rst♥s

Pr tr♠♥r st ssó♥ r s strtrs rst♥s ② q r ♥♦tr q ♥♥r ♦s rsts q ♣r♥ ♥ ♥tr③ ♥♦ ♦♥♥ ①t♠♥t ♦♥ s strtrs♠t♠áts sts qí sts ♠♣r♦♥s rst♥s ♣♥ sr rs♦s t♣♦s ❯♥t♣♦ s ♣♥ sr ♦s t♦s ♥ ♦s q ♥ ó♥ r s♣r ♥♦ ♥ ♦ ♥ r ♥ ♥t s ♥ st♦ tr♠♦♥á♠♦ ♦♠♦ s rá ♠ás ♥t s q ②♠♦t♦s ♣r q st♦s t♦s sté♥ s♠♣r ♣rs♥ts ♥ ♥ rst ♥ ♠r♦ ♦s rsts s♥sr ♦r♥ 104 s ♠ás ♠♣r♦s q ♦s q tr♠♦♥á♠ ♠♣♦♥rí ② s ♦sr♦q ♦s ♥s ♥ ♣ró♥ ♦s rsts ♥ ♣t♥ts ♥s ♣r♦♣s íss ♥trés

2.3 Ó ❳P ❯❯

♦t P rs ♥ ♦ ♥♦r♠ó♥ r ♦trs té♥s ♣r tr♠♥ó♥ s strtrs rst♥s s ♠♣♦ ét♦♦ ♠r♦s♦♣í ♠só♥ ♠♣♦ ♥ ♥♣♦r❴♠ss♦♥❴♠r♦s♦♣②

ét♦♦ ♠r♦s♦♣í tú♥ rr♦ ♥ ♥♣♦r♥♥♥❴t♥♥♥❴♠r♦s♦♣

♥ tr♠♥s r♥st♥s s ♣♦s ♦♥strr ♥ ♠♥ ♠r♦só♣ rt strtr rst ♥ ♥r s tr ♦♥ té♥s ró♥ s♦♥s ②s ♦♥ts ♦♥ s♥ ♦r♥ s♣♦ ♦s át♦♠♦s ♥ ♥ rst ♦r♥ ♣s 10−8 ♠ ♦ 1♦s rst♦s st ró♥ ♥♦s ♣r♠t♥ str strtr ♦s rsts ♦r♠ q♣♦♠♦s tr♠♥r t♠ñ♦ ♥ ♣♦só♥ ♦s ♦♥s ② stró♥ ♦str♦♥s ♥tr♦

s r♦♥s s♦♥ s q s s♦♠t só♦ s♦♥ st♥ts ♣♥♥♦ qé ♣r♦♣ís ♥trs str

s ♦♥s tr♦♠♥éts ♥tr♦♥♥ ♦♥ s rs rst ás♠♥t ♦♥♦s tr♦♥s ♠s♠♦

sí ♦s ♥tr♦♥s s♦♥ ♣rtís ♥trs ♠♥éts q ♥tr♦♥♥ ♦♥ ♦s ♥ú♦s ② ♦♥♦s tr♦♥s ♠♥ét♦s ♥ só♦ st♦ s ♦♥ ♦s q ♥ r ss ♣r♦♣s♠♥éts ② ♣♦r ♦♥s♥t s ró♥ s ♥ té♥ ♦s ♥ ♦s st♦s rsts♠♥ét♦s

st ♦♥t ♦♥ s q s ♦t♥ ró♥ r♦ λ = h/p st ró♥ ♥tr♦♥s ♣ sr r♦s t♣♦s

ást ♦r♥t ♦s ♥ú♦s ♠♦r ♥ ♦s r②♦s ❳ ♣r át♦♠♦s r♦s ② ♥♦r♠ó♥ r stró♥ ♦s ♠♦♠♥t♦s ♠♥ét♦s ♥ só♦ ást ♥♦r♥t s ② ♥ stró♥ t♦r ♦s st♥t♦s sót♦♣♦s ♦s ♠♥t♦s q ♦r♠♥ rst ♥ást ♦♥ ♦s ♦♥♦♥s rst ♦♥ ~ωfonon ∼ Eneutron

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó strtrs rst♥s

P♦r ♦tr ♣rt ♦s tr♦♥s s♦♥ ♣rtís rs q ♥tr♦♥♥ rt♠♥t ♦♥ ♠tr st♦ qr r q ♣♥tr♥ rt♠♥t ♣♦♦ ♥ rst ② ♣♥ sr úts ♣r st♦ ♠♥t ró♥ s ♣r♦♣s s s♣rs só♦s ② rsts ♠②♦s ♦ ♥ ♠r♦s♦♣♦ tró♥♦ ♣♦r rs♦ó♥ ♣ ♣r♠trst♥r ♦s ♣♥♦s rsts ♠♥rs ♦♠♦ rt♦ ♣r♦ ♥♦ ♣ t③rs ♣r ♥tr♠♥ó♥ rt ② ♣rs strtrs rst♥s tr♠♥s♦♥s s♦♥♦s

♦r ♥ ♥ ♦s ♣r♦s♦s ♥tró♥ ♦ sttr♥ ♥ s r♦♥s ♦♥ só♦♣♦♠♦s st♥r tr♦ t♣s

♥tró♥ ♦♥ ♥ tró♥

♥tró♥ ♦♥ ♥ át♦♠♦ sts ♦s t♣s ♣♥♥ ró♥

♥tró♥ ♦♥ ♥

♥tró♥ ♦♥ rst ♦♠♣t♦ sts ♦s út♠s t♣s s♦♥ ♥♣♥♥ts ró♥

② ♠♥trs q s ♥trtú♥ ás♠♥t ♦♥ ♦s tr♦♥s rst ♦s ♥tr♦♥s ♦♥ ♦♥ ♦s ♥ú♦s s ♠tr ♥♦ s ♠♥ét♦ ② ♦s tr♦♥s ♥trtú♥ ♦♥ tr♦♥s ②♥ú♦s só♦

♦♠♦ ② s ♦ ♦♥t ♦♥ s r♦♥s s q s s♥ s♦♠tr ♦s rsts sr ♦r♥ ♠♥t ♦ ♠♥♦r s♣ró♥ ♥tr ♦s át♦♠♦s Pr ♦ s út r ♥sr♦♥s ♥tr ♥rí s ♣rtís q t③r♠♦s ♦♠♦ r♦♥s ② ss ♦♥ts ♦♥♦♠♦ s ♦ ♦♥t ♦♥ s q s ♦t♥ ró♥ r♦ λ = h/p

❯ λ r♥t E

♦t♦♥sλ =

12, 4

ε❱♦ ♦ q s q♥t 1 ↔ 12, 4 ❱

tr♦♥s

λ =0, 28√ε❱

♦ ♦ q s q♥t 1 ↔ 0, 08 ❱ ↔ 4000 ♠s

tr♦♥s

λ =12√ε❱

♦ ♦ q s q♥t 1 ↔ 150 ❱↔7× 106 ♠s

② st♥ ♥tr♣♥r s ② st♥ ♥tr ♥♦s ♠ás r♥♦s s

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❯Ó ❱ ❯ P Ó ❨ ❳ P

2.4 ❯Ó ❱ ❯ P Ó ❨ ❳ P

♣r♦♠ trt♦ ♣♦r s s♥t ♥ ró♥ ♦♥ ♥♥t r sr♦♥s s ♦♥s q s♥ rst

sq♠ s ♣s s♥t

❯ sq♠ Pr rs♦r ♣r♦♠ ♠♦s s♣♦♥r q rs♣st rst s ♥ st♦ s q

ω = ω′ P♦r ♦ t♥t♦ t♠é♥ s ♠♣ q |k| = |k′|

rs ♦ rst rs ♦♥ ♣♥ q ♥ ♥ rst ♣♦♠♦s srr ♦♠♦

E(r) = E0ei(k·r−ωt).

st ♦♥ ♥ s♦r ♥ ♣♥t♦ ρ q ♣r♦ s ró♥ st♦ s ♦♥ ♦s♦♥ ♥ s♣♥t♦ ♥t♦♥s ♦r♠ ♦♥ rt s

Ed = CE(ρ)eikr

r= CE0e

ik·ρ ei(k·r−ωt)

r

♣r ♣♥t♦s ♠② ♦s ♣♥t♦ ♦só♥ ♦s t♦rs q ♥②♥ s ♣rts ♥rs ② ♦sts ♣♥t♦ ♦só♥ ♦s ♥♠♦s ♥ t♦r C ♣♦♥r♠♦s q ♠♣♦ étr♦ ♥♦ s♠♦rt ♥ t♦♦ rst st♦ s q rst s s♥t♠♥t ♣qñ♦

❯ sq♠ ♣♥t ♦sró♥

♠♦s q s st♥ ♣♥t s s♥t♠♥t r♥ q

r ≃ L− ρ cos (ρ,L)

st t♣♦ sttr♥ ást♦ s ♠♦ ♦♠s♦♥ sttr♥ ♥ást♦ s ♦ t♦ ♦♠♣t♦♥ st♦♥ s♥trs ♣r♥ts

♥ t♦♦ s♦ ♣r r ♥ ór♥s ♠♥t s ♦♥♥♥t ♦♥♦r q ♦♥t ♣♥tró♥ ♥ r②♦❳ ♥ ♥ só♦ ♣♥ ♦♠♣♦só♥ ② strtr ♠s♠♦ ② ♥rí ♦tó♥ ♣r♦ tí♣♠♥t ♣ sr ♦r♥ ♠

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó strtrs rst♥s

② ♣♦r ♦ t♥t♦

ei(k·ρ+kr) = eikLei[

k·ρ−kρ cos (ρ,L)]

② ♦ q L t♥ ♣rát♠♥t ró♥ k′ ② ♦♠♦ k = k′ ♥♦s q

ei(k·ρ+kr) = eikLei(kρ−k′·ρ) = eikLe−i(ρ·∆k) ∆k = k′ − k

②

Ed =

(CE0e

ikLe−iωt

L

)e−iρ· ∆k ♦♥ s sstt♦ r ♣♦r .

óts q ♣ré♥tss ♥♦ ♣♥ ♣♥t♦ ρ q ♣r♦ ró♥

s r♦ q s ♥♦s♦tr♦s st♠♦s ♥♦ ♥r ♥ ③ r②♦s ❳ s♦r ♥ r ♣♥t♦ r ♦♥trrá ró♥ ♠♥r q s R s ♥ t♦r

Etotal =

[CE0e

ikLe−iωt

L

]∑

R

e−iR·∆k Etotal ∝ A ≡∑

R

e−iR·∆k =∑

n1n2n3

e−i(n1a1+n2a2+n3a3)·∆k

st s♠t♦r♦ s ss A r ♥♦♠r ♠♣t ró♥

s♣♦♥♠♦s q rst s ♥t♦ ♦♥ N3 s ♣r♠ts ♥t♦♥s ni ∈ [0, N ] t♥♠♦s ①♣rsó♥

A =∑

n1n2n3

e−i(n1a1·∆k+n2a2·∆k+n3a3·∆k)

q s ♥ ♠á①♠♦ ♥♦ ♣r t♦♦ R s ♠♣ q

R ·∆k = 2π ♥tr♦ ⇔ ai ·∆k = 2π ♥tr♦ =⇒ Amax =∑

n1n2n3

1 = N3.

s ♦♥ó♥ ♥♦ s ♠♣ ♥t♦♥s A t♦♠ ♥ ♦r ♠♦ ♠♥♦r ② s rst r ♥♥t♦♥t♦♥s A = 0 s♦ q s ♠♣ q ai ·∆k = 2π ♥tr♦

P♦r ♦♥s♥t ♣r ♥ r rs ró♥ ♥ ③ r②♦s ❳ ró♥ k srá♥ qs r♦♥s k′ = k+∆k ts q

a1 ·∆k = 2π h

a2 ·∆k = 2π k h, k, l ∈ Z

a3 ·∆k = 2π h

st ♦♥ó♥ ♠ ó♥ s ♥ ♦♥ó♥ ♥sr ② s♥t ♣r ♥ r rs

rsts ♦♥ s strtr rst♥

❯ s

♣♦♥♠♦s ♦r q ② s át♦♠♦s ♦♥s ♦ ♦ qr ♥ ♦♥ ♥ ♥s tró♥ ♦♥t♥② s♣q♠♦s ♣♦r

σj = σ1ja1 + σ2ja2 + σ3ja3 j = 0, 1, 2, . . . , s

s ♦♦r♥s ♥♦s s♦s át♦♠♦s rs♣t♦ ♣♥t♦ r R = n1a1 + n2a2 + n3a3. st♦ s s ρj s t♦r ♣♦só♥ ♥♦ ♦s át♦♠♦s s

ρj = R+ σj .

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❯Ó ❱ ❯ P Ó ❨ ❳ P

❯ s

♦r ♥ ♦s tr♦♥s s str②♥ ♠♥r♦♥t♥ ♥ rst ♦r♠ q s ♠♠♦s nj ♥s tr♦♥s át♦♠♦ j st ♥s♥ ♣♥t♦ ρ nj(ρ − ρj) = nj(ρ − R − σj) st♦ ♥♦sqr r q ♥s t♦t tr♦♥s ♥ ♥ ♣♥t♦ρ ♥tr♦r rst s

n(ρ) =∑

R,j

nj(ρ−R−σj) =∑

n1n2n3

s∑

j=1

nj(ρ−n1a1−na2−n3a3−σj).

♠♣t ró♥ ♣r st s♦ ♥ q r s ♦♥t♥ s ♣ srr ♦♠♦

A∆k =

∫

todoelcristaldV n(ρ)e−iρ·∆k

=∑

n1n2n3

s∑

j=1

∫

todoelcristaldV nj(ρ− n1a1 − na2 − n3a3 − σj)e

−iρ·∆k.

♠♥♦ ρ′j = ρ−R− σj st♥ át♦♠♦ j ♣♥t♦ ρ ♥♦s q

A∆k =∑

n1n2n3

s∑

j=1

[∫

todoelcristaldV nj(ρ

′j)e−iρ′j ·∆k

]e−i(σj+R)·∆k =

∑

n1n2n3

s∑

j=1

fj e−i(σj+R)·∆k

=

[∑

n1n2n3

e−iR·∆k

]

s∑

j=1

fj e−iσj ·∆k

,

♦♥ fj ≡∫todoelcristal dV nj(ρ)e

−iρ·∆k s t♦r ♦r♠ tó♠♦ ♠ ♥ ♠♥r ♦r♠ át♦♠♦ ②

S∆k =

s∑

j=1

fj e−iσj ·∆k

s t♦r strtr s

♦♠♦ s ♥♦s q♦ ♥ ①♣rsó♥ ♥ q ♣r ♥ ♣r♦t♦

♥ ♦rt [. . . ] q t♦♠ ♦r N3 s s ♠♣ q ∆k · R = 2π ♥tr♦ ② q s♣rát♠♥t ♥♦ ♥ qr ♦tr♦ s♦

t♦r strtr s S∆k =∑s

j=1 fj e−iσj ·∆k q s ♦t♥ ♦s t♦rs

♦r♠ ♦s át♦♠♦s q ♦♥stt②♥ s strtr rst

♥ rs♠♥

A∆k = N3 S∆k s ∆k ♠♣ ó♥

♦ q ♠♣t ró♥ q ♠♦s ♦t♥♦ s ♣r♦t♦ q ♦t♠♦s ♥ts♣r ♣♦r ♥ t♦r strtr st ó♥ ♥♦s q t♦r strtr ♣♥r ♥s s r♦♥s q strí♥ ♣r♠t ♥ ♥ ♦r♠ q r♥ sts r♦♥s ♥ ♥ r♠ ①♣r♠♥t ♥♦s ♣ ♣r♠tr r qé strtr rst♥st♠♦s trt♥♦

♠ás ♦♠♦ ♥ ♥ó♥ t♦r strtr S∆k ♥tr t♦r ♦r♠ tó♠♦ fj ♣r ♥ s ②♦s át♦♠♦s t♥♥ ♥ t♦r ♦r♠ st♥t♦ ♣♦♠♦s r♥r ①st♥

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó strtrs rst♥s

át♦♠♦s st♥t♦s ♥ s s st♥ts ♥t♥ss rts q s ♦sr♥ ①♣r♠♥t♠♥t ♥♦s ♣♦s ró♥

2.5 Ó ❯Ó ❯ ❨ ❮P

• ♦♠♦ ♠♦s st♦ ♣r ♦t♥r ♥ ♣♦ ró♥ ♥ ♥ ró♥ tr♠♥ k′ = k +∆ks♥♦ k ró♥ ♥♥t rrs q

ei∆k·R = 1 ∀R ∈ RB.

Pr♦ st♠♥t s s ♥ó♥ q t♦r ∆k ♣rt♥③ r rí♣r♦P♦r t♥t♦

♥ ró♥ r②♦s ❳ ♣♦r ♥ strtr rst♥ rá r②♦s rt♦s ♥ t♦s qsr♦♥s ♥ s q ♥tr♠♦ t♦r ♦♥ ∆k s st♠♥t

♥♦ ♦s t♦rs r rí♣r♦ s♦ q ♣rr s ♦r G t♦r strtr ♠s♠ s ♥♦P♦r t♥t♦ ró♥ r②♦s ❳ ♥ rst s ♥ ♠ rt r rí♣r♦ ♠s♠♦ ② ①st♥ ①t♥♦♥s ♣r rt♦s ♦rs Gst♦ s q SG = 0 ♥♦s r ①st♥ ♥ s ♥ rst

• ♠ás ♦♠♦ k− k′ = −G t♥♠♦s q k′2 = k2 = G2 + k2 − 2k ·G =⇒ k · G =1

2G ♦ ♦ q

s

♦♠♣♦♥♥t t♦r ♦♥ ♥♥t ♦ r♦ t♦r Gs G/2 P♦r♦♥s♥t ♣r t♦♦ t♦r G ♣♦♠♦s ♦♥strr ♥ ♣♥♦ st♦r t q ♦s t♦rs k ② k′ stá♥s♦r ♦ ♣♥♦ r♦ ♦♥ r

❯ P♥♦ st♦r ♥ G

• ♥♠♥t s ♥ ♥♦ s ③♦♥s r♦♥ ♦♠♦ qs r♦♥s s♣♦ rí♣r♦♦ qq♥ ♥ ♥tr♦r ♦s ♣♥♦s st♦rs ♦s t♦rs ♥ ♦♥s♥ ♦s t♦rs k② s♦♠♥t ♦s q ♣rt♥ ♦r♥ ② tr♠♥♥ ♥ ♥ ❩ s rt♥ sú♥ t♦rk′ q s ♣rs♥t♦ ♥ r ♥tr♦r

♥t♦ t♦r strtr S ♦♠♦ ♠♣t ró♥ A ♥♦ ♥st♥ sr rs ② q ♥t♥s ♦♥rt ♦♠♦ r♦ A st♦ s A∗A ② S

Pr♦ ♣s ♥ ♣♥♦ r ♥ ♦♠♦ ♦s t♦rs s♦♥ ♣r♣♥rs ♠s ♣♥♦s rtrs ♣♥♦ r ♥♦s qé ♠s ♣♥♦s r rt♥ t♦r ♦♥ k ② ♦ ♦♥rt♥ ♥ k′

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❯ ❯Ó ❱

2.6 ❯ ❯Ó ❱

♣ ♣♥tr ró♥ ró♥ ♣♦r ♥ rst ♥ ♠♥r tr♥t Pr ♦s ♠♠♦s V (r) ♥rí ♣♦t♥ ♦♥ q rst r s♦♥ ♦ ró♥ q s ♥r s♦r é q s s♥t ró♥ t♥♠♦s q V (r+R) = V (r), ∀R ∈ RD♥ ♣r♦①♠ó♥ ♦r♥ ♣r♦ tr♥só♥ ♥ st♦ ♥ Ψk ♦tr♦ st♦ Ψk′

s ♣r♦♣♦r♦♥ r♦ ♠♥t♦ ♠tr③

Mkk′ =

∫drΨ∗k′(r)V (r)Ψk(r)

② s ♦♥sr♠♦s q ♦s st♦s ♥ ② ♥ s♦♥ ♦♥s ♣♥s

Mkk′ =

∫dr e−ik·r

∑

G

VGeiG·reik·r =

∑

G

VG

∫dr ei(k+G−k′)·r =

VG s k′ − k = G

0 ♥ qr ♦tr♦ s♦

♠♥t ♠♥t♦ ♠tr③ Mkk′ ♣♥ VG② ést t♣♦ ró♥ q ss♦♠t rst

Pr tr♦♥s rá♣♦s V (r) srá ♣♦t♥ ♦ q sr tró♥ ♥ só♦

Pr r②♦s ❳ V (r) sr ♥s tró♥ ♦

Pr ró♥ ♥tr♦♥s srá stró♥ ♦s ♥ú♦s ♥ s♣♦ ② ♦♠♦ ♥♥r ♥ rst ② st♥t♦s sót♦♣♦s rá sttr♥ ♥♦r♥t ♦♥ ♥ ♦♥♦ ró♥s♦ ② q ♦s st♥t♦s sót♦♣♦s t♥♥ r♥t só♥ ③ ♣r ♦s ♥tr♦♥s ♠ás ♥s♦♥ rsts ♦♥ ♣♦ ♠① s♦tó♣ ♦s ♥tr♦♥s s♥t♥ s st♥ts ♦r♥t♦♥s s♣í♥ ♦s ♥ú♦s q s rr sú♥ s ♣♦s♦♥s q ♦♣♥ ♥ strtr rst♥

♥ s♦ tr♦♥s ♥t♦s ♥♦s ❱ stó♥ s ♦♠♣ ② q♥ s s♦ ♦s tr♦♥s t♥♥ ♥ só♥ ③ ♠② r♥ ♥♦ ♥trtú♥ ♦♥ ♦s át♦♠♦sP♦r s r③ó♥ t♦rí ♥♠át ♥ q s ♦♥sr ♥ ú♥ s♣rsó♥ ♦ sttr♥ ② q s ♣r ♦s r②♦s ❳ ♥♦ sr ♥ ♦s ♣r♦s♦s ② ② q t♥r ♥ ♥t ♣♦s q sr♥ rs s♣rs♦♥s ♦ r♦s ♣r♦s♦s sttr♥ ♦ q ♦♥ q ♦s á♦s s♥♦♠♣♦s st♦ s♥ q ② q ♦♥srr t♠é♥ ♥tró♥ ♥tr ♦s s ♥♥ts ②rt♦s ♣♦r ♦♥s♥t rí t srr sst♠ ♠♥t ♥ t♦rí ♥á♠ ♦ s rt ♥tró♥ st té♥ s s♣♠♥t s♥s ♣♦r ♦ q rst út ♣r st♦ s strtrs s♣rs

♦s rst♦s ♦t♥♦s ♦♥ t♣♦ s♦♥ r♦ stá ♥ sr ♥tr♣rt♦s st♥t♠♥r sí ♥ st♦ rsts rr♦♠♥ét♦s ró♥ ♥tr♦♥s ♣ ♠♦strr s♥ st♥ts és ♥ r ♥ ♣♦s stró♥ ♦s s♣♥s ♥ ♥ ♠tr s t♣♦ s q s ♦t♥rí♥ ♦♥ ró♥ r②♦s ❳

❯ ♦r♥ ♠♥ét♦ ♥ ♥ ♠tr

❱és ♣rt♦ ♦rrs♣♦♥♥t ♥ s♥tr ís á♥t ♥ st s♦ ♣s t♥r♠♦s ♥ stró♥ V (r) t♣♦ ♥♦♥s t r

❯♥ ♠♣♦ tí♣♦ s s♦ ♦s rsts rr♦♠♥ét♦s ♥ ♦s q ró♥ ♥tr♦♥s ♣ ♠s♦trr s ♥st♥ts s ♦t♥s ♦♥ ró♥ r②♦s ❳

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó strtrs rst♥s

2.7 P❱ ❳P ❯ P Ó ❯

❯♥ ♥♥t ♥ ♥ rst ♦♥ t♦r ♦♥ k srá r ♥ ♣♦ ró♥ s ②s♦♠♥t s ①tr♠♦ t♦r k stá ♥ ♥ ♣♥♦ r s♣♦ rí♣r♦♦ P♦r ♦♥s♥t♣r ♥ k qr ♥♦ ①strá♥ ♣♦s ró♥ ♥ ♥r ♣♦r ♦ q ♦♥♥ sñr q♣♦s①♣r♠♥ts q ♣r♠t♥ rs k

t♠♥t s t③♥ ♦s ♣r♦♠♥t♦s

s rí ♠ó♦ k ♠♥t ró♥ ♦♥t ♥ ♥♥t

s rí ró♥ k ♣♦r ♠♣♦ r♥♦ ♠str rst♥

Pr s③r ♦♥ ♠②♦r r st♦s ♠ét♦♦s ①♣r♠♥ts s ♦♥♥♥t trr ♦♥ ♦♥stró♥ Pr ♦ ♥ s tr③ ♥ sr r♦ k ② ♥tr♥ ①tr♠♦ t♦r k ♣♦♥♠♦s q ♥ t♦r G stá s♦r sr ♥t♦♥s ú♥s r♦♥s ♣r♠ts ♥rás tr♠♥s ♣♦r ♠ ♣♥♦s rtrs♣r♣♥rs G

❯ sr

①st ♥ ♣♥t♦ q sté ♥ sr s r♦ q t♦r k q ♥ s♣♥t♦ ♦♥ ♥tr♦ sr s ♥ t♦r q ♠♣ q −∆k=k− k′ = G ∈

P♦r ♦tr ♣rt stá r♦ q ♣r ♥ t♦r qrk sr ♥♦ ♥r♥ ♥r ♥♥ú♥ ♣♥t♦ Pr♦ s r♠♦s ró♥ k ♦ s ♠♥t sí ♣♦r♠♦s♦♥sr q ♣r ♥♦ ss r♦♥s ♦ ♠ó♦s k sr ♥② ú♥ ♣♥t♦ q♥rá ♥ ♣♦s ró♥ ró♥ ♥♥t

❯

❯ ♦s srs

t③♠♦s r②♦s ❳ ♥♦♠♦♥♦r♦♠át♦s s♥♦ ♥ r♥♦ ♦♥ts ♦♥ λ1 λ0♥t♦♥s ♦s t♦rs ♦♥ sá♥♦♠♣r♥♦s ♥tr k0 = 2π/λ0② k1 = 2π/λ1 ❯t③♥♦ ♥ ③♣r♦ r②♦s s ♦t♥rá♥ ♣♦s r ♥ qs r♦♥s qsté♥ ♥s ♣♦r ♦s ♣♥t♦s q ② ♥tr s ♦s sr ♦rrs♣♦♥♥t k0 ② q stá s♦ k1

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

P❱ ❳P ❯ P Ó ❯

❯ sr q r

♥ st s♦ s ró♥ ③ r②♦s ❳ ♠♦♥♦r♦♠át♦ ♠♥trs q rst sr r♦r ♥ ♦

♥ st ♠♦♥t ①♣r♠♥t r ♦♥ rst ② ♦s ♣♦s ró♥ ♦t♥♦s s♦♥q♦s ♥ ♦s q rr ♦s ♣♥t♦s ♥trst♥ sr q r

♦s ♣♥t♦s r♥ r♦r s♦r♥ q rst r rs♣t♦ ♥ ♦ ♠♥r q ♦ ♣♥t♦s sr♥r♥r♥s és r

P P❱ ❨

♥ st s♦ ♠str s ♥ rst ♣♦rst♥♦ ♦ ♥ ♣♦♦ ♦♥ r♥♦s ♦ ♠r♦rsts ♠②r♥s s ♠r♦só♣ ② ♣♦r t♥t♦ ♦♥ ♣ ♣r rtr ♦s r②♦s ❳

♦♠♦ st♦s ♣qñ♦s rsts stá♥ ♦r♥t♦s ③r r♠ ró♥ st①♣r♠♥t♦ srá ♥ ♦♠♥ó♥ st♦ s ♥ ♣r♦♠♦ t♦♦s ♦s r♠s ♦t♥♦s ♣rqr ♦r♥tó♥ ♥ ♠♦♥♦rst P♦r ♦ t♥t♦ ♣♦♠♦s ♦t♥r s ♣♦ss r♦♥s ró♥ ♠♥r q ♥ s♦ ♥tr♦r s♦ q ♦r ♦s t♦rs ♥♦ r♥♥♥♦ ♥ r♥r♥s s♥♦ q ♥♥ ♥ sr ♦♠♣t

sí s ♠ó♦ t♦r G q st♠♦s ♦♥sr♥♦ s ♠♥♦r q 2k ♦rtrá sr ② rá ♥ ♦♥♦ r♦♥s ró♥ Pr t♦r q ♠♣q |G| < 2k s ♦t♥ ♥ ♦♥♦ ró♥ rt ♥ á♥♦ ró♥ ♥♥

sinφ

2=G

2k.

st ♦r♠ s ♦♥♦♠♦s ①♣r♠♥t♠♥t ♦s ♦rs ♦s st♥t♦s φ ♣♦♠♦s r ♦s♠ó♦s ♦s t♦rs ②♦ ♠ó♦ s ♠♥♦r q 2k

❯ ②rrr

♦♦s ♦s t♦rs G q rq♥ ó♥ rá♥ r ró♥

st ♦♥♦ ♦rrs♣♦♥ ♦rt sr ♦♥ sr ♦t♥ ♠♦r G ♣♥t♦ ♥ t♦s sr♦♥s s♣♦ r q ♠ ①♣r♠♥t stá ♣r♦♠♥♦ ♥ t♦s s r♦♥s ② q ♥ ♠str ♥ ♣♦♦ ② rstt♦s ♦r♥t♦s ♥ t♦s s r♦♥s ♣♦ss

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó strtrs rst♥s

❯ ②rrr

2.8 ❨ ❯❯

♥ ①♣rsó♥ q s s♦ ♣r ♥r t♦r ♦r♠ fj

fj ≡∫

todo el cristaldV nj(ρ)e

−iρ·∆k ♦♥ ∆k = G

♠♥r q fj ♣♥ ♥ú♠r♦ ② stró♥ ♦s tr♦♥s ♥ át♦♠♦ sí ♦♠♦ ♦♥t ♦♥ ② ♥♦ s♣rsó♥

♠♦s fj ♣r ♥ át♦♠♦ q t♥ ♥ stró♥ sér r tró♥ ♥t♦♥s

fj = 2π

∫ 1

−1d(cosα)

∫ ∞

0drr2nj(r)e

−irG cosα = 2π

∫ ∞

0drr2nj(r)

eiGr − e−iGr

irG

= 4π

∫ ∞

0drr2nj(r) sin(Gr)

Gr.

t♦♦ át♦♠♦ str ♦③♦ ♥ r = 0 ♥t♦♥s ♦♠♦ sinα/α→ 1 s α→ 0 s t♥ q

fj = 4π

∫ ∞

0drnj(r)r

2 = Z.

P♦♠♦s ♣s ♥tr♣rtr t♦r ♦r♠ tó♠♦ ♦♠♦ ró♥ ♥tr s♣rsó♥ ♣♦r ♥s tró♥ r r♥t s♣rsó♥ q rí♥ Z tr♦♥s st♦s ♥ ♦r♥ ♦♦r♥s r = 0

st sr q s♣rsó♥ r②♦s ❳ s ♣ str st♥t ♥ ♠♥t ♦s t♦rs ♦r♠ ♦s át♦♠♦s rs ♠♥r q ♥ ♠♦s á♦s ♣♦♠♦s t③r ♥ fj ♠♥t ♥s nj(r) ♥ át♦♠♦ r ♣ ♦♥ ♥ á♦ r♠♦♠s ♦ rtr♦

Pr ♠♥♦ ② ♦♥ ♥ á♦ t③♥♦ ♥s tró♥ s ♦t♥♥ ♦s rst♦s r

s srí s♦ s sinα/λ→ 0 q s ♦ ♠s♠♦ q λ→∞ ② ♥♦t át♦♠♦ ♦♠♦ s r ♥ ♣♥t♦ ♥ ♣♦♦♥ t♦ r tró♥ ♦♥♥tr ♥ é

♥ r s r♣rs♥t ♥ s sss sin θ/λ r♦ ♦♥ ♦r♠ ♠ás s srr ② ♦ ór♠ rnλ = 2d sin θ

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❨ ❯❯

t♦r ♦r♠ ♠♥♦ ② á♦

❨❲

♠♥tr t♠♣rtr rst s♠♥② ♥t♥s ♦s ♣♦s ró♥ s ♥s ♥r ♥r s ♠♥t♥ s s♥t♦ s s ♣♦s ♦t♥r ♣♦s ró♥ ♥ ♥♦s ♥rsts ♥ ♦s q ♦s ♦♥s s ♠♥ ③r tér♠♠♥t ② s stí ♥ t♠♣♦s ♦♥

❯ t♦r ❲ ♣r ♠♥♦

♥ t♦ ♠♦s str s s♦ s ♣♦s ♦♥ ♠♦♠♥t♦s ♦s ♦♥s st ♥ s♣♦ ♥trtó♠♦ t♠♣rtr ♠♥t st♦ s s s ♣♦s♦♥s ♦s ♦♥s ♥♦ s♦♥ ♦rs♦s σj s♥♦ q rí♥ ♥ t♠♣♦ σj(t) = σj + uj(t) ♦♥ uj(t) s s♣ró♥ ó♥ s♣♦só♥ qr♦ ♥ ♥str ①♣rsó♥ ♣r ♠♣t só♥

A∆k = N3S∆k = N3s∑

j=1

fj e−iσj(t)·∆k = N3

s∑

j=1

fj e−iσj ·∆k e−iuj(t)·∆k.

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó strtrs rst♥s

♦s ♦♥s ♦s♥ ♥♣♥♥t♠♥t ♥♦s ♦s ♦tr♦s ♥ A∆k ♣r♥ tér♠♥♦s q ♣♦♠♦s♣r♦♠r t♠♣♦r♠♥t ♣r ♦t♥r

〈A∆k〉 = N3s∑

j=1

fj e−iσj ·∆k

⟨e−iuj(t)·∆k

⟩.

|uj(t)| s s♥t♠♥t ♣qñ♦

⟨e−iuj(t)·∆k

⟩≃ 1− i 〈∆k · u〉 − 1

2

⟨(∆k · u)2

⟩= 1− 1

2∆k2

⟨u2 cos2 θ

⟩= 1− 1

6∆k2

⟨u2⟩≃ e−

16〈u2〉∆k2 ,

♦♥ ♠♦s s♦ q⟨u2 cos2 θ

⟩=⟨u2i⟩= 1

3

⟨u21 + u22 + u23

⟩= 1

3

⟨u2⟩ P♦r ♦♥s♥t

I = 〈A∆k〉2 = I0 e− 1

3〈u2〉∆k2 ,

s♥♦ I0 ♥t♥s q s ♦t♥rí ♥ s♦ r rí

t♦r ①♣♦♥♥ s ♠ t♦r ②❲r ❱♠♦s q ♥t♥s ♣♦ ró♥ s♠♥② ♣r♦ ♥♦ ♠♥r r♠át ♦ q ♣r♠t q ♣sr ♦s ♠♦♠♥t♦s ♦s ♦♥s s ♣♥ str s strtrs rst♥s t♠♣rtr st♥ts r♦ ♠ás t♦r ❲ s♠♥② tr ♦s ♣♦s ró♥ ♣r♦ ♥♦ s ♥r

• ❱♠♦s st♠r ♦r t♦r ❲ Pr ♦ s♣♦♥♠♦s q ♦s ♦♥s r♦♦♥ ♠♦♦ ♥st♥ s♦♥ ♦s♦rs ♥♣♥♥ts r♥ ω0 C s ♦♥st♥t r③ ♦ ♠ ♦s♦r t♥♠♦s ♣♦r t♦r♠ q♣rtó♥ ás♦

〈Ep〉 =1

2C⟨u2⟩=

1

2Mω2

0

⟨u2⟩=

3

2kBT =⇒ Iclasico = I0e

−(kBTG2/Mω20)

s st♠♦s t♠♣rtrs s♥t♠♥t ts

♥ ①tr♠♦ ♦♥trr♦ ♠② s t♠♣rtrs T = 0 r♦ s♦t♦ ♣♦♠♦s r ♥ó♥ ♣r♦①♠ t♦ ♥rí ♣♥t♦ r♦

〈Ep〉 =3

4~ω0 =⇒ Ipunto cero = I0e

−(~G2/2Mω0),

♦♥ ♠♦s ♦♥sr♦ q ♥rí ♣♥t♦ r♦ s ~ω0/2

sts ♥t♥ss s ♣♥ st♠r ♥♦ ♦rs ♣♦ss s ♠♥ts q ♣r♥ ♥s ①♣rs♦♥s ♥ ór♥s ♠♥t s t♦♠♠♦s G = 109 ♠−1 ω0 = 1014 s−1 M = 10−22

♦t♥♠♦s q Ipunto cero ≃ 0,9 ② q Iclasico = 0,96 sts st♠♦♥s ♠② ♣r♦①♠s ♥♦s♥♥ q ♥ ♣r♥♣♦ s ♣♥ ♠r ♣♦s ①♣r♠♥ts ró♥ ♥ ♥♦s

t♦ össr ♦s r♠♥t♦s ♣r str t♦r ❲ t♠é♥ s ♣♥sr ♣r ♠só♥ s♥ rtr♦s♦ ♦ ást ♥ ♥és r♦ss ♥ r②♦ γ ♣♦r♥ú♦s ♦s rst ♠♦ t♦ össr

♥ st ♠só♥ s♣tr♦ ♣ ♣rs♥tr í♥s ♠② strs ♠só♥ q ♥♦♣r♥ r s♦ ♥s♥s ♣♦r ♠♦♠♥t♦ tér♠♦ ó♥ q ♦r♠ ♣rt ♥ú♦ ♥ r Pr r♦♥s uq ♣qñs ró♥ s♣tr♦ ♥ ss ♠s♦♥s

ásts ♦ s♥ rtr♦s♦ s ♣r♦♣♦r♦♥ 1− 12

⟨(kγ · uq)

2⟩ ♦♥ kγ s t♦r

♦♥ ♦tó♥ γ st♦ ♥♦s r ♦tr♦ t♦r ❲ ♦♠♦ q s st♦ ♣r

st s ♠♦ ♠♦♦ ♥st♥ q s strá ♥ t♠ ♥á♠ r rst♥ s á♦ tst♠♣rtrs

r♦ ♦♥ á♦ r③♦ ♣r t♦r ❲ ró♥ r②♦s ❳ ♥ í♠t ♠♣ts ró♥s♥t♠♥t ♣qñs

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❨ ❯❯

♦s r②♦s ❳ q s♠♥r ♥t♥s ♣♦ ♠só♥ γ s♥ rtr♦s♦ ♣r♦q ♥♦ ♦ ♥s♥ ♣r♠♥t

st rst♦ s ♣ ♥tr♣rtr ♦♠♦ q ♥ r♥ ♣rt ♠♦♠♥t♦ q át♦♠♦♦ ó♥ qr ♠tr ♥ r②♦ γ s tr♥sr ♥tr♠♥t rst ♦♠♦ ♥ t♦♦ ♣qñ ♣rt rst♥t s tr♥s♦r♠ ♥ ♦♥♦♥s ♦r♠ q tr♥sr♥ ♥rí♥s♥ í♥ ♠só♥ γ t♦r ❲ s♦♦ ró♥ r ♠ ♥ ♠♥r ♣r♦♣♦ró♥ ♠s♦♥s γ s♥ rtr♦s♦ q ♠t rst

♦ q t♦ össr é í♥s t♥ strs s ♠② út ♣r st♦ ♠♣♦s ♠♥ét♦s ♥ ♠♦és ② só♦s

❯❯

t♦r strtr ♦♥sr ♦♠♦ ♥ ♦♥ s ♦♣♦♥♠♦s q r ♦♥str♠♦s ♦♠♦ ♦♥ ♥ s ♦s át♦♠♦s s♥♦ ♥ (0, 0, 0) ② ♦tr♦ ♥ (12 ,

12 ,

12) ♦r♥♦ q

SG =

s∑

j=1

fj e−iG·σj bi · aj = 2πδij

♥♦s q q ♥ st s♦

SG =s∑

j=1

fj e−iG·σj = f

[1 + e−iπ(h+k+l)

]= Shkl,

♣♦r ♦ q

s h+ k + l = ♠♣r e−iπ impar = −1 Shkl = 0

s h+ k + l = ♣r e−iπ par = −1 Shkl = 2f

♦ strtr t♥ ①t♥♦♥s ♣r h + k + l = ♠♣r ♦♠♦ ♠♣♦ s ♣♣♥sr ♥ ♣♥♦ (100) ♦♥ r♥ s ♥tr ♥♦ ♦s r②♦s rt♦s ♣♦r ♣♥♦ rst♥♦ s π

❯ ró♥ ♣♥♦

♥ ♠♥r ♠ás ♦r♠ s ♣♦rí r q ♥ ♥ ♣r♦s♦ srt♦ ♣♦r r ♦r♦ t♦rí ♣rtr♦♥s♣♥♥ts t♠♣♦ és s♥tr ís á♥t ♥ q st♦ ♥ sst♠ ② ♠♥♦s ♥ ♣rtí ♥ ♦♥t♥♦ ♥rís ♣r♦ tr♥só♥ ♣♦r ♥ t♠♣♦ s ♣r♦♣♦r♦♥ ♣r♦t♦ r♦ ♠♥t♦ ♠tr③ tr♥só♥ ♣♦r ♥s st♦s ♦s st♦s ♥s rst♥ts♥ ró♥ r②♦s ❳ ♣♦r rsts ② ♥ t♦ ösr ♥rí r②♦ ❳ s♥t s ♠♦ ♠②♦r q s ♥rís ①tó♥ r ♠♥r q ♥ s♣♦ ss ② ♠② ♣♦ r♥ ♥tr s♣tr♦ ást♦ ② ♥ást♦ ♥♦♥s♥ ♥♦ ② ♥ ♥s♥♠♥t♦ ①s♦ ♦s ♣♦s s ♥ ss ♥t♥ss s♠♥②♥

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❮ Ó strtrs rst♥s

t♦r strtr ♦♥sr ♦♠♦ ♥ ♦♥ s tr♦ át♦♠♦s♣♦♥♠♦s q r ♦r♠♠♦s ♣rtr ♥ ♦♥ ♥ s tr♦ át♦♠♦ss ♥♦ ♥ (0, 0, 0) ② ♦tr♦s ♥ (0, 12 ,

12) (

12 , 0,

12) ② ♥ (12 ,

12 , 0) ♥ st s♦

SG = Shkl =s∑

j=1

fj e−iG·σj = f

[1 + e−iπ(k+l) + e−iπ(h+l) + e−iπ(h+k)

].

P♦r t♥t♦

s h, k, l = ♣rs ♦ ♠♣rs Shkl = 4f

s h, k, l = ♦s ♣rs ② ♦tr♦ ♠♣r Shkl = 0

s h, k, l = ♦s ♠♣rs ② ♦tr♦ ♣r Shkl = 0

♦ strtr t♠é♥ t♥ ①t♥♦♥s t♦s s r①♦♥s ♥ s q ♣r (h, k, l) s st♥t

s♦ s ♦sr ♥ r ♥ q ♣r♥ ♦s s♣tr♦s ró♥ r②♦s ❳ ② r ♠♦s ♥♥ strtr ♦♠♦ ♦s ♦♥s − ② + t♥♥ strtr tró♥ r s ♦♠♣♦rt ♦♠♦ ♥ strtr ② ♦s♥t♥t r s a/2 r♥t ♦s r②♦s ❳

ró♥ r②♦s ❳ ② r

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♣ít♦

❯ P PÓ

r♦ ♦♥ ♦ q ♠♦s ①♣st♦ ♥ ♥tr♦ó♥ ♠♦s trtr só♦ ♦♠♦ ♥♦♥♥t♦ ♥ts q ♣♦♠♦s s♣rr r♠♥t ♥ ♦s r♣♦s

♦s ♦♥s q s ♥♥tr♥ r♦s ♣♦st♠♥t ② ② stró♥ s♣ rst sstrtr ♣ró

♦s tr♦♥s q s ♠♥ ♥tr♦ ♣♦t♥ r♦ ♣♦r ♦s ♦♥s ② ♣♦r ♦s ♠s♠♦s

r♠♦s ♦s s♣♦s♦♥s ♥♠♥ts

♦s ♦♥s stá♥ qt♦s ♥ ss ♣♦s♦♥s qr♦

♦s tr♦♥s t♥♥ ♠♦♠♥t♦s ♥♣♥♥ts ♥♦s ♦tr♦s

❱r♠♦s ♠ás ♥t q ♣r♦①♠ó♥ ♦r♥♣♣♥♠r ♦ át st s♣♦só♥ ② q r♥ ♠ss ♥tr ♦s ♦♥s ② ♦s tr♦♥s ♥♦s ♣r♠trá s♣rr② r ♥♣♥♥ts ♠♦s ♠♦♠♥t♦s

s♣t♦ ♣♥t♦ ② q r q ♣r♦ s ♠s♠♦ ♦r♥ ♠♥tq ♦♥t ♦♥ r♦ ♦s tr♦♥s ♥ ♠♦♦ tr♦♥s rs sí ♣s s♥sr ♥ sr♣ó♥ á♥t st ♣r♦ P♦r ♦tr ♣rt ♦ ú♥♦ q ♣♦♠♦s rrs♣t♦ ♣♦t♥ t♦ U(r) q tú s♦r ♦s tr♦♥s s q s rst s ♣rt♦ ♣♦t♥ stsr ó♥ ♣r♦

♠s ♦♥♦♥s ♥♦s ♥ q ♥str♦ st♦ só♦ ♥trrs ♥ rs♦r ♣r♦♠ ♥ tró♥ ♣r♦①♠ó♥ ♠♦♥♦tró♥ ♥ ♥ ♣♦t♥ U(r) q t♥ ♣r♦ r rs st♦ s q

U(r+R) = U(r) ♣r t♦♦ R ∈

3.1

♦s tr♦♥s ♥♣♥♥ts q sts♥ ó♥ rö♥r ♠♦♥♦tró♥

− ~2

2m∇2Ψ(r) + U(r)Ψ(r) = εΨ(r)

♠♣♥♦ U(r) ó♥ s ♠♥ tr♦♥s ♦

Ps ♥ ♦s tr♦♥s ♦ ♣♥ sr srt♦s ♣♦r ♥ ♥ó♥ ♦♥ Ψ ♣r♦t♦ ♥ ♦♥ ♣♥ ♣♦r ♥ ♥ó♥ u(r) q t♥ ♣r♦ r st♦ s

Ψnk(r) = eik·runk(r) ♦♥ unk(r+R) = unk(r) ♣r t♦♦ R ∈

st s ♠♦ t♦r♠ ♦ q ♥♦s q ♦s tr♦♥s ♦ stá♥ srt♦s ♣♦r ♥♦♥ ♣♥ ♠♦ ♣♦r ♥ ♥ó♥ unk ② ♣r♦ s r

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❮ Ó tr♦♥s ♥ ♥ ♣♦t♥ ♣ró♦ ♦r♠ ♦

tr ♠♥r r♦ s ♦s t♦st♦s H ♣♥ s♦rs ♦r♠ q Ψ stás♦ ♥ t♦r ♦♥ k t q

Ψnk(r+R) = eik·RΨnk(r) ♣r t♦♦ R ∈

♠♦stró♥ t♦r♠ ♦♥♠♦s ♥ ♦♣r♦r trsó♥ TR ♣r t♦r R st♦ s q ♠♣q

TRf(r) = f(r+R).

♦♠♦ ♠t♦♥♥♦ s ♣ró♦

TR [H(r)Ψ(r)] = H(r+R)Ψ(r+R) = H(r)Ψ(r+R) = H(r)TRΨ(r),

q s ♥ q s ♠♣ ♣r t♦ ♥ó♥ Ψ P♦r t♥t♦ s t♥ q

TRH(r) = H(r)TR

② ♣♦r ♦♥s♥t ♦♣r♦r TR ♦♥♠t ♦♥ H[TR, H

]= 0

♦♠♦ TRTR′ = TR′ TR = TR+R′ ♥t♦♥s ♦♥♥t♦ ♦♣r♦rsH; TR, ∀R ∈

s ♥ ♦♥♥t♦ ♦♣r♦rs q ♦♥♠t♥ ② ♣♦r ♦ s ♣♥ s♦r t♦st♦s H q ♦ s♦♥ s ③ t♦♦s ♦s ♦♣r♦rs TR ♥ rs♠♥ ①st ♥ ♦♥♥t♦ t♦st♦s Ψ q r♥

H(r)Ψ(r) = εΨ(r)

TRΨ(r) = c(R)Ψ(r) =⇒ TRΨ(r) = Ψ(r+R) = c(R)Ψ(r) ∀R ∈ RD

s♥♦ c(R+R′) = c(R)c(R′) ② q TRTR′Ψ = c(R)c(R′)Ψ = TR+R′Ψ = c(R+R′)Ψ♦s c(ai) s♦♥ ♥ú♠r♦s ♦♠♣♦s q ♣♥ srrs ♦♠♦ c(ai) = e2πixi xi ∈ R P♦r♦ c(R) = |c(a1)|n1 |c(a2)|n2 |c(a3)|n3 s t♦r r rt R s R = n1a1+n2a2+n3a3 ♦r ♥♠♦s t♦r k = x1b1 + x2b2 + x3b3 ♦♥ ♦s t♦rs bi s♦♥ ts qbi · aj = 2πδij s ♠♣ q c(R) = eik·R

♠♦s ♠♦str♦ t♦r♠ ② q ♠♦s ♠♦str♦ q ♦s tr♦♥s ♦ st♦s ♦s q r♥ q H(r)Ψ(r) = εΨ(r) ♣♥ sr srt♦s ♣♦r ♥ ♥ó♥ ♦♥Ψ q ♠♣ q

Ψ(r+R) = TRΨ(r) = c(R)Ψ(r) = eik·RΨ(r).

3.2

• • t♦r♠ ♦ ♥tr♦ ♥ t♦r k q t♥ ♥ r♥ ♣r♦ ♦♥ t♦r ♦♥ ♦♥ ♣♥ q r♣rs♥t ♥ó♥ ♦♥ ♦s tr♦♥s rs ♦r ♥ ♠♥trs q ♠♦♠♥t♦ ♥ p s ~ ♣♦r ♦ t♦r ♦♥ ♥ s♦ ♦s tr♦♥s rs ♥♦ t♥s ①♣rsó♥ ♣r ♦s tr♦♥s ♦ ② q ♠t♦♥♥♦ ♥♦ t♥ ♥r♥ trs♦♥♦♠♣t s♦♠♥t t♥ r

♥ t♦ ♣r t♦♦ R ∈ s t♥ q∫

|Ψ(r+R)|2dr = 1 =∫

|c(R)|2 |Ψ(r)|2dr = |c(r)|2

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♥ t♦

pΨ =~

i∇Ψnk =

~

i∇(eik·runk(r)

)= ~kΨnk + eik·r

~

i∇unk(r)

② ♦sr♠♦s q Ψ ♥♦ s ♥ t♦♥ó♥ ♦♣r♦r ♠♦♠♥t♦

P♦r ♦♥s♥t ♥q ~k s ♦♥♦ ♦♠♦ ♠♦♠♥t♦ rst♥♦ ② q t♥r q ♥t q ♥♦s ♠♦♠♥t♦ ♥ ② q k s ♥ ♥ú♠r♦ á♥t♦ q rtr③ sst♠ ♥ st s♠trítrs♦♥ r ♣♦t♥ ♣ró♦ q tú s♦r ♦s tr♦♥s

• • t♦r k q ♣r ♥ t♦r♠ ♦ ♣ rrs ♣r♠r ③♦♥ r♦♥P❩

♥ t♦ s k′ /∈ P❩ ♥t♦♥s ♣♦♠♦s r q ①st ♥ t♦r G ∈ t q k′ = k+G♦♥ k ∈ P❩ ♦♠♦ s ♠♣ q eiG·R = 1 ♣♦r ♣r♦♣ ♥ó♥ r rí♣r♦ s t♥♠♦sq

Ψ(r+R) = eik′·RΨ(r)

♣r ♥ó♥ ♦♥ Ψ ♥t♦♥s t♠é♥ s ♠♣rá q

Ψ(r+R) = eik·RΨ(r)

♣r ♠s♠ Ψ

♦♠♦ rst♦ qr tró♥ ♦ ♣ sr rtr③♦ ♣♦r ♥ t♦r k r♦ P❩

• • í♥ n q ♣r ♥ t♦r♠ ♦ s ♦ q ♣r ♥ k ♦ ①st♥ ♥♥tss♦♦♥s ó♥ rö♥r

♥ t♦ sq♠♦s t♦s s s♦♦♥s ♥str q t♥♥ ♦r♠ Ψk(r) = eik·ruk(r) ssttr ♥ ♠♦s

H(r)u(r) =

[~2

2m

(1

i∇+ k

)2

+ U(r)

]uk(r) = εkuk(r).

st s ♥ ó♥ t♦♦rs ②s s♦♦♥s ♥ stsr ♦♥♥t♦ ♦♥♦♥s ♦♥t♦r♥♦ uk(r+R) = uk(r) ♣r t♦♦ R ∈

♥ ♦♥s♥ ♠♦s ♦ ♥ ♣r♦♠ t♦♦rs ♥♦ ♥ ♦♠♥ ♥t♦ ♥ú♥ ♣r♠t r ♥ s ♠ás s ♣r♠ts s rá rs♦r ♥ s♠r♥♠♦s ♣s q ♣r k rá ♥ sr ♥♥t ② srt t♦♦rs q ♥♦s♦tr♦srtr③♠♦s ♦♥ í♥ ♥ n ♠♦s tqtr ♣s s ♥♦♥s ♦♥ ♦♠♦Ψnk(r) ② unk(r) ② ♦s t♦♦rs ♦♠♦ εnk

rs ♥♦tr q ♥ ó♥ t♦r k ♣r ♦♠♦ ♥ ♣rá♠tr♦ ♠t♦♥♥♦ ♠♥r q ♣♦♠♦s s♣rr q s♣tr♦ s s♦♦♥s εnk rí ♦r♠♦♥t♥ rr ♣r♦♣♦ k r♣rs♥tá♥♦s ♣♦r εn(k)

• • ♥♠♦s sí ♥ sr♣ó♥ ♦s ♥s ♠♦♥♦tró♥♦s ♥ ♥ ♣♦t♥ ♣ró♦ ♥ tér♠♥♦s ♥ ♠ ♥♦♥s εn(k) q ♥♦♠♥r♠♦s strtr ♥s só♦

Pr ♦r n ♦♥♥t♦ ♥s tró♥♦s s♣♦s ♣♦r εn(k) s ♠ ♥

♥rí

♦♠♦ ② ♠♦s ♦♠♥t♦ s ♥♦♥s ♦♥ rst♥ sr s ♣r k ② k+G ♣r t♦♦G ∈ ♥t♦♥s s t♥ q

Ψn k+G(r) = Ψnk(r)

εn(k+G) =εn(k)

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❮ Ó tr♦♥s ♥ ♥ ♣♦t♥ ♣ró♦ ♦r♠ ♦

• • ❱t♦rs k ♣r♠t♦s❱♠♦s ♦r str s s ♦♥♦♥s ♦♥t♦r♥♦ q rst♥ s♠trí trs♦♥ r ♥♦s ♥ ♦ r ♦s ♦rs ♣♦ss ♦s t♦rs k q rr③♥ s s♦♦♥s Prs♠♣r ② s♥ q t ♦s rst♦s q ♦t♥r♠♦s ♠♦s s♣♦♥r q t♥♠♦s ♥ rst♥t♦ ♦♠♥ V ♥ ♦r♠ ♣r♣í♣♦ ♦♥ Ni s ♦ r♦ s r♦♥s ai q ♥♦s♥♥ r rs ♥ú♠r♦ t♦t s ♣r♠ts ♥ rst s N = N1N2N3

♠♣♦♥♠♦s ♦♥ó♥ ♦♥t♦r♥♦ s♥t ♦s tr♦♥s ♦

Ψ(r+Niai) = Ψ(r) i = 1, 2, 3, . . .

❯t③♥♦ t♦r♠ ♦ ó♥ ♥tr♦r ♥♦s q

Ψn k+G(r+Niai) = eiNik·ai Ψnk(r) i = 1, 2, 3, . . .

② ♦♠♣r♥♦ ♦♥ ó♥

eiNik·ai = 1, k = x1b1 + x2b2 + x3b3) =⇒ e2πiNixi = 1 i = 1, 2, 3, . . .

♦♥ ♦ q ♦♠♥t t♥♠♦s q Nixi = mi ∈ Z ② q

k = x1b1 + x2b2 + x3b3 =m1

N1b1 +

m2

N2b2 +

m3

N3b3.

♥ rs♠♥ ♦s t♦rs k ♣r♠t♦s s♦♥ q♦s q ♠♣♥ ó♥ ♦♥ mi ∈ Z ♦♠♥ s♣♦ rí♣r♦♦ s♦♦ ♥ t♦r ♣r♠t♦ s ♣♦r t♥t♦

|∆k| = b1

N1·(b2

N2× b3

N3

)=

1

N· ♦♠♥ ♣r♠t

s♠♥♦

♣r ♦r í♥ ♥ n ♥ ♥ ♣r♠t ② t♥t♦s t♦rs k

♣r♠t♦s ♦♠♦ ♥ú♠r♦ s ♣r♠ts ② ♥ r st♦ s ♦♠♦ ♣♥t♦s ② ♥

♦ q ♦♠♥ ♣r♠t s (2π)3 /(V/N) t♥r♠♦s q ♦♠♥ s♣♦ rí♣r♦♦ s♦♦ ♥ t♦r ♣r♠t♦ s |∆k| = (2π)3 /V

3.3 ❯P

st♦ ♥♠♥t ♥ ♦♥♥t♦ N tr♦♥s ♦ s ♦♥str② ♦♣♥♦ t♦♦s ♦s♥s ♠♦♥♦tró♥♦s ♥rís εn(k) ♦♥ εn(k) <εmax s♥♦ n í♥ ♥ ② k ♠♦♠♥t♦rst♥♦ tró♥ ② st♥♦ k r♦ P❩ ♦r εmax s rqr♥♦ q ♥ú♠r♦ t♦t ♥s tró♥♦s ♦♣♦s s ♥ú♠r♦ t♦t tr♦♥s sst♠

P ♦rrr q tr♠♥r ♦♣r ♦s ♥s ♠♦♥♦tró♥♦s ♦♥ ♦s tr♦♥s rsts ♥♥ ♥ rt♦ ♥ú♠r♦ ♥s ② s rst♥ts q♥ ís r♥ ♥ ♥rí ♥tr t♦♣ ♥ ♥ ♣rt s♣r♦r ♠s♠ ② ♣rt ♥r♦r ♥ í s ♠ ♥♥és ♥r② ♣ ② r ♥ s♣ñ♦ ♥♦♠r ③♥ ♦ r ♥rí ♦ ♥tr♦ ♣r♦♦ ♥rís ♠♦s r q ♥ú♠r♦ ♥s ♥ ♥ ♦♥rt s ♥ú♠r♦ s ♣r♠ts rst ♦♠♦ ♥ ♠♦♥♦tró♥♦ t♥ ♥ró♥ ♦s ♣♦r s♣í♥

st ♦♥ó♥ ♦♥t♦r♥♦ s ♠s♠ q ♦r♥♦♥ r♠♥ q s srá ♥ sr♣ó♥ ♥ ♠t ♦♥ ♠♦♦ r♦♠♠r

st rst♦ ♦♥ ♦♥ q s ♦t♥ ♥ ♦trs r♠s ís ♥♦ s ♠♣♦♥♥ ♦♥♦♥s ♥t♦r♥♦ ♣♦r♠♣♦ ♥ ís stíst

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❱

s♦♠♥t ♣ ♣rr ♥ ♣ ♥rís ♥♦ ♥ú♠r♦ tr♦♥s ♣♦r ♣r♠t s♣r

P♦r ♦tr ♣rt ♦♣r ♦s ♥s ♣♥ qr ♥ ♦ rs ♥s ♣r♠♥t ♥s ♦♥♥t♦ s s♣rs ② ♥rí s ♥rí r♠ s ♠ s♣r r♠

rst ② s ♠ r♠ n s♣r r♠ q ♦rrs♣♦♥ s ♦r tr♠♥♦ n ♥ st s♦ ♦r εmax r ♥♦♠r ♥rí r♠ εF q s ♥rí qs♣r út♠♦ ♥ ♦♣♦ ② ♣r♠r s♦♣♦ só♦

♦♠♦ s ó♦ ♦s só♦s ♦♥ ♣ ♥rís s♥ts ② s♠♦♥t♦rs ♥♦ t♥♥ s♣r r♠ ② q qr ♥rí q sté ♥ ♣ ♠♣ ♦♥ó♥ s♣rr út♠♦ ♥♦♣♦ ② ♣r♠r s♦♣♦ só♦ P♦r ♦tr♦ ♦ ♦s ♠ts sí q ♠♣♥ s ♦♥ó♥♣♦r ♦ q t♥♥ ♥rí r♠ ② s s ♣ ♠r só♦s ♦♥ s♣r r♠

3.4 ❱

♥♦ ♣r r③r á♦ ♥ ♣r♦♣ ís ② q r ♥ s♠ s♦r ♦st♦rs ♣r♠t♦s k s ♣ r s s♥t ♣r♦①♠♦♥s

♥ó♥ Fn(k) ♥♦ rí ♣r♠♥t ♥ ♥ ró♥ t♠ñ♦ ∆k ♦♥ ∆k s ♦♠♥ s♣♦ rí♣r♦♦ s♦♦ ♥ t♦r ♣r♠t♦ ② s ♥ó♥ t♥ ♣r♦ ♣r♦①♠ó♥ r③r ♦♥sst ♥

2∑

n,k

Fn(k) −→V→∞

2V

(2π)3

∑

n

∫

Pdk Fn(k)

♦♥ P s♥ ♣r♠t ② ♥ ♥ró♥ s♣í♥

♠ás Fn(k) ♣♥ n ② k trés strtr ♥s εn(k) ♣♦♠♦s ♥r♥ ♥s st♦s ♦ ♥s Dn(ε) t ♠♥r q

2∑

n,k

Fn(k) =∑

n

∫

Pdε Dn(ε) Fn(ε)

s♥♦ Dn(ε) ♥s ♥s ♥ ♥ q s ♣ r ♦♠♦

Dn(ε) =2V

(2π)3

∫

Pdk δ (ε− εn(k)) .

①st ♦tr ♠♥r r ♥s st♦s Dn(ε) ♥ t♦ ♣♦r s ♣r♦♣ ♥ó♥

Dn(ε) dε = 2× ♥♠ st♦s k ♣r♠t♦s ♥ ♥ n ♦♥ ♥rí ♥tr ε ② ε+ dε.

♥ú♠r♦ t♦rs k ♣r♠t♦s ♥ ♥ n ♥tr ε ② ε+ dε s ♦♠♥ s♣♦ k

♦♠♣r♥♦ ♥tr s s♣rs q♥réts ε ② ε + dε ♦ ♣♦r ♦♠♥ ∆k q ♦♣ k ♣r♠t♦ ♥t♦♥s s ♠♣ q

Dn(ε) dε =2V

(2π)3

∫

Pdk ×

1 s ε ≤ εn (k) ≤ ε+ dε

0 ♥ qr ♦tr♦ s♦

st♦ s ♦♥♥t♦ ♣♥t♦s q ♠♣♥ εn(k) = εF ♣r t♦s s ♥s n

ísst♦ó♦

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❮ Ó tr♦♥s ♥ ♥ ♣♦t♥ ♣ró♦ ♦r♠ ♦

♣rs q♥réts

♠♦s ♦r s s♣rs q♥réts Sn (ε) ②Sn (ε+ dε) s♣♦♥♥♦ q stá♥ ♥ ♥tr♦r ♣r♠t

♥s ♥s s ♣ r ♥t♦♥s ♦♠♦és r

Dn(ε) dε =

∫

P

2V

(2π)3dk δ (ε− εn(k)) dε

=2V

(2π)3

∫

Sn(ε)dS δk (k) .

Pr♦ ε+ dε = ε+ |∇εn(k)| δk (k) ♠♥r q ♥t♦♥sδk (k) = dε/|∇εn(k)| ② ♥♦s q

Dn(ε) =2V

(2π)3

∫

Sn(ε)dS

1

|∇εn(k)|

q ♥♦s ♥ str ró♥ ♥tr strtr ♥s εn(k) ② ♥s ♥s Dn(ε)♥ ♥ ♦♥♥ ♥♦tr q s ró♥ ♥ ♠② ♥♥ ♣♦r s rs εn(k) trés tér♠♥♦ 1/|∇εn(k)|

• ♥rs ♥s st♦s

♦♠♦ ♣r n εn(k) s ♣ró ♥ ② ♦t rá ♦rs k ♥ P ♥ ♦s q|∇εn(k)| = 0 P♦r ♦♥s♥t ♥tr♥♦ Dn(ε) r ♣r♦ s ♣ ♠♦strr q ♥ trs♠♥s♦♥s sts s♥rs s♦♥ ♥trs ② ♥ ♦rs ♥t♦s Dn(ε)

P♦r ♦tr ♣rt ② ♣♥t♦s ♥ ♦s q r D′n(ε) s ♥♥t sts s♥rs s ♠♥s♥rs ♥ ♦ ♦♠♦ ♥ á♦ s ♣r♦♣s íss s t♠♣rtrs♥tr♥ Dn(ε) ② ss rs sts s♥rs ♥ ♦ ♣♥ ♥r ♥ ♦♠♣♦rt♠♥t♦tró♥♦ s♣♠♥t ♦s ♠ts t♠♣rtr

♥s st♦s

♠♦s ♥ rís♠♦ ♦sq♦ s♦s♣♥t♦s rít♦s ♥♦s ♣♦r |∇εn(k)| = 0q ♣♦rt♥ rtrísts ♥ ♥s ♥s st♦s Dn(ε) ② ss rs s♦ ♠ás s♥♦ s ♦s ♠á①♠♦s ②♠í♥♠♦s ♥ ♥ ②s ♣r♦①♠ss ♣ srr ♥♦ ♠♥t ♦ss ♥rí ♥ ♦♠♦ ♥①♣rsó♥ rát t♣♦

εn(k) = εm + γ1κ21 + γ2κ

22 + γ3κ

23,

♥ tér♠♥♦s t♦r κ = k − k0 s♥♦ k0 ♣♥t♦ ♦♥ stá ♠á①♠♦ ♦ ♠í♥♠♦ ♦♥ ♦rεm ♦t ♦s s♥♦s ♦s ♦♥ts γi s♦♥ ♥t♦s ♦ ♣♦st♦s ♣♥♥♦ s ♣♥t♦ k0 s ♥♠á①♠♦ ♦ ♥ ♠í♥♠♦ ♥

st♦ ♦♥ rt♠♥t ♥ ♥s st♦s ♦r♠

Dn(ε) ∝

(ε− εm)1/2 s k0 s ♥ ♠í♥♠♦

(εm − ε)1/2 s k0 s ♥ ♠á①♠♦

♦ q s♥ q ♥ s♦s ♣♥t♦s ∂Dn(ε)/∂ε s ♥♥t

❯♥ ♠♣♦ s♥♦ s t♣♦ s♥rs ♥ ♥trs ♥♠♥s♦♥s s ♥tr∫ 10

1√xdx ②♦ ♥tr♥♦

r ♣r♦ q s ♥tr

ísst♦ó♦

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3.5

• • ❯♥ ♥♥♦ ♠ás ♣rs♦ t♦r♠ srí♦ ♠t♦♥♥♦ − ~2

2m∇2 + U(r) ♦♥ U(r+R) = U(r) ♣r t♦♦ R ∈ s s♦♦♥s ó♥ rö♥r ♠♦♥♦tró♥

− ~2

2m∇2Ψn(r) + U(r)Ψn(r) = εnΨn(r)

q ♥♦ s♥ ♥rs ② ♦♠♥♦♥s ♥s s s♦♦♥s ♥rs s♦♥ t♠é♥t♦♥♦♥s ♦♣r♦r TR ♣r t♦♦ R ∈ ♦♥ t♦♦rs eik·R s t♦♥♦♥s Ψn(r)♦ s♦♥ ♠t♦♥♥♦ ② TR ♣♦r ♦ q s tqt♠♦s ♦♠♦ Ψnk(r) ② ♦s t♦♦rs εn s♦♥♥♦♥s t♦r k st♦ s εn(k).P♦r ♦♥s♥t ♣r ♥ó♥ ♦♥ Ψn(r) ② s♦♦ ♥ t♦r k q ♥♦s ♦s t♦♦rseik·R t♦♥ó♥ Ψn(r) rs♣t♦ t♦♦s ♦s ♦♣r♦rs TR ♣r t♦♦ R ∈ st ♠♥rΨn(r) stá s♦ ♦ rtr③♦ ♣♦r t♦r k Ψnk(r)

• • U(r) = ♦♥st♥t ♥t♦♥s Ψ(r) = Ceik·r ② ♥ í♠t tr♦♥s rs t♦r k t♥ sr t♦r ♦♥ tró♥

• • G ∈ ♥t♦♥s ♣r qr Ψn(r) s ♠♣ q

TRΨnk+G(r) = ei(k+G)·RΨnk+G(r) = eik·RΨnk+G(r),

P♦r ♦ t♥t♦ ♥ ♥ó♥ ♦♥ Ψnk(r) q s t♦♥ó♥ H ♥♦ s♦♠♥t ② ♥ t♦r ks♦♦ s Ψnk(r) s♥♦ q t♦♦s ♦s t♦rs k+G ♣r t♦♦ G ∈ stá♥ t♠é♥ s♦♦s ♥ó♥ ♦♥ Ψnk(r) ② ♣r t♦♦s ♦s t♦♦r Ψnk(r) rs♣t♦ TR s eik·R♥ rs♠♥

Ψnk(r) ≡ Ψnk+G(r); ♣r t♦♦ G ∈

② t♦♦s ♦s ♣♥t♦s k+G ♣r t♦♦ G ∈ s♦♥ q♥ts ♥ sr♣ó♥ ♦s tr♦♥s só♦st♦ qr r q ♥ ③♦♥ r♦♥ s s♥t ♣r sr ♥ tér♠♥♦s ♥t♦r k s s♦♦♥s ♥ ♣♦t♥ ♣ró♦

• • s ♠ü ♦r k s♥♦ ♥ó♥ ♦ Ψnk(r) ② rs ♦r♠s♦♥♥♦♥s r st s♥ó♥sts s♦♥s s ♠♥ sq♠s ③♦♥ ② r ♥♦ ♦tr♦ sq♠ stá s♦♥♦r♠♠♥t ♥ qé ♣r♦♣s strtr ♥s s qr♥ rstr

• ♠♣♠♦s ♣♦r ♥ ♠♥só♥

sq♠ ③♦♥ ①t♥

♥ sq♠ ③♦♥ ①t♥

s s♥ k q ♦rrs♣♦♥ ♠ás♣r♦①♠♠♥t ♦♠♣♦rt♠♥t♦ tr♦♥s rs ♦ ♠♦♦ ♠ás ♥r ♦♠♣♦rt♠♥t♦ s ♦♥s ♥ ♥ ♠♦♦♥t♥♦ s♥ strtr ♣ró ♦s♦♥s

♥ st sq♠ ♣r♠r ③♦♥ s ró♥ s♣♦ rí♣r♦♦ q stá ♥tr♦ ♣r♠r s♦♥t♥ ♥rét s♥③♦♥ s ③♦♥ q stá ♦♠♣r♥♥tr ♣r♠r ② s♥ s♦♥t♥s

ísst♦ó♦

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❮ Ó tr♦♥s ♥ ♥ ♣♦t♥ ♣ró♦ ♦r♠ ♦

t r q s ♥s ♥rí s♦♥ ss ♥tr♦s ♥rí ♥ ♦s q ♥♦ ②s♦♥t♥s

♥ sq♠ ③♦♥ r s t③ t♦r♠ ♦ ♣r q t♦♦s ♦s t♦rsk sté♥ ♥tr♦ P❩ st ♦r♠ s q ♥ ♥rís ♠ás stá ♥ ♣r♠r③♦♥ s♥t ♥ ♥ s♥ ③♦♥ t

s q εn(k+G) = εn(k) ② q t♦r k+G r♣rs♥t ♦ s ♠s♠♦ tró♥ q t♦r k ♥ sq♠ ③♦♥ r♣t ♦ ♣ró s ♣r♠t q ró♥ ♣ró εn s ①t♥ t♦♦ s♣♦ rí♣r♦♦ ♥t♦♥s ♥ st sq♠ ♥ ③♦♥ s rr ♥ ♥ ♥rís ♠ás q ♥ ró♥ ♦♥rt

sq♠ ③♦♥ r

rs sq♠s ③♦♥ ♦ r♣t tt s♣ñ♦ ♥ ♥és

• Pr ♦s ♠♥s♦♥s s ♠♦r t ③♦♥ r ② r♣t ♥ t♦ ♦♥str②♠♦s ♣r♠r ② s♥ ③♦♥s r♦♥ ♥r ♠♥s♦♥ és r s ③♦♥s ♠ást♦ ♦r♥ s♦♥ ♦♠♣s ② sq♠ ③♦♥①t♥ ♥♦ s út

♥ r s ó♠♦ ♠♦s trs♦ s♥ ❩ ♣rts r②s ♥ r♦♦ s♦r P❩t③♥♦ ♣r ♦ ♦s t♦rs ±a1 ±a2 ② ±(a1±a2)♦s ss r♦♥s s ♠♦s trs♦s ♣♦r t♦rs ② ♦♠♦ ♦♥s♥ ♥ ♦s s♠♥t♦s qs♣r♥ ss r♦♥s trss ♥ó♥ εn(k) sr ♦♥t♥ ② ∂εn(k)/∂k sr ♥ ♣r

❱és ♥ ♣rt♦ q ♣r ♥♠t♠♥t ♦ ♦rró♥ ♥tr s s♦♥t♥s ② s ③♥s ♥rí♣r♦s

ísst♦ó♦

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q ♥ó♥ εn s ♣ró P♦r ♦♥s♥t ♥ s♥ ③♦♥ strtr εn(k) s ♠ás♦♠♣ q ♥ P❩

• • P♦r t♦r♠ ♦ s s q

TRΨ∗nk(r) = e−ik·RΨ∗nk(r)

② q t♠é♥TRΨ

∗n −k(r) = e−ik·RΨ∗n −k(r).

♦♠♦ t♦r♠ ♦ ♥♦s s ♥ó♥ ♦♥ ♦♥ t♦r k s r♦ q Ψnk(r) s Ψ∗n −k(r) ♦♠♦ ♠t♥♥♦ H s r H = H∗ ♥t♦♥s ♥♦♥s ♦♥ Ψnk(r) ② Ψ∗nk(r)stá♥ ♥rs ♦ q qr r q t♠é♥ ♦ stá♥ Ψn −k(r) ② Ψnk(r)P♦r t♥t♦ t♦r♠ r♠rs

εn(k) = εn(−k).

• • s ♥s ♥rí ♣♥ s♦♣rs ♦r♠q ♦s st♦s ♥rí ♠ás t ♥ n stá♥♣♦r ♥♠ ♦s st♦s ♥rí ♠ás ♥ n+ 1

♥ ♦s ♠ts ♥rí r♠ stá♥ ♣r ♥ ♥ ♦ ♣r ♥ ♦♥♥t♦ ♥sq s s♦♣♥ ♦♥ rst♦ q s♣r r♠ rst♥t q ♣ t♥r ♣♦r ♦ t♥t♦ ♥♦r♠ ♠② ♦♠♣ tr♠♥rá s ♣r♦♣s ♠t qr t♠♣rtr

• • ♥s ♥ ② ♦♥ó♥

♥♦♠♥ ♥ ♥ ❱ ♥ ♥s ♥rís ♠ás t♦s q stá♥♦♣♦s ♣♦r tr♦♥s ♥ r♦ s♦t♦♦♠♦ ② s ♦ ♥ s♠♦♥t♦rs ② s♥ts ♣r ♥ ♥ ♣r♦ ♦ ♣ ♣♦r ♥♠ ♥ ♥ s ♥ ♥ ♦♥ó♥ ♥rís s♣r♦rs♥ ♦s ♠ts ♣♦r ♦♥trr♦ ♥♦ ② ♥♥ú♥ ♥tr♦ ♥rís ♣r♦s ♥tr ♥s ♥ ② ♦♥ó♥♥ ♦ rt♦ s ♣r♦♣s ♦♥ó♥ ♥ ♦s s♠♦♥t♦rs str♠♦s ♥trs♦s ♥ strtr ♥s r ♦s ①tr♠♦s s ♥s ♥ ①tr♠♦ s♣r♦r ② ♦♥ó♥①tr♠♦ ♥r♦r st♦ s q ♦♠♦ ♥♦ ② tr♦♥s t♠♣rtr r♦ ♥ ♥ ♦♥ó♥ ② q ♥ ♥ stá ♦♠♣t♠♥t ♦♣ ♣r ♥ ♠♣♦ étr♦ ♦str♦♥s ♥♦ ♣♥ ♥r♠♥tr s ♥rí ♥♦ r st♦s s♣♦♥s ♥ ♥ ♥♥ ♠r♦ ♣r ♣r♦♣s ó♣ts s ♥sr♦ ♥ ♦♥♦♠♥t♦ t♦ rs ♥s ②q ♦s tr♦♥s st♥ ♥s ♦trs ♥ st♦s ♣r♦s♦s sts ♣r♦♣s ♣s s♦♥ ♠♣♦rt♥t♥ s♥ts ② s♠♦♥t♦rs

t♠♣rtrs st♥ts r♦ rá tr♦♥s q s ①t♥ tér♠♠♥t s ❱ ② ♣r♦ r♦ s ♣r♦♣♦r♦♥ t♦r ♦t③♠♥♥ ♦rrs♣♦♥♥t tr♥só♥ ♥rét

e−(Eg/kBT ).

♥♥♦ ♥ ♥t q kBTamb ≃ 1/40 ❱ ② q ♦s ♦rs tí♣♦s ♣ s♦♥ ESCg ≃ 1 ❱s♠♦♥t♦rs ② Eaislantes

g ≃ 10 ❱ q r ♥♦r♠ r♥ ♥tr ♦s t♦rs ♦t③♠♥♥

♣♠♦s ♦♣r♦r ♦♥ó♥ ♦♠♣ ♥ó♥ ♦ Ψnk(r) = eik·runk(r) ❯♥ ③ ♦t♥ ♦♥ó♥♣♠♦s ♦♣r♦r TR

s ♥rí s♣r ♦s st♦s ♦♣♦s ② ♦s í♦s

ísst♦ó♦

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❮ Ó tr♦♥s ♥ ♥ ♣♦t♥ ♣ró♦ ♦r♠ ♦

♣r ♠♦s t♣♦s ♠trs st ♠♥r t♦rí ♥s ♣r♦♣♦r♦♥ ♥ rs♣st ♥♠t ♥♦r♠ r♥♦ ró♥ ♦s ♦rs ♦♥t étr q ♣rs♥t♥ ♦s só♦s

• • ♥♦♥s ❲♥♥r ♠♥r q ♠♦s rtr③♦ ♥ó♥ ♦♥ ♦♥ ♥ t♦r k s ♣♦s t③r♥♦♥s s♦s s ♣♦s♦♥s ② ♣♦r ♦♥s♥t tqtrs ♦♥ s♦s t♦rsPr ♦ ♠♦s ♥♦tr q ♥ ♥ó♥ ♦♥ ♦ Ψnk(r) ♣r r ♦ t♥ ♣r♦ P♦r ♦♥s♥t s ♣ srr♦r ♥ sr ♦rr

Ψnk(r) =∑

R

fnR(r)eiR·k

♦♥ s ♥♦♥s fnR(r) q r♥ ♥♦♠r ♥♦♥s ❲♥♥r s♦♥ ♦s ♦♥ts ♦rr srr♦♦ st♦ s

fnR(r) =1

ΩP❩

∫

P❩dk e−iR·kΨnk(r).

♦r ♥ fnR(r) ♣r♠♥ ♥tr s ♥ ó♥ s s♣③♥ r ② R ♥ st♥R0 ∈ ② ♣♦r t♥t♦ fnR(r) ≡ fn(R, r) = fn(r−R) ♥ ♦♥s♥ ♥ s fnR(r) s♦♦♣♥ st♥ s ♣♥t♦ r ♦♥ ♠♦s ♥ó♥ ♣♥t♦ R ♦♠♦ s ♥♦♥s ❲♥♥r fn(r−R) st♥t♦ ♦r n ② st♥t♦ R s♦♥ ♦rt♦♦♥s ♥tr sí② ♠♦s r q qr ♥ó♥ ♦ ♣ ①♣rsrs ♦♠♦ ♦♠♥ó♥ ♥ s♦♥stt②♥ ♥ ♦♥♥t♦ ♦rt♦♦♥ ♦♠♣t♦ ♥♦ n r♦rr N ② R r rs s ♥♦♥s❲♥♥r s♦♥ ♣♦r t♥t♦ ♥ s út ♥♦ st♠♦s ♥trs♦s ♥ srr ♣r♦♣s tró♥s♥ s q t♥♥ ♠♣♦rt♥ ♦③ó♥ tr♦♥sP♦r ♦tr ♣rt s ♥♦♥s ❲♥♥r st♥ ♦r♠♠♥t t③ó♥ ♠ét♦♦ rsrts q r♠♦s ♥ r ♥ tr♠♥s r♥st♥s ♥ t♦ s♠t ♦r♠♥ q s ♣ srr ♥ó♥ ♦ ♠♥t s ♥♦♥s ❲♥♥r ② ♥ ♠ét♦♦ rsrts ♣♥sr ♥ q s ♥♦♥s ❲♥♥r ♥ sr ♥♦♥s ♦③s ♥ s s♦ rí♥ stó♥ ♦r♠ s♦ rs rts ♥q s s q s ♥ ♥rís ♥♦ ss♥t♠♥t str s ♥♦♥s ❲♥♥r ♣♥s t♥♥ ♣r♦ ♦♥ ♦s ♦rts tó♠♦s

• • ①t♦♥s ♠♥ts♥q ♠♦s ♦ s♦♠♥t tr♦♥s ♥♦ s ♣♥t♦ t♦r♠ ♦ ♠ss s②♥ts s♦♥ ♣s ♦tr♦s ♥ó♠♥♦s q s ♣rs♥t♥ ♥ ♦s só♦ss ♥ ♣♥t♦ st ♦♥♣t s♣rr q ♦s tr♦♥s rsts ♦♥ ♥s ♠tá♦ ♦♦♥t stá♥ ♦♠♣rt♦s ♥tr ♦s át♦♠♦s ② ♣♦r t♥t♦ srr♦s ♦♠♦ s♦③♦s ♥ t♦♦ rst t♦r♠ ♦ s ♠② ♥trP♦r ♦tr ♣rt s ♥ ó♥ r s ♣rt s ♣♦só♥ qr♦ ② r ♦♣♠♥t♦♥tr ♦s ♦♥s rá q ♥rí r♦♥ s ①t♥ ② ♣♦r t♦♦ rst ② ♥①tó♥ r♦♥ ♥ t♦♦ rst ss r♦♥s ♥ sr ♥t③s s s sr♠♦s ♥♥ ♦r♠s♠♦ á♥t♦ ② ♣rtí s♦ r ♥♦♠r ♦♥ó♥st sr♣ó♥ á♥t s ♣♦ss ①t♦♥s ♥ ♥ rst ♥♦s s♣rr q ♣♦♠♦ssrr t♦s s ♥ tr♠♥♦s ♦♥s ② s♦r ♥ ♣rtí ♥t♦ ♥rí sí♠ás tr♦♥s ♦♥♦♥s ② ♦t♦♥s ♣♦r♠♦s r ♠♥♦♥s ♣s♠♦♥s t ♦s sts

♥ó♥ ♦ ①♣rs ♥ tér♠♥♦s s ♥♦♥s ❲♥♥r q ♠♦s ♥♦ q ♦♠♦

Ψnk(r) =∑

R

fn(r−R)eik·R,

s♠r ♠♥r q s ♦t♥ ♥ó♥ ♦ ♣rtr ♦rts tó♠♦s ♥ ♣r♦①♠ó♥ rs rts

ísst♦ó♦

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❩ ❯

①t♦♥s ♦ s♦♥ rst ♥ s ♦♥♥t♦ ♦ q ♣r♠t str ss ♣r♦♣s ♥rs ♠♥r q s ♣♥t♦ ♣r t♦r♠ ♦ ② t♦s s strá♥ rtr③s♣♦r ♥ t♦r k

♥ ♥ r ♥t ♥ú♠r♦ ①t♦♥s t♠é♥ ♦ s ♦s ♦rs k ♦r♠♥ ♥ ♦♥♥t♦♥t♦ st srt③ó♥ ♥tr♦ ♣♦r s ♦♥♦♥s ♦♥t♦r♥♦ q s ♠♣♦♥♥ ♥ s♣r sst♠ ♥t♦ ♥♦ s r ♥ ♥♥♥ s ♣r♦♣s íss s♦ q t♠ñ♦ rst s ♠② ♣qñ♦

st♦ ♣♦r ♠♣♦ s ♦sr♦ ♥ ①♣r♠♥t♦s ♦♥ ♣ís rr♦♠♥éts s ♦♥ ♥tró♥ ♠♥♦♥s ♥② ♥ trs♠só♥ trés ♣í s ♦sr♥♥t♦♥s ♠♥♦♥s srt♦s ♦♠♦ s ♠str ♥ r

3.6 ❩ ❯

①tó♥ ♠♥t tró♥ ♦♥ó♥ t rtr③ r♦ ♦♥ t♦r♠ ♦♣♦r ♥ t♦r k t♥ ♠s♠ ♠♣t ♥ rst ♣rt♦ |Ψnk(r)|2 = |Ψnk(r+R)|2♣r t♦♦ R ∈

①st ♥ ♠♣ró♥ ♦③ ♥ só♦ ♣r r♦ q ró♥ ♥ ♣♦r ♠♣ró♥ ♥ s Ψnk(r) strá rstr♥ ♦s r♦rs ♠s♠ ♠♥r q ♦r♠ ♦ ♦s st♦s ①t♥♦s s rá ♣♦♦ t P♦r ♦tr ♣rt ♣♥ ①str①t♦♥s ♦♥♥s ró♥ ♠♣rt ② ♠♣ t♥ ♦r ♣r s♦♠♥t ísts ①t♦♥s ♥ sr ♠② st♥ts s ①t♥s ♥ rst ♣rt♦ ② q ♥♦ t♥♥♥♦ ♥ t♦r k q s rtr

♥ ♥r st♦s st♦s ♥♦s s♦♦s s ♠♣r♦♥s ♦③s srá♥ st♦s♦③♦s ♦ ♦s ② s ♥rí strá r r♥♦ s ♥s ♥rí rá s♥ ♠r♦ú♥ s♦ ♥ q s ♥rí ♦♥rá ♦♥ ♥ ♥ ♦r♠ q s ♦♣♠♥t♦ st♦♦③♦ ♥ s é s ♣ ♥tr♣rtr ♦♠♦ ♥ st♦ ♥ ② ♠♣t s ♥♦ts♥♦ ♣ ♥ s ♠②♦r r ③♦♥ ♠♣ró♥ ♥ st♦ rs♦♥♥t

♣rs♥t ♥ sq♠ ♠♦s t♣♦s st♦s ♦s ② rs♦♥♥ts ♥ r ♦♥♥r ♥♦tr q ♥♦ ♠♦s st♦s ♦s ♦s st♠♦s ♦♠♣r♥♦ ♦♥ ♦s s♦③♦s ♥ ♣♦r ♦ q ♥rí s♦s st♦s ♦s str ♣♦r ♦ ♠í♥♠♦ ♥ st♦s s♦③♦s

❱♠♦s ♦♥rtr ♥ ♣♦♦ ♦ q ♠♦s ♦ st ♦r ♦s r♠♦s ♥ ♦s t♦s q♥tr♦ ♥ s♣tr♦ tró♥♦ ♥ ♠♣ró♥ ♦③ ♥ ♦s r♦rs ♣♥t♦ r0 qtr ♦♠♥t ♣r♦ sst♠ ♣ sr ♥ át♦♠♦ t♣♦ st♥t♦ ♦s r ♥♥t t

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❮ Ó tr♦♥s ♥ ♥ ♣♦t♥ ♣ró♦ ♦r♠ ♦

Pr rst ♦s st♦s st♦♥r♦s ♠t♦♥♥♦ tró♥♦ H0(r) = H0(r + R)∀R ∈ ♣♥ srrs ♦♠♦ ♥♦♥s ♦

Ψnk(r) =1√Veik·runk(r)

② stá♥ rtr③♦s ♣♦r ♥ s♠♦♠♥t♦ ~k ② ♥ ♥rí εn(k)

♣rtró♥ ♠♥♦♥ ♣ srrs ♠♥t ♥ ♥rí ♣♦t♥ W (r − r0) ♥♦ ♠t♦♥♥♦ s H0(r) +W (r− r0) ② ó♥ rö♥r ♦rrs♣♦♥♥t srá

[H0(r) + W (r− r0)

]Φn(r) = EnΦn(r)

♥ ♥ó♥ ♦♥ sst♠ ♣♦♠♦s ①♣rsr ♦♠♦ ♦♠♥ó♥ ♥ s ♥♦♥s♦ ♥tr♦rs ♥ ♦♥ ♥ ♥rí εn(k)

Φn(r) =∑

k

akΨnk(r).

ssttr ♥ ó♥ s ♦t♥ ♥ sst♠ N ♦♥s ♥ ♣r t♦r ♥ n q st♠♦s ♦♥sr♥♦ ♥♦ ♠③♠♦s ♥s ② q ♥♦ s♠♠♦s s♦r n

[En − εn(k

′)]ak′ −

∑

k

Wnkk′ak = 0 ∀k′

♦♥Wnkk′ =

∫dr Ψ∗nk′(r)W (r− r0)Ψnk(r)

s ♠♥t♦ ♠tr③ ♣♦t♥ ♣rtr♦r ♥tr s ♥♦♥s ♦ k ② k′ ♦♠♦ ♠♦s♥t♦ ♥ts W (r−r0) s ♣r s♦♠♥t ♥ ♥ ♦♠♥ Ω0 r♦r r0 ♠♣ró♥stá ♦③ ♣♦♠♦s srr q

Wnkk′ ≃ Ψ∗nk′(r′0)W0Ω0Ψnk(r′0),

s♥♦ r′0 ∈ Ω0 ② ♥♦ ♥♦ W0Ω0 ≡∫dr W (r − r0) ♦r ♣r s♠♣r s ①♣rs♦♥s

♠♦s r′0 = 0 ② s rsr ♦♠♦

[E − εn(k)] ak′ −[∑

k

akΨnk(0)

]

︸ ︷︷ ︸Φn(0)

Ψ∗nk′(0)W0Ω0 = 0 ∀k′

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② s♣♥♦ ♦t♥♠♦s

ak′ =W0Ω0Ψ

∗nk′(0)Φn(0)

E − εn(k′).

♣rtr st s♦ó♥ ♣♦♠♦s r s ♥rís ② s ♥♦♥s ♦♥

• ♥rís

stt②♥♦ ♥ ó♥ ①♣rsó♥ ak′ q ♠♦s ♦t♥♦ ♥ ♥♦s q q

1

W0Ω0=∑

k

|Ψnk(0)|2En − εn(k)

≃ C∑

k

1

E − εn(k)

út♠♦ ♣s♦ ♣ rs s♦♠♥t ♥ s♦ ♥ q Ψnk(0) ♥♦ rí ♣r♠♥t ♦♥ k

st s ó♥ q ♥♦s s ♥rís En s ♥ ó♥ r♦ N ② q t♦r k

r♦rr N ♦rs ♣♦r ♦ q t♥ N rís ♦r♥♠♦s s ♥rís εn(k) ♥ ♦r♥ r♥t ε1ε2 ε3 εN ♣♦♠♦s r♣rs♥tr ♠♦s ♠♠r♦s ♥ r

s s♦♦♥s s♦♥ s En q sá♠♦s ♦♠♦ s♦♦♥s ó♥ t♦♦rs ♦s st♥t♦s ♦rs En s♦♥ ♣rt♠♥t s ♦s εn rr q ♦s ♦rs ε1ε2 ε3 εN ♦r♠♥ ♥ ♥ s♦♥t♥ ♥rís s s ①♣tú♥ ♦s ♥s E1 ♥♦W0 < 0 ♦ EN ♥♦ W0 > 0 ♥ rs♠♥ s♦♠♥t ♥♦s ♦s N st♦s s♦③♦s ♥ n s ♥♦ ♦r♠ ♥♦t ♣♦r ♣rs♥ ♣♦t♥ ♣rtr♦r W (r− r0)

• ♥♦♥s ♦♥

Pr r s ♥♦♥s ♦♥ st♦s ♥s sstt♠♦s ♦ ♦t♥♦ ♥ ♥ ①♣rsó♥♣r Φn(r)

Φn(r) =∑

k

akΨnk(r) =∑

k

W0Ω0Ψ∗nk(0)Φn(0)

E − εn(k)Ψnk(r)

=W0Ω0Φn(0)∑

k

Ψ∗nk(0)

E − εn(k)Ψnk(r)

≃W0Ω0Φn(0)∑

k

Ψnk(r)

E − εn(k)

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❮ Ó tr♦♥s ♥ ♥ ♣♦t♥ ♣ró♦ ♦r♠ ♦

♠♣♦

♣q♠♦s ♦ ♥tr♦r ♥ ♠♣♦ ♥♠♥s♦♥ s♥♦ ♦s t♦rs k ♣r♠t♦s ♣♦r s♦♥♦♥s ♦♥t♦r♥♦ ís s♦♥ km = 2πm/Na ♦♥ a s ♣rá♠tr♦ r

♦♥sr♠♦s s♦♠♥t ♥ ♥ ♥rís sr♠♦s ♦♠♦

ε(k) = ε0 +∆(1− cos ka)

qí 2∆ > 0 ♥♦s ♥r ♥ ♦t♥♠♦s ♣r ó♥

1

W0Ω0≃ B

∑

km

1

A(Ei) + cos(kma)

s♥♦ A(Ei) = (Ei − ε0 −∆)/∆

♦♠♦ ♦s ♦rs km ♦r♠♥ ♥ s♦♥t♥♦

1

W0Ω0≃ NB

2π

∫ 2π

0

dx

A(Ei) + cosx= NB

A(Ei)

|A(Ei)|1√

|A(Ei)|2 − 1

s |A(Ei)| > 1

á♥♦♥♦s ♥ st♦ ♥rí Ei ♠ás q ♥♦t♠♦s ♣♦r E0 t♥r♠♦s

E0 = ε0 +∆

1 + W0

|W0|

√

1 +

(W0Ω0NB

∆

)2

q ♥♦s r♠♥t q sW0 < 0 ♥ E0 s s♣r ♦ ε0 st s♣ró♥ ♣♥ ♥r ♥ ♠②♦r ∆ ♠②♦r s♣ró♥ ② W0B

♦t st♦s rst♦s ♣r♥ ♥ st♦ ♦s ♥s ♠♣r③ ♥ ♥ s♠♦♥t♦r♦♥ ①st ♥ ró♥ ♥tr ♦♦ó♥ s♦s ♥s ② ♦♥st♥t étr ♠trq ③ ♥② ♥ ♥r t♥t♦ ♣ ♦♠♦ s ♥s s♠♦♥t♦r

♦r ♥♦s ♠♦s ♥ ♥ó♥ ♦♥ t③♥♦ ó♥ ♣r ♥ E0 ②s♣♦♥♥♦ q s ♠♣ q uk(z) ≃ uk(0) t♥r♠♦s

Φ(0)(z) = Φ(0)(−z) ∝∑

k

eikz

A(E0) + cos(ka)

∝

[√[A(E0)]

2 − 1− |A(E0)|]z/a

[A(E0)]2 − 1

s |A(E0)| > 1

♦♥ s ♦ ♥tró♥ ♠s♠ ♠♥r q ♥tr♦r♠♥t

st ♦♥ó♥ ♠♣ |∆+ ε0 − Ei| > ∆ st♦ ♥♦s q s W0 > 0 Ei > ε0 + 2∆ q s ♦♥ó♥ ♣r st♦ ♠ást♦ ② q st♠♦s ♣♦r ♥♠ t♦ ♥ ❨ s W0 < 0 Ei < ε0 q s ♦♥ó♥ ♣r st♦ ♠ás ♦ ♣♦r ♦ ♦r ε0

W0 > 0 q E0 = ε0 +∆

[

1 +

√

1 +(

W0Ω0NB∆

)2]

② ♦rt s ♠②♦r q 2

W0 < 0 ①♣rsó♥ s E0 = ε0 +∆

[

1−

√

1 +(

W0Ω0NB∆

)2]

② ♦rt s ♥t♦

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❩ ❯

♣rs♥t♠♦s ♥ r ♥ó♥ ♦♥ Φ(0)(z) q ♦♠♦ s r♠♥t stá ♦③♥ s ♣r♦①♠s ♣rtró♥♥ rs♠♥ s s ♠♣ q |A(E0)| > 1 q ♥ ♥ st♦ ♦♥ ♥rí E0 r ♥ ♥ó♥ ♦♥ stá ♦③ r♦r ③♦♥ ♠♣ró♥

ísst♦ó♦

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ísst♦ó♦

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♣ít♦

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♦s ♠ts ♦♥stt②♥ ♥ ♣rt ♠♣♦rt♥t ♦s só♦s rst♥♦s ♠ás s ♣rts ♦s ♠♥t♦s rst③♥ ♥ ♦r♠ ♠tá ♦s ♠ts t♥ s s♥ts rtrísts ♣r♥♣s

♦♥t étr ② tér♠ ♥s ♣♦rt♦rs r ② ♥rí

r♦ ♥♦ ♣♥tr ♥ ♦s ró♥ s♣tr♦ s

t ② ♠ s♥ q ♥ ♠tá♦ s ♣♦♦ r♦♥

♠♦♦ r♦♠♠r s ♥ ♠♦♦ ♦r♥ r♦ ♣r srr ís ♦str♦♥s ♥ ♠t s♠♣

♦s ♠ts s♦♥ só♦s ♥ ♦♥♦♥s ♥♦r♠s s♦♥ ♥♦s ♦♥t♦rs ♦r ② tr② ♠á♥♠♥t s♦♥ úts s ♣♥ rr ♣r ♦r♠r ♠rs ♦ ♦s ② ♠s s ♣♥①t♥r ♥ ♣♥s ♦ á♠♥s ♠ás s♥ sr r♦s ② t♥r ♥ ♣♥t♦ só♥ t♦ ♦r♠♥strtrs ♦♠♣ts ♦♥ ♦ q t♥♥ ♥ss ts ② s ♣♥t♦ st ó♣t♦ s♥sr r♥ts ② ♥♦ tr♥s♣r♥t♥ ③

st ♣r♠r ♣s♦ ♥ sr♣ó♥ ♦s ♠ts ♦♥sst ♥ s♥ ♥ t③r ♥ ♠♦♦ s tr♦♥s q stá ♦♠♣♥s♦ étr♠♥t ♣♦r ♥ r ♣♦st ♥♦r♠ q r♣rs♥t ♦s ♦♥s

♠♦♦ ② s rtrísts ♦♥stt ♥s t③r ♥ stíst á♥t s ♥ ♥á♠ ♣sr ás P♦r ♦♥s♥t st♦ ♥♠♥t s ♥ sst♠ r♠♦♥s ♦str♦♥s s♦♥ ♣s ①t♦♥s s st♦ ♥♠♥t ② t♥♥ ♥ t♠♣♦ tr♠♥♦ ♣♦r ♥ ♣r♦①♠ó♥ t♠♣♦ ró♥ τ

♦♥t σ0 str rt♠♥t r♦♥ ♦♥ s ♣rá♠tr♦ trés ♠♥♦ r ♠♦ l = vF τ r♠♦s ♦r s♣í♦ tró♥♦ str♠♦s t♦ s tr♥s♠só♥ s ♦♥s tr♦♠♥éts ♥ ♠t sí ♦♠♦ ♦♥t tér♠ ② s ró♥♦♥ ♦♥t étr

♠♦♦ r♦♠♠r s ♥ ♣r♦①♠ó♥ ♦r♥ r♦ ♠t♦♥♥♦ ás♦ qsr só♦ H = Hion + He− + He−−ion ♥ q s s♣♦♥ q

♥♦ ①st ♥tró♥ tró♥ó♥ tr♦♥s rs ♥ ♥tró♥ tró♥ tró♥ tr♦♥s♥♣♥♥ts

♦s ú♥♦s ♦♥stt②♥ts ♠t q ♥ r ♣r♦♣s íss s♦♥ ♦s tr♦♥s ♥ ♦s át♦♠♦s ♠t q r♥ ♥♦♠r tr♦♥s ♦♥ó♥ ♦s♥♠♥t tr♦♥s

♠rr♦ s ♥ ①♣ó♥ st♦ s♥ q s ♥ ♠♦♦ ♥á♠♦ ♥♦ ♣♦r ♣rá♠tr♦ τ

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❮ Ó ♦♦ r♦♠♠r

♦♠♣♦rt♠♥t♦ ♦s tr♦♥s stá r♦ ♣♦r t♦rí ♥ét ♥tr♦ ♣r♦①♠ó♥ t♠♣♦ ró♥ τ

s ♦♠s♦♥ sr tró♥ r ♦r♠ s ♠♦♦ ás♦ ♣r s tr♦♥s tr♦r ♣r♦♣♦♥ s ♠♦♦ ♣r át♦♠♦

st ♠♦♦ s ♣s s♥♠♥t ♣ó♥ t♦rí ♥ét ♦s ss ♦♥♥t♦ ♦s tr♦♥s ♥ ♥ ♠t s♠♣ st♦ s q♦s ♠ts ②♦s tr♦♥s ♦♥ó♥♣r♦♥ ♣s tó♠s s ② p

4.1 ❱ ❳P

❱♠♦s ♣s s ①st ♥♠♥tó♥ ①♣r♠♥t r③♦♥ ♣r t③r ♥ ♠♦♦ t♥s♥♦ ó♠♦ s ♣♦s s♣♦♥r q ♦s tr♦♥s ♥ ♠t ♣♥ sr ♦♥sr♦s ♦♠♦rs ♥♣♥♥ts ♦s r♠♦s ♣r♠r♠♥t ♥ ♥♦s rst♦s ①♣r♠♥ts q ♣r♥str ♦♥ó♥ tr♦♥s ♦s ♠ts ♦♠♦ ♥♦ ♦s tr♦♥s rs

• ♦♥t étr rsst ♥ ♠t t♠♣rtr rt♠♥t t T ∼ 300 ♠♥t ♥♠♥t ♦♥T ♠♥trs q s t♠♣rtrs T < 10 t♥ ♥ ♦r ♦♥st♥t st ♦r ♣♥r♥♠♥t ♠str rst q s trt ② ♣ r sr ♠♦ ♠♥♦r 4 q rsst t♠♣rtr ♠♥t st rsst s t♠♣rtrs stá s♦ ♦st♦s r ② s ♠♣r③s q ♦st③♥ ♣s♦ ♦s tr♦♥sP♦r ♦tr ♣rt s s r tts♥ q s ♥ú♠r♦ ♠♣r③s ♥ ♠str s♣qñ♦ rsst ρimp ♠♣r③s s ♥♣♥♥t t♠♣rtr

♥ ρ ♦♥ T t♠♣rtrs ts ♣ ①♣rs s rsst s tó♥ tér♠ r ♦s ♠♦s ♦♥♦♥s ♥ t♦ t♠♣rtr ♠♥t ró♥ ♦s ♦♥s r ♣ trtrs ás♠♥t P♦r ♦♥s♥t t♦r♠ ás♦ q♣rtó♥ ♥rí ♥♦s q ♥rí ró♥ ♦s ♦♥s r st♠♣rtr s ♣r♦♣♦r♦♥ kBT P♦r ♦tr ♣rt ♥rí ró♥ s ♣r♦♣♦r♦♥ 〈u2〉 s♥♦ u s♣③♠♥t♦ ó♥rs♣t♦ s ♣♦só♥ qr♦ ♦r ♥ 〈u2〉 s ♠ás ♥ ♠ só♥ ③tr♥srs ó♥ r♥t s r á s t♠ñ♦ q t♥ ó♥ q r sú♥ ♦ tró♥ ② ♥ ♥ ♣r♠r ♣r♦①♠ó♥ s ♣ ♣♥sr q rsst s ♣r♦♣♦r♦♥ só♥ ③

ρfonones ∝ 〈u2〉 ∝ kBT (a Tamb).

Pr ♣s r③♦♥ ①trr ♦♥só♥ q ♥ú♠r♦ ♣♦rt♦rs r ♥ ♠t♥♦ ♠♥t ♦♥ T

t♠♣rtrs s ♦s ♠♦♦s ró♥ r stá♥ ♦♥♦s ② ♦s ♦stá♦s qs ♣rs♥t♥ ♦s tr♦♥s s♦♥ ♦s t♦s r ♦ s ♠♣r③s q ♣ ♦♥t♥r rst② q ρfonones → 0 ② ♦♥tró♥ ♦s ♦♥♦♥s rí sr ♥ t♦s s ♠strs♦r ♥ ♦♥st♥ ρimp rr t♠♣rtr sr q ♥s ♣♦rt♦rs r s ♥♣♥♥t T ♥ ♦♥tr♣♦só♥ ♦♥ ró♥ ①♣♦♥♥ q ♣r ♥ ♥s♠♦♥t♦r t♦r ♦t③♠♥♥ q r ♣r♦ ①tó♥ tér♠ ♥ ♥ ♦♥ó♥

s ♣♦r ♠♣♦ q ♥ ♥áss ás♦ ♦r s♣í♦ ♣r♦♣♦r♦♥ ♥ ♦r cv ♦♥st♥t q stá r♦♦♥ ② ♠♣ír ♦♥ ② Ptt

❱és s♥tr r♠♦♥á♠

ísst♦ó♦

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❱ ❳P

• s t♠é♥ q ♦s ♠ts ♥♦s ♠ts íq♦s t♥♥ ♥ ♦♥t étr♥♦ ♠② r♥t ♦s só♦s r s ♣♥t♦ só♥ Pr ♥t♦♥s q ♥q ♥ só♦♦s ♦♥s r♥ r♦r ♦s ♣♥t♦s s r ② ♥ íq♦ ♠♦♥ ♦♥t♥♠♥t s ♣♦só♥st♦s ♦s ♠♦♠♥t♦s tér♠♦s ♦s ♦♥s t♥♥ ♥ t♦ ♣r♦ ♣r s ♦s♦♥s ♦ ♥tr♦♥s♦♥ ♦s tr♦♥s st♦ ♣ ♥tr♣rtrs ♦♠♦ q ♦♥t ♥♦ stá ♥♥ ♣♦r s♣♦s♦♥s ♥st♥tá♥s ♦s ♦♥s r és ♠ás ♥t rst♦s ①♣r♠♥ts

• P♦r ♦tr ♣rt ♣r ♥t q ♥ s♦♣♠♥t♦ s ♥♦♥s ♦♥ ♦s tr♦♥s q♣rt♥♥ ♦♥s ♥♦s ♦rrá ♦♥t ♠tr q s trt ② q s♦ s♥ s♦③ó♥ ♦s tr♦♥s ♥ só♦ ♥ s♥t♦ ♣ ♦s ♦rts ♠♦rs qsr♥ ♦s tr♦♥s s ♠♦és

s q ♥ ♥ s♠♦♥t♦r ♦♥t♠♥♦ ♦♣♦ ♦♥ ♠♣r③s ♦♥♦rs ♦s tr♦♥s sts ♠♣r③s s ♦♠♣♦rt♥ ♦♠♦ s r♥ tr♦♥s ♥ át♦♠♦ r♦♥♦ r♦r r ♣♦st ♠♣r③ ♣♥t ♣♦r rst♦ ♠tr s♠♦♥t♦r st♦ r

♦ ♣♦♠♦s s♥ ♠r♦ s♣rr ♦s ♦♥s ♥ rst ♣r r s ② ♥ ró♥ t ♦ ♠♥r ♠ás s♠♣ s♦♥ qs ♠♣r③s q ♣♦rt♥ tr♦♥s s♠♦♥t♦r ♦ q ss át♦♠♦s

t♥♥ ♥ ♠②♦r ♥ú♠r♦ tr♦♥s q ♦s át♦♠♦s s♠♦♥t♦r és♣

ísst♦ó♦

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❮ Ó ♦♦ r♦♠♠r

q s♦s tr♦♥s ♦♥♦s r♥ órts ②♦s r♦s s♦♥ ♠② r♥s ♦r♥ ♦ q ♥♦s qr r q s ♥♦♥s ♦♥ s♦s tr♦♥s ♠♣③rá♥ s♦♣r ♣rtr ♥ ♦♥♥tró♥ ♣r♦①♠♠♥t ♥s 1018 ♠♣r③s♠3♦t Pr ♣♦r sr s s♦♥♥tró♥ ♠♣r③s s r♥ ♦ ♥♦ rr q ♥ú♠r♦ ♦r♦ s ♦r♥ 1022

♦s rst♦s ①♣r♠♥ts ♣r ♠str♥ q t♠♥t ♣r ♥ ♦♥♥tró♥ ♠♣r③s s ♦r♥ ♠♥t ② ♥ r♥ ♠♦ ♥ ♦♥t ♠♥t rá♣♠♥t♦♥ ♥s ♠♣r③s ② s st♥t ♥♣♥♥t t♠♣rtr ♠ás ♦♥t ♥♦s q ♣♦r ♥♠ s ♦♥♥tró♥ rít ♥ú♠r♦ ♣♦rt♦rs ♥♦♣♥ T ② s ♣r♦①♠♠♥t ♥ú♠r♦ ♠♣r③s qr♥♦ ♦♥trt♦ rátr ♠tá♦ ♣♦r t♦♦ ♦

• st ♦r ♠♦s st♦ ♠♦ ♦s tr♦♥s s♦ s s ♦♥ó♥ tr♦♥s ♥♦♦s tr♦♥s s♦③♦s ♦ rs ♣♦r sí r♦P♦♠♦s ñr t♠é♥ ♥ ♥ ①♣r♠♥t s♦r ①st♥ tr♦♥s♥♣♥♥ts st♦ s q ② tr♦♥s ♦♠♦ ♥ts rs ♣r♦ s♥ q sr♥ t♦s♠♣♦rt♥ts ♦ ♥tró♥ tró♥tró♥ ♦♥ rst♦ ♦s tr♦♥s ♥ t♦ ♣♦rí♦rrr q tr♥s♣♦rt r ♥sr♦ ♣r ①♣r t ♦♥t étr ♥♠t r ♦ ♥ st♦ ♦t♦ ♠♦s tr♦♥s ♦♣♦s ♦r♠ q t♠♣♦ ró♥ τ s ♦ sí ♦♠♦ t♠♣♦ ♦ st♦ ♦t♦

♥ ♠r♦ s ♣ ♠r á s ♥rí ♥sr ♣r ñr ♦ qtr ♥ tró♥ ♦♥ó♥ ♥ ♠t st♦ s ♦♥s ♦♥ r②♦s ❳ ♥♦s ♥rí q ①♣s♥ ♥tró♥ ♥ ♥ ♣r♦♥♦ ♦ ♥♦ ♦s ♦♥s rst ♥♦ ♥ tró♥ ♦♥ó♥st ♣r ♦♣r st♦ q ♦ í♦ tró♥ ①♣s♦ ♣♦r r②♦ ❳ s ♠t ró♥❳

♥♦s ♣r♠t ♠r ♥s ♣♦rt♦rs r ♦ str♠♦s ♠ás ♥t rátr ♠tá♦

s T → 0 rsst ρ ② ♦♥t σ s ♥ ♦♥st♥tss T →∞ ρ ∝ T ② σ ∝ 1/T

ísst♦ó♦

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♦s tr♦♥s ♥♦ r♥ ♥♣♥♥ts ♠ τni ♦s tr♦♥s srí ♠② ♣qñ② ♣♦r ♦♥s♥t s ♥rí ♥♦ strí ♥ ♥ ♥ ♠r♦ s ♠s ①♣r♠♥ts♠str♥ q ♣♦r ♠♣♦ ♥ ♥rí ♦s tr♦♥s ♦♥ó♥ ♠ás t ♥rí♦s q stá♥ ♥ ♦s út♠♦s ♥s ♦♣♦s stá ♥ ♥ ♣s ♥♦ s ♣rs♥t ♠só♥ r②♦s ❳ ♣r ♠②♦rs ♥rís é♥s s rs

♠ ♦s tr♦♥s ♥ ♦♥ó♥ s ♣♦r t♥t♦

tm >~

kBT∼ 10−34

5× 10−21∼ 2× 10−14 s.

❱r♠♦s ♠ás ♥t q t♠♣♦ ♥tr ♦s♦♥s ♥ tró♥ ♥ ♥ ♠t s sr τ ∼ 10−15 − 10−14 s ② ♦ q tm > τ ♣♦♠♦s ♦♥r q ♦s tr♦♥s s♦♥ ♣rtís ♥♥s ♥tr ♦s♦♥s

4.2

s♣és ①♣♦♥r sts ♥s ①♣r♠♥ts ♠♦s srr ♠♦♦ r♦♠♠r ♣r trt♠♥t♦ ♦s ♠ts s♠♣s

♥ st ♠♦♦ s s♣♦♥ q ú♥♠♥t ♦s tr♦♥s ♥ r s ♣r♦♣s íss ♠t ♥ st s♥t♦ ♦s ♦♥s ♠t sr♥ ú♥♠♥t ♣r ♥tr③r r ♦str♦♥s ♦♠♦ r♠♦s ést s s♣♦♥ ♥♦r♠ ♥ s♥ ♠♣♦s ①tr♥♦s ② ♣♦r ♦ t♥t♦ ♠♦♦ s♣♦♥ q t♠é♥ r ♣♦st stá ①t♥ ♠♥r ♥♦r♠ ♥ t♦♦ ♠t sí♥ s♥ ♠♣♦s ①tr♥♦s ② ♥tr ♦ r ② ♥ t♦♦s ♦s s♦s ② ♥trt♦t r

♦r ♣r♥t q sr s s ♠t s ♦♥sr ♦♠♦ ♥ s tr♦♥s rs ♥♣♥♥ts ♥tr③♦ ♣♦r r ♣♦st ♦s ♦♥s á s ♣rá♠tr♦ q ♥♦s ♣r♠tst♥r ♥ ♠t ♦tr♦ rs♣st s q ♠♥r q ♥ t♦rí ♥ét ás ♥ s ú♥♦ ♣rá♠tr♦ ♣♦s s ♥s ♦s tr♦♥s

Zc ♥ú♠r♦ tr♦♥s q át♦♠♦ q ♦r♠ ♣rt rst s tr♦♥sst ♥ú♠r♦ srá Zc = Z− tr♦♥s ♦s rt♠♥t ♥ú♦ ♦s♦tr♦s ♠♦s s♣♦♥r q♦s tr♦♥s ♦s s♦♥ ♦s tr♦♥s ♥ át♦♠♦ q s♦♥ ♦s q stá♥ ♠♥♦s ♦s ♥ú♦ st s♣♦só♥ s st♥t rt ♣♦rq ♥ r ♥♦ s♠♦s qé s ♦ q s♥♦s rt♠♥t ♣r♦ s rá ♠ás ♥t q s ♠② r③♦♥ ♥ s ♦s rst♦s①♣r♠♥ts

♥s tr♦♥s ♣♦r ♠3 s

n =N

V= NA

ZcρmA

♦ ♥ ♥rt♠r ♦r♥ kBT q s ♣r♦①♠♠♥t ❱ t♠♣rtr ♠♥t

ísst♦ó♦

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❮ Ó ♦♦ r♦♠♠r

s♥♦ NA ♥ú♠r♦ ♦r♦ ρm ♥s ♥ r♠3 ② A ♠s tó♠ ♥ r ♠♥t♦ st ♥s s ♦r♥ 1022 − 1023 ♠−3

♥♠♦s ♥ ♦♠♥ ♠♦ ♣♦r tró♥ ♥ sr r♦ r0 s t♥rá

1

n=V

N=

4

3πr30 =⇒ r0 =

[3

4πn

]1/3rs ≡

r0a0

s♥♦ a0 = ~2/me2 = 0,529 × 10−8 cm = 0,529 r♦ órt ♦rt s át♦♠♦ ró♥♦ r♦ ♦r

♦s ♦rs q t♦♠ ♣rá♠tr♦ rs ♥ ♦s st♥t♦s ♠ts s rs ≃ 2 rs ≃ 6 s♣s♥♦ ♣♦r rs ≃ 4 ♣r ♦s ♠ts t♥♥ ♦rs rs ♥tr ②

s ♥ss tr♦♥s ♥ ♦s ♠ts s♦♥ ♣s ♠ás s ♠②♦rs q s ♥s ás♦ ♥ ♦♥♦♥s ♥♦r♠s ♣rsó♥ ② t♠♣rtr

♦s s♣st♦s ás♦s ♠♦♦ r♦♠♠r s♦♥

s♣r ♥tró♥ tró♥tró♥ st♦ s st♠♦s ♥ ♥ ♣r♦①♠ó♥ tr♦♥s♥♣♥♥ts ② t♠é♥ ♥tró♥ ♥tr ♦♥s ② tr♦♥s ♣r♦①♠ó♥ tr♦♥srs ♥tr ♦s♦♥s ♥á♠ ♦s tr♦♥s stá r ♣♦r s ②s ♠♦♠♥t♦ t♦♥ ♦ ♦s ♠♣♦s ①tr♥♦s s♥ ♥r r♦ stá ♦s t♦s ♦s ♦♥s ♥ ♦str♦♥s ♠t

①st ú♥ ♠♥s♠♦ ♦só♥ sttr♥ ♦s tr♦♥s ♦♥ ♦s ♦♥s q ♣r♠ttr♠③r sst♠ ♥á♦♦ ♥ t♦rí ♥ét ♦s ss srí ♦só♥ tró♥tró♥ ♣r♦ r♠♦s♣♦str♦r♠♥t q ♠♣♦rt♥ sts ♦s♦♥s s ♣qñ ♥ ♦♠♣ró♥ ♦♥ ♦tr♦s♠♥s♠♦s ♥tró♥

♣r♦ q ♥ tró♥ ♦s♦♥ r♥t ♥ ♥tr♦ dt s dt/τ ♦♥ τ s t♠♣♦

ró♥ q s ♥♣♥♥t ♣♦só♥ r ② ♦ v ♦s tr♦♥s ♠♦s ③r ♥ tró♥ tr♥srrrá ♥ ♣r♦♠♦ ♥ t♠♣♦ τ ♣r q ♦s♦♥ ② tr♥srr♦ ♥ ♣r♦♠♦ ♠s♠♦ t♠♣♦ τ s q ♦ ③♦ út♠ ③

qr♦ tér♠♦ ♦s tr♦♥s sst♠ s ♦♥s ♠♥t s ♦s♦♥s ② q ♥♦② ♥tr♦♥s tró♥tró♥ ♥ tró♥ó♥ ♥ s q tró♥ s ♦ t♦s ♥tr♦r s ♦r♠ ró♥ s ♦s tr♦♥s s♣és ♦só♥ st♦r ② ♠♥t ♦ ♥♠t♠♥t s♣és ♦só♥ ♣♥ s♦♦ t♠♣rtr ♦ |v|(T )

4.3 ❯❱ ❯

s♦♠t♠♦s ♠t ♥ ♠♣♦ étr♦ E ② s j s ♥s ♦rr♥t q trs♦ ♣rs♥ ♠♣♦ ♠t s ♥ rsst ρ ♠♥t

E = ρj

♥ ♠♦ rsst ρ s ♥ sr ② q ♥♦ ② r♦♥s ♣rs

s t♥ ♥ ♦♥♥t♦ n tr♦♥s ♣♦r ♥ ♦♠♥ q s ♠♥ t♦♦s ♦♥ ♦v ♥s ♦rr♥t s j = −nev ♥ ♥ ♠t ♦♥sr♦ ♦♠♦ ♥ s tr♦♥s tró♥ ♥ ♦ st♥t ♥ s♥ ♠♣♦ ①tr♥♦ v t♥ ♥ ♣r♦♠♦ ♥♦

(st♦

♥♦♠r t♠é♥ t♠♣♦ ♦só♥ ♦ t♠♣♦ r ♠♦

ísst♦ó♦

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❯❱ ❯

s vm(E = 0) = 0) v0 s ♦ s ♥ tró♥ ♥ ♦só♥ ♥♦ t = 0

♦ ♦ tró♥ ♣♦str♦r♠♥t s

ve(t) = v0 −eEt

m

② ♦ ♣r♦♠ srá

vm = 〈ve〉 = 〈v0〉 −eE

m〈t〉 = −eE

mτ.

st s ♦ rrstr ♦s tr♦♥s ♣♦r ♠♣♦ étr♦ P♦r ♦ t♥t♦ ♠♣♦♠♥ét♦ ♥♦

j =ne2τ

mE =

E

ρ= σE.

qí s q

τ =m

ρne2

② s♥♦ ♦s t♦s ①♣r♠♥ts ρ ♣r r ♦r τ ♥tr♦ ♠♦♦ r♦♠♠r s ♦t♥ τ ∼ 10−14 − 10−15 s

• ♣rtr ♦r τ q s ♦ s♥♦ ♦s ♦rs ρexp s ♣ t♠é♥ r ♦r ♠♥♦ r ♠♦ l = vmτ

♠♦s ♠♥r ás ♦ ♠ ♦s tr♦♥s ♠♥t s♦ t♦r♠ q♣rtó♥ 1

2mv2m = 3

2kBT s ♦t♥ ♥ ♦r l t♠♣rtr ♠♥t ♥tr ② q ♦♥r ♠♥r tt ♦♥ ♠♦♦ s s ♣♥sr q ♠♥s♠♦ ♦só♥ ♦s tr♦♥s s ♦♥ ♦s ♦♥s ♠t

♥ ♠r♦ s ♦ ♠ s ♠♥t ♥ stíst á♥t s ♦t♥♥♠♥♦s rs ♠♦s ♦r♥ l ≥ 103 t♠♣rtrs ♠② s♠ás ①♣r♠♥t♠♥t s ♥ ♦t♥♦ ♦♥ ♠strs s♣♠♥t ♣r♣rs ② T ♠②s ♥♦s ♦rs st l ≃ 108 st♦ s ♠ ♥ ♠♦s s♦s ♦s rst♦s r♥ ♥r r♥ ♦♥ só♥ ♦r♥ r ② ♣♦r t♥t♦ s ♥tr♣rtó♥ τ r q s ♦s♦♥s ♦rr♥ ♥tr ♦s tr♦♥s ② ② q ♦♥r q ésts s♦♥ s ♦tr♦s♠♥s♠♦s

Pr ♦r t ♥tr♣rtó♥ τ ② ♣♦r ♥ l ♠♦s ♦r♥♦s s s♥♦② s♦♠♥t ♦♠♣rr♠♦s ♦♥ rst♦s ①♣r♠♥ts s♠♣r q ♦t♥♠♦s ①♣rs♦♥s ♥ sq ♥♦ ♣r③♥ ♥ t♠♣♦ ró♥ τ ♥ ♠♥♦ r ♠♦ l

❯♥ ♣r ♦♠♥tr♦s ♠♣♦rt♥ts ③ s ♣r♦①♠♦♥s ás ♦ á♥t♥ ♥á♠ ② ♥ stíst ♥ sst♠ ís♦

s ♥ ♣♥t♦ st stíst ♦s tr♦♥s ♣r q ♣r♦①♠ó♥ ás s á ♥rt♠r ♥ ♣♦só♥ ♥ tró♥ sr♠♥♦r q ♦♠♥ ♠♦ ♣♦r tró♥

(∆x)3 .1

n.

❯♥ só♥ ♥sr♠♥t ♠② s♥ s st♦ ♥ ssó♥ stró♥ r♠r ♥ s♥tr♥♠♥t♦s ís ② s rá ♠ás ♦r♠♠♥t ♥ s♥tr á♥ stíst

ísst♦ó♦

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❮ Ó ♦♦ r♦♠♠r

♦♠♦∆x·∆p ≃ h ② ♦ q ♥ ♠♦♦ s♦♦ ② ♥rí ♥ét ♥rt♠r♥ p ♥♦s r ♥ ♦♥tró♥ ♥rí ♦r♥ (∆p)2 /2m ② ♣♦r t♥t♦

∆x ≃ h√2m〈Ec〉

=⇒ h3

(2m〈Ec〉)3/2.

1

n=⇒ 〈Ec〉 &

h2

2mn2/3.

♥ ♣r♦①♠ó♥ ás 〈Ec〉 ≃ kBT ♣♦r r♦ rt ♥ ♦♥só♥

T & 10−10n2/3

s ♦♥ó♥ ③ ♣r♦①♠ó♥ ás ♣r ♥ s ♥s n

ts n ≃ 1022 − 1023 ② T & 105 t♠♣rtr ♠♥t s ♦r♥ 102 ♠♥r q ♣r♦①♠ó♥ ás ♥♦ s ♣

♠♦♥t♦rs n ≃ 1012 − 1014 ② T & 0,01 − 0,1 ② ♣♦r t♥t♦ sí ♣t③rs ♣r♦①♠ó♥ ás

st ssó♥ s♦♠♥t s rr s♣t♦ stíst♦ ♣r♦①♠ó♥ ásst♦ s q ♠♦s ♥ú♠r♦ ♣rtís ♥ ♥ tr♠♥♦ ♦♠♥ ② s♦s ♦ ♠s♠♦ q r ♥ú♠r♦ ♣rtís ♥ ♥ s♣♦ ás♦ ♠♦s♦♠♣r♦ st♥ ♥tr♣rtís r0 ♦♥ ♦♥t r♦ λB ♦s tr♦♥s

P♦r ♦tr ♣rt ♣r r s ♥á♠ ♦s tr♦♥s ♣ sr srt♣♦r ♣r♦①♠ó♥ ás ♠♦s ♦♠♣rr r♦rr♦ r ♠♦ l ♦♥ λB ♥rt♠r ♥ ♣♦só♥ tró♥ ♥② ♥ ó♥ ♦s♦♥ ①t♠♥t ♣r♦ ♥♦♥ ♥á♠ ♥tr ♦s♦♥s

❯t③♥♦ ♥ stíst á♥t s s q és ♥♦ ♥♦t ♣ ♣á♥♥tr♦r λB ≃ 1− 10 ♠♥trs q l ≃ 100 t♠♣rtr ♠♥t ♥♠♦s ♣sq λB ≪ l ♣♦r ♦ q ♦s tr♦♥s s ♠♥ r♠♥t ♥tr ♦s♦♥s ② ♣♦r♠♦st③r ♥ ssó♥ ás ♥á♠ ♦s tr♦♥s ♦ ♦tr ♠♥r ♠♦sq ♣rsó♥ ♥sr ♥ ♣♦só♥ tró♥ λB ≪ l ♥♦ ♥ ♥rt♠r♥♣t ♥ ♠♦♠♥t♦ ♥ tró♥

P♦r ♦♥s♥t ♠♦s sr ♥t ♥ sr♣ó♥ ♦s ♠ts s♠♣s♠♥t ♥ ♣r♦①♠ó♥ ás ♣r ♥á♠ ♦s tr♦♥s r ♣r♦t③♥♦ ♥ stíst á♥t ♣r ♥ó♥ stró♥ ♦♠♠r

tr♦s ♦♠♥tr♦s í♠t ás♦ s ♣ srr ♦r♠♠♥t ♦♠♦ í♠t ♥ q ó♥♠á♥ S s ♠♦ ♠②♦r q ~ ♥ ♥str♦ s♦ ♦ q ♥ó♥ r♥ ♥ ♣rtí r s L = 1

2mv2o s t♥ q

S =

∫ τ

0Ldt ≃ 1

2mv20τ =

1

2pl.

♦♥t ♦♥ r♦ s λB = h/p ♣♦r ♦ q ♥t♦♥s ♥♦s q S = 12hl/λB

② s s ♠♣ q l ≫ λB rst q S ≫ ~

♥ s♦ ♥ ♠♣♦ ①tr♥♦ ♦♥ ♦♥t ♦♥ λ r♠♥t♦ s s♠r♥q ♦r ♦ rrstr ♥ ♣♦r ♠♣♦ ∼ eEt/m s s ♥ ♠♣♦

♥ ♠r♦ sí ② q sr ♦s♦s ♦♥ ♦♥t ♦♥ ♦s ♠♣♦s q s ♥♥ ♥ sst♠ Pr st stró♥ á♥t ♦ tí♣ ♦s tr♦♥s s vF ② s ♠♦♠♥t♦ ~kF P♦r t♥t♦ λB = h/(~kF ) =

2π/kF ∼ 1− 10

ísst♦ó♦

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étr♦ s ♠♦ ♠♥♦r q ♦ ♣r♦♣ ♦s tr♦♥s v0 ∼ vF st ♠♥r ♣♦♠♦s ♠r τ ♣♦r τλ/l ♥ ①♣rsó♥ ♥tr♦r ② s t♠é♥ λB ≪ (λ/l) l = λ

4.4

♣♦♥♠♦s ♥ tró♥ ♦♥ ♠♦♠♥t♦ p(t) s♦♠t♦ ♥ r③ F(t) r♥srr♦ ♥ ♥tr♦ t♠♣♦ dt tró♥ t♥rá ♥ ♣r♦ ♥♦ r ♦s♦♥♦ 1 − dt/τ ② s ♥♦ ♦s♦♥♦ ♥ ♠♦♠♥t♦ ♥ p(t+ dt) = p(t) +F(t)dt s s♦♠♥t ♦♥sr♠♦s st♣r♠r ♦r♥ ♥ dt

♦♥tró♥ ♠♦ ♠♦♠♥t♦ ♦s tr♦♥s q ♥♦ ♥ ♦s♦♥♦ ♥ ♥ s♥tr♦ t♠♣♦ s

pncm (t+ dt) =

(1− dt

τ

)(pm(t) + Fm(t)dt

)= pm(t) + Fm(t)dt−

dt

τpm(t),

♦♥ s ♥ ♥♦ ♦s ♦rs ♠♦s ♠♥t sí♥s ♠

P♦r ♦tr ♣rt ♦s tr♦♥s q sí ♥ ♦s♦♥♦ s♦♥ (dt/τ × numero total de electrones) ② ♦♥tró♥ ♥♦ ♦s ró♥ p s Fm(t)dt ♣s ró♥ ss♣és ♦só♥ s t♦r st♦ ♥ q s ♦♥tró♥ t♦t s s♥♦ ♦r♥ ♥ dt ②♣♦♠♦s s♣rr s st♠♦s ♦♥sr♥♦ s ♦ss st ♦r♥ dt

♥ rs♠♥ ♥♦ í♠t ①♣rsó♥ ♥tr♦r ♥♦ dt→ 0

dpm(t)

dt= Fm(t)−

1

τpm(t).

♦♠♦ ♠♦s s ♦s♦♥s ♥tr♦♥ ♥ tér♠♥♦ ♠♦rt♠♥t♦ s♥♦ ♠♠r♦r♦ q ♣♥ rt♠♥t τ

4.5

❱♠♦s ♣r ó♥ ♠♦♠♥t♦ ♦s tr♦♥s st♦ t♦ Pr ♦s t③ ♠♦♥t ①♣r♠♥t s♥t

rr ♥ ♦rr♥t j

♥ ró♥ ❳ ② s s♦♠t ♠t ♥ ♠♣♦ ♠♥ét♦ B ♦r♦s tr♦♥s tú ♥ r③ ♦r♥t③ − e

cv × B ② ♦ q −v s ♠s♠♦ s♥t♦ q j st r③s ❨ ♥t♦ st♦ ♣r♦♥ ♠ó♥ r ♥t♥ s ♣rt ② ♣♦r ♦♥tr ♥ r ♣♦st ♥ ♦ ♦♥trr♦♥ ♦♥s♥ ♣rrá ♥ ♠♣♦étr♦ ♦ r♦ ❨ q ♥ qr♦ ♦rá q ♦rr♥t s♦♦ ② ♥ ró♥ ❳

♠♥♦ Ey st ♠♣♦ q qr r③ ♦r♥t③ s t♥

−eEy = −ecv ×B

② s ♥ ♦♥t ♠♥t ①♣rsó♥

RH ≡ EyjxB

.

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó ♦♦ r♦♠♠r

② q rs q s♥♦ ♦♥t Rh ♥♦s s♥♦ r ♦s ♣♦rt♦rs ♥ ♠t ♥ s♦ tr♦♥s q s q st♠♦s Ey < 0 ② RH < 0 ♦s ♣♦rt♦rs r♥♣♦st♦s Ey > 0 ② RH > 0

❱♠♦s r RH ♥ ♥ó♥ ú♥♦ ♣rá♠tr♦ ♠t s ♥s n ó♥ ♠♦♠♥t♦ ♣r♦♠ ♣rtr ♦r ♠♥♠♦s ♦s sí♥s m ♥ tró♥ srá

dp

dt= −e

(E+

p

mc×B

)− p

τ

♥ st♦ st♦♥r♦ ♥♦ ② ♣♥♥ t♠♣♦r s j ♥ s p ②

dpxdt

= 0 = −eEx − ωcpy −pxτ

dpydt

= 0 = −eEy − ωcpx −pyτ

s♥♦ ωc = eB/(mc) r♥ ♦tró♥ ♦s tr♦♥s rs ♥ ♥ ♠♣♦ ♠♥ét♦

t♣♥♦ ♣♦r −neτ/m ② s♥♦ q j = −nev = −nep/m ♥♦s q

σ0Ex = ωcτjy + jx

σ0Ey = −ωcτjx + jy

−1/(RHnec)

♥ qr♦ s♠♦s q Ey ♦♠♣♥s sst♠ ♦r♠q s jy = 0 ♣♦r ♦ q

σ0Ex = jx

σ0Ey = −ωcτjx

② ♦t♥♠♦s

RH = − 1

nce.

sí ♣s ♠ ♦♥t ♣ strr ③ ♠♦♦ q t③♠♦s ② q ♣♥ ú♥♠♥t ♥s tr♦♥s ♥ ♠t

❯t③♥♦ ♠♣♦s ♦r♥ 104 ss t♠♣rtrs♠② s ② ♦♥ ♠strs ♠② ♣rs s ♦t♥♥ ♦s rst♦s①♣r♠♥ts t

♦♥ s♦s rst♦s ♣♦♠♦s ♦♥r q ♦r n sí♦t♥♦ s ♣r♦①♠♠♥t q t♥rí ♥ s tr♦♥srs ♥ q át♦♠♦ s ♥ tró♥ ♣r ♦r♠r ♦s st♦ ♥ q ♥s ♦s ♣♦rt♦rs st s st♠♥t ♠② t ♦ q ①♣ ①♥t ♦♥t st♦s ♠ts

♥ ♠r♦ ♦s rst♦s ①♣r♠♥ts s♦♥ ♠ás ♦♠♣♦s♥♦♥trá♥♦s q RH ♣♥ ♦r ♠ó♦ ♠♣♦ ♠♥ét♦ RH t♠é♥ ♣♥ t♠♣rtr T ② trt♠♥t♦ q s s♦♠t♦ ♠str s♥♦ RH ♣ rr sú♥ rí ♠♣♦ ♠♥ét♦ ♥ r s ♣rs♥t♥ ♦s rst♦s♣r q ♠str♥ ó♠♦ rí ♦♥t ♠♥tr B

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

P❯ ❯ P

❯ RH ♣r

② sr♥ ♣rs♥ ♥ ♣♦rt♦r ♣♦st♦ ♣♦r át♦♠♦

♦t ♠♥t♦rsst♥ tr♥srs s ♥ ♦♠♦

ρM (B) =Exjx.

♥ ♥str♦ s♦ ρM (B) = cte = 1/σ0 ② ♥♦ ♣♥ ♦r B st♦ s ♦ q ♦sró ♥ ♠r♦ ♥♦ ♠s ♦ss s ♥♥tr q ρM (B) sí rí ♦♥ Bs ♠♦s ♠ts ② s ♦r♠ r♠át

4.6 P❯ ❯ P

♦♠t♠♦s ♠t ♥ ②♦ ♠♣♦ étr♦ ♦s♥t s

E(t) = ℜ[E(ω)e−iωt

]

♦r♠ q ó♥ ♠♦♠♥t♦ s

dp

dt= −eE− p

τ.

s♠♦s ♥ s♦ó♥ st♦♥r

p(t) = ℜ[p(ω)e−iωt

]

st ♠♦♠♥t♦ stt②♥♦

−iωp(ω) = −eE(ω)− p(ω)

τ=⇒ p(ω) =

−eE(ω)

1/τ − iω

♦rr♥t ♥ sj(t) = ℜ

[j(ω)e−iωt

]

♥♦s q

j(ω) = −nep(ω)m

=−ne2/m1/τ − iω

E(ω) =−ne2τ/m1− iωτ

E(ω)

♦♥ ♦ q ♦♥t ♦♠♣ s

σ(ω) =σ0

1− iωτ−→ω→0

σ0

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó ♦♦ r♦♠♠r

❯ ♦♥t σ(ω) ♦♠♣

♦ts ♠♦s ♦♠t♦ st♦ s ♦ q ♥tr t♦ étr♦ ② ♠♥ét♦② s♠♣r ♥ t♦r v/c ♥s♦ ♣r ♥ ♦rr♥t t♥ r♥ ♦♠♦ j = 1 ♠♠2 ♦ s♦♠♥t s v ≃ 0,1 ♠s ♠♥r q r③ó♥ v/c s ♦r♥ 10−10

♦t s τ ∼ 10−14 s ② E ∼ 104 ♦t♠ ♥t♦♥s ♦ rrstr seEτ/m ∼ 10 ♠s = 103 ♠s

rí♠♦s r ♥♦ ró♥ t♠♣♦r ♥ ♠r♦ λ ≫ l♣r ró♥ s ②s ♦♥ts ♦♥ stá♥ ♥ r♥♦ λ ∼ 103 − 104 ②♣♦r t♥t♦ ♣r♦♦ ró♥ t♠♣♦r s ≪ τ st♦ ♥ q r③ qtú s♦r tró♥ ♥ ♣♥t♦ r s eE(r, ω) st♦ t♠é♥ ♠♣ q ♥s ♦rr♥t ♥ r stá tr♠♥ ♣♦r E(r, ω)

j(r, ω) = σ(ω) E(r, ω) (respuesta local)

4.6.1 Pr♦♣ó♥ ♥

♣♦♥♠♦s ♦r q tr♥srss q ♥ s♦r ♠t s tr♥s♠t ♣r♦ ♥♦ s♣r♦ r ♥

s ① ♥ st s♦ s♦♥

∇ ·E = 0 ∇ ·H = 0

∇×E = −1

c

∂H

∂t∇×H = −4π

cj+

1

c

∂E

∂t

② s♠♦s ♥ s♦ó♥ ♦st♦r ♥ q ♦s t♦rs j E ② H s♦♥ ♣r♦♣♦r♦♥s e−iωt

ó♥ ♦♥s s ♦t♥ ♦♠♦

∇×(∇×E

)= −∇2E =

iω

c∇×H =

iω

c

[4πσc

E− iω

cE]

st♦ s s r ♥ ♥ ♣♥t♦ r sr♠♦s ♦♠♦ n(r, t) = n0 + δn(r, t) ♥t♦♥s δn(r, t) = 0 st♠♦s s♣♦♥♥♦ q ε = µ = 1 ♥ s♦ ♥r ∇×H = (−4pi/c)j+ (ε/c)∂E/∂t

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

P❯ ❯ P

♦

−∇2E =ω2

c2

(1 +

4πiσ

ω

)E

q s ♦ ♦♥s ♥

−∇2E =ω2

c2ε(ω)E,

♦♥

ε(ω) = 1 +4πiσ

ω

♦♠♦ σ(ω)/ω = σ0/(ω(1− iωτ)

) s s ♠♣ q ωτ ≫ 1 s t♥ q

ε(ω) = 1− 4πσ0ω2τ

.

stt②♥♦ σ0 = ne2τ/m ♥♦s q

ε(ω) = 1−ω2p

ω2

♦♥ ω2p = 4πne2/m ♠ r♥ ♣s♠

❨ qé s♥♦ t♥ st ωp s ♥ ♦♥ ♣♥ E = E0 ei(k·r−ωt) s ♠♣ q∇2E = −k2E ② ♣♦r ♦ t♥t♦ q k2c2 = ω2ε(ω)

P♦r ♦♥s♥t

s ε(ω) < 0 ⇒ k2 < 0 ⇒ ①♣♦♥♥ r ⇒ ♥♦ ② ♣♥tró♥

s ε(ω) > 0 ⇒ k2 > 0 ⇒ ①♣♦♥♥ ♠♥r ♣r ⇒ ② ♣♥tró♥

♣♦r ♦ q

s ω > ωp s♦ó♥ s ♦st♦r s ♣r♦♣ ♥ ♠t

s ω < ωp s♦ó♥ s ♥s♥t ② s r

❯ t

♦♦ st♦ r♦ stá srá á♦ s s♣♦só♥ ωτ ≫ 1 s rt ♣r ω = ωp ♦r ωp s ♦r♥ ❱ ♥q ♣♥ ♠t ♥ stó♥ q ♦rrs♣♦♥ ♥ ♣r♦♦T ≃ 3 × 10−16 s ❨ ♦ q τ s ♦r♥ 10−14 s ♠♦s q sí s rt♦ q s ♠♣ qωpτ ≫ 1 ♥ ♠♥r ♥ér

♥ ♦♥só♥ sú♥ ♠♦♦ ♦s♠ts ♣rs♥t♥ ♥ tr♥só♥ ♦♣ tr♥sr♥ ♣r r♥s ♦r♥ ωp

s♦ s ♥ rst♦ ①♣r♠♥t ♥♦♥♦♦ ♦s ♠ts ♥♦s s t♦r♥♥tr♥s♣r♥ts ♥ tr♦t Pr ♠♦♥t①♣r♠♥t s t③♥ ♣ís s ♠t

♥ s♦ ♠ás ♥r srí ε(ω) = ε+4πiσ(ω)

ω

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó ♦♦ r♦♠♠r

❯ ♥ó♥ étr ε(ω)

♦♠♣ró♥ ♥tr ♦s rst♦s ①♣r♠♥ts ② ♦r tór♦ ωp ♦ ♣rtr ♥s n ♣r st♥t♦s ♠ts s ♠str ♥ t

×103 s

λteo

λexp

❱♦rs λT ② λexp ♦s ♠ts ♥♦s

❯ ró♥ s♣rsó♥ω(k)

♦♠♦ s s♣rr s tr♦♥s s s♣rs♦♣r tr♥s♠só♥ ② ró♥ s♣rsó♥♥♦ s ♥ és ó♥ ♥ t♦ s ωτ ≫ 1

② ró♥ s♣rsó♥ ♥♦s q c2k2 = ω2(1 − ω2

p

ω2

)

st♦ sω2 = ω2

p + c2k2,

♦♠♦ s ♠str ♥ r

4.6.2 ♥s ♥s r

❱♠♦s ♣♥tr ♦r ♦tr ♣♦s s r ♥ ♥ ♠t ♥♦ s ♥

á s ó♥ ♠♦♠♥t♦ r

s ♣♦s ①st♥ ♦s♦♥s ♦♥t♥s r t♦♠♥t♥s ①t♥s♥ t♦♦ ♠t ② q s ♦♠♣♦rt♥ ♦♠♦ eiωt

st♦ s n(r, t) = n0 + δn(r, t) 6= n0

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

P❯ ❯ P

♣♦♥♠♦s q sí ♣♥ ①str sts ♦s♦♥s ó♥ ♦♥t♥ r s

∇ · j = − ∂

∂t(δn) =⇒ ∇ · j(ω) = iω δn(r, ω)

② ② ss ♥♦s q∇ ·E = 4π δn(r, ω).

Pst♦ q j(ω) = σ(ω) E(ω) s ♦t♥ ♠♥r rt

∇ · j(ω) = σ(ω) ∇ ·E(ω) = 4πσ(ω) δn(r, ω).

♥♦ ♦♥ó♥ ♣r q ♥ s tr♦♥s s ♣r♦③♥ ♦s♦♥s r t♠♣♦rsst♦ s q rí♥ ♦♠♦ eiωt sr q

1 +4πiσ(ω)

ω= 0.

♦r♥♦ q ε(ω) = 1 + 4πiσ(ω)/ω st ♦♥ó♥ rst sr é♥t q ε(ω) = 0 st♦s♥ q ♥ s♦ ♥ q ωτ ≫ 1 ♣r s r♥s ♥trés ♦♥ó♥ ♥tr♦r ♥♦s ♦♥r q s ♣r♦♥ ♦s♦♥s ♥ r sst♠ s s r♥ s ωp s ♠♣rq ωpτ ≫ 1

♦r ♥ ♥ó♥ ♥ó♥ étr ♥♦s ♣r♠t srr V tot = V ext/ε ♦♥ V ext s ♣♦t♥ ①tr♥♦ q sr s tr♦♥s ② V tot s ♣♦t♥ q tú s♦r ♦s tr♦♥s

Pr♦ s ε(ω) = 0 st♠♦s ♥ ♥ s♦ ♣r q ♣♦t♥ q sr♥ ♦s tr♦♥s ♣ srst♥t♦ r♦ ♥q s ♥♦ V extst♦ ♥♦s stá ♥♦ q s s ♣r♦ ♥ ♦só♥ r ♥ s tr♦♥s st ♦só♥s t♦♠♥t♥ ♣s ♥♦ ♥st ♠♣♦ ①tr♥♦ ♣r ♠♥t♥rs

ó♠♦ s ♣♦s ①tr sts r♦♥s r r♥ ωp ❨ r♠♦s ♠ás ♥tq ♦s tr♦♥s sst♠ ♥♦ ♣♥ ①trs ♦ q s ♣rs ♠ ♥rí ♣r ♦≃ 10 ❱ ♥ ♠r♦ s t ①trs ①♣r♠♥t♠♥t ♠♥t ♦♠r♦ tr♦♥srá♣♦s s♦r ♣s s ♠t ❨ st♦ s q ♦s tr♦♥s ③ q ♦♠r ♣ ♠tá ♥tr♦♥♥ rt♠♥t ♦♥ ♦s tr♦♥s ♠t ② ② q s ♥rí s t s♣♦s q ♣rtr♥ ♥s r ♦ ♠♥r q ①t♥ ss ♦s♦♥s

❯ ①tó♥ ♣s♠♦♥s

s♥ ♠r ♦♥s ♥s r st♦s ♣♦t♥s ♥♦ ♦♥♥ ② q ♦s tr♦♥s stá♥ r♦s ② r♦♥♥ r♥t ♣♦t♥ ①tr♦r ② st ró♥

♥♦t♥ s ③ rst♦ ♦s tr♦♥s ♠é♥ r ♥♦♠r ♦só♥ ♣s♠ ♣♦r s♠t sts ♦s♦♥s r♥ ♥♦♠r ♣s♠♦♥s

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó ♦♦ r♦♠♠r

st♦ s♣tr♦ ♥rís q ♣r♥ ♦s tr♦♥s q trs♥ ♠t ♣r♠t s♦r♦s ♣♦s ♦ s♣tr♦ ♦♥ ①tó♥ ♦s♦♥s r ♥ á♠♥ ♥ t♦ ♣r♥r♦s ♣♦s ♦♥st♦s ♦♥ ♥ s♣ró♥ ♥rét ♥ ♥ ♥tr ♦s ♦rrs♣♦♥♥t ωp tr ♦s ♣♦s s♠♥②♥♦ ② q s ♠ás ♣r♦ ①tr ♥ ♦só♥ r q♦s t

♥ t s ♣rs♥t ♥ ♦♠♣ró♥ ♥tr ♦s rst♦s t♦rí ② ♦s ①♣r♠♥ts

❱

ωexpp

ωteop

❱♦rs ωp ♥♦s ♠ts

• s ♥ ♣♥t♦ st ás♦ ② ♦♥ á♥♠♦ r ♥ ♥tt ó♠♦ s ♣r♦♥s ♦s♦♥s ♥tr♦rs ♣♦♠♦s str ♥♣♦s ♠♥r str st ♠♦♠♥t♦ r ♦♥ ♥ r♥ ωp ♥ ♠ts ♣rtr t ♠♥r q r s s♣③♦ ♥ ♣qñ st♥ δ rs♣t♦ stró♥ r ♣♦st ♦s ♦♥s és r ♣r ♥ ♥s s♣r r

σ = ne δ

st ♥s s♣r ♣r♦ ♥ ♠♣♦ étr♦ E = 4πσ = 4πne δ ó♥ ♠♦♠♥t♦ t♦♦s ♦s tr♦♥s s♥ ♦s♦♥s s

Nmd2δ

dt2= −Ne|4πσ| = −4πne2N δ

q s ♥ ♠♦♠♥t♦ r♠ó♥♦ s♠♣ r♥ ω2p = 4πne2/m ♦♥♥♦ ♦♥

á♦ ♥tr♦r

• ♦♠♦ s st♦ r♥ ωp t♥ ♦s s♥♦s

s r♥ ♦rt ♣r tr♥s♠só♥ s ♦♥s tr♦♠♥éts tr♥srss ♥ ♠t

s r♥ q s ♣r♦♥ ♦s♦♥s ♦♥t♥s r ♥ ♠t ♥ s♦♥♥t♦

4.7 PP

Pr str sts ♣r♦♣s ② r♦ ♦♥ ♦s ♦♠♥tr♦s q ② s ♥ ♦ ♥ só♥ t③r♠♦s stró♥ r♠r ♣r srr stíst ♦s tr♦♥s

Pr r ♥ ♥tt á♥ r♥t s stró♥ stró♥ ás ①♦t③♠♥♥ t♠♣rtr ♠♥t ② ♥ r♥♦ ♥ss ♠tás ♣rs♥t♠♦s s♥t rá ♣r ♠s str♦♥s

♦ ♦♥srr s ♦s♦♥s s ó♦ ♣s s♠♦s q ωpτ ≫ 1 ❯♥ ♥tr♦ó♥ s♦♠r st stró♥ s ♦ r ♥ s♥tr ♥♠♥t♦s ís ② s strá ♥

t ♥ s♥tr á♥ stíst

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

PP

❯ ②

• ♦♥t♥♠♦s s♣♦♥♥♦ q ♦s tr♦♥s s♦♥ rs ♥♣♥♥ts ② ♣♠♦s ♦♥♦♥s ♦♥t♦r♥♦ ♣rós ♦ ís ♦ ♦r♥ ♦♥ r♠♥ ♥ ♦♥♥t♦ N tr♦♥s ♥ ♥ ♦ ♦♠♥ L3 P♦r ♦♥s♥t ♠♦s rs♦r ② s ♦♥♦♥s ♦♥t♦r♥♦

− ~2

2m

( ∂2∂x2

+∂2

∂y2+

∂2

∂z2

)ψ(r) = εψ(r)

ψ(x+ L, y, z) = ψ(x, y, z)

ψ(x, y + L, z) = ψ(x, y, z)

ψ(x, y, z + L) = ψ(x, y, z)

♦♥ ε s ♥rí tró♥ srt♦ ♣♦r ♥ó♥ ♦♥ ψ s q ♠♦s ♠♣st♦♣r♠t♥ trtr ♦s tr♦♥s ♦♠♦ s r♥ ♦♥s rs ♥ t♦♦ s♣♦ s♥ q s r♥♥ s♣r ♦

s s♦♦♥s ② s s♦♥

ψ(r) =1√Veik·r ε(k) =

~2k2

2m

eikxL = eikyL = eikzL = 1 =⇒ kx =2πnxL

, ky =2πnyL

, kz =2πnzL

, ni ∈ Z.

P♦r ♦ t♥t♦ s♦♠♥t ♥♦s t♦rs s♣♦ rí♣r♦♦ stá♥ ♣r♠t♦s ♣♦r s ♣r srt♦rs k q ♣♥ rtr③r ♥ó♥ ♦♥ ψk(r) st ♥ó♥ ♦♥ s t♦st♦ ♦♣r♦r ♠♦♠♥t♦ p = −i~∇ ♦♥ t♦♦r ~k ♦♠♥ s♣♦ rí♣r♦♦ s♦♦ ♥♦ st♦s k ♣r♠t♦s s (2π)3/L3 ♠♦♦q s ♦♠♥ s♣♦ rí♣r♦♦ q ♥♦s ♥trs s ♠♦ ♠②♦r q s ♦♠♥ (2π)3/L3

♣♦r♠♦s ♦♥srr ♥str♦ ♦♥♥t♦ t♦rs k ♣r♠t♦s ♦♠♦ s♦♥t♥♦❨ st♦ s ♦ q ♦rr ♥ ♥ ♠t t♠ñ♦ ♠r♦só♣♦ ♥♠♦s ♥t♦♥s ♦r♥ 1022

t♦rs k ♣r♠t♦s ♣s ♦s tr♦♥s s♦♥ r♠♦♥s ② s♦♠♥t s ♣ ♦♣r ♦♠♥t st♦ ♣♦r ♣r♥♣♦ ①só♥ P ♣♦r ♦ q ♣♦r♠♦s ♣r♦①♠r ♣♦r ♥ ♦♥t♥♦ ♥ ♥♦♠♥ Ω s♣♦ rí♣r♦♦ ② Ω/(2π/L)3 st♦s ♣r♠t♦s ♦ ♦tr♦ ♠♦♦ ♥s t♦rs k ♣r♠t♦s ♥ s♣♦ rí♣r♦♦ s

L3

8π3=

V

8π3.

♦ q s sté♥ st♥♦ ♣r♦♣s s♣r sst♠ ♠♦s t♥r r♦ q s q s ♠♣♦♥♥ s♦r s♣r ♦ ♥ t♥r ♠② ♣♦ ♠♣♦rt♥ srr ♥ ♣rtí q sté ♥ ♥tr♦r ♦ st♦ s ♦ q ró♥ ♣rtís ♣ró①♠s s♣r s ♦r♥ λL2/L3 = λ/L s♥♦ λ ♦♥t ♦♥ r♦ s ♣rtís ♦♠♦ λ≪ L ♥ ♥ sst♠ ♠r♦só♣♦ st ró♥ s s♣r

ísst♦ó♦

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❯

❯

❯

❯

❯

❯

❮ Ó ♦♦ r♦♠♠r

♦♠♦ ♦s tr♦♥s ♥♦ ♥tr♦♥♥ st♦ ♥♠♥t sst♠ s ♦♥str② ♦♦♥♦♥ ♥ ♠♦♥♦tró♥♦ ♥ ú♥♦ tró♥ s ♥ ♣r k rá ♦s tr♦♥s ♣♦r ♥ró♥ s♣í♥ ♥♦ ♥ú♠r♦ tr♦♥s N s r♥ ró♥ s♣♦ k q♥rá♥ ♦s tr♦♥s srá ♣rát♠♥t ♥ sr ② q ♥rí ♦s tr♦♥s s ♣r♦♣♦r♦♥ k2 st sr st♦s k ♦♣♦s r ♥♦♠r sr r♠ ② s r♦ kF stá♥♦ ♣♦r ♦♥ó♥ q ♥ ♦♠♥ 4

3πk3F ② ♥ t♦t N/2 ♥s k ♣r♠t♦s P♦r

t♥t♦4

3πk3F

V

8π3=N

2=⇒ k3F = 3π2

N

V= 3π2n.

♥♦♠♥tr r sr r♠ s ♥ ♣♦♦ rr sr r♠s♣r sr r♠ ♥rí r♠ εF k r♠ kF ♦ r♠vF

♦♠♦ st♥ ♥trtró♥ r0 s ♥ ♦♠♦ 43πr

30 = 1

n s t♥♥ ♦s♦rs

kF =(9π/4)1/3

r0=

1,92

r0=⇒ kF =

3,63

rs−1

vF =4,20

rs108 ♠s ≃ 1% c st♠♦s T = 0 ás♠♥t 〈v〉300 K ∼ 107cm/s)

εF =50,1

r2s108 ❱ ∼ 1,5− 15 ❱ ♦r♥ ♥rís r tó♠s

TF ∼ 105

• ♥rí t♦t sst♠ ♥ st♦ ♥♠♥t s ♣r♠♥t ♥ét

Etotal = 2∑

k<kF

~2k2

2m.

♠♦ s♠t♦r ♣♦r ♥ ♥tr ♦♠♦ ♥ts s ♦♠♥t♦

∑

k<kF

· · · = 1

∆k

∑

k<kF

· · ·∆k ≃ V

8π3

∫ kF

0· · · dk

♣♦♠♦s r ♥rí t♦t ♦♠♦

Etotal = 2∑

k<kF

~2k2

2m=

2V

8π3

∫ kF

04πk2dk =

V

π2~2k5F10m

.

♥t♦♥s ♦r ♠♦ ♥rí ♣♦r tró♥ s

ε =Etotal

N=

3π2

k3F

~2k5F10π2m

=3

5εF ∼ 1− 10 eV.

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

PP

♦♠♦ ♦♠♣ró♥ rr q ♥rí ♥ét ♥ ♣rtí ♥ s ás♦ s32kBT ♦ q ♥♦s ♥ ♦r ♦r♥ 1/40 ❱ t♠♣rtr ♠♥t

• ♦s ♠♦s ♥trr ♦r ♥ ♦r s♣í♦ s tr♦♥s qí ♥♦ ♠♦s ♥trr♥ t á♦ s♥♦ q s♦♠♥t str♠♦s rst♦ ♦ ♠♣♦rt♥t s t♥r ♥♥t q ♥rí r♠ εF ∼ 10 ❱ s ♠♦ ♠②♦r q ♥rí tér♠ sst♠ t♠♣rtr ♠♥t kBT ≃ 1/40 ❱ ② s s á♦ ♠♥r ①t s ♦t♥ s♥t ♦r ♣r ♦r s♣í♦ tró♥♦

celect =(π22nkB

) T

TF.

P♦♠♦s ♦r r r♦s ♦♠♥tr♦s

♦♠♣r♠♦s ♦♥ ♦r s♣í♦ ♥ s ás♦ ♦t♥♠♦s

cv(ás♦) = 32nkB

celect =(π2

2 nkB

)TTF

=⇒ celectcv

≃ T

TF≃Tamb

10−2.

st rst♦ ①♣ ♣♦rqé t♠♣rtr ♠♥t ♥♦ s ♦sr ♦♥tró♥ ♦r s♣í♦ ♦s r♦s rt tró♥♦s

q celect s ♣r♦♣♦r♦♥ t♠♣rtr T s ♥ ♦♥s♥ rt stíst q st♠♦s s♥♦ ♥ t♦ t♠♣rtrs kBT ≪ εF stró♥ FFD(T ) s r♥ FFD(T = 0) s♦♠♥t ♥ ♥ r♥♦ ♣qñ♦ ♥rís ♦r♥ kBT r♦r ♥rí r♠ ♦♠♦ ♦♥s♥ ♥ú♠r♦ tr♦♥s q ♣♥ ①trs tér♠♠♥t s t♠é♥ s ♦r♥ ♠♥t

❨ ♦♠♦ ♦♥s♥ ♦r s♣í♦ ♦♥st♥t cv = 32nkB

s♦ srá st♦ ♥ t ♥ s♥tr á♥ stíst r q cv(ás♦) stá s♦♦ ♦s ♠♦♦s ró♥ r q sí s♥ ♥ ró♥ ás ♦♥st♥t

ts t♠♣rtrs

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó ♦♦ r♦♠♠r

♠ás ♥rí tér♠ q qr♥ ♦s tr♦♥s q ♣♥ sr ①t♦s T 6= 0 s ♦r♥ kBT ♠♦ ♠ás ♣qñ q εF P♦r ♦ t♥t♦ ♥r♠♥t♦ ♥rí ♥rá ♦ ♣♦r

∆E ≃ D(εF )× kBT︸︷︷︸num elect pueden excitarse

× kBT︸︷︷︸energia adquirida/elect

,

♣♦r ♦ q ♥ st♠ó♥ rt ♦r s♣í♦ ♥♦s

Celect ≃ 2D(εF )k2BT =⇒ celectv ≃ 3nkB

T

TF,

♥ ♥ t♠♣rtr T ② ♦♠♦ ♠♦s r♥♦ ♦r ①t♦ ♥ts t♦

Pr ♦♠♣rr st rst♦ ♦♥ s ♠s ①♣r♠♥ts ② qt♥r ♥ ♥t t♠♣rtrs s♥t♠♥t s T ≪ ΘDebye ♦♥tró♥ ♦r s♣í♦ r♦♥s r s ♣r♦♣♦r♦♥ T 3

♦♥tró♥ tró♥ ♣r T ≪ TF rst sr ♣r♦♣♦r♦♥ T

♦ s Cv = γT + AT 3 ♥t♦♥s cv/T = γ + AT 2 ② s ♣ ♦t♥r ♦r γ①tr♣♦♥♦ ♦s ♦rs Cv/T ♣r t♠♣rtrs T ♠② ♣qñs

❯ st ①♣r♠♥t γ

10−4 ♠♦2 s ♥

γteo

γexp

ΘDebye

❱♦rs γ t♦ ② ①♣ ② ΘDebye ♣r ♥♦s ♠ts

♦s ♦rs tí♣♦s ♣r só♦s ♠ t♠♣rtr ② ΘDebye stá♥ ♥tr ② ❱és t ♠ás♦

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❯❱ ❯

• t③ó♥ stró♥ ♣r stíst ♦s tr♦♥s ♥② s♠♥t♥ t♦s qs ♣r♦♥s ♥ s q s t③ stró♥ s ♦s ♦ q ♥♦s s τ s ♥♣♥♥t v q t ♠♥r rt s ♠♥ts íss q ♣♥♥ ♦ ♠ vm ② ♠♥♦ r ♠♦ l

♥ t♦ t♥♠♦s q ♣r♥tr♥♦s r á s ♦ ♠ ♦s tr♦♥s q♣♥ ①trs ♥ ♥ sst♠ srt♦ ♣♦r stró♥ ♥t♠♥t ♣♦r s ♦♥♦♥sq ♠♣♦♥ stró♥ s♦♠♥t s ♣♥ ①tr q♦s tr♦♥s ② ♥rí sr♥ εF s♦ ♥rt♠r ♦r♥ kBT ♦ q kBT ≃ (10−3 − 10−4) εF s t♥ q ♦ ♠ st♦s tr♦♥s srá ♠② ♣r♦①♠♠♥t vF

P♦r ♦tr ♣rt ② ♠♦s q t♠♣♦ ró♥ τ s ♣♦í ♦t♥r s ♠s rsst ♠♥t τ = 0,22 r3s 10−14/ρ ♥ s♥♦s s ρ s ①♣rs ♥ µΩ♠ ♦♠♦ vF =4,2/rs × 108 ♠s ♥♦s q ♥ ♠♥♦ r ♠♦

l = vF τ = 92 r2s1

ρ.

t♠♣rtr ♠♥t ρ ∼ 1− 20 µΩ♠ ② ♦♥♠♦s q ♠♥♦ r ♠♦ q s ú ♣rtr s ♠s rsst s ♦r♥ l ≃ 100 − 1000 ♣r T ≃ Tamb st s ♦r l t♥ r♥ q ♦♠♥t♦ ♥ só♥

4.8 ❯❱ ❯

s ♦♥♦♦ q ♦s ♠ts ♦♥♥ ♦r ♠♦ ♠♦r q ♦s s♥ts st♦ sr q ♦str♦♥s q s♦♥ ♦s ♣♦rt♦rs r s♦♥ t♠é♥ ♦s ♣♦rt♦rs st ♥rí tér♠

♠♦♦ ♣r♠t str ♣♦rqé s tr♥s♣♦rt ♥rí s ③♦♥s ♠②♦rt♠♣rtr s ③♦♥s ♦♥ ♠♥♦r t♠♣rtr ♥ t♦ ♦ s ♦s tr♦♥ss♣és ♥ ♦só♥ ♣♥ s♦♠♥t t♠♣rtr ♦ r ♦só♥ P♦r t♥t♦♦s tr♦♥s srá♥ ♦♥ ♠②♦r ♥rí qs ♦s♦♥s q ♦rr♥ ♥ ♣♥t♦s ♠ás ♥ts♦ q ♣r♦rá ♥ ♦ ♥t♦ ♥rí s s r♦♥s ♠②♦r t♠♣rtr s ♠♥♦rt♠♣rtr

♠♦♥t ①♣r♠♥t s

s ② ♦rr q ♦ ♥rí ♣♦r ♥ ár st♦ s ♦rr♥t ♦rs ♣r♦♣♦r♦♥ r♥t t♠♣rtr s ést s s♥t♠♥t ♣qñ♦

Q = −κ∇T,

s♥♦ κ ♦♥t tér♠ ♠tr

♦♥sr♠♦s ♥ ♣♥♦ ♦♥ x = cte ♦s tr♦♥s q trs♥ ♣♥♦ s r ♥♥♦♥ ♠♥♦r ♥rí q ♦s q ♦ trs♥ s ③qr ♥ ♠♦s s♦s tró♥ rá

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó ♦♦ r♦♠♠r

♦s♦♥♦ ♥ ♣r♦♠♦ ♥ ♥ t♠♣♦ ♥tr♦r t tactual tactual − τ tró♥rá r♦rr♦ ♥ s♣♦ vxτ ♥ts trsr ♣♥♦ ② s ♥rí srá ε

(T (x± vxτ)

)

♦ ♣rtís q trs ♣♥♦ ③qr r s (1/2)nvx ♠♥trs q ♦ q trs ♣♥♦ r ③qr s (1/2)nvx ♣s ♥ ♣r♦♠♦ rá ♠s♠♦♥ú♠r♦ tr♦♥s ♠♦s ♦s ♦ ♥rí trés ♣♥♦ srá

1

2nvx

[ε(T (x− vxτ)

)− ε(T (x+ vxτ)

)]

ró♥ t♠♣rtr ♥ st♥ vxτ s ♣qñ st ♦ ♥rí s ♣ srr♦♠♦

1

2nvx

dε

dT

dT

dx(−2vxτ) = −v2xτn

dε

dT

dT

dx

② ♦♥t tér♠ s

κ =1

3v2mτcv

♦♥ ♠♦s s♦ q n(dε/dT ) = (1/V )(d(Nε)/dT

)= cv

♠♦s st ♦♥t tér♠ ♣♦r ♦♥t étr ♦t♥♠♦s ♥ú♠r♦

♦r♥t③

L ≡ κ

σ0T=

1

3v2mτcv

ne2τ

mT

.

s♠♦s q v2m ≃ v2F = 2εF /m ② q cv = (π2/2)(kBT/εF ) nkB ♥t♦♥s ♠♦s

L ≡ κ

σ0T=π2

3

k2Be2

= 2,44× 10−8 watt− Ω/K2

st rst♦ s ♥♣♥♥t ♦s ♣rá♠tr♦s ♠t ② ♣♥ s♦♠♥t ♦♥st♥ts♥♠♥tsst ♥ú♠r♦ ♦r♥t③ stá ♠② r♦ ♦♥ ♦s t♦s ①♣r♠♥ts κ/(σ0T ) ♦t♥♦s t♠♣rtrs ts T ≃ 273 − 373 ♦s ♦rs ①♣r♠♥ts L stá♥ ♥tr 2 ②3,5 10−8 watt− Ω/K2♦t ♦r ①♣r♠♥t L s♥ ♣r t♠♣rtrs ♠② s ♦ q ♦s t♠♣♦s ró♥ étr♦ ② tér♠♦ ♥ sr s st♦ s rá ♠ás ♥ t ♥♦ s st ♣♥♥ rsst tér♠ ♦♥ T

qí ♠♦s ♣s♦ ♣♦r t♦ ♦ q ♠♦♦ r♦♠♠r ♣♥t r♥t♦ s ♦s tr♦♥s s♣és ♥ ♦só♥ s r♦ q rá ♥ ♠ó♥ tr♦♥s ♥ ♣rt ♠ás rí ♠t st ♠ó♥ srá ♠♥tr ♥ ♠♦♠♥t♦ ♥q ♠♣♦ étr♦ ♣r♦♦ ♣♦r s ♠ó♥ ♥tr t♦ ♥tr♦r st ♥ó♠♥♦ s♦sr ①♣r♠♥t♠♥t ② s ♥♦♠♥ t♦

♠♦s ♠♦ vx ♦ ♠ ♦s tr♦♥s ♥ ró♥ ❳ ♥t♠♥t 〈v2x〉 = 〈v2y〉 = 〈v

2z〉 = (1/3)v2m

st♠♦s s♣♦♥♥♦ q sst♠ stá ♥ qr♦ ♥ ♦♥rt♦ s ♥♣♥♥t ♥s q s ú♥♦ ♣rá♠tr♦ ♦s ♠ts ♥ ♠♦♦

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

ÓÓ

♠♣♦ q ♣r s ♦♣st♦ r♥t t♠♣rtr ② s ♠ ♠♣♦ tr♠♦étr♦ ♥st♦ st♦♥r♦ ♦ rrstr ♠♣♦ tr♠♦étr♦ ② ♦ tér♠ ♣♦r sí♠r ♥ ♥rs st út♠ ♦ ♣ rs ♦♠♦ ♦r ♠♦ s ♦s ③qr ② r ♦ tér♠ srá ♥ s s♦

v0x =1

2

[v(x− vxτ)− v(x+ vxτ)

]= −τvx

dvxdx

= −τ ddx

(12v2x

)

s♠♣r ② ♥♦ T rí ♣♦♦ ♥ st♥ 2τvx ♥ trs ♠♥s♦♥s st♦ s ♣ srr ♦♠♦

vQ = −τ6

dv2

dT∇T.

st♦ s ♦ tér♠ s ♣r♦♣♦r♦♥ r♥t t♠♣rtr

P♦r ♦tr ♣rt ♦ rrstr ♠♣♦ tr♠♦étr♦ s ♣r♦♣♦r♦♥ ♠♣♦ étr♦

vE = −eEτm

② ♦♠♦ ♠♦s ♦ q ♠♣rs q vQ + vE = 0 ró♥ ♥tr ♠♣♦ étr♦ ② r♥t t♠♣rtr ♦♥t ♦ ♣♦t♥ tr♠♦étr s

Q ≡ E

∇T= − 1

3e

d

dT

(mv22

)= − cv

3ne.

stt②♥♦ cv ♥ Q

Q = − 1

3ne

π2

2

kBT

εFnkB = −π

2

6k2B

T

eεF= −1,42

kBT

εF10−4 ♦t

♦♠♦ t♠♣rtr ♠♥t kBT ≃ 10−2εF t♥r♠♦s q |Q| ≃ µ♦t♦ ①♣r♠♥t♠♥ts ♥♥tr q |Q| ≃ µ♦t♦ ♥q ♥ ♥♦s s♦s rst q s♥♦ Q s ♣♦st♦

4.9 ÓÓ

♦♠♦ st♠♦s ♥♦ ♦s tr♦♥s ♦♥ó♥ ♥ ♠ts q s ♥♥tr♥ s♣r♦s ♥♣r♦♠♦ ♥ st♥ ♥ s♥ ♦s♦♥r rs st♥s sí Tamb ♠♥♦ r♠♦ ♦s tr♦♥s s ♠♦ ♠②♦r q s s♣ró♥ l ∼ 104 ② s ♦ ♦srr♦rs l ∼ 108 ♥ ♠strs s♣♠♥t ♣r♣rs 1

t♦r q ♠ás ♠♣♦rt♥ t♥ ♥ ♦ q ♦s tr♦♥s ♥♦ ♦s♦♥♥ ♣sr srt ♥tró♥ ♦♠♥ s ♣r♥♣♦ P ♣♦r ♦♠♦ ♥ ssó♥ ♥♠♦s r♦ ♥rís ♥ ♥rí r♠ ♥ t♠♣rtr T s♦♦ ② tr♦♥s ①t♦s ♦♥♥rí ♣♦r ♥♠ ♥ ♥rí r♦ ♥ ♥ r♥♦ ♦r♥ ε1 ∼ kBT

st♦ ♣♥tr s s♦r s♥♦ ♦s ♣♦rt♦rs r ♥ ♦s ♠ts rr q s t♠é♥ s♥♥tr q RH > 0

② ♥ ♠♣♦ ①tr♦r q ♣rtr sst♠ tró♥♦ r♥♦ ♥rís srá ♥tró♥ ♦♥ ♦ ♠♣♦

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó ♦♦ r♦♠♠r

♣♦♥♠♦s q t♥♠♦s ♥ tró♥ ①t♦ ♣♦r ♥♠ ε ≡ 0 s r♦ q s ♦s♦♥ ♦♥♦tr♦ tró♥ ② ε2 < 0 ♥rí ♦s tr♦♥s ② rst♥ts ♦só♥ sr ♥♠♦s s♦s ♠②♦r q r♦ ② q t♦♦s ♦s st♦s ♠♥♦r ♥rí ♦♥ ε < 0 stá♥ ♦♣♦s♥ s t♣♦ ♦só♥ ♣r q s rq ♦♥sró♥ ♥rí ε1 + ε2 = ε3 + ε4 t♥ q ♠♣rs q|ε2| < ε1 és r ♦r♠ q s♦♠♥t ♥ ró♥ tr♦♥s ♦r♥ ε1/εF ♣♦s♦♥r ♦♥ tró♥ ♠ás ♦♥srrs ♠♦♠♥t♦ ♥ t♦t ♦ q s♥ q s♦♦ ♥♦s st♦s ② r♥ t ♦♥ó♥ ♦s ♠♦♠♥t♦s p3 ② p4 ♥ str s♦r sr ♣qñ r❨ s♦ ♥tr♦ ♠♥r t♦♠át ♦tr♦ t♦r ε1/εF

P♦r ♦♥s♥t só♥ ③ t st♦s ♣r♦s♦s tró♥tró♥ s ♣r♦①♠♠♥t

Sel−el ≃(kBT

εF

)2

S0

♦♥ S0 s só♥ ③ tór ♥tró♥ tró♥tró♥ ♣♥t

♦♠♣♦rt♠♥t♦ t♠♣rtrs ts T ∼ Tamb ♦r ♦ ♣r ♠ts tí♣♦s s S0 ∼ 10−15 ♠−2 ♦♠♦ kBTamb/εF ∼ 10−2

Sambel−el ∼ 10−19 ♠2 =⇒ lamb

el−el ∼1

nS∼ 104 & 10 lel−fon︸ ︷︷ ︸

102−103

st ♠♥♦ r ♠♦ s ♥s ③ s ♠②♦r q s ♦s♦♥s tró♥♦♥ó♥ ♦r♠ q t♠♣rtr ♠♥t Tamb ♦s ♣r♦s♦s ♦só♥ ♦s tr♦♥s stá♥ ♦♠♥♦s♣♦r s ♦s♦♥s ♦♥ ♦s ♦♥♦♥s

♦♠♣♦rt♠♥t♦ t♠♣rtrs s T → 0 s t♠♣rtrs T ∼ THe liquido s ♥ ♦sr♦ ♥ ♥ ② ♥ rssts ♣r♦♣♦r♦♥s T 2 ♦♥sst♥ts ♦♥ ó♥ P♦r ♦tr ♣rt s ♥ ♠♦ ♠♥♦s rs ♠♦s ♥ ♥ st ♥♦s ♠ ♦ q s ♦♥♦r♠ ♦♥ ♦ q ♠♦s ♦ ♥ ♣rt♦ r ♦sr♥s ♠♥♦s rs ♠♦s q s ♣♥ ♦t♥r r t♠♣rtr

r q st♠♦s ♥♦ st♦s ♥rét♦s ♦s tr♦♥s q r♠♦s ♥♦ tr♦♥s

ísst♦ó♦

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❯

♣ít♦

❯P PÓ ❨ ❯

❯

t♦r♠ ♦ ♥♦s q ♥rí ♥ tró♥ ♥ ♥ ♣♦t♥ ♣ró♦ rq

ε(k+G) = ε(k) ∀G ∈

st♦ s t♥ s♠trí

♠♦s t♠é♥ q st ró♥ ε(k) s ♣ r♣rs♥tr st♥ts ♠♥rs ♦s ♠♦ssq♠s ③♦♥ ② ♦r ♠♦s str ♥trs♦s ♥ sq♠ ③♦♥ r

5.1

♣♦t♥ rst♥♦ t♥ r♦ s t♥ ♥ ♦♥♥t♦ tr♦♥s rs ♦s q st♠♦s♥trs♦s ♥ r♣rs♥tr ♥ó♥ ε(k) ♥ P❩

• Pr ♠♣③r ♣♥s♠♦s ♥ ♥ sst♠ ♥ ♠♥só♥Pr r s ♥s ♥rí ♥ sq♠ ③♦♥ r ♣♦♠♦s sr ♦s ♠♥♦s

♠♦s ♣rá♦ ε(k) ♥tr ♥ ♦r♥ st♦ s ♣r k = G = 0 ♣r♠r♥ stá r♣rs♥t ♥ st ♣r♦①♠ó♥ ♣♦r ♣rá♦ ♦♥t♥ ♥ P❩ st♦ s ♣r|k| ≤ π

a

♥♦ |k| > πa s t♥ q t♦r k /∈ P❩ ② s qr♠♦s srr s ♥s rs

P❩ ♣♦♠♦s t♥r ♥ ♥t q ε(k+G) = ε(k) ∀G ∈

sí s s♠♠♦s ♥ t♦r G1 =2πa ♠ás ♣qñ♦ ♦s t♦rs k st♦s ♥ ♥tr♦

(−2π

a ,−πa

) ♦ q ♠♦s

s trsr ♥ó♥ ε(k) s s♥ ❩ P❩ ♦t♥♥♦ ♥ ♦rrs♣♦♥♥t í♥s♦♥t♥ ♥ r

♦r rst♠♦s t♦r G1 =2πa ♦ s♠♠♦s G′1 = −2π

a ♦st♦rs k st♦s ♥ ♥tr♦(πa ,

2πa

) trs♠♦s ♥ó♥ ε(k)

s s♥ ❩ P❩ ②♦t♥♠♦s ♥ r♣rs♥t ♣♦r♥ í♥ r♦ s♦♥t♥ ♥ r

st ♦♥stró♥ rí♠♦s ♣rt♦♦s ♦s t♦rs ② sí♦t♥rí♠♦s strtr ♥sr P❩

ísst♦ó♦

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❮ Ó ♥s tr♦♥s tr♦♥s ♥ ♥ ♣♦t♥ ♣ró♦ é ② ♠ét♦♦ rs rts

P♦r ♦tr ♣rt ♣♦♠♦sr ε(k) ♥ t♦♦s ♦s ♣♥t♦s G ♦ q ε(k + G) = ε(k)∀G ∈ s t♥ q t♦s sr♦♥s ε(k) s♦♥ s ♦qrq sté ♥tr ró♥ q♦♥sr♠♦s

♥ st stó♥ ♥♥tr ♥ ♦r♥ st♦ s♥ G = 0 ♦♥tr② ♦♥ ♥ ♥ ♥rís ε(k) = ~2k2/(2m) ♦♥ k ∈P❩

Pr |k| > πa ♦s t♦rs k ♥♦ stá♥ ♥ P❩ ② ♣♦r t♥t♦ ♥rí s♦ s♦s t♦rs ♥♦

♦r♠ ♣rt P❩ ♦ ♣r♠r ♥

♣rá♦ ♥tr ♥ G1 ♦♥tr② ♦♥ ♥ ♥ ♥rí ε(k) = ~2(k+G1)2

2m ♦♥ k ∈ P❩ ②♥♦t ♣♦r 2+ ♠♥trs q ♥ ♣rá♦ ♥tr ♥ −G1 ♦♥tr② ♦♥ ♥ ♥ ♥rísε(k) = ~2(k−G1)

2

2m ♦♥ k ∈ P❩ ② ♥♦t ♣♦r 2−

P♦r ♦ t♥t♦ ♦s ♦s ♠♥♦s ② ♥♦s ♥ ♦t♥r s ♠s♠s ♥s

• ♥ trs ♠♥s♦♥s stó♥ s ♣r

P♦r ♠②♦r ♦♠♦ s♦♠♦s ♥s ♥ s q ~2/2m = 1

r♠♦s r s ♥s ♥rí tr♦♥s rs ♥ r ú s♠♣ ♦ r♦ ró♥ kx

♦♥sr♠♦s ♥ tró♥ t♦r K = k+G ♦♥ k ∈ P❩ ② G ∈ s t♥ q

ε(kx, ky, kz) = ε(Kx,Ky,Kz) =~2

2m(k+G)2 =

~2

2m

[(kx +Gx)

2 + (ky +Gy)2 + (kz +Gz)

2]

♦♠♦ ♥ s ♦tr s t♥q ♦ r♦ kx P❩ srá ♦♠♣r♥♥tr −π

a ② πa s♥♦ a ♣rá♠tr♦ r

P♦r t♥t♦ ♣r♠r ♥ ♦t♥r♠♦sr♥♦ k ♥ P❩ ε (kx, 0, 0) = k2x

♦s t♦rs ♠ás ♣qñ♦s ♣r♦s kx s♦♥ 2π

a (±100) q ♥ r s ♥rís

ε (kx, 0, 0) =(kx ± 2π

a

)2 ♥s ② r♥r♦r q s♦♥ s ♥ s♥③♦♥ ♥ s♦ ♥♠♥s♦♥ óts q ♥ró♥ sts ♥s s ② q só♦ ②♥ t♦r G q ♥♦s r sts ♥s t♦r k +G r ♥ ② k −G ♥

óts q ♥ st s♦ 2− ♦s t♦rs k ♥ s −π/a st π/a ♠♥trs q ♥ s♦ ♥tr♦r 2+ ♥ ♥ π/ast −π/a ♦ q ♥ ♥ s♦ s s♠ G1 ② ♥ ♦tr♦ s rst ♥ ♥rí s ♠s♠ ♥q ♥♦ ró♥ ε(k)

ísst♦ó♦

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❯

P♦r ♦tr ♣rt r♦ ♦♥ ó♥ ♦s tr♦ t♦rs ♣r♣♥rs kx ♠ás♦rt♦s ♥ r ♠s♠ ♥ ♥ st s♦ ♣♦r t♥t♦ ♦♥ ♥ ♥ró♥

2π

a(010)

2π

a(010)

2π

a(001)

2π

a(001)

ε (kx, 0, 0) = k2x +

(2π

a

)2

♥s

Pr ♦s t♦rs G s♥ts ♥rí s t♠é♥ ♠s♠ ♥ró♥

2π

a(110)

2π

a(101)

2π

a(110)

2π

a(101)

ε (kx, 0, 0) =

(kx +

2π

a

)2

+

(2π

a

)2

♥s

♠♥t ♣r

2π

a(110)

2π

a(1010)

2π

a(110)

2π

a(101)

ε (kx, 0, 0) =

(kx −

2π

a

)2

+

(2π

a

)2

♥s ♥ró♥

♦♦ st♦ stá r♣rs♥t♦ ♥ r ♦♥ s s♣ t♠é♥ ♥ró♥ ♥

ísst♦ó♦

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❮ Ó ♥s tr♦♥s tr♦♥s ♥ ♥ ♣♦t♥ ♣ró♦ é ② ♠ét♦♦ rs rts

r♣rs♥tár♠♦s ε(k) s♥♦ k ♥ t♦r ♦s ♠♥s♦♥s s ♣rá♦s s ♦♥rt♥ ♥♣r♦♦s ② r♠♥t ② ♣♥t♦s P❩ ♥ ♦s q ② ♥ró♥ ♥ s ♥s ♥rí ♦ ♥ ♥str♦ s♦ ② ♥s ♣r♦♦s ♦♥ ♥ró♥ ♠út♣ ♥ t♦♦s ♦s ♣♥t♦s

• sts s♦♥ s ♥s tr♦♥s rs ❨ ♦r sr ♦r ♥ ♣r♥t ♥ró♥ qt♥♥ s ♥s ♥ s♦s ♣♥t♦s s ♦ ♥trí♥s♦ ís ♣r♦♠ ♦ ♣r s♠♣♠♥t♣♦r ♣r♦①♠♦♥s t③s srr ♦s st♦s tró♥♦s

Pr ♥t♥tr r ♥ rs♣st st ♣r♥t ♠♦s ♥tr♦r ♥ ♣rtró♥ ♣qñ♥ sst♠ tr♦♥s rs q ♠♦s st♦ s♣♦♥♠♦s q ♣♦t♥ rst♥♦ qtú s♦r ♦s tr♦♥s s ♣qñ♦ ♠ás ♥t r♠♦s s st♦ s ♣s ♦ ♥♦ ② ♥ qé s♦s♦ s ás s♦♥ s ♥s ♥rí ♦s tr♦♥s s♣st♦s ♥♣♥♥ts ♣r♦ ② ♥♦ rs

5.2 ❯

❱♠♦s trtr ♦r ♣s s strtr ♥s tr♦♥s s♦♠t♦s ♥ ♣♦t♥ é♦ q s s♣♦♥ q ♣♦t♥ rst♥♦ s ♣ró♦ sr♠♦s

U(r) =∑

G 6=0

U(G)eiG·r,

P♦r ♦tr ♣rt ♦s tr♦♥s ♦ t♥♥ ♥ ♥ó♥ ♦♥

Ψk(r) =1√Veik·ruk(r) =

1√Veik·r

∑

G′

uk(G′)eiG

′·r.

stt②♥♦ st ♥ó♥ ♦ ♥ ó♥ rö♥r

[− ~2

2m∇2 + U(r)

]Ψ(r) = εΨ(r)

♦t♥♠♦s

∑

G′

~2

2m(k+G′)2 ei(k+G′)·r uk(G

′) +∑

G 6=0

U(G)∑

G′

uk(G′) ei(k+G+G′)·r

= ε(k)∑

G′

uk(G′) ei(k+G′)·r,

q s ♥ ó♥ r q♥t

♦♠♦ st♠♦s s♠♥♦ s♦r t♦♦s ♦s t♦rs G′ ♥ s♥♦ s♠♥♦ ♣♦♠♦sssttr

∑G′ ♣♦r ♥ s♠

∑G′′ s♦r t♦♦s ♦s t♦rs G′′ ♦♥ G′′ = G+G′ ♥t♦♥s

q s♠t♦r♦∑

G′′ uk(G′′ − G)ei(k+G′′)·r ② ♦ q G = G′′ − G′ ♥♦s q ♥♠♥t

s♥♦ s♠♥♦ ♦♠♦ ∑

G 6=G′′

U(G′′ −G′)∑

G′′

uk(G′′ −G)ei(k+G′′)·r

♥ s♠ ♥♦ ♠♦s ♥♦ G = 0 ② q r♣rs♥t ♦r ♠♦ q s♦♠♥t sr ♣r r ♦r♥ ♥rís② s♦ ♥♦ t♥ ♠♣♦rt♥ ♥ ssó♥

qí uk(0) r♣rs♥t ♦♥tró♥ Ψk ♦♥ ♣♥ eik·r q s ♥ó♥ ♦♥ tró♥ s str r ♣rtró♥ s ♣qñ ♣♦rí♠♦s s♣♦♥r q uk(0) ≈ 1 ② q uk(G′)≪ uk(0) ♣r t♦♦ k

ísst♦ó♦

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❯

Pr♦ í♥ s♠t♦r♦ s ♠♦ ♣♦r ♦ q ♣♦♠♦s ♠r ♥♠♥t∑

G′′ ♣♦r ♥∑

G′

② ♥♦♥tr♠♦s q

∑

G′

~2

2m(k+G′)2 uk(G

′) ei(k+G′)·r +∑

G′

∑

G 6=G′

U(G′ −G) uk(G′) ei(k+G′)·r

= ε(k)∑

G′

uk(G′) ei(k+G′)·r

♦ q s ①♣♦♥♥s s♦♥ ♥♦♥s ♥♠♥t ♥♣♥♥ts ♣♦♠♦s r tér♠♥♦ tér♠♥♦ ♦s tér♠♥♦s q ♣r♥ ♠♦s ♦s ♦♥ ♠s♠ ①♣♦♥♥ ei(k+G′)·r

② ♦t♥♠♦s[~2

2m(k+G′)2 − ε(k)

]uk(G

′) +∑

G 6=G′

U(G′ −G) uk(G) = 0 ∀G′ ∈

♦ tr♥t♠♥t[ε0(k+G′)− ε(k)

]uk(G

′) +∑

G 6=0

U(G) uk(G′ −G) = 0 ∀G′ ∈ .

♦♥ ② t♥ts ♦♥s ♣r uk(G′) ♦♠♦ G′ ② ♥

ó♥ r s ♥♦♠r ó♥ ♥tr ② s ①t ♥ s♥t♦ q♦♥stt② ♥ ♦♥♥t♦ ♥♥t♦ ♦♥s rs ♦♥♥t♦ q s ♥tr♠♥t q♥t ó♥ r♥ rö♥r q ♠♦s ♣rt♦

Pr sst♠s ♠tá♦s ♥ ♦s q s♣rr q ♣rtró♥ ♥ ♣♦r r ♣ sr♣qñ s t♥rá q uk(0) ≃ 1 ② q uk(G′) ≪ uk(0) ♣r t♦♦ t♦r k ♠ás ♥ ♥ ♣r♠r♣r♦①♠ó♥ ♣♦♠♦s srr ♥rí ♦♠♦

ε(k) ≃ ε0(k) =~2k2

2m,

st♦ s ♥rí ♥ tró♥ rP♦r ♦tr ♣rt ② q r ♥♦tr q ♦s ♦♥ts U(G′) ♣♦t♥ rst♥♦ s♠♥②♥s ♦r ♦♥ r♣③ ♠♥tr ♠ó♦ G s♦ ♥♦s ♣r♠trá ♣r♦①♠r s ♦♥s ♦♥sr♥♦ s♦♠♥t q♦s U(G′ −G) q s♥ ♦r ♣r ♣r G′

❱♦rs ♦s ♦♥ts uk(G)

♦♠♦ ♣r♠r ♣s♦ ♣r r uk(G′) t♦♠♠♦s uk(0) = 1 ♥ ♦♥♥t♦ ♦♥s ♦s♠ás uk(G) q ♣r♥ ♥ s♠t♦r ♦s s♣♦♥♠♦s ♥♦s r♦r♠♦s q s♠♣r ♣r♥♠t♣♦s ♣♦r U(G′ −G) ♥t♦♥s s♣♥♦ ♦t♥♠♦s ①♣rsó♥

uk(G′) ≃ U(G′)

~2

2m(k+G′)2 − ε0(k)

=U(G′)

~2

2m

[(k+G′)2 − k2

] ,

q ♥♦s ♦s ♦rs uk(G′) ♣r G′ 6= 0

❨ s s ♠♣ q |k+G′| ≫ k ♠♦s q t♠♥t s r q uk(G′) s ♣qñ♦ ♦♠♦s♣♦♥í♠♦s Pr♦ s ♣r ♥ tr♠♥♦ G′ rst q (k+G′)2 ≃ k2 ♦r uk(G′) ♦ ♣♦r ② ♥♦ s ♣qñ♦ ② s♣♦só♥ ♥tr♦r ♥♦ s á

P♦r ♦♥s♥t ♦s ♦♥ts uk(G′) q t♦♠♥ ♥ ♦r ♣r s♦♥ uk(0) ② ♦suk(G

′) ♦rrs♣♦♥♥ts ♦s t♦rs r rí♣r♦ G′ q rq♥ (k+G′)2 ≃ k2st♦ s ts q k sté r♥♦ ♦s ♣♥♦s r s♦♦s s♦s G′

sí ♥ s♦ ♥ ♣♦t♥ ♦♠♥♦ ♣♦r ♠♣♦ t♥♠♦s ♥ ♦♠♣♦rt♠♥t♦ U(G) ∝ 1/G2

ísst♦ó♦

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❯

❯

❯

❯

❯

❯

❮ Ó ♥s tr♦♥s tr♦♥s ♥ ♥ ♣♦t♥ ♣ró♦ é ② ♠ét♦♦ rs rts

♦t ❯♥ ♠♥r ♥tr♣rtr ♣♦rqé st♦ s s♥t ❯♥ ♦♥ ♣♥♥tr♦ ♥ rst s rt ♣♦r ♦s ♣♥♦s rst♥♦s s♦♦s G′ s♠♣rq s ♠♣ q (k+G′)2 ≃ k2st♦ s♥ q ♥ tró♥ ♦ ♥② t♦s s r♦♥s ♥s ♣♦r r ②q

Ψk(r) =1√Veik·r

∑

G′

uk(G′)ei(k+G′)·r,

♦♥ uk(G′) 6= 0 ∀G s ♠③♠♦s ♦♥ ♣♥ eik·r ♦♥ t♦s s rts ei(k+G′)·r

♦t♥♠♦s ♥ó♥ ♦s ♣♦r st♦ q tró♥ ♦ s ♥ st♦ st♦♥r♦ ♠t♦♥♥♦ ♦ ♣s♥② t♦s s r♦♥s ♣♦ss ♦♥ r rst♥

❱♦rs uk(G) ♣r k r♥♦ ♥ r♦♥tr ③♦♥ r♦♥

♦♥ó♥ ♥tr♦r ♥♦s q ♥♦ srá♥ ♣qñ♦s q♦s uk(G′) q rq♥ (k+G′)2 ≃ k2

st♦ s q t♦r k sté r♥♦ ♣♥♦s r s♦♦ G′ ♥ ♣s ♣♦r r♦s ♦♥ts ② ♥rí ♦s tr♦♥s ②♦ t♦r k sté st♦ r ♥ r♦♥tr ③♦♥ r♦♥ ② q ♦s ♠ás ② stá♥ ♦s trés ó♥

♣♦♥♠♦s q k stá s♦r ♥ ❩ ♥ ♣♦r ♣♥♦ st♦r G ∈ ♥t♦♥s t♥t♦uk(0) ♦♠♦ uk(G) ♥ sr s♥t♠♥t r♥s ② ó♥ ♥♦ sr á ♣r ♦s♦♥ts

P♦r t♥t♦ ② q ♣♥tr s ♦s ♦♥s ♥trs ó♥ ♥ ♣r uk(0) ② ♦tr ♣ruk(G) ② s♣♦♥♥♦ ♥♦s rst♦ ♦s ♦♥ts ♥ s♠t♦r♦

∑G 6=G′ U(G′ − G) uk(G)

s♦♠♥t ♣♥ ♣rr uk(0) ② uk(G) ② ② q ♠♣r sP♦r ♦♥s♥t s ♥♥ts ♦♥s ♥trs ♥♦s q♥ rs s ♦s s♥ts

G′ = 0 =⇒[~2

2mk2 − ε(k)

]u(0) + U(−G) u(G) = 0

G′ = G =⇒[~2

2m(k+G)2 − ε(k)

]u(G) + U(G) u(0) = 0.

♦♠♦ U(r) s r ♥t♦♥s s ♠♣ q U(G) = U∗(−G)

Pr q t♥ s♦♦♥s ♥♦ trs tr♠♥♥t ♦s ♦♥ts s ♥rst♦ s

∣∣∣∣∣∣∣

~2

2mk2 − ε(k) U∗

U~2

2m(k+G)2 − ε(k)

∣∣∣∣∣∣∣= 0.

♥ ♦♥s♥ s ♠♣r q

ε2(k)− ~

2m

[(k+G)2 + k2

]ε(k) +

[(~

2m

)2

(k+G)2 k2 − |U |2]= 0

♦tr q s k stá ♥♦ s♦♠♥t s♦r ♥ ❩ s♥♦ ♥ rs ❩ ♣ ♥ ♥ ért ♥ ❩ ♥ú♠r♦ ♦♥tsst♥t♦s r♦ ♠ás uk(0) sr ♥ú♠r♦ ♣♥♦s r

st♦ qr r q ♥ srr♦♦ ♥ó♥ ♦ Ψk(r) s ♦♥sr♥ st♥t♦s r♦ s♦♠♥t uk(0) ② uk(G)♠♥trs q s k stá s♦r rs ❩ ② q ♦♥srr t♦♦s ♦s uk(G) ♦rrs♣♦♥♥ts ♦s ♣♥♦s r

ísst♦ó♦

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❯

❯

❯

❯

❯

❯

Ó ❯❯ ❨ ❯❯

② ♦t♥♠♦s ♦s s♦♦♥s

ε(k) =1

2

~

2m

[(k+G)2 + k2

]±

√1

4

[((k+G)2 − k2

]2( ~

2m

)2

+ |U |2

ε(k) =1

2

(ε0(k) + ε0(k+G)

)±√

1

4

(ε0(k)− ε0(k+G)

)2+ |U |2.

sts ♦s rís sr♥ ♦s ♥s rs♣ts ♥rí♠♦s q ♦s tr♦♥s rs t♥♥ ♥s ♥rís ♥rs ② ♠♦s ♦r q ♣♦r t♦ ♣♦t♥ rst♥♦ ε(k) t♥ ♦s s♦♦♥s ② ♥ ♥r r♦♠♣ s ♥ró♥ ♥ ❩ q ♠♦s ♦♥sr♦

• ❱♠♦s r ♥ ♥ ♠♥só♥ qé ♣s ♦♥s ♥s tr♦♥s srs r k = 0 r♥ ♥tr ♥rís ♥♦♣rtrs ε(k) + ε0(k + G) s ♠② r♥ ♥♦♠♣ró♥ ♦♥ |U | ♠♦♦ q ε(k) ≃ ε0(k)② ♥rí tró♥ ♦ s ♣rát♠♥t ♥ tró♥ r

P♦r ♦tr ♣rt ♣r k = 12G ♥ ♥ ♠♥só♥

s ♦t♥

ε±(k) = ε0(k)± |U |.

t♥♠♦s ♣s q s♣r ♥ró♥ ♥ tr♦♥s rs ♥ ❩ ♦♥ ♥rí t♥í ♦r ε0(k) r♦♠♣é♥♦s ♥ ♥ s♦ó♥ ε0(k) + |U | ② ♥ ♦tr ε0(k)− |U |

P♦r ♦♥s♥t ♥♦ ♦♥sr♠♦s ♥ ♣♦t♥ rst♥♦ é ♦s t♦rs k q sts♥ ♦♥ó♥ r t♥♥ r♦t s♦ ♣♦r ♥ ♦tr s♠trí ♥ró♥ s ♥s ♥rí tr♦♥s rs rst♥♦ q s♣ró♥ ♥tr ♦s ♥s ♥rét♦s s ♣r♦♣♦r♦♥ ♦rrs♣♦♥♥t ♦♠♣♦♥♥t ♦rr ♣♦t♥ rst♥♦

♦t ♦♠♦ s s♦ ♦s ♠ts ♥♦s rst③♥ ♥ r ♣st♦♥ ♦♠♣♦rt♠♥t♦ tr♦♥s rs ♦r kF s ♠♥♦r q kPZB ♦ q s♥ q ♥♥r ♥♦ rá ♥♥ú♥ tró♥ ♥ ♠t q rq q |k+G|2 ≃ k2P♦r ♦ t♥t♦ s ♣♦t♥ rst♥♦ s ♣qñ♦ st♦s ♠ts s ♦♠♣♦rtrá♥ ♦♠♦ ss tr♦♥s rs ♦ q st q ♠♦♦ r♦♠♠r sr ♦♥ r③♦♥ ♣rsó♥♥ ss ♣r♦♣s

5.3 Ó ❯❯ ❨ ❯❯

♣♦♥♠♦s q st♠♦s st♥♦ ♥ rst ♥ q ♦s ♠♥t♦s s q ♦r♠ strtr rst♥ s♦♥ s rst stá ♦r♠♦ ♣♦r ♥ ♠s♠♦ t♣♦ át♦♠♦s

♦♠♦ ② s ♦♠♥t♦ ♥ts s k stá ♥ rs ❩ ♥ú♠r♦ ♦♥ts uk(G) st♥t♦s r♦ ♠ás uk(0)s ♥ú♠r♦ ♣♥♦s r ② ♦♠♦ ♦♥s♥ ♣rrá ♦ ♥ú♠r♦ ♠ás ♥♦ s♦♦♥s ε(k)

♥ trs ♠♥s♦♥s ♥ s ♣r♦①♠s k = 0 r♥ ♥tr ♥rís ♥♦ ♣rtrs ε(k) + ε0(k +G) s t♠é♥♠② r♥ ♥ ♦♠♣ró♥ ♦♥ |U | ♠♦♦ q ε(k) ≃ ε0(k) ② ♥rí tró♥ ♦ s ♣rát♠♥t ♥ tró♥r

r q ♥ú♠r♦ t♦ tr♦♥s ♣♦r át♦♠♦ ♣r rsts ♣r q sr r♠ s t♥♥t P❩ s ♣r♦①♠♠♥t ♠♥trs q ♦s ♠ts ♦s ♥♦s s♦♠♥t ♥ tró♥át♦♠♦ s tr♦♥s rs

ísst♦ó♦

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❯

❯

❯

❯

❯

❯

❮ Ó ♥s tr♦♥s tr♦♥s ♥ ♥ ♣♦t♥ ♣ró♦ é ② ♠ét♦♦ rs rts

♥ s s♦ ♥ t♦r strtr SG = SG =∑s

j=1 fj(G)e−iG·σj s ♣♥ sr ♦s t♦rs ♦r♠ fj(G) ♦♠♦ t♦r ♦♠ú♥ ② ♥r t♦r strtr ♦♠étr♦ SG

SG =

s∑

j=1

e−iG·σj .

♣♦t♥ rst♥♦ ♥ st s♦ s

U(r) =∑

R

∑

j

Vat(r−R− σj)

♣♦r ♦ q s ♦♠♣♦♥♥t ♦rr ♣♦♠♦s srr ♦♠♦

UG =1

Ω

∫

CPdr∑

R

∑

j

Vat(r−R− σj) =1

Ω

∫

todo el espaciodr∑

j

Vat(r− σj).

♠♠♦s r′ = r− σj ♥♦s q

UG =∑

j

1

Ω

∫

CPdr′ eiσj ·GVat(r

′)e−iG·r′=

1

Ω

(∑

j

eiσj ·G)∫

todo el espaciodr Vat(r)e

−iG·r

=1

ΩS∗G Vat,G.

P♦r ♦ t♥t♦ ♥♦ t♦r ♦♠étr♦ strtr s ♥ ♣r ♥ t♦rtr♠♥♦ G t♠é♥ ♦ ♦♠♣♦♥♥t ♣♦t♥ rst♥♦♦rrs♣♦♥♥t s t♦r G r rí♣r♦ s♦ ♥♦s ♥ q s ♥♦♥tr♠♦s q ♥♦② ♣♦s ró♥ r②♦s ❳ ♣r tr♠♥♦s ♣♥♦s r ♥str♦ rst ♣♦♠♦sr♠r q s ♦♠♣♦♥♥ts ♦rr ♣♦t♥ rst♥♦ ② ♣♦r t♥t♦ ♦s♣s ♥rí s♦♦ s♦s ♣♥♦s s♦♥ ♥s

st♦s rst♦s ②♥ t♠é♥ ♥t♥r ♠♦r s r③♦♥s ♣♦r s q ♠♦♦ s ♣③ ①♣r ♦♥ s♦r♣r♥♥t é①t♦ tr♠♥s ♣r♦♣s ♦s ♠tss♠♣s♠♦s q ♠♦♠♥t♦ q ♥ ♥ tró♥ ♥tr ♦s♦♥s s ♥ ♣r♦♠♦ ∆k = eEτ/~ ∼107m−1 = 10−3 −1 ♦♥ ♠♦s ♦♥sr♦ q τ ∼ 10−12 s ② q E ∼ 100 ♦t♠ ♦♠♦ P❩ t♥ ♥ t♠ñ♦ ♦r♥ π/a ∼ 1 −1 ♣s a ∼ 3 − 4 rst q tró♥ r♦rr♥tr ♦s ♦s♦♥s ♥ s♣♦ ♥ ♠♦ ♠♥♦r q t♠ñ♦ P❩ ♦s tr♦♥s ♥♦♥ ♥trrs ①st♥ ♣♥♦s r ② s ♦♠♣♦rt♥ ♦♠♦ tr♦♥s rs ♥tr♦s♦♥s

P♦r ♦tr ♣rt ♥tró♥ tró♥ó♥ s ♠② rt s s♣ró♥ s ♣qñ Pr♦ ♦str♦♥s ♦♥ó♥ ♥♦ ♣♥ rrs s r♥ís ♦s ♦♥s ♣♦rq ♣r♥♣♦ ①só♥ P ♥♦ s ♦ ♣r♠t ♦s st♦s tró♥♦s ♥rí ♠ás ♦s ♦♥s stá♥♦♣♦s ♣♦r ♦s tr♦♥s q stá♥ ♦s ♦s ♠♥r q s ♦s tr♦♥s ♦♥ó♥s r♥ ♠♦ ♦ ♦♥s ♥rí t♦t sst♠ ♠♥t ♥♦r♠♠♥t P♦r ♦♥s♥t ♣r♦♠♦ ♥tró♥ tró♥ó♥ ♣ sr ♠♦ ♠♥♦sr ♦ q ♥ ♣r♥♣♦ ♣♦rí♠♦ss♣rr

tr ♠♥r r st s♥t♦ s ♣♥sr ♥ q ♠♦ ♦s tr♦♥s ♦♥ó♥♣♥t rá♣♠♥t ♣♦t♥ r♦ ♣♦r ♦s ♦♥s ♣♦st♦s r ♦♥ rst♦ q ♣♦t♥ rst♥♦ t♦ q ♥♠♥t tú s♦r ♦s tr♦♥s ♦♥ó♥ q s♠♥♦♣♥t♦ ② s t♦ ♦ s ♠♦ ♠♥♦s ♥♦t q s ♦s tr♦♥s ♦♥ó♥ ♥♦r t♥ ♠ós

♥ rs♠♥• Pr ♥t♥r ♣♦rqé q ♣r♦①♠ó♥ tr♦♥s rs ♣ r r rst♦s

ísst♦ó♦

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❯

❯

❯

❯

❯

❯

❯ ❯ ❯

♣ts t♥♠♦s ♦s r♠♥t♦s ♣r♠r♦ s q ♥s t③r ♥ stíst á♥t ♣r ♦s tr♦♥s ♦♥ó♥♦ s♥♠♥t s ♥s ♠♦ ♠ás t q ♥ ♥ s ás♦ q só♥③ t ♦s ♣r♦s♦s tró♥tró♥ s ♠♦ ♠♥♦r q ♣r ♥tró♥♦♠♥ ♥tr ♦s ♣s t♦♠ ♦r

Sel−el ≃(kBT

εF

)2

S0

♦♥ S0 s só♥ ③ tór ♥tró♥ ♦♠♥ tró♥tró♥ ♠ás ♦s tr♦♥s s ♦♠♣♦rt♥ ♦♠♦ tr♦♥s rs ♦ t♦ ♣♦t♥ rst♥♦② q r♦rr♦ r ♠♦ ♥tr ♦s ♦s♦♥s ♥ s♣♦ rí♣r♦♦ sr ♠♦ ♠♥♦r q t♠ñ♦ P❩ q ♦s tr♦♥s ♥♦ ♥♦t♥ ①st♥ r st♦ s t♦ ♦s ♣♥♦s r ② s ♦♠♣♦rt♥ ♦♠♦ tr♦♥s rs• P♦r ♦tr♦ ♦ ♣r ♥ ♣♦s ♠♦r sr♣ó♥ ♦s tr♦♥s ♥tr♦ ♣r♦①♠ó♥♠♦♥♦tró♥ t♥♠♦s ♣♦s t♦♠r ♥ ♦♥sró♥ ♥tró♥ ♥tr ♦s tr♦♥s♦♥ ♦ q strí♠♦s ②♥♦ ♠ás á ♣r♦①♠ó♥ tr♦♥s ♥♣♥♥ts

5.4 ❯ ❯ ❯

♣♦♥♠♦s ♥ ♦♥♥t♦ át♦♠♦s ② s♣ró♥ s r♥ ♥ ♦♠♣ró♥ ♦♥ ♦♠♥ st♦s át♦♠♦s ♦s tr♦♥s strá♥ ♥ ♥s tó♠♦s ♦③♦s r♦r ♦s ♣♥t♦s s♣♦♥ ♦s q stá♥ ♦s át♦♠♦s s♣♦♥r♠♦s q st♦s ♣♥t♦s ♦r♠♥ ♥ rPr♦ s s♠♥♠♦s s♣ró♥ ♥tr ♦s át♦♠♦s ♦ ♦♥st♥t r ♦s ♥s ♥rí ♦s tr♦♥s rá♥ ♦♥r ♦♥ ♦s ♥s tó♠♦s ♥ ♥t♦ q st♥ ♥trtó♠s ♦♠♣r ♥ s ♥♦♥s ♦♥ tó♠s ♣s ② q tró♥ ♥trtrá ♦♥♦s át♦♠♦s ② tr♦♥s ♥♦s

r♠♦s ♦r rr♥♦s ♣r♦♠ ó♠♦ srr ♦s tr♦♥s ♥ só♦ ♥♠♥r tr♥t ♣r♦①♠ó♥ tr♦♥s srs ♥ ♥♦ ♣r♦♠♥t♦ sr♠♦s♥ ♥♦q ♠ás r♥♦ ♠♥r ♥ q s ♦♥str②♥ s ♥♦♥s ♦♥ s ♠♦és ♣rtr s ♥♦♥s ♦♥ tó♠s

5.4.1 ♥tr♦ó♥ ♦sq♦ ♥ sr♣ó♥ ♥ ♠♦é s♥

♥♦ ♦♥ sts s ♠♦s srr ♣r♠r♠♥t ♥ ♠♦♦ q st ♦s ♥s ♥rí ♠♦é ró♥♦ ♦♥③ +

2 és r

♥ó♥ ♦♥ r♦♥♦ tró♥r♦r ♣r♦tó♥ Pi s

ϕi =1√πa30

e−ri/a0

♦♥ a0 r♦ ♦r

❯♥ ♣♦s ♣r♦①♠ó♥ ♥ó♥ ♦♥ ①t ψ tró♥ ♥ ♠♦é +

2

♦♥sst ♥ r ♥ ♠ ♥♦♥s ♦♥ ♦♠♥ó♥ ♥ ♦s ♦rts tó♠♦s ♦s ♣♦r ó♥ st♦ s

ψ = c1ϕ1 + c2ϕ2,

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó ♥s tr♦♥s tr♦♥s ♥ ♥ ♣♦t♥ ♣ró♦ é ② ♠ét♦♦ rs rts

♦♥ ♣rá♠tr♦s ♦ ♦♥ts c1 ② c2 rs s ♠ ♥♦♥s s s♦rí ♥ó♥ ♦♥ ó♣t♠ s r ♠♥♦r ♥rí

♠t♦♥♥♦ sst♠ s

H = − ~2

2m∇2 − e2

r1− e2

r2+e2

R.

♠♠♦s Sij = 〈ϕi|ϕj〉 s ♥trs s♦♣♠♥t♦ ♥tr s ♥♦♥s ♦♥ ϕi ② Hij =⟨ϕi|H|ϕj

⟩ ♦s ♠♥t♦s ♠tr③ ♠t♦♥♥♦ ssttr ♥ rö♥r ② ♠t♣r

♣♦r 〈ϕi| s ♦t♥(H11 − ES11) c1 +(H12 − ES12) c2 = 0

(H21 − ES21) c1 +(H22 − ES22) c2 = 0.

P♦r ♥♦r♠③ó♥ t♥♠♦s q S11 = S22 = 1 ② ♣♦r s♠trí q S12 = S21 = S ♥♦ ♦sá♦s ♥sr♦s ♦t♥♠♦s

S =

[1 +

R

a0+

1

3

R2

a20

]e−R/a0

H11 = H22 = −EH +e2

R− C

H12 = H21 = −EHS +e2S

R−A.

qí

C =

⟨ϕ1

∣∣∣e2

r2

∣∣∣ϕ1

⟩= EH

2a0R

[1− e−2R/a0

(1 +

R

a0

)]

A =

⟨ϕ1

∣∣∣e2

r2

∣∣∣ϕ2

⟩= 2EHe

−R/a0

(1 +

R

a0

)

s♥♦ EH ♥rí st♦ ♥♠♥t át♦♠♦ ró♥♦

♠♠♦s E ≡ ε EH A ≡ ∆ EH ② C ≡ Γ EH tr♠♥♥t ♦s ♦♥ts ♥♦s q∣∣∣∣∣∣∣∣∣∣

−1 +2a0R

− Γ− ε

(−1 +

2a0R

)S −∆− εS

(−1 +

2a0R

)S −∆− εS −1 +

2a0R

− Γ− S

∣∣∣∣∣∣∣∣∣∣

= 0

② s♦ó♥ s

ε± = −1 +2a0R

± ∆∓ Γ

1∓ S

P♦r ♦♥s♥t ♥ tó♠♦ ú♥♦ át♦♠♦ ró♥♦ (ε = −1) s s♦♦ ♦♥srr ♠♦é ró♥♦ ♦♥③ +

2 r ♦t♥♦ ♦s s♦♦♥s ε± ♣r ♦s ♥s

♠♦é

s♥♠♥t s ♥ ♣r♦♠♥t♦ r♦♥ ♦s q s ♥ srt♦ ♥ s♥tr ís á♥t s ♠♣♦rt♥t r ♥♦tr q ♦♠♦ ♦s ♠♥t♦s ♠tr③ Hij ♥♦ ♥③♥ ♥♥ú♥ ♠í♥♠♦ ♥ ♠♦r s ♥

♣r♥♣♦ ♦ s♦♣♠♥t♦ S ♣r ♥ ssó♥ ♠ás t és t♠ ♦és s♥tr ís á♥t ♠é♥ ♦♥♥ r♦rr q s t③♦ ♣r♦①♠ó♥ ♦r♥♣♣♥♠r q s♣♦♥ q ♠t♦♥♥♦ sst♠s ♥ tró♥ st♦ ♥tró♥ ♦♥ ♦s ♣r♦t♦♥s ♦s st♦ s q ♦ tró♥ s ♠♦ ♠②♦r q ♦s ♣r♦t♦♥s

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❯ ❯ ❯

s♦♠♥t♦ ♥ ♥ tó♠♦ ♦♠♦ q ♠♦s r ♣r ♥ ♥♠♥t át♦♠♦ ró♥♦ s r s♠♣r q st♥ q s♣r ♦s ♥ú♦s ♦s át♦♠♦s s♦ s♥t♠♥t ♣qñ ♣r q ①st s♦♣♠♥t♦ ♥tr s ♥♦♥s ♦♥ ♦s tr♦♥s ♦s ♠s♠♦s

♥rís ♦s ♥s ♠♦é +2

r♥ ♥tr s ♥rís ♥s♦♦ ♦♥ ♥rí ♠ás ② ♥s♦♦ ♦♥ ♥rí ♠②♦r ♣♥ Rés r ♣r +

2 ② q s st♥ ♥tr ♦s ♥ú♦s q ♥ s♦♣♠♥t♦ s ♥♦♥s ♦♥ ♣r♦♣r ♥ st♥ R ♥♦ ♣♥s♥t♠♥t ♥ú♠r♦ át♦♠♦sq ② ♥ sst♠ ♥ ♥ tó♠♦tr♠♥♦ s s♦rá ♥ ss ♥s ♥♥ sst♠ ♦♥stt♦ ♣♦r ss át♦♠♦s ♣r♦ ♥r t♦t s s♦♠♥t♦ ♣♥s rrá s sst♠ s ♦♦ ♦ ③ át♦♠♦s s♣r♦s ♠s♠ st♥ R

5.4.2 ♣r♦①♠ó♥ ♦ ♠ét♦♦ rs rts

sí s ♦ss ♠♦s ♦r ♣rs♥tr ♥ ♠ét♦♦ á♦ strtr ♥s ♦s só♦s q ♥ ♠♥r r♦ s s q ♠♦s ①♣st♦ ♠♦s st♦ q ♣♦rí♠♦s♦♥srr ♥ ♠t ♦♠♦ ♥ ♦♥♥t♦ tr♦♥s srs ♣rtr♦s ú♥♠♥t ♣♦r ♥♣♦t♥ ♣ró♦ é ♦r ♥ s ♦♣t♠♦s ♣♥t♦ st ♦♥trr♦ ♥ q s ♥♣é ♥ s♣t♦ t♦♠íst♦ ♥ só♦ rst♥♦ ♣♦rí♠♦s ♦♥srr♦ s ♠t ♦ s♥t♠♦r st♦ út♠♦ ♦♠♦ ♥ ♦♥♥t♦ át♦♠♦s ♥tr♦s q ♥trtú♥ é♠♥t st♥♦ ♦str♦♥s ♦s rt♠♥t ♦s ♦♥s

② q str q st ♠ét♦♦ rs rts s t③ s♣♠♥t ♦♠♦ ♥ ♠ét♦♦♣r ♣r♠tr③r s ♥s ♥rí ♠♥r q ♣♥ r③rs á♦s ♣r♦♣stró♥s s♥ ♥s r ♣r♦♠♥t♦s ♠ás s♦st♦s ♣r ♦♥♦r s r♦♥s s♣rsó♥ s ♥s εn(k)

• ♣r♦①♠ó♥ q qí s ♣♥t s ♥ ♣r♦①♠ó♥ q ♦♥sr ♠♣♦rt♥t s♣t♦t♦♠íst♦ ♦s só♦s ② s♣♦♥ q ♥tró♥ ♥tr ♦s tr♦♥s s s ♣r♥♣♠♥t ♥ s♦♣♠♥t♦ s ♥♦♥s ♦♥ ♦s tr♦♥s s♦♦s q♦s ♦♥s q s♦♥ ♥♦s♣ró①♠♦s P♦r ♦ t♥t♦ s ♥ ♠ét♦♦ á♦ ♥♦ s r♥ ♦s ♦♥♦♥s

s♦♣♠♥t♦ s ♥♦♥s ♦♥ s s♥t♠♥t r♥ ♦♠♦ ♣r ♠♦r át♦♠♦s s♦s

st s♦♣♠♥t♦ s s♥t♠♥t é ♣r ♥♦ ♥♦♥r sr♣ó♥ t♦♠íst ♦♦③ ♦s tr♦♥s só♦

ás ♥ ♦♥rt♦ ♥tr s ♥♦♥s ♦♥ ♦♥ s♦s ♥ tó♠♦ q st♠♦s st♥♦ ♦t ♠♣♦rt♥t qí s♦♠♥t s rá trt♠♥t♦ ♠ét♦♦ ♣r s♦ ♦s rsts rs

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó ♥s tr♦♥s tr♦♥s ♥ ♥ ♣♦t♥ ♣ró♦ é ② ♠ét♦♦ rs rts

sts ♦s ♦♥♦♥s st♥ s♦ ♣r♦①♠ó♥ ♣r r strtr ♥s ♥rí ♥ s♦ ♠ts ♦♥ ♣s ♣r♠♥t ♥s ② s♥ts ♥ ♥r

• ♥ ♠r♦ s í ①♣r ♦♠♣♦rt♠♥t♦ ♦s tr♦♥s ♦ ♥ s ③♦♥s ♥tró♥s só♦ ♦♥ ♥ ♣r♦①♠ó♥ rs rts st♦ s q ♥ sts ③♦♥s ♣♦t♥rst♥♦ stá ♣♦r ♦ ♦s ♥♦s tó♠♦s ♥rí ② ♠ás s ró♥ s ♠② ♥t ♥♦♠♣ró♥ ♦♥ ♣♦t♥ tó♠♦ ó♠♦ s ♣♦s srr ♥♦s tr♦♥s ♥ ♦♥♥♦♥s ♦♥ ♦③s ♦♠♦ s rá ♠ás ♥t rs♣st st ♣r♦ stá ♥ s♥♦♥s ❲♥♥r

• ♥ ♠ét♦♦ rs rts s s♣♦♥ ♣r♠r♠♥t q ♥ s r♥ís ♣♥t♦ r ♠t♦♥♥♦ ♣ró♦ H ♣ ♣r♦①♠rs ♣♦r ♠t♦♥♥♦ Hat ♥át♦♠♦ ♦③♦ ♥ ♦ ♣♥t♦ r♠é♥ s s♣♦♥ q s♣tr♦ Hat s ♦③♦ st♦ s q s Hatϕn = εnϕn s r q t♦♥ó♥ ϕn t♦♠ ♦rs s♣rs st♥s ♠②♦rs q ♦♥st♥t r

s♦ ♥♦s ♣r♠t srr ♠t♦♥♥♦ rst♥♦ ♦♠♦

H = Hat +∆U(r),

♦♥ ∆U(r)s ♥ ♣♦t♥ q ♦♥t♥ s ♦rr♦♥s ♥srs ♣r q ♣♦t♥ t♦t t♥ ♣r♦ r

sr♠♦s ♥ s♦ó♥ ♦ ó♥ rö♥r ♣r H ♦r♠

ψk(r) =∑

R

eik·Rφ(r−R)

s♥♦ φ(r) ♥ ♥ó♥ ♣r ♥ó♥ ϕn(r) ② q ♥② ♦s st♥t♦s ♥s n q t♥ ♠t♦♥♥♦ Hat sí

φ(r) =∑

n

bnϕn(r)

ψk(r) =∑

R

eikR∑

n

bnϕn(r)

rs♦r s

Hψk(r) =(Hat +∆U(r)

)[∑

R

eik·R∑

n

bnϕn(r−R)

]= ε(k)

(∑

R

eik·R∑

n

bnϕn(r−R)

)

= ε(k)ψk(r)

t♣♥♦ ♣♦r ϕ∗m(r) ♥tr♥♦

(ε(k)− εm

)∫dr ϕ∗m(r)ψk(r) =

∫dr ϕ∗m(r)∆U(r)ψk(r)

óts q s ♣s s♥ str ♠② ♦③s r♦r ó♥ s q stá♥ s♦s ♦♠♣r ♣♦r ♠♣♦ ♥ s♦ s ♥♦♥s ♦♥ r♦♥♦s ♦s ♥s q t♥♥ s ♥♦♥s ♦♥ s ②

s r♦ q s ϕn(r) s ♥ t♦♦r Hat t♠é♥ ♦ srá ♠t♦♥♥♦ H s s ♠♣ q ∆U(r) = 0 ♥♦ϕn(r) 6= 0 st ♦♥ó♥ s ♣♦♦ ís ♣r♦ s ♣r♦①♠ ♥ q♦s s♦s ♥ ♦s q ♠♦♦ rs rts s ♣

❱ríqs q s ♥ ♥ó♥ ♦

ísst♦ó♦

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♦♥ s t③♦ q⟨ϕ(r)|Hat|ψk(r)

⟩= εm

⟨ϕm|ψ

⟩

♦r s♠♦s ①♣rsó♥ ψk(r) ② r♦r♠♦s q 〈ϕm|ϕn〉 = δmn ♦t♥♠♦s

(ε(k)− εm

)bm =−

(ε(k)− εm

)∑

n

bn∑

R 6=0

(∫dr ϕ∗m(r)ϕn(r−R)

)eik·R

+∑

n

bn

(∫dr ϕ∗m(r)∆U(r)ϕn(r)

)

+∑

n

bn

∑

R 6=0

∫dr ϕ∗m(r)∆U(r)ϕn(r−R)

eik·R.

st s ♥ ó♥ r q♥t s♣st♦s ♥ ∆U(r) ♦ ② s♠s ♥♥ts∑n ②

∑R

♦r ♥ ás s♦♥ s bn q ♥ str ♥ srr♦♦ ♣r q s trt rs♦r ó♥ r ♥tr♦ s ♠t♦♥s ♠♦♦

♥ t♥♠♦s ♥trs s♦♣♠♥t♦∫dr ϕ∗m(r)ϕn(r−R) q s♣♦♥r♠♦s ♠♦

♠♥♦rs q ♥ ② q ♠♦♦ ♣rs♣♦♥ ♥♦♥s ♦♥ tó♠s ♠② ♦③sP♦r ♦ ♣r♠r♦ ② trr s♠♥♦ ♠♠r♦ r♦ s ♣♥ ♦♥srr ♣qñ♦s

♠é♥ s♥♦ s♠♥♦ s ♣qñ♦ ② q ♠♦s s♣st♦ q ∆U(r) ♦ s ♥♦ϕn(r) 6= 0

s♣r♠♦s t♦♦s s♦s tér♠♥♦s ♦♠♦ ♥ ♣r♠r ♣r♦①♠ó♥ ♥♦s q q

(ε(k)− εm

)bm ≃ 0 ∀m ∈ espectro de Hat

st♦ qr r q s

ε(k) ≃ εm =⇒ en general bm 6= 0, m ∈ espectro de Hat

ε(k) ≇ εm =⇒ bm ≃ 0.

② ♣♦r ♦ t♥t♦ ♥♦ ♥rí ε(k) tró♥ ♦ stá ♣ró①♠ εn ♥ ♦rt tó♠♦só♦ ♦♥t bn s ♣r

st♦ ♥♦s ♣♥sr q ♣r r ♦rs ♠♥♦s tt♦s st ssó♥ ♥ s♠ sr ♦♥♥♥t ♦♥srr t♦s qs ♥♦♥s ♦♥ ϕn(r) q t♥♥ ♥ ♥rí tó♠r♥ ε(k) q st♠♦s ♦♥sr♥♦ ② ♣♦r t♥t♦ s εm sí s st♠♦s ♥trs♦s ♥ ♥ ♥rís ♣r♦♥♥t ♦s tr♦♥s s ♦s át♦♠♦s ♥ rst ♥t♦♥s ♣♦♠♦s ♦♥srr s♦♠♥t ♥ ó♥ ♦♥ bs 6= 0 Pr s ♥s ♣ rá trs ♦♥s ♦♥ bpx 6= 0 bpy 6= 0 ② bpz 6= 0 Pr ♦s ♥s rí ♥ ♣r♥♣♦ ♥♦ ♦♥s ♣r♦ ♥ ♥r s trt♠♥t♦ s ♠ás♦♠♣♦ ♣♦rq s sr t ①st♥ ♥s s ♥rí s♠r ♣♦r ♦ q r♦ ♦♥ ssó♥ q ♠♦s t♥r s ♥sr♦ r③r ♦ ♠③r t♦♦s ♦s

♥ st ♣♥t♦ ② ♦♠♦ rr♥ ♦♥♥ r♦rr ó♠♦ rí♥ ♦s ♥s ♥rét♦s ♦sát♦♠♦s sú♥ rí ♦r Z ♥ú♠r♦ tó♠♦ st♦ ♥♦s ♣r♠trá str qé ♥s tó♠♦s♥ ♥trr ♥ á♦ ♥s ♥ ♠tr tr♠♥♦

ísst♦ó♦

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❮ Ó ♥s tr♦♥s tr♦♥s ♥ ♥ ♣♦t♥ ♣ró♦ é ② ♠ét♦♦ rs rts

♥ ♥r ♥ ♦r♠ s♠♣r ♠♦ ♦s á♦s ♠ét♦♦ s s♣♦♥r q s ♥♦♥s ♦♥ tó♠s t♥♥ ♥ ♥ s♥t♠♥t ♣qñ♦ ♣r q s♦♦ s♥ st♥t♦s r♦♦s tér♠♥♦s q ♣r♦♥♥ ♦s ♥♦s ♠ás ♣ró①♠♦s st♦ s ♦♥ R ♠ás ♣qñ♦ st♣r♦①♠ó♥ s s ♠r ♣r♦①♠ó♥ ♣r♠r♦s ♥♦s

♠♣♦ ♥ s ♦r♠ ♣♦r ♥ ú♥♦ ♦rt tó♠♦ ó♥ ♦♥ φ(r) = bsϕs(r) ♣♦♠♦s ♦t♥r ♥ ró♥ ♣r ♥rí

ε(k) = εs −β +

∑R γ(R)eik•R

1 +∑

R α(R)eik•R

s♥♦

β =−∫dr∆U(r)|φ(r)|2

α(R) =

∫dr φ∗(r)φ(r−R) (solapamiento)

γ(R) =−∫dr φ∗(r)∆U(r)φ(r−R)

♦♠♦ s s♣♦♥ q α(R) s ♣qñ♦ r♥t ♥♦ sú♥ s ♦♥♦♥s ♣ó♥ ♠ét♦♦ ♥♦s q

ε(k) = εs − β −∑

R

γ(R)eik·R ≃ εs − β −∑

MC

γ(R)eik·R,

♦♥ ♥ ♥♦s ♠ás r♥♦s♠♦s q φ(r) s ♥ ♦rt s ♣♦r ♦ q s ♥ ♥ó♥ r q s♦♠♥t ♣♥

|r| ♠ás s♠trí ♥rsó♥ r rs ∆U(−r) = ∆U(r) ② st♦ rst q γ(−R) = γ(R) ♦♦ ♦ ♥♦s ♣r♠t srr

ε(k) ≃ εs − β −∑

MC

γ(R) cos(k ·R).

ísst♦ó♦

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• ♣q♠♦s ♦r st rst♦ s♦ ♥ r q t♥ ♦ ♥♦s ♠ás♣ró①♠♦s st♦s ♥ és r

R =a

2(±1,±1, 0) ;

a

2(±1, 0,±1) ;

a

2(0,±1,±1) .

sr♠♦s k = (kx, ky, kz) t♥r♠♦s k ·R = a2 (±ki±kj), (i, j) = (x.y) ; (y, z) ; (z, x)

♦♠♦ φ(r) ♣♥ s♦♦ |r| ② ∆(U(x, y, z) t♥ t♦ s♠trí r γ(R) =γ, ∀R ∈ ②

ε(k) ≃ εs − β − 4γ

[cos

kxa

2cos

kya

2+ cos

kya

2cos

kza

2+ cos

kza

2cos

kxa

2

]

s♥♦ γ = −∫dr φ∗(x, y, z)∆U(x, y, x)φ(x− a

2 , y − a2 , z)

P♦r ♦♥s♥t ♠♦s ♦t♥♦ ♦s rst♦s ♥rs ♥r ♥ s ♣r♦♣♦r♦♥ γ ♦ s 12γ P♦r t♥t♦ s ♥s rs rts s♦♥ ♥s strs ② ♥t♦ ♠♥♦r s s♦♣♠♥t♦ ♠ás strs ♥ Pr ka→ 0 s t♥ q

ε(k) ≃ εs − β − 12γ + γk2a2,

♦r♠ q ♦r♠ ♥ ♥rí s ♥♣♥♥t ró♥ k ② ss♣rs q♥réts s♦♥ sérs st♦ r s♣rr ② q s k → 0 tró♥♥♦ s♥t r rst♥ ② ró♥ s♣rsó♥ sr rát ♥ k

5.4.3 ♦♠♥tr♦s r ♠t♦♦ rs rts

❯♥ rtríst ♥r ♠ét♦♦ s ró♥ ♥tr ♥r s ♥s ② s ♥trs s♦♣♠♥t♦

γij(R) = −∫dr φ∗i (r)∆U(r)φj(r−R),

♦r♠ q s ♦s ♣rá♠tr♦s γij(R) s♦♥ ♣qñ♦s t♠é♥ ♦ s♦♥ s ♥rs s ♥s♥ ♥r ♠♥tr ♥rí ♥ ♥ tó♠♦ ♠♥t s ①t♥só♥ s♣ ② ♣♦r t♥t♦s♠♥② ♠ét♦♦ ♣r ♥s ♥rí ♠ás st♦ s ♣rtr♠♥trt♦ ♥ ♠ts

♥ó♥ ♦♥ rs rts ψk(r) =∑

R eikR

∑n bnϕn(r) stá ♦♥str

♦♥ ♦rts tó♠♦s ϕn(r) ♦③♦s ♣sr ♦ ♥ tró♥ ♦ t♥ ♠s♠♣r♦ str ♥ qr rst ② q s ♥ó♥ ♦♥ ♠ s♦♠♥t

ísst♦ó♦

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❮ Ó ♥s tr♦♥s tr♦♥s ♥ ♥ ♣♦t♥ ♣ró♦ é ② ♠ét♦♦ rs rts

♥ ♥ s eik·R ♥♦ s s♣③ ♥ st♥ R❯♥ ♥ó♥ ♠ás ♣r r q ♦s ♦rts rs rts t♥♥ ♥ rátr t♥r♥t sq s♠♦s q v(k) = (1/~) ∂ε/∂k ② q s ①st s♦♣♠♥t♦ ♥ ε(k) ♥♦ s ♦♥st♥t♥ P❩ st♦ ♥♦s q s ①st s♦♣♠♥t♦ ♥♦♥s ♦♥ ♦s tr♦♥sést♦s s♦♥ ♣s ♠♦rs ♣♦r t♦♦ rst s♠♥r s♦♣♠♥t♦ ♦ s♠♥♦r ♣r♦ ♥♦ s ♠♥r ♠♦♠♥t♦

só♦ ♥♦ s ♥ rst rs ♣r♦①♠ó♥ rs rts s ♠ás í r ❯♥ ♠♥r trtr ♥t♦♥s ♣r♦♠ s ♦♥srr s ♦♠♦ ♥ ♠♦é ②t③r ♦s ♦rts ♠♦rs ♦rrs♣♦♥♥ts ♣r ♦♥strr ♥ó♥ ♦♥ ♦

♥ ♦s ♠♥t♦s ♠ás ♣s♦s ♦♣♠♥t♦ s♣í♥órt és s♥tr ísá♥t s ♠♣♦rt♥t ♣r tr♠♥r ♦s ♥s tó♠♦s ② ♥rs ♥ trt♠♥t♦ rs rts sí ♣♦rí♠♦s ♥tr♦r ♥ ∆U(r) s ♥tró♥ ②t③r ♦♠♥♦♥s ♥s ♦rts ② ♥♦♥s s♣í♥ ♦s ♦♥ts sts♦♠♥♦♥s ♥s ♥ ♣♥r t♠é♥ k

rr s q ♣r♦①♠ó♥ tr♦♥s ♥♣♥♥ts ♥♦ r ♥s♣r♠♥t ♥s q ♣r♦♥♥ ♥s tó♠♦s ♥ ♦③♦s ②s ♥trs s♦♣♠♥t♦ s♦♥ ♣qñs ♣ ♦s ♥s s ♣s ② ♦s ♠tsst ♦ ♣r ♠♣r ♣s♦ ♦♥t♥♦ sr♦ ♣♦r ♠ét♦♦ rs rts s ♥st♦ ♠tá♦ tó♠♦ ♠ q ♠♥t♠♦s st♥ ♥trtó♠ ❨ ♥ t♦ ♥á♦ ♠ás ♦♠♣t♦ q ♠ás á ♣r♦①♠ó♥ tr♦♥s ♥♣♥♥ts ♣rq ♣r rt♦ ♦r ♣rá♠tr♦ r ♦♥t st r♣t♠♥t r♦ ② ♠tr s s♥t st tr♥só♥ s ♠ tr♥só♥ ♦tt ② s♦ ♦sr♥ rt♦s ó①♦s ♠ts tr♥só♥ q s ♥ sút♠♥t s♥ts ♣♦r ♥♠ rt t♠♣rtr

5.5 ❯ ❲

❱♠♦s ♠♦strr ♦r q s ♠ét♦♦ rs rts s á Pr ♦ ② qr ♥♦tr q ♥ ♥ó♥ ♦♥ ♦ ψnk(r) ♣r ♥ r ♦ t♥ ♣r♦ rrí♣r♦ P♦r ♦♥s♥t

ψnk(r) =∑

R

fn(R, r)eiR·k,

♦♥ s ♥♦♥s ❲♥♥r fn s♦♥ ♦s ♦♥ts ♦rr srr♦♦ st♦ s

fn(R, r) =1

ΩPZB

∫

PZBdke−iR·kψnk(r).

♦r ♥ fn(R, r) ♣r♠♥ ♥tr s s s♣③♥ ♥ t♥t♦ r ♦♠♦ R ♥ st♥R0 ∈ RD ② ♣♦r t♥t♦

fn(R, r) = fn(r−R)

sts ♥♦♥s s♦♦ ♣♥♥ st♥ ①st♥t ♥tr ♣♥t♦ r ♦♥ ♠♦s ♥ó♥② ♥ ♣♥t♦ R r rt

♦♠♦ s ♥♦♥s ❲♥♥r fn(r−R) s♦♥ ♦rt♦♦♥s ② qr ♥ó♥ ♦ ♣①♣rsrs ♦♠♦ ♦♠♥ó♥ ♥ s ♦r♠♥ ♥ ♦♥♥t♦ ♦rt♦♦♥ ♦♠♣t♦ ♥♦ n ∈ N ②R ∈ RB

s ❲ st♥t♦ n ② R s♦♥ ♦rt♦♦♥s ♥tr sí

ísst♦ó♦

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❯ ❲

s♠t ♦r♠ s ❲ ② s ♥♦♥s ♦♥ ♠ét♦♦ ♦♥r♥ q s ❲ s♥ t♠é♥ ♥♦♥s ♦③s ♥ ♠r♦ s ♥ ♥♦ s ♥ ♥ s♥t♠♥t str ♥♦ ♥t♥t♠♦s srr tró♥ ♦ ♦♥ ❲ ésts ♥♦ t♥♥ ♣♥s♣r♦ ♦♥ s ♥♦♥s ♦♥ tró♥s ♦s át♦♠♦s s♦s

s ❲ ♣♥ ♦♥sttr ♣♦r ♦ t♥t♦ ♥ s út ♥♦ st♠♦s ♥trs♦s ♥ srr♣r♦♣s tró♥s ♥ s q t♥♥ ♠♣♦rt♥ ♦③ó♥ ♦s tr♦♥s ② st♥ t③ó♥ ♠ét♦♦ ♥ tr♠♥s r♥st♥s

• ♦♠♦ ♠♣♦ ♠♦s ♣r sts ❲ st♦ ♥ tró♥ ♥ ♥ r ♣rtr♣♦r ♥ ♣♦t♥ H1 q rí ♥t♠♥t rs♦r s

Hψnk(r) = (H0 + H1)ψnk(r) = εnψnk(r),

♦♥ H0 s ♠t♦♥♥♦ rst ♦ q s ❲ ♦♥stt②♥ ♥ s ♣r srr♦rs rst

ψnk(r) =∑

R

fn(r−R)eik·R,

♣r ♦ srr q ψnk(r) sst♠ ♣rtr♦ ♣ srrs ♦♠♦

ψ(1)nk (r) =

∑

R

fn(r−R)ηn(R),

♦♥ ηn s ♥ ♥ó♥ ♠♦♦r q ♥♦ ♦♥♦♠♦s ♥q s♣♦♥r♠♦s q rí ♥t♠♥t♦♥ R

stt②♥♦ ♥ ♥tr♦r ♠t♣♥♦ ♣♦r f∗n(r−R) ♥tr♥♦

∑

R

∫dr[f∗n(r−R)

(H0 + H1 − εn

)fn(r−R)ηn(R)

]= 0.

H1 rí ♥t♠♥t ♥ ♥

∑

R

∫dr[f∗n(r−R)H0fn(r−R)ηn(R)

]+ H1(R

′)ηn(R′) = εnηn(R

′).

s q ró♥ s♣rsó♥

ε0(k) =

∫dr ψ0

nk(r)H0(r)ψ0nk(r)

s ♣ró ♥ ② s ♣ srr♦r ♥ sr ♦rr

ε0(k) =∑

R′′

A(R′′)e−ik·R′′

con A(R) =∑

k

ε0(k)eik·R.

st♦ q ♣r♠r tér♠♥♦ ♣ rrs

∑

k

∑

R

eik·(R′−R)ηn(R)ε0(k) =

∑

R

ηn(R)A(R′ −R)

stt②♥♦ ♥ s ♦t♥

ψnk(r) =∑

R

fn(r−R, )eiR•k,

s♠r ♠♥r ó♠♦ ♦t♥♠♦s ♥ó♥ ♦ ♣rtr ♦rts tó♠♦s ♥ ♠ét♦♦ rs rts P sr ♣♦r ♠♣♦ ♥ ♠♣♦ ①tr♥♦ q t♥ ♥ ♦♥t ♦♥ λ ♠♦ ♠ás r♥ q ♣rá♠tr♦ r

ísst♦ó♦

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❮ Ó ♥s tr♦♥s tr♦♥s ♥ ♥ ♣♦t♥ ♣ró♦ é ② ♠ét♦♦ rs rts

② s ♦♥rt ♥ ∑

R

A(R)ηn(R′ −R) + H1(R

′)ηn(R′) = εnηn(R

′)

ró♥ ηn ♦♥ s r♠♥t♦ s ♥t ② s ♣ srr♦r

ηn(R′ −R) ≃ ηn(R

′)−∇ηn(R′) · (R′ −R) ≃[e−(R

′−R)·∇]ηn(R

′)

♦♥ ♦ q q [∑

R

A(R)e−(R′−R)·∇ + H1(R

′)

]ηn(R

′) = εnηn(R′)

② ♠♥♦ rs ♦♥t♥s[ε0(i∇) + H1(r)

]ηn(r) = εnηn(r).

st ó♥ s s♠r ♠♥♦ k ♣♦r i∇ ② ♥ ♣r ♦♠♦ ♣♦t♥ ♣♦t♥♣rtr♦r H1

♦♦ ♦ ♥tr♦r s♥ q ♣r♦♠ q♦ rst♦ ♠♥t s♦ó♥ ψ(1)nk (r) ♥ tér♠♥♦s s♦♦♥s ♥♦ ♣rtrs ♠♦s ♣♦r s ♥♦♥s ηn(R) q ♥♥

tr♠♥s ♣♦r ♥♦trs q ♦r s trt♥ s ♥♦♥ts ηn(r) ♦♠♦ ♥♦♥s ♦♥t♥s r ② ♥♦ s♦♠♥tstá♥ ♥s ♣r ♦s ♣♥t♦s ② q ♦♠♦ s ♥ s ♥tró♥ s ♠② s♠r ♥tró♥ ó♥ r♥ ♥ó♥ ♦♥ ♥ tró♥

♦♥ s ♥♦♥s ❲♥♥r fn(r−R)

ísst♦ó♦

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❯

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❯

❯

❯

♣ít♦

6.1 PP ❯Ó εN (K)

• ♥ó♥ εn(k) s ♦♥t♥ ② r♥ ♥ P❩

srr♦♠♦s ♥ó♥ ψn(k, r) ♣r ♥ ♣♥t♦ k + g P❩ r♥♦ ♣♥t♦ k t③♥♦♦♠♦ ♥♦♥s s ♦♥♥t♦

χn(k+ g, r) ≡ eig·rψn(k, r)

♦rt♦♦♥ s χn(k+ g, r) ♣♦♠♦s srr

ψn(k+ g, r) =∑

m

Bnm(k+ g) χm(k+ g, r) = eig·r∑

m

Bnm(k+ g) ψm(k, r).

♣♦♥♠♦s q ♥ó♥ εn(k) ♥♦ s ♥r ♥ ♣♥t♦ k ♣♥♦ ♠t♦♥♥♦ ♥ó♥ ♦ ψn(k+ g, r) t♥♠♦s Hψn(k+ g, r) = εn(k+ g) ψn(k+ g, r)

P♦r t♥t♦

Hψn(k+ g, r) =

[− ~

2m∇2 + U(r)

](eig·r

∑

m

Bnm(k+ g)ψm(k, r)

)

= eig·r∑

m

Bnm

[− ~

2m∇2 + U(r) +

~2

img · ∇+

~2g2

2m

]ψm(k, r)

= eig·r∑

m

Bnm

[εm(k) +

~2

img · ∇+

~2g2

2m

]ψm(k, r)

= εn(k+ g) ψn(k+ g, r)

t♣q♠♦s ♦r ♣♦r e−ig·rψ∗m′(k, r) s ♦s í♥s s út♠s s ♥tr♦rs ♣♥út♠

∑

m

Bnm

[εm(k) +

~2

img · ∇+

~2g2

2m

]ψm(k, r)ψ

∗m′(k, r)

st ♦♥♥t♦ s ♦rt♦♦♥ ♣s∫

χ∗n′ (k+ g′)χn(k+ g′′)dr =

∫

ei(g′′−g′)·rψ∗

n′ (k, r)ψn(k, r)dr

=

∫

ei(g′′−g′)·ru∗n′ (k, r)un(k, r)dr

♦r ♥ s ♥♦♥s u t♥♥ ♣r♦ r ② s ♣♥ srr♦r ♥ sr ♦rr st♦ ♥♦s ♥ q t♠é♥ ♣r♦t♦ u∗

n′ (k, r)un(k, r) t♥ ♣r♦ r ② s ♣ srr

u∗n′ (k, r)un(k, r) =∑

G

An′nG e−iG·r.

♦∫

χ∗n′ (k+ g′)χn(k+ g′′)dr =

∑

G

An′nG

∫

ei(g‘−g′−G)·rdr = V∑

An′nG δg′′,k′+G

s♥♦ V ♦♠♥ rst♦r ♥ s G 6= 0 s t♥ q t♥t♦ g′ ♦♠♦ g′′ t♥♥ ♠ó♦ ♠♦ ♠♥♦r q G P♦r ♦ t♥t♦ s♦♠♥t s ♣♦s s♦rG = 0 ♥ s♠t♦r ② ♥♠♥t ♥♦s q

∫

χ∗n′ (k+ g′)χn(k+ g′′)dr = V An′n

0 δg′′,g′ = δg′′,g′

∫

u∗n′undr = δg′′,g′

∫

ψ∗n′ψndr = δg′′,g′δn,n′

q s ♦ q s qrí ♠♦strr

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó ♥á♠ ♦s tr♦♥s

② út♠ s sr ♦♠♦

εn(k+ g)ψn(k+ g, r)ψ∗m′(k, r)e−ig·r = εn(k+ g)

[eig·r

∑

m

Bnm(k+ g)ψm(k, r)

]ψ∗m′(k, r)e−ig·r

= εn(k+ g)∑

m

Bnm(k+ g)ψm(k, r)ψ∗m′(k, r)

♥tr♠♦s ♠s ♣rts s♦r dr ♥ t♦♦ s♣♦ ♥♦s q

εn(k+ g)Bnm′(k+ g) =

[εm(k) +

~2g2

2m

]Bnm′ +

∑

m

Bnm~

mg · pm′m

=

[εm(k) +

~2g2

2m+

~

mg · pmm

]Bnm′ +

∑

m 6=m′

Bnm~

mg · pm′m

♦(εn(k+ g)− εm′(k)

)Bnm′ =

~2g2

2mBnm′ + summBnm

~

mg · pm′m.

st ó♥ s ♦r♠♠♥t é♥t ♦t♥ ♥ t♦rí ♣rtr♦♥s rr ♦s♦♥t♥♦s s♥tr ís á♥t s ①♣rs♠♦s ♦♠♦

(εn(k+ g)− εm′(k)

)Bnm′ =

∑

m

Vm′mBnm,

s♥♦

Vm′m′ =~2g2

2m+

~

mg · pm′m′

Vm′m =~

mg · pm′m m′ = m.

P♦r ♦♥s♥t s♦ó♥ ó♥ ♣♦♠♦s ♦t♥r ♦r♠♠♥t ♠♥t t♦rí ♣rtr♦♥s t♥♥♦ ♥ ♥t tér♠♥♦ ♣rtr♦r Vnm sr♠♦s ♣s rt♠♥t q

εn(k+ g) = ε(k) +~

mg · pnn +

~2g2

2m+

~2

m2

∑

m 6=n

|g · pnm|2εn(k)− εm(k)

s♥♦

pnm = −i~

∫ψ∗m(k, r)∇ψn(k, r)dτ

♠♥t♦ (n,m) ♠tr③ ♦♣r♦r ♠♦♠♥t♦

P♦r ♦ t♥t♦ εn(k) s ♦♥t♥ s s ♥ ♥ó♥ ♥♦ ♥r ♥ k s ♥t s s g → 0♥

• ♥s♦r ♠s t

srr♦♠♦s εn(k+ g) ♣r |g| s♥t♠♥t ♣qñ♦ s t♥ q

εn(k+ g) = εn(k) + g · ∇kεn +1

2

∑

αβ

∂2εn∂kα∂kβ

gαgβ .

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

PP ❯Ó εN (K)

sr♠♦s ♦s tér♠♥♦s ♦r♥ ♥ g ♥tr♦r srr♦♦ ♣rtr♦♥s ♣rεn(k+ g) ♦♠♦

~2g2

2m+

~2

m2

∑

m 6=n

|g · pnm|2εn(k)− εm(k)

=~2

2m

∑

αβ

gαgβδαβ +~2

m2

∑

m 6=n

∑αβ gαgβp

αnmp

βmn

εn(k)− εm(k).

P♦r ♦tr ♣rt tér♠♥♦ s♥♦ ♦r♥ s

1

2

∑

αβ

∂2εn∂kα∂kβ

gαgβ .

♥♦ ♦r ♠♦s tér♠♥♦s s ♦t♥ q

1

2

∂2εn∂kα∂kβ

=~2

2mδαβ +

~2

m2

∑

m 6=n

pαnmpβmn

εn(k)− εm(k).

Pr♦ p s ♥ ♦♣r♦r r♠ít♦ st♦ s ♠♣ q pαnm = pαmn t ② ♥♦s q

1

~2∂2εn∂kα∂kβ

=1

mδαβ +

1

m2

∑

m 6=n

pαnmpβmn + pβnmpαmn

εn(k)− εm(k).

♦♠♦ ♣r ♥ tró♥ r ♥♦ ♣rrí s♥♦ s♠♥♦ ♥ ♠♠r♦ r ♣♦♠♦s ♥tr♣rtr ó♥ ♥♦ q tró♥ ♦ ♥♦r♣♦r ♥ ss♣r♦♣s st♦ s ♥ s ♥ó♥ ♦♥ ♥tró♥ ♦♥ r rst♥ ♠♠r♦ r♦ st ó♥ r ♥♦♠r t♥s♦r ♠s t ② q sstt② ♠s sr tró♥ r

• ♥ t s♥t s ♣rs♥t ♥ r♦ ♦♠♣rt♦ ♥tr ♦s tr♦♥s rs ② ♦s tr♦♥s♦

♦♠♣rt ♥tr tr♦♥s rs ② tr♦♥s ♦

tr♦♥s rs tr♦♥s ♦

♥ó♥ ♦♥ eik·r ψn(k, r) = eik·run(k, r)

ú♠r♦s á♥t♦ss♥ s♣í♥

k ~k♠♦♠♥t♦ s♥♦ k ♥t♦r q r s ♦r♥♦♥r♠♥

k, n ~k♠♦♠♥t♦ rst♥♦ ní♥ ♥

t♦♦r H ε(k) = ~2k2/2m εn(k) ♦♥ εn(k+G) = εn(k)

❱♦ v = ~k/m = (1/~)∂ε/∂k vn(k) = (1/~)∂εn(k)/∂k

(1/~2)(∂2εn/∂kα∂kβ

) 1

mδαβ

1

mδαβ +

1

m2

∑m 6=n

pαnmpβmn + pβnmpαmn

εn(k)− εm(k)

qí s s q

|g · pnm|2 = (g · pnm) (g · pnm)∗ =

(

∑

α

gαpαnm

)(

∑

α

gαpβmn

)

.

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó ♥á♠ ♦s tr♦♥s

6.2 Ó P

♥ sts ♦♥♦♥s qr♠♦s str ♥ó♠♥♦s tr♥s♣♦rt s♦♦s ♥á♠ ♦str♦♥s ♦ ♦♥♥rí s♥ ♠r♦ rs♣♦♥r ♥ts ♥ ♥ ♣r st♦♥s ♣rs

á s ♥tr③ s ♦s♦♥sr s♣s♦ q ♦s tr♦♥s ♦s♦♥♥ ♦♥ ♦s ♦♥s ♦s ② ♣s♦s ♣r♦ st s♣♦só♥ ♥♦stá r♦ ♦♥ ♦s r♥s r♦rr♦s ♠♦s ♠♦s ♥ ♠tsP♦r ♦tr ♣rt t♦rí ♦ ①② s ♦s♦♥s ② q s ♥♦♥s ♦♥s ♦s♦♥ s♦♦♥s st♦♥rs ó♥ rö♥r ó♥ q ♥② ♣♦t♥♣ró♦ ♦s ♦♥s ♥ ♥ ♦ ψnk t♥ ♥ ♦ ♠ ♥♦ ♥ ♥♦ ♣♦♠♦sr♠r q s ♦s♦♥s ♦♥ ♦s ♦♥s s♠♥②♥ s ♦ ② q t♦s s ♥tr♦♥s tró♥ ♦♥ stró♥ ♣ró ♦s ♦♥s s ♥ t♥♦ ♥ ♥t ♥ ó♥ rö♥r q ψnk s s♦ó♥ ♦r♠ q ♦♥t ♥ rst ♣rt♦ sr ♥♥t r③ó♥ q ♦s ♠ts t♥♥ ♥ rsst♥ étr str ♥ q ①st♥ ♠♣r③s♥ts ② ♦trs ♠♣r♦♥s q ♠t♥ ♦♥ó♥ t♠♣rtrs ♠② s sts s♦♥s ss rsst♥ ♠ás t♠♣rtrs ♥ts ①st♥ r♦♥s r q♠♦♥ ♣r♦ ♣rt ♠s♠ ② ♦♠♦ s st♦rs♦♥s rs♣t♦ ♥ r ♣♥♥ t♠♣rtr sí ♦ rsst♥P♦r ♦♥s♥t s♥ sr ó♠♦ s ♠♥s♠♦ ♦só♥ ♥t♥tr♠♦s sr ♦♥s♥s ♦ q t ♠♥s♠♦ ①st ② ♦ srr♠♦s ♠♥t ♥ ♣r♦①♠ó♥ t♠♣♦ ró♥ τ

ó♠♦ s ♠♥ ♦s tr♦♥s ♦ ♥tr ♦s♦♥s❱♠♦s str ♣r rs♣♦♥r st ♣r♥t ♥ ♠♦♦ s♠ás♦ ♥á♠ tr♦♥s stó♥ st ♠♦♦ s r♠♥t í ♣r♦ s ♣ r♠♥tr ♣♦str♦r q ♠♦♦ rs ♦♥ é①t♦ ♠♦s ♦s ♠str♦s ♥♦♠ís t♦rí tr♦♥s rs♠♦s q ♦ ♠ ♥ tró♥ ♥ ♥ ♥ ♦ ψnk s

vn(k) =1

~

∂εn(k)

∂k.

♦♥str②♠♦s ♦r ♥ ♣qt ♦♥ ♥s ♦

ψn(r, t) =∑

k′

g(k′)ψnk′(r)eiεn(kt~

♦♥ g(k′) ≃ 0 s |k′ − k| > ∆k ② s♥♦ ∆k ♣qñ♦ ♥ ♦♠♣ró♥ ♦♥ t♠ñ♦ ❩P♦r t♥t♦ εn(k) rrá ♣♦♦ ♥ t♦♦ r♥♦ s♠t♦r♦ st ♣qt ♦♥s srá ♥♥ sr♣ó♥ ♥ tró♥ s ♥♦ s ♥sr♦ s♣r s ♣♦só♥ ♥ ♠ás t q ①t♥só♥ ♣qt ♦♥s ♦♠♦ ∆k s ♣qñ♦ ♥ ♦♠♣ró♥ ♦♥ s ♠♥s♦♥s ❩ ♥t♦♥s ♦ q ①t♥só♥ ♣qt ♦♥s ♥ s♣♦ s ♦r♥ ∆R ≃ 1/∆k rst sr ♠♦ ♠②♦r q ♣rá♠tr♦ r a ór♠ ♦ ♣♦r s ♦r♠♠♥t ♠s♠ q ♦ r♣♦ ♥ ♣qt ♦♥s

♠♦♦ s♠ás♦ trtr ♥t♦♥s ♦♥ ♣qts ♦♥s ② ①t♥só♥ s ♠♦ ♠②♦rq a s♦♠t♦s ♠♣♦s ①tr♥♦s ②s ♦♥ts ♦♥ s♦♥ s ③ ♠♦ ♠②♦rs q ①t♥só♥ ∆R ♦s ♣qts ♣r♦♠ s q s♥ ♠r♦ ♣♦t♥ ♣ró♦ rrí ♥ st♥s ♣qñs ♦♠♣rs ♦♥ ①t♥só♥ ∆R ② ♣♦r ♦ t♥t♦ ♥♦ ♣♦♠♦s trtr♦

P r♣sr ♦ q s trt♦ s♦r st♦ ♥ s♥tr r♠♦♥á♠ r s♥♦ ♣qt ♦♥ s s♥trs ís á♥t

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

ás♠♥t ♠♦♦ s♠ás♦ trtr ♦s ♠♣♦s ①tr♥♦s ♠♥r ás ♣r♦ ♥♦ sí ♣♦t♥ ♦s ♦♥s ♣ró♥ ♠♦♠♥t♦ t♦r ♣♦só♥ r ② t♦r ♦♥ k tró♥ s ♥ ♠♦♦ s♠ás♦ ♥ s ú♥♠♥t strtr ♥s εn(k) ♠t

P♦r t♥t♦ s ♣ t③r ♠♦♦ ♣r ♦s ♦ss r ♣r♦♣s tr♥s♣♦rt ♣rtr strtr ♥s ♣r♠♥t ♥ ♠t r s rtrísts strtr ♥s ♥ ♠t rsts s ♣r♦♣s trs♣♦rt ♦srs

6.3

• ❯♥ ③ q s ♥ rs♣♦♥♦ s ♣r♥ts ♥tr♦rs ♣r♦♣♦♥♠♦s ♠♦♦ s♠ás♦ ♣r ♥á♠ ♦s tr♦♥s ♦ s♦ ♥ ♦s s♥ts ♣♥t♦s

í♥ ♥ n s ♥ ♦♥st♥t ♠♦♠♥t♦ ♦r♠ q ♠♦♦ s♠ás♦ ♥♦r ♣♦s tr♥s♦♥s ♥tr♥s s ♦s ♠♣♦s ①tr♥♦s q trt♠♦s ♦♥ st♠♦♦♦t ♥ qr♦ s ♥s ♦♥ ♥rís ♠② s♣r♦rs εF + kBT s ♥s♦♣s st ♠♦♦ s♦♦ ② q ♦♥srr ♣♦rt♦rs ♥ qs ♥s ♥rí② ♥rí s ♠♥♦r q εn(k) 6 εF + mkBT ♦♥ ♥ ♦r ♣qñ♦ m ♠ás ♦♠♦s ♠♦strrá q s ♥s ♥s s♦♥ ♥rts ♠♣♦s ①tr♥♦s s♦♠♥t s♦♥ ♥srs ♥♥ú♠r♦ ♣qñ♦ ♥s ♣r srr s ♣r♦♣s tr♥s♣♦rt ♥ só♦♦t ♥ st ♠♦♦ ♦s ♠♣♦s ①tr♥♦s ♥♦ ♣r♦♥ tr♥s♦♥s ♥tr♥s ♣♦r ♦ q ♥♦sr ♣r str ♥ó♠♥♦s ♦♠♦ s♦ró♥ ó♣t ♦♥ sí s ♣rs♥t♥ sts tr♥s♦♥s♥ s s♦ ♦s ♠♣♦s ①tr♥♦s ♥♦ s♦♥ ♦♥t ♦♥ r

s ♦♥s ♠♦♠♥t♦ ♣r r ② k s♦♥ ♥ s♦ ♥

d

dtr = vn(k) =

1

~

∂εn(k)

∂k

~d

dtk = −e

[E(r, t) +

1

cvn(k)×B(r, t)

]= Fext(r)

óts q ó♥ s st♥t s dp/dt = Ftotal ② q ♣♦r ♥ ♣rt Ftotal =Fext − ∇U(r) 6= Fext ② ~k ♥♦ s ♠♦♠♥t♦ ♥ ♣r♦ s r③♦♥ q sí s ② q ♥tró♥ ♦ ♥② ♥ s ♥ó♥ ♦♥ t♦ r rst♥ ♣♦r ♦ q s♣rrq ró♥ k s ú♥♠♥t ♠♣♦ ①tr♥♦ ② ♥♦ t♦t stó♥ ♠♣♦rt♥t s q st ó♥ s rr♦s ♥♦ s rr ♣s ♥ ♦r ♠♦ k ♣r♦♠♦ ♥ ♥ ♣qt ♦♥s s♥♦ q s rr qr st♦ ♦♥ ♥ ♦r k r♦ q ♣r♠♥ ♦♠♦ t st♦ ♦ ó♥ ♠♣♦ ①tr♥♦ ♦♥ ♥ ♦rtr♠♥♦ ② ♣♥♥t t♠♣♦ k r♦

♠♦stró♥ ♥♠♥s♦♥ r♦♠r P ♣♦♥♠♦s ♥ rst ♥♠♥s♦♥ ♥ |nk〉 ② TR ♦♣r♦r trsó♥♦rrs♣♦♥♥t ♥ t♦r R ∈

TR|nk〉 = eikR|nk〉.

♠♠♦s H0 ♠t♦♥♥♦ ♦ sst♠ ② Fext ♥ r③ ①tr♥♥♦r♠ q tú s♦r ♦s tr♦♥s rst

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó ♥á♠ ♦s tr♦♥s

♠♣ q[H0, TR

]= 0

[H0 − Fextx, TR

]= FextRTR.

r t♠♣♦r ♦r ♠♦ TR s

d

dt〈TR〉 =

i

~

⟨[H, TR

]⟩=

i

~FextR〈TR〉 =⇒ 〈TR〉 = T0 e

ik(t)R

s♠♣r q

~d

dt

(k(t)

)= Fext.

P♦r t♥t♦ ♦r ♠♦ 〈TR〉 t♥ ♥ ①♣rsó♥ ♣r t♦♦r TRsstt②♥♦ ♦r k(t) k ② ♣r♥♦ T0 st ♦r ♠♦ 〈TR〉 ♥♦ s s♦♠♥t st♦ |nk〉 s♥♦ t♠é♥ qr st♦ tró♥♦ ♣♦s q s♠♣rs ♣ srr ♦♠♦ ♥ ♦♠♥ó♥ ♥ |nk〉

|ψ〉 =∑

nk

ank|nk〉.

P♦r t♥t♦ ♦r k(t) ♣r ♥ ♥st♥t ♦ ♣ ♥t♥rs ♦♠♦ ♣r♦♠♦ ♦s k q ♣r♥ ♥ |ψ〉 ♥ ♦ ♥st♥t Pr♦ eikR ∈ C ② |eikR| = 1 ♣♦r ♦ q ♦r ♠♦ strá ♥tr♦ ír♦ ♦♠♣♦ r♦ ♥ ② s♦♠♥t strás♦r r♥r♥ ♥ s t♦♦s ♦s k q ♣r♥ ♥ s♠ q ♥ |ψ〉t♥♥ ♠s♠♦ ♦r r♦

♥ rs♠♥ s ♣rt♠♦s ♥ st♦ ♦♥ ♥ t♦r k ♥ ♥♦ st♦ s ♥st♦ |nk〉 ♦r k ♦♦♥ ♥ t♠♣♦ ♦♠♦ ♥ st♦ ♦♥ ♥ t♦r k(t)♥ ♥♦ q r q

~dk(t)

dt= Fext

♦ ó♥ ♥ ♠♣♦ ①tr♥♦ ♥♦r♠

Prt á♥t ♠♦♦ ♦s ♥ú♠r♦s á♥t♦s n rk ② n rk+G s♦♥ q♥ts♣r sr♣ó♥ ♠s♠♦ tró♥ ♦♦s ♦s k ♥ ♥ stá♥ ♥ ♥ ú♥ ♣r♠t ♥ qr♦ tér♠♦ t♠♣rtr T ♦♣ó♥ ♥ st♦tró♥♦ ♥rí εn(k) ♥ ♥ n ♦♥ t♦rs ♦♥ ♥ ♥ ♠♥t♦ ♦♠♥dk s

f(εn(k)) =dk

4π3=

dk/(4π3)

e[εn(k)−µ]/kBT + 1.

♥ ♠r♦ ♣♦r s♠♣ sr♠♦s ♣♦r stró♥ qr♦ ♠s♠ q T = 0P♦r t♥t♦ ♣r εn(k) < εF s t♥ q f(εn(k)) = 1 ② ♣r εn(k) > εF f(εn(k)) = 0 st♦s ♦ q ♣r ♦s ♠ts ♦s t♦s q t♠♣rtr ♥♦ s ♥ s♥ t♥r ♥♥♥ s♣r ♥ s ♣r♦♣s q ♠♦s str

♥ ♣rs♥ Fext t♦♠ ♦r eikR ♥ t♦ ó♥ s ♠♣ ♣r qr n ② k ♣♦r ♦ q ó♥ t♠é♥ s r ♣r st♦ |ψ〉

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

• s ♥s ♥s s♦♥ ♥rts

♥s ♥s ♥ ♥ ♥ ♥ ♦♥ ♥rí ♠♥♦r q εF s dk/4π3 q ♥♦ s ♠ásq ♦♥ó♥ ♠♦♦ s♠ás♦ T = 0 ② ♦♥ εn(k) < εF P♦r t♥t♦ ♥ ♥ ró♥ dr② dr dk/4π3 tr♦♥s ♦♥ s t♦r ♦♥ ♥ dk sí ♣s t♥♠♦s ♥ sr♣ó♥ s♠ás ♥ ♥ ♥ ♥♦ q ♥s tr♦♥s ♥ s♣♦ ás♦ r,k s 1/4π3

♠♥r q ♥ s♣♦ ás♦ ás♦ r,p t♥♠♦s ♥ t♦r♠ ♦ q q s ♥ ró♥ Ωt s♣♦ ás♦ ♥ ♥ t♠♣♦ t ♣♦r ♥ ♦♥♥t♦ t♦rs r ② k ♠♠♦s r′ ② k′ ♦s t♦rs ♥ ♦s q t♠♣♦ t′ ♥ ♦♦♥♦ ♥♦ ♦s t♦rs r

② k ♦♠♥ Ωt′ st♦s t♦rs ♥ s♣♦ ás♦ s ♦♠♥ Ωt

st♦ qr r q ♥s tr♦♥s 1/4π3 ♥ ♥ ♥ s ♦♥sr ♥ t♠♣♦ ♦r♠ q ♠♦♠♥t♦ s♠ás♦ ♥tr ♦s♦♥s ♥♦ tr ♦♥ró♥ ♥ ♥♥

♦r ♥ ♥s ♦rr♥t étr ② ♦rr♥t ♥rí s♦♥

j = (−e)∫

CP

dk

4π31

~

∂ε

∂kjε =

∫

CP

dk

4π3ε(k)

1

~

∂ε

∂k=

1

2

∫

CP

dk

4π31

~

∂ε2(k)

∂k

② ♦ q ♥tr s♦r ♥ P r♥t ♥ ♥ó♥ ♣ró s r♦ ♠s ♦rr♥tss♦♥ ♥s

♥ rs♠♥ s♦♠♥t ♦s tr♦♥s ♥s ♣r♠♥t ♥s ♦♥tr②♥ ♦s ♥ó♠♥♦s tr♥s♣♦rt ② ♥ú♠r♦ tr♦♥s ♦♥ó♥ s ♥ú♠r♦ tr♦♥s ♥ sts ♥s

• s t ♥á♠ st♦s r♥♦s ♥ ♠á①♠♦ ♥ ♥

r ♠á①♠♦ ❱ ♥rí s ♣ srr♦r ♦♠♦

ε(k) ≃ ε(k0)−A(k− k0)2 A > 0,

♦♥ k0 ♦rrs♣♦♥ ♦ ♠á①♠♦ ♣♥t♦ k0 t♥ ♥ s♠trí s♥t♠♥t t st♦ ss ♥ ♣♥t♦ ❩ s♠trí t ② s ♥♠♦s ♠s t m∗ < 0 ♠♥t ró♥~2/2m∗ = −A t♥♠♦s q

v(k) =1

~

∂ε

∂k≃ (k ≃ k0) ≃ +

~(k− k0)

m∗=⇒ a =

d

dtv(k) =

~

m∗d

dtk

♦ q ♥♦s q ró♥ s ♣r dk/dt

♣r♦①♠ó♥ ♠s t ♥♦s ♣r♠t str ♥ ró♥ rt ♥tr ró♥ ♦ v(k) ♥ t♠♣♦ ♦♥ r③ ①tr♥ Fext ú♥ ♥ s♦ ♥ q ró♥ s♣rsó♥ ε(k) ♥♦ s ♣ró ♦♠♦ ♥

♥ t♦ ♥ ♣r♥♣♦ ♥á♠ s♠ás ♥ tró♥ ♦ ♥ tr♠♥ ♣♦r

v(k) =1

~∇kε(k) ♦ r♣♦ ♣qt ♦♥s q r♣rs♥t ♥ tró♥

~dk/dt = Fext Fext s r③ ①tr♥ s♦r rst

♣r♦ ♥♦ ② ró♥ ♥tr ♦ v(k) ② k

• ♥ ♥ ♣ró ② s♣♦♥♠♦s ♥ ♠s t m∗ sr ♥♠♦s

m∗δv ≡ ~δk =⇒ m∗dv

dt= Fext,

P♦r ♦♥s♥t ♥ só♦ ♦♥ ♥ ♥ú♠r♦ ♣r tr♦♥s ♣♦r P ♣ t♥r ♥s ♥ts ♥s ♥s ② rst♦ ís ♦r♠ q s ♥ s♥t étr♦ ② tér♠♦ ♠♥♦s ♥ ♦ rt♦ tr♥s♣♦rt tér♠♦ ♣♦r tr♦♥s st s ♥ ♦♥ó♥♥sr ♣r q rst s ♥ s♥t ♣r♦ ♥♦ s ♥ s♦t♦ ♥ ♦♥ó♥ s♥t

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó ♥á♠ ♦s tr♦♥s

s♥♦ 1/m∗ ≡ (1/~2) ∂2ε/∂k2 ♥ ♣rá♠tr♦ ♥♦ ♣♦r rtr ♦ ♥

• ♥ ♥ s♦ ♠ás ♥r ♣r♦①♠ó♥ ♠s t q m∗ ♥♦ s ♥sr s♥♦ ♥ t♥s♦r P♦r ♦♥s♥t δv ♥♦ s ♣r δk s♥♦ q ♠♣ ró♥

δvi = ~∑

i

(1

m∗

)

ij

δkj =⇒(

1

m∗

)

ij

=1

~2∂2ε

∂ki∂kj=⇒ dvi

dt=∑

j

(1

m∗

)

ij

Fext,j .

6.4 Ó ❯❱ ❨ ❯P

♣♦♥♠♦s ♥ ♠t ♦ t♦ ♥ ♠♣♦ étr♦ ② ♦♥sr♠♦s ♠t ♦♠♦ ♥ s tr♦♥s rs ♥ ♣r♦①♠ó♥ t♠♣♦ ♦só♥ ♥ st♦ st♦♥r♦ ♦ rrstr s♦ ♦rr♥t étr q ♥ s ♠♣♦ s ♦rrs♣♦♥ ♦♥ ♥ ♠♦ ♥ t♦r ♦♥ tró♥ sr r♠ δk q ♣♦♠♦s r ♦♠♦ δk = (m/~)v =−(eτ/~)E P♦r ♦♥s♥t sst♠ t♥rá ♥ ♥ó♥ stró♥ ♣r ♦s tr♦♥s t qs sr r♠ strá rí♠♥t s♣③ ♣♦r st ♦r δk

t♦♠♠♦s ♥ ♥t s ♦s♦♥s ó♥ ♠♦♠♥t♦ s ♣ rsrr ♥tr♦ ♣r♦①♠ó♥ t♠♣♦ ró♥ ♦♠♦

~

(dk

dt+δk

τ

)= eE,

❯ s♣③♠♥t♦ sr r♠

st ♠♥r δk s q♥t ♥ ♦ rrstr ♠♣♦ étr♦ ♥ s♥t♦ q δk s s♣③♠♥t♦ s♣r r♠ ♦♥rs♣t♦ s ♣♦só♥ ♥ qr♦ tér♠♦ ♥♦ E = 0

r♠♦s ♦r ♦♥t étr s♦ st ♣rtró♥ s♣r r♠ s♥ t③r ♦♥♣t♦ ♠s t

♦♥sr♠♦s ♥ ♠♥t♦ ár dSF s♣r r♠ q s s♣③ ♥ st♥ δk = eEτ/~ ♦ ó♥

♦♦s ♦s tr♦♥s q stá♥ ♥ ♦♠♥ dSF · δkt♥♥ ♣r♦①♠♠♥t ♥ ♦ vF ② ♥ ♥♦rr♥t

dj =2

(2π3)evF (dSF · δk)

♥ ♥ s♠♦♥t♦r r t♦♣ ❱ s ♦t♥ m∗BV < 0 ♠♥trs q ♥ s ♣r♦①♠s ♠í♥♠♦

t♥r♠♦s m∗BC > 0

♥ ♣♦r v = −(eτ/m)E ♦♠♦ s♠♣r q t♠♣rtr ♥♦ s ♥ ② q ♣♦r ♦♥s♥t ♥ó♥ stró♥ tr♦♥s ♥♦ s

①t♠♥t ♥ sr r♠ ♣s t♥ ♥ ③♦♥ r♦r εF ♥r ♣r♦①♠ kBT ♥ q ② t♥t♦ st♦s♦♣♦s ♦♠♦ s♦♣♦s ♥♦ s r♥t ♣s s♠♦s q kBT/εF ≪ 1 ♣r♦①♠♠♥t 3/1000 t♠♣rtr ♠♥t♦♠♦ ② s ó ♣r ♦rr♥ts ♦r♥ 107 ♠2 ♦ rrstr ♥ ♥ ♠t s ♣ st♠r ♦♠♦ 1 ♠sq s 10−8vF s♣③♠♥t♦ sr r♠ s ♣s r♠♥t ♣qñ♦ ② ♠♦ ♠♥♦r t♠é♥ q s♥♦q♦ ♥rí tér♠

st♦ s rsr♥♦ ó♥ ♠♦♠♥t♦ ♠♥r ♥á♦ ó♥ ♣r ♦ ♥ ♠♦♦

m∗(

dv

dt+

v

τ

)

= eE.

♠♥t st s♣③♠♥t♦ rrá s♦r s♣r s t♠♣♦ ró♥ rí ♦♥ k st♦ s s τ(k)

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ ❨ ❯Ó P❳Ó P

② ♦rr♥t t♦t sst♠ s ♦t♥ ♥tr♥♦ s♦r s♣r r♠

j =1

4π3

∫

SFe dSF (vF · δk)

=e2

4π3~

∫

SFτdSF (vF ·E)

=e2E

4π3~

∫

SFτvF · E dSF ≡ σE

♦♥ s s♦ q vF = (1/~)∇kε|ε=εF ② s ♥♦ ♥ ♦♥t sr tσef = |j|/|E| ♣♥♥t ró♥ E

σefec =e2

4π3~

∣∣∣∣∫

SFτvF · E dSF

∣∣∣∣ .

♣♦♥♥♦ q τ s ♦♥st♥t ② q ♦s tr♦♥s t♥♥ ♦ r♠ st ♦♥ts ♣ srr ♦♠♦

σefec =

(e2

4π3~

)lelec S

efecF ,

♦♥ lelec s ♠♥♦ r ♠♦ ♣r♦♠♦ ♥ sr r♠ ② SefecF s ár t ♦ r

♥ ♦♥s♥ ♥ ♥ ♠t ♦♥ s♣r r♠ sté rstr♥ ♣♦r ♦♥tt♦♦♥ ♣♥♦s ❩ ♦♥t ♥ tr♠♥ ♣♦r ár r ♠ás q ♣♦r ♥ú♠r♦ t♦t tr♦♥ssí ♥ ♦s ♠♥t♦s ♣♥t♥ts ♦♠♦ rsé♥♦ ♥t♠♦♥♦ ♦ s♠t♦ q ♣♦r s strtrrst♥ t♥♥ ♥ s♣r r♠ s ♠② r r♥t sr r♠ tr♦♥srs ♣rs♥t♥ ♦♥ts étrs ② tér♠s ♦♠♣rt♠♥t s s♦♥ s♠♠ts

6.5 ❮ ❨ ❯Ó P❳Ó P

♣r♠r st srr ♥ ♠t ♦♠♦ ♥ ♦♥♥t♦ tr♦♥s srs ♦♥s♦ ♦♠♦ ♥ sst♠ tr♦♥s ♦ ♥♣♥♥ts ♣r ♥ ♣r♦①♠ó♥ ①s♠♥t♦♣t♠st sí ♥♦ ♥s st♠♦♥s s♥s s ♠♦s ór♥s ♠♥t ♥rí♥ét ♦s tr♦♥s s ♦r♥ εF ♥ s tr♦♥s rs ♠♥trs q ♥tró♥ ♦♥ r s ♦r♥ ♥tró♥ ♥ tró♥ ♦♥ ♥ ó♥ ♣♦st♦ ♥ st♥ ♦r♥ ♥tró♥ ♥tr tr♦♥s t♠é♥ ♣ ♦♥srrs ♦r♥ ♥ ♥tró♥♦♠♥ ♦s tr♦♥s s♣r♦s ♣♦r ♥ st♥ s♠r ❨ t♦s sts ♥rís s♦♥ ♦r♥ ♣♦♦s ❱ ♠♥r q ♣r ♣♦♦ sr♦ s♣rr qr s r♥t s ♦trs

• ♥rí ♥tró♥ ♥tr tr♦♥s s ♠② í trtr ② q s ♥ s ♥♥t♦♥ ♠r♦ st r♦ ♥ ♥tró♥ t♥ t♦s ♠r♦só♣♦s ♦r♠ q s ♥♦② ♥tr ♦ r ♦s t♦s ♣♦rá♥ ♦srrs r♥ s st♦ ♣ s♠♣r trt♠♥t♦ ♣r♦♠♥ t♦ r♦r♥♦ s ♦s♦♥s r ♣s♠♦♥s s q s ó trtr ♠♦♦ s s♣♦♥♠♦s q ♦s tr♦♥s ♥ ♠♦♠♥t♦ ♦t♦ ♥♦ sr♥ ♦s♦♥s ♥ ♥ ss

érs q stró♥ qr♦ ♦♥ E = 0 ♥♦ r ♦rr♥t P♦r t♥t♦ vF s ♣r dSF ② ♣ rs ♥tr♠♦ t♦rs q s ♦ st♦ qr r q r rst♥ ♦ ♣♦t♥ ♣ró♦ r q s♣r r♠ t s ♠②

st♥t sr r♠ tr♦♥s rs st♦ ♥ tr♠♥♦ s♥♠♥t ♣♦r stíst á♥t q r s ♣r♦♣s stísts ♦s tr♦♥s

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó ♥á♠ ♦s tr♦♥s

♦s♦♥s ♣♦♠♦s srr

∇ ·E = 4π(n− n0)

mdv

dt= eE ♦♥ v s ♦ rrstr ♠♣♦ E

② ♣♦r ó♥ ♦♥t♥

∂n

∂t+∇ · (nv) = 0 =⇒ n− n0 ≪ n0 (linealizacion) =⇒ ∂n

∂t+ n0∇ · v = 0.

r♥♦ ♥♦ ② s♥♦ ó♥ ♠♦♠♥t♦

∂2n

∂t2= −n0

∂

∂t∇ · v = −4πn0e

m(n− n0)

② st♦ ♥♦s ♥ q ♥s ρ r③ ♥ ♠♦♠♥t♦ ♦st♦r♦ r♠ó♥♦ rs♣t♦ ρ0 r♥ s ♠♦♠♥t♦ s

ωp =4πn0e

2

m≃ 1016 s−1 =⇒ ~ωp ≃ 10 ❱

♥óts q st ♦r ωp s ♦r♥t ♦♥ ♥♦ r ♥♦ s ♦s♦♥s ♥ ssó♥

s ♥t q ♦♥ s ♦r t♥ r♥ ~ωp s í ①tr tér♠♠♥t ♣s♠♦♥s ② sst♠ tr♦♥s strá ♥ s st♦ ♥♠♥t ② ♥♦ ♣rrá♥ ♠♦♠♥t♦s rtró♥ ♦♥ ♥ ♦♥t ♦♥ ♠② r

trt♠♥t♦ s♦ ♥t♠♥t ♣r♦①♠♦ ♣r♦ str t♦ q qr♣rtró♥ r tró♥ rs♣t♦ qr♦ ♦♥ ♥ ♦♥t ♦♥ ♠② r♥♣r♦♦rí ♥ ♦só♥ r♥ ♥ q ♦srí ♦♥ ♥ r♥ ωp

• ♦♥t vF /ωp s ♣r♦♣♦r♦♥ n−1/6 ② q vF ∝ n1/3 ② ωp ∝ n1/2 ♥t♦♥s s n ss♥t♠♥t r♥ ♦s tr♦♥s ♣♦rí♥ r ♣♥tr ♣♦t♥ ♦s ♦♥s ♣♦st♦s s ♠♥r rst ♣♦s q ♣r③♥ st♦s tró♥♦s s♦③♦s ♥ rst ♥ s♠♦♠♥t♦ só♦ s ♦♥rtrí ♣♦r sí r♦ ♥ ♥ ♠t st♦ s ♥ ♣r♦s♦ ♦t♦ ② t♠♥t♥♦ ♥ ♠②♦r ♥ú♠r♦ tr♦♥s ♠②♦r ♣♥t♠♥t♦ ♦ q st ♥ ♠♥r tr♥só♥ rs ♠t s♥t q ♦rr ♥ rt♦s ó①♦s ♠tá♦s ♠♥tr t♠♣rtrtr♥só♥ ♦tt

tr ♠♥r r st♦ s q ♣r q ♥ só♦ sté ♥ ♥ st♦ ♠tá♦ t♥r ♥♥s tró♥ s♥t♠♥t t ♣s ♣r ♥ss ♠ás s ♦s tr♦♥s qrá♥♦③♦s ♥ ♦s ♦♥s

• P♦r ♦tr ♣rt s♦♦ s ♦♥sr♦ ♦ rrstr ♣♦r ♠♣♦ ①tr♥♦ ♣r♦ ♥♦ ♦ ♠ ♥rí ♥ét ♦s tr♦♥s q s ♦r♥ vF st♦ s♥q q ♣rt ♥tró♥ ♦♠♥ ♥tr tr♦♥s ♣♦r ♣♥tr q ♦rrs♣♦♥♥t st♥s ♦r♥ vF /ωp ♣♦r t♥t♦ ♦rt♦ ♥ ② q vF /ωp ≃ 106/1016♠ ∼ 1

Pr s ♣♦ss ♦s♦♥s ♥tr ♦s tr♦♥s s ♣ ♦♥srr ♥ só♥ ③ Seficaz ♣♦rtró♥ ♦r♥ 2 ♦ q ♣r♠t srr ♥ ♠♥♦ r ♠♦ ② ♥ t♠♣♦ ró♥

l =1

nSeficaz= vF τ =⇒ τ =

1

nSeficazvF.

r ♥♦ s ♥♠♥t ♥ stró♥ ♥♦r♠ ♦t♥rí♠♦s ♥ r♥ r ♦♥ k ωp(k) ♥ ③ ωp s ♦♥♣t♦ ② s stó ♥ st♥t t r s♦♣♠♥t♦ s ♥♦♥s ♦♥ ♥♦ trt♠♦s ♠ét♦♦

tr♦♥s rt♠♥t ♦s Pst♦ q ♦s tr♦♥s s s♣③♥ ♥ st♥ ♦r♥ vF /ωp ♥ ♦só♥ ♣s♠ s ♦s♦♥s

♣s♠ ♣♥t♥ s♦♠♥t ♣rt r♦ ♥ ♠♦ ♠②♦r q st♥ vF /ωp ♥ s♦ ♥ q s st ♣♥t♠♥t♦ trés ♥ ♥ó♥ étr ǫ(q) rérs s♥♦ kFT ♣

② ♦r q t♦♠

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❯

st t♠♣♦ τ r ♥ ♥s♥♠♥t♦ ♥ ♦s ♥s ♠♦♥♦tró♥♦s ♦r♥

∆ε = ~/τ = nSeficaz~vF ≃ nSeficazkF

εF ≃ εF ,

♦♥ s s♦ q (nSeficaz/kF ) s ♦r♥ ♥ ♥s♥♠♥t♦ ∆ε s ♣s ♦r♥ ♣r♦♣ ♥rí r♠ ♦♥ ♦ q ♣♦rí♠♦s ♦♥r q sr♣ó♥ ♠♦♥♦tró♥♥♦ t♥ s♥t♦ ♣sr q ♥ ♣rt ♠② ♠♣♦rt♥t ♥tró♥ tró♥tró♥ stá♣♥t ♣♦r ♦s ♣s♠♦♥s

Pr♦ ♦♠♦ ② s ♦♠♥tó ♥tr♦r♠♥t s♣tr♦ ①♣r♠♥t ♠só♥ r②♦s ❳ ♦s♠ts ♠str q ♥rí ♦s tr♦♥s ♥♦ t♥ s ♥rt♠r t♥ r♥ ∆ε ≃ εF ♥ ♦♥s♥ t♠♣♦ ró♥ r ♦s ♠ts sr ♠②♦r q q qí s st♠♦

♥ r t♦ ssó♥ ♥tr♦r stá ♣♦rq ♥♦ st♠♦s ♦♥sr♥♦ ♣r♥♣♦ P ② q ♥♦ s t♦♠ ♥ ♥t stíst á♥t s♦♠♥t ♥ ró♥ ♦r♥ T/TF ♦s tr♦♥s ♣♥ ♦s♦♥r s ró♥ q ♦rrs♣♦♥ q♦s tr♦♥sq ♣♥ ♥tr♠r ♥rí q s♦♥ ♦s q stá♥ ♥ ♥ ♥t♦r♥♦ ♥rí kBT r♦r εF P♦r t♥t♦ ♥ú♠r♦ ♦s♦♥s ♣♦r s♥♦ ② ♣♦r ♥ ♦♠♥ ♣sr sr n/τ = n2SeficazvF sr n2 (T/TF )

2 SeficazvF ♦♠♦ rst♦

τ =(T/TF )

2

nSeficazvF=⇒ ∆ε ≃ ~

τ≃ nSeficazεF

kF

(T

TF

)2

≃ kBT

(T

TF

).

Pr t♠♣rtr ♠♥t T ≃ 300 ∆ε t♦♠ ♦r ∆ε ≃ 5 × 10−23 ≃ 3 × 10−4 ❱ ②♥t♦♥s τ ≃ 2× 10−12 s

st t♠♣♦ ró♥ rst sr ♥♦ ♦ ♦s ór♥s ♠♥t ♠②♦r q ♦s ♦t♥♦s①♣r♠♥t♠♥t ♠♥♦ rssts étrs ♣♦r ♦ q ♠♦s ♥tr♣rtr q s♦s♦♥s tró♥tró♥ s♦♥ s♣rs r♥t s ♦trs ♣rtr♦♥s q sr♥♦s tr♦♥s ♣♦r ♠♣♦ ♥tró♥ ♦♥ ♦s ♦♥s ♥ ♠♦♠♥t♦ ♦♥♦♥s

6.6 ❯

♦♥tró♥ t♦♦s ♦s tr♦♥s ♥ ♥ ♥s ♦rr♥t s

j = −e∫

ocupados

dk

4π3v(k),

♣r♦ ♦♠♦

1

4π3

∫

zonadk v(k) = 0 =

1

4π3

∫

ocupadosdkv(k) +

1

4π3

∫

desocupadosdk v(k)

♥♦s q ∫

vacıos

dk

4π3v(k) = −

∫

ocupados

dk

4π3v(k)

② ♣♦♠♦s ♦♥r q ♦rr♥t ♣r♦ ♣♦r ♦s tr♦♥s q ♦♣♥ ♥♦s ♥s tr♠♥♦s ♥ ♥♥ s ♠s♠ q s ♣r♦rí s ♦s ♥s str♥ s♦♣♦s ② t♦♦s ♦s♦tr♦s ♥s ♥ str♥ ♦♣♦s ♦♥ ♣rtís r +ests ♣rtís ts q ♣♦rí♥ r ♦rr♥t s ♠♥ ♦s

s♥♠♥t s ♥ r qí ♦s ♠s♠♦s r♠♥t♦s ② st♦s ♥ só♥ ♦s ♦rs ①♣r♠♥ts s♦♥ ♦r♥ τ ≃ 10−14 − 10−15 s

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó ♥á♠ ♦s tr♦♥s

P♦r ♦ t♥t♦ tr♥s♣♦rt ♦rr♥t ♥ ♥ ♥ ♣r♠♥t ♥ ♣ ①♣rs ♦s♠♥rs tr♥ts ♦ ♥ ♠♥t ♦s tr♦♥s ♦ ♥ ♠♥t ♦s ♦s ♥ ♣r♦ s♥♠③r ♠♦s ♠♥s♠♦s trtr ♠s♠ ♥

• Pr♦♣s ♦s ♦s

❱♠♦s srr s ♣r♦♣s ♦s ♦s q ♣♦rí♠♦s ♥r ♦♠♦ ♦s ♥s ♥rét♦sí♦s ♥ ♥ ♦♥ ♥♠♥ts ♣r srr ís ♦s s♠♦♥♥t♦rs ② str♦s ts tró♥ st♦ só♦ ♥ t♦♦ s♦ ♣♥srs t♠é♥ q st t♣♦ ♣♦rt♦rs ♣♥ ♣rr t♠é♥ ♥ ♠ts

♠♦♥t♦rs ② ♠ts s♠♦♠♥t♦ kh ♦ s ♦♣st♦ ke tró♥ q ♣r♦♦ ♥t ♥ ♥ ♥ t♦ ♣♥s♠♦s ♥ ♥ ♥♥ ♥♦ ②♦s tr♦♥s st ♥ s♥t sr ①t♦ ♣♦r ♥ ♦tó♥

❯ ①tó♥ rt ♥ tró♥

s♠ s♠♦♠♥t♦s ①t♥ t♦ ♥ r♦∑

banda k = 0 ♦ s♠trí ♥rsó♥ ❩ q t♥ ♣♦rq t♠é♥ t♥ Pr q s s♦♥sr♥♦

∑banda k = 0 s♣és ①tó♥ ② q trr ♥ s♠♦♠♥t♦ −ke

♦ rst♦ s ó♦ ♣s s ♦ ♠s♠♦ q r q ♥ ♥ ♦♥ ♥ tró♥ ♠♥♦st♥ ♥ s♠♦♠♥t♦ t♦t −ke

kh = −ke.

♠♦♥t♦rs ② ♠ts s ♠ ♥rí rs♣t♦ t♦♣ ❱ ♥s♠♦♥t♦rs ♦ εF ♥ ♠ts ♥rí ♦ s

εh(kh) = −εe(ke).

♥ t♦ s♣♦♥♠♦s r♦ ♥rí ♥ ♠á①♠♦ ♥ ♥ ♦ ♥ εF sr♦ q ♥t♦ ♠ás s ♥rí tró♥ ♠ás ♥rí t♥rá sst♠ ♥s♣és q tró♥ s ①t ♦tr ♠♥r s ♥sr♦ ♠ás tr♦ ♣r r str♥ tró♥ ♦♥ ♥rí q ♦♥ ♥rí ♠②♦rP♦r t♥t♦ s ♥ t♥ s♠trí ♥rsó♥

εe(ke) = εe(−ke) = −εh(−kh) = −εh(kh).

srr ♥ t t ♣á♥

r♦rrá♥ ♦r ♥♦s rst♦s ①♣r♠♥ts q s str♦♥ ♠ r ♠♦♦ ♦♥ s♣♥t q ♥ ♥s ♦s♦♥s ♣rí q ♦s ♣♦rt♦rs r r♥ ♣♦st♦s

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❯

♠♦s r♥ ♥rís ♥ rst ♥♦ s ♥ ①t♦ tr♦♥s ❱ ♠♠♦s E ♥rí t♦t ♥ ② E0 ♥

E − E0 =∑

k

E(k)nkσ −∑

k<kF

E(k)

♦♥ E(k) s ♥rí ♦s tr♦♥s ♥ ♥ k st♦ s ♣ rsrr ♦♠♦

E−E0 =∑

k>kF

E(k)nkσ+∑

k<kF

E(k)nkσ−∑

k<kF

E(k) =∑

k>kF

E(k)nkσ−∑

k<kF

E(k)(1−nkσ)

♦♥ ♣ré♥tss (1 − nkσ) s r♦ s ② tró♥ ♥ st♦ k ② s ♥♦ s ②♦ ♥ ♦ st♦

r♥ ♥rís ♥♦ rí s s♦♠♦s r♦ ♥rís ♥ ♥rí r♠ εF ♦r♠ q ♦r ♠♠♦s ε(k) = E(k) − εF s ♥♦ s ♣r ♥rεF s t♦♠ ♦r ❱ ♦♠♦ ♦r♥ s ♥rís ♥t♦♥s

E − E0 =∑

k>kF

ε(k)nkσ −∑

k<kF

ε(k)(1− nkσ)

♦♠♦ ε(k) > 0 ♣r k > kF ② ε(k) < 0 ♣r k < kF ♥♦s q

E − E0 =∑

k>kF

ε(k)nkσ

︸ ︷︷ ︸tr♦♥s

−∑

k<kF

|ε(k)|(1− nkσ)

︸ ︷︷ ︸♦s

♦s ♦s ♦rs q ♣ t♦♠r (1− nkσ) st♦ s ♣ rsrr ♦♠♦

E − E0 =∑

ε(k,σ)>0

ε(k, σ)nkσ +∑

♦s

ε(khσ)

♣♦r ♦ q εh(kh) = −εe(ke) > 0

❯ ♥ ♥ tr♦♥s ② ♥ ♦s

♠♦♥t♦rs ② ♠ts ♦ ♦ s ♦ tró♥ ♣r♦ ♣♦r ♥

vh(kh) = ve(ke),

♣s r♦ ♦♥ r ♥tr♦r s t♥ q ∇εh(kh) = ∇εe(ke)

ísst♦ó♦

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❮ Ó ♥á♠ ♦s tr♦♥s

♠♦♥t♦rs r ♠á①♠♦ ❱ ♥rí s ♣ srr♦r ♦♠♦

ε(k) ≃ ε(k0)−A(k− k0)2 A > 0,

♦♥ k0 ♦rrs♣♦♥ ♦ ♠á①♠♦ Pr r ♠ás á ssó♥ s♣♦♥♠♦s q ♣♥t♦ k0 t♥ ♥ s♠trí s♥t♠♥t t st♦ s s ♥ ♣♥t♦ ❩ s♠trí t s ♥r ♥t♦♥s ♥ ♥t m∗ ♠♥t ró♥

~2

2m∗= −A =⇒ v(k) =

1

~

∂ε

∂k≃ (k ≃ k0) ≃ +

~(k− k0)

m∗m∗ < 0

② ♣♦r ♦ t♥t♦

a =d

dtv(k) =

~

m∗d

dtk ∝ Fext

♦ q ♥♦s q ró♥ s ♣r dk/dt♦♠♦ ♠♦s ♠s t ♦ s ♣♦st ♥♦ stá r ♠á①♠♦ ❱ ②q m∗e < 0 ♣♦r sr 1/m∗ ∝ −A ♠s t ♦ t♥ s♥♦ ♦♣st♦ tró♥

m∗h = −m∗e.

♦t ❨ ♠♦s st♦ q ♠s t s ♥rs♠♥t ♣r♦♣♦r♦♥ rtr d2ε/dk2

♥ ② ♥ ♦s t♥ rtr ♦♣st tr♦♥s ❱

♠♦♥t♦rs ó♥ ♠♦♠♥t♦ q s ♣rí tró♥ ①t♦ srí

~dkedt

= −e(E+

cve ×B

).

sstt♠♦s ke ♣♦r −kh ② ve ♣♦r vh ♥t♦♥s ♥♦s q ó♥ ♠♦♠♥t♦ ♣r ♦ q s♠ tró♥ ♣r♦ ♥

~dkhdt

= e

(E+

cvh ×B

),

q s ó♥ ♠♦♠♥t♦ ♥ ♣rtí r +eóts ♣s q sstt♠♦s ♥á♠ t♦ ♥ ①♣t♦ tró♥ ①t♦♣♦r ♥ ♣rtí r ♣♦st q s ó♥ ♠♦♠♥t♦

s ♦♥s ♠♦♠♥t♦

d

dtr = vn(k) =

1

~

∂εn(k)

∂k

~k = −e[E(r, t) +

1

cvn(k)×B(r, t)

]= Fext(r)

s♦♥ ss ♦♥s ♥s ♦♥ ss ♥ó♥ts ♦ q ♥♦s q s♦ó♥ s ♥í♦ ♣r♥s ♦♥♦♥s ♥s s r(t0) = r0 ② k(t0) = k0 st♦ qr r q ♦s tr②t♦rsr,k ♥♦ s r③♥ ② ♣♦r ♦♥s♥t q ♦s st♦s í♦s t♥♥ s ♠s♠s ♦♥s ♦ó♥ q ♦s ♥♦s s♥ q s ♠③♥ ♦s ♥♦s ♦♥ ♦s ♦tr♦s ♥ ♦ó♥ t♠♣♦r

❯♥ ③ q ♠♦s ♥③♦ s ♣r♦♣s ♦s ♦s ♣♦♠♦s r♠r q ♥á♠ ♦s tr♦♥s ♦ ♠♣♦s ①tr♥♦s s ♣ str ♠♥r ♦♥ ♥s♣rtís ts q ♠♠♦s ♦s ② q t♥♥ r ♦♣st ♦s tr♦♥s qs♦♥ ♦s ís♠♥t rs♣♦♥ss ♦♥ó♥

ísst♦ó♦

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❯

❯

❯

❯

❯

❯

❱ ❯ ❯ ❨

♥ s♦s tr♠♥♦s ♣ s s ①tr ♥ tró♥ ♥ ♥ ♥ ♣♦r ♦t♦♠só♥ ♣♦r①tó♥ tér♠ ①st♥ ♠♣r③s t s ♠ás s♥♦ trr ♦♠♦ s ♦♥ó♥ r ♦ q tró♥ q ♦♠♦ s r ♥ sst♠ N − 1 tr♦♥s ♥

qí ♥♦ ♥tr♠♦s ♥ ss♦♥s ó♠♦ ♦ qé ♣r♦ st♦ tró♥ ♥ ♦tr ♥ ②♠♦s r ♥♣é ♥ q s♦♠♥t ♣♦♠♦s t③r ♥ sr♣ó♥ tr♠♥ ♦ tr♦♥s♦ ♦s str ♦♥ó♥ ♥ ♥

s ♦♥s ♠♦♠♥t♦ srts ♥ ♣♥t♦ ♥tr♦r ♥♦s ♥♥ q ♥ ♥str ♥á♠s♠ás ♦s ♦s s♥ s ♠s♠s ♦♥s ♠♦♠♥t♦ q ♦s tr♦♥s s♦ q ♦ ó♥ ♠♣♦s ①tr♥♦s s r③s q sr♥ ♦rrs♣♦♥♥ ♣rtís s♥♦ ♦♥trr♦

♦r ♥ ♥♦ ♦ s ♣r♦ ♥ ♦s st♦s ♥rí ♠ás t ♥♥ ♥á♠ ♥ s♣♦ r ♦s ♦s s ♦♠♣t♠♥t ♥á♦ ♦str♦♥s ② q t♥♥ r ♦♣st ② t♠é♥ ♠s t ♦♣st st♦ s m∗h > 0 ♣s ♥s s♦ a = dv(k)/dt = (~/m∗) dk/dt = Fext/m

∗

♥ ♥♦s s♦s ♥ ♦♥ó♥ ♠tá stá st♥t ♥ ② s ♣rr str s♥á♠ ♠♥t ♦s P♦r s♦ ♦s ♦♥ts s♥♦ ♣♦st♦ q s ♠♥ ♥ ♥♦s♠ts ♦♠♦ rs♣st ② ♥♦r♠s ② ♣r♣♥rs ♣♥ ♥tr♣rtrs ♦♠♦ ♦s q s ♥s ♥ s s♦ ♣♥ sr ♠♦r srts ♠♥t ♦s

6.7 ❱ ❯ ❯ ❨

♦ ó♥ ♥ ♠♣♦ étr♦ E ó♥ ♠♦♠♥t♦ ♣r k s

k(t) = k(0)− eEt

~.

♦rr♥t tr♥s♣♦rt ♣♦r ♥ tró♥ s ♣r♦♣♦r♦♥ s ♦ ♦ v (k(t)) ♥tró♥ t♥ ♣r♦ ② ♣ rs ♥ t♠♣♦ ♦♠♦

v (k(t)) = v

(k(0)− eEt

~

).

❯ ♥rí ε(k) ② ♦ v(k)

♣♦♥♠♦s q v s ♣r♦♣♦r♦♥ E ♣♦r♠♣♦ s tró♥ st ♥ r♣♦s♦ ♥ k =0 ♥ts ♣r ♠♣♦ s♦♠♦s ró♥ k ♣r E ② r♣rs♥t♠♦s ♥ ♥rí ε(k) ② ♦ v(k) tró♥ ♦ r♦ ró♥ és r ♦♠♦ s ♦sr ♣r tr♠♥♦s♦rs k ♦ v(k) s ♣♦st♠♥trs q ♣r ♦tr♦s ♦rs k s ♥t st ♠♥r ♦ ♦s tr♦♥s ♥ rst ♦ ♥ ♥♦♠ ② stát♦ ♥♦♠♥t ♥♥♠♥t ♦♠♦ srí ♥ s♦ ♦s tr♦♥s rs

Pr q s s ♦♥♦♥ s ♦♥♦♥s ♥s tró♥ s ♣♥ r ♣♦só♥ ② s♠♦♠♥t♦ ♥ st♦♣r qr ♥st♥t t♠♣♦

② ①♣rsó♥ s RH = 1(

nportadores c qportadores)

♦ v (k(t)) tró♥ s ♣s ♦t ♥ t♠♣♦

ísst♦ó♦

❯

❯

❯

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❯

❮ Ó ♥á♠ ♦s tr♦♥s

♥ ♦♥s♥ ♠♦♠♥t♦ ♦s tr♦♥s ♥ s♣♦ r s ♠♦ ♠ás ♦♠♣♦ q♣r tr♦♥s rs ② ♥ s♦ t♥ s♥♦ ♦♠♦ q st♠♦s ♥♦ s tr♥ ró♥ ♦ sú♥ t♦r k r♦rr ❩

P♦r ♦ t♥t♦ s ♥♦ ①str♥ ♠♣r③s t♦s ♦♥♦♥s ♦ qr ♦tr♦ ♠♥s♠♦ s♣rsó♥ st♦ s s τ → ∞ ♥t♦♥s ♥♦ rí ♦♥ó♥ ♣♦r tr♦♥s ♦ Pr♦s τ s s♥t♠♥t ♣qñ♦ tró♥ ♥♦ rá t♥r ♥ ♦ ♣ró ② ♣ ①str♥ ♦♥ó♥ ♣r♦♦ ♣♦r

❨ s♦ s ♦ q r♠♥t ♦rr ♣s s τ = 10−14 s ② |E| = 10 ♦t♠ ♥t♦♥s ró♥ k ♥ t♠♣♦ ró♥ τ s eEτ/~ ≃ 10 ♠−1 ♠♥trs q 1/a ≃ 108 ♠−1 st♦ qr rq r♦rr♦ k tró♥ ♥tr♦ P❩ s ♠♦ ♠♥♦r q t♠ñ♦ ③♦♥♦♥ ♦ q ♥ó♠♥♦ ♣r♦ v(k) ♥♦ ♣r ② s♥ ♣ ①str ♦♥ó♥ r étr ♥ rst

♦♦ ♦ ♥tr♦r str ♠♣♦rt♥ s r♦♥trs ❩ ♥ ♥á♠ ♦s tr♦♥s ♦♥ ♠♣♦ ①tr♥♦ ♥ ♥ rst ♣rt♦ ♠s♠ ♣rát♠♥t q ♥ s ss♦♥s q ② s♥ t♥♦ r ♦s st♦s tró♥♦s st♦ s s ♥s ♥rí

♥ t♦ t ♦s♦♥s s ♠♣♦ E s ♣r♦ ♥ t♦r ♦ tró♥ v (k(t)) s ♥ ♥ó♥ ♦st♦r ② q ♣r t = t1 t = t2 t♦♠rá ♦s ♦rsv (k(0)−G1) v (k(0)−G2) P♦r t♥t♦ r♥ ♦s tr♦♥s rs ♦ ♦str♦♥s ♥ ♥ rst r ♥♥ ♣♦t♥ ♣ró♦ r trés ♦r♠♥♦♥ εn(k)

6.8 ❱ ❯ ❯ ❨

♣ ♦t♥r r♥ ♥t ♥♦r♠ó♥ s♦r s ♣r♦♣s tró♥s ♦s só♦s♠♥t ♠s rs♣st ♦s ♠s♠♦s ①♣r♠♥t♦s t♦s ♥ ♣rs♥ ♥ ♠♣♦♠♥ét♦ ♥♦r♠

❯ r②t♦r ♥ ♣♦rt♦r r ♦ ♥ ♥♦r♠ s♣♦ k

s ♦♥s ♠♦♠♥t♦ s♠ásss♦♥

d

dtr = v(k) =

1

~

∂ε(k)

∂k

~d

dtk = −e1

cv(k)×B

ó♥ s t♥ q dk/dt s♣r♣♥r B ♣♦r t♥t♦ ♣r♦②ó♥ k s♦r ró♥ B ♥♦ rí ② s ♥♦♥st♥t ♠♦♠♥t♦

P♦r ♦tr ♣rt sr r③ ①tr♥ s♦r tró♥ ♣r♣♥r s ♦ ♥♦ sr tr♦ s♦r ♠s♠♦ ② s ♦♥sr s♥rí ε(k) P♦r ♦♥s♥t s tr②t♦rstró♥s ♥ s♣♦ k srá♥ rst♦ ♦rtr s s♣rs q♥réts ♦♥ ♣♥♦s♣r♣♥rs B st♥ ♣♥♦ ♦r♥ s♣♦ k srá ♦♠♣♦♥♥t k qs ♦♥sr

Pr tr♦♥s rs ♦ v s ♣r♦♣♦r♦♥ k ② r ♥♠♥t ♦♥ t♠♣♦

ísst♦ó♦

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❯

❯

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❱ ❯ ❯ ❨

s♥t♦ órt ♥ s♣♦ k ♥♦ ♣♦r ♦ v q s ♣r♦♣♦r♦♥ ∇ε(k)

♥ r s s♣st♦ q v(k) s r ♣s♦ s♣♦♥♠♦s q s♣rq♥rét ε(k) = cte s ♥ ♣s♦ r♦ ♦♥ ♥ ♣r♦①♠ó♥ ♠s t

♥ s♦ q s ♦♥srr♥ ♦s ♦♠♦ ♦s ♣♦rt♦rs r t♦♦ srí s♦ q♦ q s r s ♣♦st r♦ s♦r s♣r ε(k) = cte s s♥t♦ ♦♥trr♦

st ♦r♠ ♦♥ ♥ ♠♣♦ ♠♥ét♦ s ♦t♥♥ tr②t♦rs ♥ s♣♦ k q s♦♥ s♦♥s s s♣rs ♥rí ♦♥st♥t ♠♥r q s r♠♦s ró♥ B s ♦t♥ ♥♦r♠ó♥r s st♥ts s♦♥s s♣rs st♦ t♥ ♠♦ ♥trés ♣♦r ♠♣♦ ♥ s♦ s♣r r♠ ② q s♠♦s q s s♣r q ♦♠♥ ♥ ♦s ♥ó♠♥♦s tr♥s♣♦rt♥ ♠ts ② r♠♦s ♣♦str♦r♠♥t q t♠é♥ ♥ ♦s ó♣t♦s

② q t♥r ♥ ♥t qí q s s♣rs q♥réts ♣♥ sr rts ♦ rrsPr r♦ ♠♦r ♥ ♦s ♠♥s♦♥s ♣♦♠♦s r

❯ r②t♦rs rts ② rrs② ♦♠♦ sts tr②t♦rs ♠♥s♦♥s s♦♥ ♦rts s s♣rs tr♠♥s♦♥s ♠♦sr♠♥t ♠♥r ♥ s q ♣r♥ tr②t♦rs rts ② rrs

❯ á♦ ♣r♦♦ ♥tr②t♦r rr ♥ s♣♦ k

♠♦s ♣r♦♦ s órtsrrs ♦s ♣♦rt♦rs r ♥ s♣♦ k

♦♥sr♠♦s ♥ órt s♦r s♣r ♥rí ε ② ♥ ♥ ♣♥♦ ♣r♣♥r B t♠♣♦ q tró♥ ♣sr k1 k2

s

t2 − t1 =

∫ t2

t1

dt =

∫ k2

k1

dk

|dk/dt|② ♦♠♦ ♣♦♠♦s s♣r dk/dt s ♦♥s ♠♦♠♥t♦ ♦t♥♠♦s

t2 − t1 =~2c

eB

∫ k2

k1

dk∣∣∣(∂ε/∂k

)⊥

∣∣∣

♦♥ (∂ε/∂k)⊥ s ♦♠♣♦♥♥t ♦ ♣r♣♥r ♠♣♦ st♦ s q stá ♥ ♣♥♦ órt

♦♠♦ ② s st♦ s ♣ srr és r q |∂ε/∂k)⊥|∆(k) = ∆ε ② ♣♦r t♥t♦

t2 − t1 =~2c

eB

1

∆ε

∫ k2

k1

∆(k)dk,

P♦r ♠♣♦ r ♥s ♥s tró♥♦s

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó ♥á♠ ♦s tr♦♥s

♦♥ ♥tr s ár q ② ♥tr s ♦s s♣rs ② ♥tr ♦s ♣♥t♦s k1 ② k2 ♠r♠♦sA1,2(ε, kz) s s ár

∆ε→ 0 ♥♦s q

t2 − t1 =~2c

eB

∂

∂ε

(A1,2

)

s♥♦ A1,2 ár rr ♣♦r tr②t♦r ♥tr k1 ② k2 ♠♥tr ε ♥ ♥t dε óts qqí ♣♦♥♠♦s ∂(A1,2)/∂ε ② q s♣r A1,2 ♣♥ t♥t♦ ε ♦♠♦ kz

st♠♦s ♥ ♥ órt rr ♣r♦♦ ♠♦♠♥t♦ srá t♠♣♦ q tró♥r ♥♦ k2 = k1 P♦r ♦ t♥t♦

T (ε, kz) =~2c

eB

∂

∂εA(ε, kz)

♦♥ A(ε, kz) s s♣r ♥rr ♣♦r órt ♥ ♣♥♦ kz ♦♥sr♦

❯ ♣r ♥rr ♣♦r tr②t♦r rr

P♦r ♦tr ♣rt s♠♦s q ♥ tró♥ rsr ♥ órt rr ♥ s♣♦ k ②r♥ s ωc = eB/(mc) st♦ s♥ qTlibre = 2πmc/(eB) Pr ♣rs♥tr ♦s rst♦s①♣r♠♥ts s ♦st♠r ♥r ♥ ♠s

t ♦tró♥ q ♣r♠t r T (ε, kz)♦♠♦ s ♦♥ Tlibre

m∗c(ε, kz) =~2

2π

∂

∂εA(ε, kz)

st ó♥ ♥♦s ♣r♠t r T (ε, kz) ♠s♠ ♠♥r q Tlibre ♠♥♦ m ♣♦r m∗c ♠s t m∗ ♥tr♦ ♦♥ s ♥s ♥rí ② st m∗c stá♥ r♦♥s ♦♠♦ ♣♦rárs ♥ t♠ s♦r s♠♦♥t♦rs

Ps ss s♣rs q♥réts s♦♥ srs ②♦s ♦rts ♦♥ qr ♣♥♦ ♥ ♣rtr ♦♥ ♥♦ ♣r♣♥r Bs s♠♣r ♥ r♥r♥

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

♣ít♦

7.1 ❯Ó

♥♦ ♠♦s st♦ ♥ s ss♦♥s s♦r s ♣r♦♣s ♦s r♣♦s rst♥♦s s♦♥s♥s ♥tr♦r ♣♦t♥ ♣ró♦ ♦s ♦♥s s st♦ q st ♣♦t♥ ♥ ♣♣ s♦ ♥ trt♠♥t♦ s ♣r♦♣s tró♥s ♥ só♦ ♦♠♦ s st♦ ♠♥r ♠② r ♥♦ s ♦ ó♠♦ ♥t♥r ①st♥ ♦♥t♦rs ② s♥ts

Pr♦ s♣♦só♥ q ♦s ♦♥s q ♣r♦♥ ♦ ♣♦t♥ rst♥♦ sté♥ ♦s ♥ s♣♦s ♥ s♣♦só♥ ♠② rt ② q ♠s ♦s ♦♥s ♥♦ s ♥♥t ② ♥♦ ①st♥ t♠♣♦♦r③s ♥♥ts q ♦s ♠♥t♥♥ ♥♠ós

st ♠♥r ♥♦ s ♠s q s♦ ♣r rsts ♦s q ♠♦s ♥ trt♠♥t♦ ás♦ T = 0 ♠♦♦ r stát s á♦ sí ♣♦r ♠♣♦ ♥♦ ♥t♥t♠♦s srrs ♣r♦♣s íss ♦s s♥ts ♠♥t ♥ ♠♦♦ r stát ♥♦s ♥♦♥tr♠♦s qést♦s s♦ ♥ ♦s ♥ó♠♥♦s q ♥ ♥rí s♥t ♣r ①tr tr♦♥s trés ♣♣rs♥t♥ ♦♠♣♦rt♠♥t♦s ♠② ♠t♦s ♦s s rs ♣r♦♣s q s s ♥♥tr①♣r♠♥t♠♥t

♥ ♦♥rt♦ s t♥ ♥ st ♦s ♥ ①♣ó♥ ♠s ♣r♦♣s ♥♠rr♠♦s♥s s

♦r s♣í♦ts ①♣r♠♥t♠♥t ♥♦ s ♥♥tr q ♦r s♣í♦ s ♣r♦♣♦r♦♥ T cv ∝ T s♦ ♣r T < 10 ♥ ♠♦ s ♦sr ♥ ♦♠♣♦rt♠♥t♦ ♣r♦♣♦r♦♥ T 3

t♠♣rtrs ♠ás ts ② q ♦r s♣í♦ t♥ ♥ ♦♥st♥t ♥ ♥ r♥♦ t♠♣rtrs 102 103 s♥ts ♦♠♥t ♦♥trrí♥ ♦r s♣í♦ q♦s tr♦♥s ①t♦stér♠♠♥t ②♦ ♥ú♠r♦ s ♣ st♠r ♦♠♦ ♣r♦♣♦r♦♥ exp (−Eg/kBT ) s♥♦ Eg ♥r ♣ ♥rís st ♦♠♣♦rt♠♥t♦ ①♣♦♥♥ r sst♥♠♥t ♦♠♣♦rt♠♥t♦ ♦sr♦ ♥ T 3 ♥♦ T → 0 ② ♥ ♦♠♣♦rt♠♥t♦ ♦♥st♥t ♥♦T ∼ Tamb

①♣♥só♥ tér♠ ② só♥ ♠♦♦ r rst♥ stát ♥♦ ①♣ st♦s ♥ó♠♥♦s

♠♣♦ ró♥ ♦s tr♦♥s ♣♥♥ ♥ T 5 rsst st♠♣rtrs ② ♥ T ts s s ♦s♦♥s ♦s tr♦♥s ♦♥ ♠♦♠♥t♦ ♦s♦♥s

♣r♦♥t ♦ rí s♣r♦♥t♦rs q s ♦♥♦♥ ♠♥♦s ♦str♦♥s s ♥♦ ①str♥ s r♦♥s r

♦♥t tér♠ ② s♦♥♦ ♥ ♥ ♠♦♦ r stát ♥♦ ①strí♥ sts♣r♦♣s tr♥s♣♦rt ♥ ♦s s♥ts

t ♦s rsts ó♥♦s Pr ~ω ≪ Eg ♦s rsts ó♥♦s t♥♥ ♥ ♠á①♠♦♣r♦♥♥♦ ♥ s rt q s♦♠♥t s ♣ ♥t♥r ♦♥ ♥ ♠♦♦ r ♥á♠

ró♥ r②♦s ❳ ② ♥tr♦♥s Pr ①♣r ♠♦s ♦s rst♦s①♣r♠♥ts ② q r ♥ sr♣ó♥ ♥ r q ♣ ♣rs♥tr r♦♥s

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó ♥á♠ r rst♥

7.2

P♦r ♦♥s♥t s ♠♣♦rt♥t ♦♥strr ♥ ♠♦♦ r rst♥ ♥á♠ ♣r r s ♥r ♠♦♠♥t♦ ♦s ♦♥s ①♣ ♣r♦♣s ♠♣♦rt♥ts ♦s só♦s ♦s♦tr♦s ♠♦s sr st ♠♦♦ ♥á♠♦ ♥♠♥t♠♥t ♥ ♦s ♣♥t♦s

♣♦só♥ ♠ qr♦ ♦s ♦♥s só♦ s ♥ ♣♥t♦ r rss♦ rsts rs r♦

s♣ró♥ ♦s ♦♥s s ♣♦só♥ ♠ s ♣qñ ♥ ♦♠♣ró♥ ♦♥ s♣♦ ♥tró♥♦

♣r♠r s♣♦só♥ ♠♣ trt♠♥♦ só♥ ó♥ ② q ó♥ s ♠ r♦r ♠s♠♦ ♣♥t♦ r s♠♣r P♦r t♥t♦ s st♠♦s r ♣♥t♦ só♥ st ♣r♠rs♣♦só♥ srá ①s♠♥t rstrt

s♣♦só♥ s♥ ♣♦r ♦tr ♣rt s ú♥♠♥t ♥ ♣r♦①♠ó♥ q ♣r♠t ♥ trt♠♥t♦♠ás á s ♣♥t♦ st ♥ít♦

Pr s♠♣r ♥str♦ trt♠♥t♦ ♠♦s ♦♥srr ú♥♠♥t rs rs ♠♦♥♦tó♠s ♠♦♦ q ♣♦♠♦s s♥r ó♥ t♦r R q ♥ ♣♥t♦ s♣♦ r♦r q ♦sst ♦só♥ ♣♦♠♦s srr ♦♠♦

r(R) = R+ u(R).

s♣♦♥♠♦s q Φ(r, r′) s ♥rí ♣♦t♥ ♥tró♥ ♥tr ♦s ♦♥s st♦s ♥ ♦s ♣♥t♦sr ② r′ ② q ♥rí ♥tró♥ ♥tr ♦s ♦♥s ♣♥ s♦♦ r♥

r′ − r ♥rí ♣♦t♥ t♦t rst srá

Utotal =1

2

∑

R,R′

Φ(r(R)− r(R′)

)=

1

2

∑

R,R′

Φ(R−R′ + u(R)− u(R′)

).

t♦♦s ♦s s♣③♠♥t♦s ♦s ♦♥s u(R) s♦♥ ts q u(R) − u(R′) rst♥ sr ♣qñ♦s♣r t♦♦s q♦s ♣rs ♦♥s ② ♥tró♥ Φ(r, r′) s ♣r ♣♦♠♦s r ♥rí♣♦t♥ ♦♠♦ ♥ srr♦♦ ②♦r r♦r qr♦

Utotal =N

2

∑

R

Φ (R) +1

2

∑

R,R′

(u(R)− u(R′)

)·(∇Φ

(R−R′

) )

+1

4

∑

R,R′

[(u(R)− u(R′)

)· ∇]2Φ(R−R′

)+O(u3).

Pr r s s r③♦♥ st s♣♦só♥ ♠♦s r ♥s st♠♦♥s ♥rí ás ♣♦r r♦ rt ♥ sst♠ ♥ qr♦ tér♠♦ t♠♣rtr T s sú♥ t♦r♠ q♣rtó♥1/2kBT P♦r t♥t♦ ♥ rst t♥rá ♥ ♥rí 3kBT ♣♦r ó♥ ② q ② ss r♦s rt ♣♦s♦♥s ② ♠♦♠♥t♦s♥s ♠♦s ♦r ♥ st♠ó♥ ás ts t♠♣rtrs ♣r ♣♦r ♣r t♦r♠ ♥ ♥ rst ♥♠♥s♦♥ ♥rí tér♠ r♦♥ ♣♦r át♦♠♦ s kBT s rst stá t♠♣rtr T ♠♠♦s Ec ♥rí ♦só♥ ♣♦r át♦♠♦♥♦ kBT ≃ Ec sst♠ ♠♣③rá s♦♥rs ② ♠♦♠♥t♦ ♦s át♦♠♦s srá r♥ ♣♦r ♠♣♦ s ♠♣t b srá ♦r♥ ♠t s♣♦ ♥trtó♠♦ b ≃ a/2 Pr♦ kBTamb ∼

140

eV ≪ Ec ∼ eV ② ♦♠♦ ♥ ♥ ♦s♦r r♠ó♥♦ ♠♣t ró♥ s ♣r♦♣♦r♦♥ rí③ r ♥rí

b ≃a

2

√

kBTamb

Ec≃

a

30≪ a,

♦♥ ♠♦s s♣st♦ q Ec ≃ 5 ❱ ② q T = 300 s ♣s ♥ ♥tró♥ ♣rs

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

Pr♦ ∑

R′

∇Φ(R−R′

)

s r③ q tú s♦r át♦♠♦ R t♦♦s ♦s ♠ás át♦♠♦s rst ② ♥ss ♣♦s♦♥s qr♦ s r③ s ♥ ② ♣♦r t♥t♦

Utotal ≃ U eq + Uarm

U eq =N

2

∑

R

Φ (R)

Uarm =1

4

∑

R,R′

[∆ux∇x +∆uy∇y +∆uz∇z]2Φ(R−R′

)

=1

4

∑

R,R′

∑

µν

[uµ(R)− uµ(R

′)] ∂2Φ (R−R′)

∂ (R−R′)µ ∂ (R−R′)ν

[uν(R)− uν(R

′)].

♥♠♥t ♠t♦♥♥♦ q ② q rs♦r s

H =∑

R

P2(R)

2M+ Utotal,

s♥♦ P(R) ♠♦♠♥t♦ ♥ ó♥ ② ♣♦só♥ qr♦ s R

P♦r t♥t♦ ♣♦♠♦s r ♦ ♣rt U eq trtr ♦s ♣r♦♠s ♥á♠♦s ② q ♥♦♣♥ ♦s P ♥ ♦s u

♣r♦①♠ó♥ r ♥♦♠r ♣r♦①♠ó♥ r♠ó♥ ♦s tér♠♥♦s trr ②rt♦ ♦r♥ ♥ u r♥ ♥♦♠r ♥r♠ó♥♦s ♦s ♥ó♠♥♦s ís♦s s♦♥ ①♣s ♥ ♣r♦①♠ó♥ r♠ó♥ ② ♥tr♦ ♥♦s qr♠♦s ♣♦r ♠♦♠♥t♦

♥♦♠♥♠♦s

Φµν(r) ≡∂2Φ (r)

∂rµ∂rν,

s ♣ srr ♣rt r♠ó♥ ♥rí ♣♦t♥ ♦♠♦

Uarm =1

2

∑

R,R′,µ,ν

uµ(R)Dµν(R−R′)uν(R)

s♥♦Dµν(R−R′) = δR,R′

∑

R′′

Φµν(R−R′′)− Φµν(R−R′

).

st ♠♦ q ♥♦s ♣r♠t r ①♣rsó♥ ♣r tér♠♥♦ r♠ó♥♦ Utotal s ② q ♥ ♥r ♥rí ♥tró♥ ó♥ó♥ ♥♦ s ♥ s♠ ♥tró♥ ♥tr

♣rs Φ(R−R′) ② ♥ s srt♦ Uarm ♥ ♠♥r ♠ás s♠étr ♥ ♥ó♥ s Dµν s♦ ♥♦ t q á♦ sts ♠♥ts Dµν s ♠② í ♣s t♥t♦ stró♥tró♥ rst ♦♠♦ s ♦♥tró♥ ♥rí t♦t ♣♥ ♣♦só♥ ♦s ♦♥s ♥ ♠♥r ♠② ♦♠♣ ♦ q ♣rát♠♥t ♠♣♦s ♣r♦①♠r ♦♥ ♣rsó♥

• qí s ♦♥ ♦r s ♥tr♦ ♣r♦①♠ó♥ át q s♣♦♥ q ♠♦♠♥t♦ ♦s ♦♥s s ♦ s♥t♠♥t ♥t♦ ♦♠♦ ♣r q ♦s tr♦♥s sté♥ s♠♣r ♥ st♦ ♥♠♥t♦rrs♣♦♥♥t ♦♥ró♥ ♥st♥tá♥ ♦s ♦♥s st♦ s ♣ ♥t♥r ♥ ♠♥r

r q t♦♦ st♦ s á♦ ú♥♠♥t ♣r rsts rs st ♦♦ ♥♦ ♣♦♠♦s r♣tr♦ trtr ♥rí ♦só♥ st♦ s ♦s tr♦♥s s ♦♠♦♥ t♥ rá♣♠♥t ♠♦♠♥t♦ ♦s ♦♥s q ♣♦♠♦s trtr st♦ tró♥♦ ♥

♥ ♥st♥t tr♠♥♦ ♦♠♦ s r ♥ ♥ó♥ ♥í♦ s ♦♦r♥s ♦s ♦♥s ♥ s ♥st♥t

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó ♥á♠ r rst♥

♥tt r♦r♥♦ q s ♦s tí♣s ♦s tr♦♥s s♦♥ ♦r♥ ♦ r♠ vF ∼ 108 ♠s ♠♥trs q s ♦s ♦♥s s♦♥ ♦r♥ 105 ♠s st ♠♥r s♠♦ ♠♣ír♠♥t ♥tró♥ ♥tr ♦s ♦♥s ♦♥ tér♠♥♦s q ♥♦s ♥ ♣♥♥ ♦♥ ♣♦só♥ ♦s ♦♥s ♥rí tró♥ ♦♥sr♥♦ ♦♠♦ ♣rá♠tr♦s ♦s ♠♥t♦s ♠tr③ D q s st♥ ♦s rst♦s ①♣r♠♥ts

❯ cv ♦s ss ♥♦s

♦t ♠♣♦rt♥t♦♠♦ ② s s♦ ♦r s♣í♦ ♦s só♦s ♥ r s ♠str♥ ♦srst♦s ①♣r♠♥ts r r ② ❳ qs♦♥ só♦s s♣♠♥t s♥♦s s ♣♥rs♠r s♥t♠♥t ♦♠♦

Pr T ∼ 100 ♦r s♣í♦s ♣rát♠♥t ♦♥st♥t ② r♥♦ ♦rás♦ ♦ ♣♦r ② ♦♥ ② Ptt cDP

v

T ≪ 100 ♦r cv s♠♥②r♥♠♥t ♠♣é♥♦s q cv → 0 ♥♦T → 0

q ts T ♥♦ s ♦t♥ ①t♠♥t ♦r cDP

v ♣ rs q ♥ ♣rt ♣r♦①♠ó♥ r♠ó♥ Pr♦ ♦s rst♦s s t♠♣rtrs ♥♦ ♣♥ rs s♦ ♠♥r q s ♥sr♦ r ♥ sr♣ó♥á♥t ♥á♠ r ♣r ①♣r ♦ ♦♠♣♦rt♠♥t♦

• ♥ ♠r♦ ② ♥ r♥st♥ ①♣♦♥ ♥ ♣r♦①♠ó♥ r♠ó♥ t♥t♦♥ trt♠♥t♦ ás♦ ♦♠♦ ♥ á♥t♦ ♠t♦♥♥♦ r♠ó♥♦ s rát♦ ♥ ♦ss♣③♠♥t♦s u(R) ② ♥ ♦s ♠♦♠♥t♦s P(R) ♦s ♦♥s ② r♣rs♥t ♥ s♦ s♣ ♣r♦♠ s ♦s♦♥s ♣qñs q ♣ rs♦rs ♠♥r ①t

st ♠♦♦ s♦ó♥ ♣r ♥str♦ ♦♥♥t♦ N ♦♥s s ♥ ♦♠♥ó♥ ♥ ♦s 3N♠♦♦s ♥♦r♠s sst♠ ♥♦ ♦s s stá rtr③♦ ♣♦r ♥ r♥ ♥rs♣í ω ❨ ♥♦ st♦s ♠♦♦s s ♥♣♥♥t ♦s ♠ás❨ ♦♠♦ s ♦ st♦ s t♠é♥ rt♦ ♥ s♦ á♥t♦ ♣r s♦ rst r♠ó♥♦ ♠♥r q t♥r♠♦s 3N ♦s♦rs ♦♥ ♥rís

(n+ 1

2

)~ω n ∈ N ② ♥rí t♦t sst♠

srá3N∑

i=1

(ni +

1

2

)~ωi.

• érs q ♥ s♦ ♦s tr♦♥s s s ♥ ♥á♠ s♠ás ♣♦rq s ♠♣íq λe ≪ l ♥ st s♦ s♥ ♠r♦ st♠♦s ♥trs♦s ♥ t♦♦s ♦s r♥♦s ♦♥t ♦♥ s r♦♥s ② s ♣♦r ♦ q ♥st♠♦s r ♥ sr♣ó♥ á♥t ♣r♦ ♦♥ ♥t q ♣r♦①♠ó♥ r♠ó♥ ♥♦s ♣r♠t srr sst♠ rt♠♥t ♥ tér♠♥♦s ss♠♦♦s ♥♦r♠s

♥ s í♠t ♦s ♠♦♠♥t♦s tó♠♦s ♥ t♥r ♠② ♣qñ ♠♣t ♣♦r ♦ q ♣r♦①♠ó♥ r♠ó♥ sr♠② ♥

st♦ s st♦ ♣r♠♥t ♥ ♦trs s♥trs r♦♥s ♦♥ á♥ ♣♦r ♠♣♦ ♥ ❱r♦♥s ② ♥s

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❯ ❱ ❯

7.3 ❯ ❱ ❯

❯ rst rs

♦♥srr♠♦s ♥ ♦♥♥t♦ N ♦♥s ♠sM s♦r ♥ r rs ♥♠♥s♦♥ s♣ró♥ a ♠♠♦s un s♣ró♥ ó♥ st♦ ♥ ♣♦só♥ na rs♣t♦ s♣♦só♥ qr♦

st♠♦s ♥trs♦s ♥ srr ♦s ♠♦♦s ♦♥t♥s sst♠ ♥tró♥ ♥tr ♦s♦♥s s♦♦ s r③ ♥tr ♣r♠r♦s ♥♦s ♥rí ♣♦t♥ r♠ó♥ s

Uarm =1

2K∑

n

(un − un+1)2 s♥♦ K = Φ′′(a),

② ♦♥ Φ(x) s ♥tró♥ ♥tr ♦s ♦♥s s♣r♦s ♥ st♥ x ó♥ ♠♦♠♥t♦ sst♠ s

Mun = −∂Uarm

∂un= −K

(2un − un−1 − un+1

),

q s st♠♥t ó♥ q ♠♣rí un s ó♥ str ♦♥t♦ ss ♥♦s ♠♥t♠s s♥ ♠s ② ♦♥st♥t K

♦♠♦ ② ♠♦s ♦♠♥t♦ ♦trs s ♥♦ st♠♦s ♥trs♦s ♥ t♦s s♣r qí ♦r ♦ ①tr♠♦ ♥ át♦♠♦s ♠♥r q ♣♦♠♦s s♦r ♠♥r ♠t♠át♠♥t♠ás ó♥ ♣r trtr ♦s ♦♥s q stá♥ st♦ ♥ ♦s ①tr♠♦s P♦r s♦ s ♦♥♦♥s ♦♥t♦r♥♦ ♦r♥♦♥ r♠♥ s♦♥ ♣ráts

uN+a = u1 u0 = uN .

s♠♦s ♥ s♦ó♥ ①t♥ t♦ ♥ t♣♦ ♦♥ r exp[i (q · r− ωt)

] ♦ q

s♥ ♥ st s♦ q ó♥ n st♦ ♥ na s ♠ sú♥ ②

un(t) ∝ ei(qna−ωt)

② s ♦♥♦♥s ♦♥t♦r♥♦ ♥♦s ♥ q ♥♦ t♦♦s ♦s ♦rs q s♦♥ á♦s ♣s ♥ ♠♣rq eiqNa = 1 ♦ ♦ q s

q =2π

a

m

Nm ∈ Z.

♦♠♦ un(q, t) = un(q ± 2πa , t) ♥t♦♥s s♦♦ ② N ♦rs q ♣r♠t♦s ♣♦r s q

♥ r s♦♦♥s st♥tss♦r♠♦s q♦s ♦rs q q stá♥ ♥tr −π/a ② π/a st♦ s t♦♦s ♦s q stá♥ ♥tr♦ P❩

♦r sstt♠♦s un(t) ∝ ei(qna−ωt) ♥ ♠♦♠♥t♦

−Mω2ei(qna−ωt) = −K(2− e−iqa − eiqa

)ei(qna−ωt) =⇒Mω2 = 2K (1− cos qa)

ω(q) =

√2K(1− cos qa)

M=⇒ ω(q) = 2

√K

M

∣∣∣ sin qa2

∣∣∣

q ♥♦s ró♥ s♣rsó♥ sst♠

trt♠♥t♦ ♠t♠át♦ s é♥t♦ ♣r ♠♦♦s ♦♥t♥s ② ♣r ♠♦♦s tr♥srss s♦♠♥t ② q ♦♥srrq s ♦♥st♥ts r③ srá♥ r♥ts

♥ ♥ rst tr♠♥s♦♥ st♦ s♥rí q qr♠♦s srr s r♦♥s ♦s ♦♥s q s ♣r♦♣♥ ♥ qsr♦♥s ♥ s q ♣♦r③ó♥ ♦♥ s ♦ ♥ ♣r♠♥t tr♥srs ♦ ♥ ♣r♠♥t ♦♥t♥ ♥ r♦♥s ♠ás♥rs ♣r♦♣ó♥ ♣♦r③ó♥ ♥♦ s ♥ ♥ ♦s ♥ ♦tr st♦ ♥♦s r s r♦♥s [100] [110] ② [111] ♥rst ú♦ ♥ s q ♣r♦♣rs ró♥ ② ♣♥♦s ♥tr♦s ♦♥s q r♥ ♥ s ♦r♠ q ♠♦♠♥t♦ s♥♠♥s♦♥

s ♦r♠ q t♥ ♥ó♥ ω(q)

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó ♥á♠ r rst♥

❯ ó♥ s♣rsó♥

♦♠♦ ② N ♦rs ♣r♠t♦s q ② ♦s ♣♦ss ♠♦♠♥t♦s ♦s ♦♥s ♣r qt♥♠♦s ♥ t♦t 2N s♦♦♥s ♥♣♥♥ts ♦♠♦ ♣♦♠♦s s♣r N ♣♦s♦♥s ♥s ② N♦s ♥s stá r♦ q ♣♦♠♦s ♦♥strr ♥ ♦♠♥ó♥ ♥ s 2N s♦♦♥sq s rq♥ ② ♥♦♥trr s♦ó♥ ♦♠♣t ♣r♦♠ ♣♥t♦ ♥ ♠r♦ s♦♠♥t② N ♠♦♦s ♥♦r♠s ② q ♦s♥♦ ② s♥♦ r♥ s♦♦ ♥ ♥ s

♦ s ♦♥ s c = ω/q ② s ♦ r♣♦ s ♠♥t∂ω/∂q ♠s ♣♥♥ ró♥ s♣rsó♥ ② ♥ ♥r ♥♦ s♦♥ ♦♥st♥ts ♥ r s♣rsó♥ ♥ r ♦♥ s ♦sr q

∂ω

∂q= 0 ♣r q = ±π/a

ω(q) = a√K/M |q| ♥ í♠t ♥ q q ≪ π/a

st út♠ ró♥ ♥ ♥♦ q ≪ π/a s ró♥ s♣rsó♥ t ♣r ♦♥s ♠♥♦ss② ústs ♥ ♠♦s ♦♥t♥♦s

P♦♠♦s ♣s ♦♥r q s q sst♠ q st♠♦s trt♥♦ s srt♦ ② ♥♦♦♥t♥♦ ② ♥♦ ①st ♥ ω ∝ q ② s ♣r♦ s♣rsó♥

• ♥♠♦s ♥tr♦♥s ♦♥ ♠ás ♥♦s st♦ s ♠♦s ♠ás á ♣r♦①♠ó♥ ♣r♠r♦s ♥♦s ró♥ ω(q) s ♠ás ♦♠♣ ♣r♦ s ♥♦ N ♠♦♦s ♥♦r♠s un(t) ∝ei(qna−ωt) rá♥♦s t♠é♥ q ♣r í♠t q → 0 ró♥ s♣rsó♥ s s♥♦ ♥ω(q) ∝ q ② q ♣r q = ±π/a ♥t♦♥s ∂ω/∂q = 0

• ♠s♠♦ ♠♦♦ s trt♥ s r♦♥s tr♥srss q s♥ t♥r ♥ r♥ ♠s ♦♥st♥t ♠ s ♠ás ♥ ♥ s s♦

7.4 ❯ ❯

st st♦ ♦♠♦ t♠é♥ s ♦ ♥ s♦ ♥tr♦r s á♦ t♠é♥ ♣r rststr♠♥s♦♥s ♥ ♦s q ♥♦ ♦s ♣♥♦s ♣r♣♥rs ró♥ ♣r♦♣ó♥♦♥t♥♥ s♦♠♥t ♥ t♣♦ ♦♥s st♦s ♣♥♦s s ♥♥tr♥ ♣♦r ♠♣♦ ♣r ró♥ [111] strtr ② ró♥ [100] strtr s

trt ♦s ♠♦♠♥t♦s ♥♠♥t ♥♣♥♥ts cos (qna− ωt) ② sin (qna− ωt) ♥ ♥ sst♠ tr♠s♥♦♥ ♥ ró♥ ♥r r ♣s s ♣ srr ♦♠♦

∑

q>0

(

Aqeiq·R +A∗

qe−iq·R

)

,

♦♥ Aq ∝ e−iωt ♦t sts ♦s ①♣♦♥♥s s♦♥ t♠é♥ ♥♠♥t ♥♣♥♥ts ② ♣♦r t♥t♦ sr♥ ♦♠♦ ♥ s ♦♠♦♦ r♥ ♥ts sin ② cos

q = ±π/a s ♦t♥ ♥ ♦♥ st♦♥r ríqs q s rt♦ ② ♣♦r ♦ t♥t♦ s ♦t♥ ∂ω/∂q

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❯ ❯

❯ ♦♥ s ♦

♣♦♥♠♦s ♣s ♥ r rs ♦♥ s ♦s ♦♥s st♦s ♥ qr♦ ♥ na ② ♥na+ d rs♣t♠♥t ♥ s♦ ♥r ♦s ♦♥s s♦♥ st♥t♦s ② d ≤ a/2 ♠♥r q r③♥tr ♦s ♦♥s ♣♥ s s♣ró♥ ♥tr ♦s s d ♦ a−d Pr s♠♣r s♥ ♠r♦ ♠♦s s♣♦♥r q ♦s ♦♥s t♥♥ ♠s♠ ♠s ② q s♦♠♥t ② ♥tró♥ ♥tr ♣r♠r♦s

♥♦s ② ♣s ♦s ♦♥st♥ts r③ G ② K ♦♥ K ≥ G

♣♦t♥ r♠ó♥♦ rst s

Uarm =K

2

∑

n

[u(1)n − u(2)n

]2+G

2

∑

n

[u(2)n − u

(1)n+1

]2,

♦♥ u(1)n s s♣③♠♥t♦ ó♥ ② ♣♦só♥ qr♦ s na ♠♥trs q u(2)n s ó♥ ♦♥ ♣♦só♥ qr♦ na+ d

s ♠♦♠♥t♦ s♦♥

Mu(1)n = −∂Uarm

∂u(1)n

= −K[u(1)n − u(2)n

]−G

[u(1)n − u

(2)n−1

]

Mu(2)n = −∂Uarm

∂u(2)n

= −K[u(2)n − u(1)n

]−G

[u(2)n − u

(1)n+1

].

qr♠♦s ♥ s♦ó♥ q s ♣r♦♣ ♣♦r t♦ r s♦ó♥ sr ♥ ♦♥ q t♥♥ r♥ ♥r ω ② ♥ t♦r ♦♥ q s s ♠s♠ ♦♥ r ♣r t♦ rt♥t♦ ♣r u(1)n ♦♠♦ ♣r u(2)n

u(1)n = ǫ1ei(qna−ωt) u(2)n = ǫ2e

i(qna−ωt).

qí ǫ1 ② ǫ2 s♣♥ ♠♦♠♥t♦ rt♦ ó♥ ♥ ♣r♠t ♣s ♦ q s♣ró♦ s ♣r♠t

stt②♥♦ ♥ s ♠♦♠♥t♦[Mω2 − (K +G)

]ǫ1 +

[K +Ge−iqa

]ǫ2 = 0

[K +Geiqa

]ǫ1 +

[Mω2 − (K +G)

]ǫ2 = 0

q s ♥ sst♠ ♦♠♦é♥s ♥ ǫ1 ② ǫ2 q ♥ t♥r ♥ s♦ó♥ ♥♦ tr st♦ sst♥t r♦ st♦ ♦ r r♦ tr♠♥♥t ♦s ♦♥ts ♦t♥é♥♦s q

[Mω2 − (K +G)

]2= |K +Ge−iqa|2 = K2 +G2 + 2KG cos qa

♦ ♥♦ ♥ ♣♦♦ ár(Mω2

)2 − 2mω2 (K +G)− 2KG (cos qa− 1) = 0.

s ♦♥st♥ts r③ ♦rrs♣♦♥♥ s s♥s rs Φ′′(a− d) ② Φ′′(d) ❯♥♦ strí t♥t♦ srr

u(2)n = ei(q(na+d)−ωt) = u

(1)n eiqd

q s ♥ ①♣rsó♥ ♣r u(2)n = ǫ2ei(qna−ωt) ♥ ♠r♦ ② ♦ q qmax = π/a ② d < a ①♣♦♥♥ eiqd ♥♥♣ r −1 st♦ s q qd = π ♣♦r ♦ q ♦s ♦♥s ② ♥♦ ♣♥ rr ♥ ♦♣♦só♥ s P♦r t♥t♦ ♥tr♦ó♥ ǫ1 ② ǫ2 ♣r♠t r ♠ás ① ♦♥ r rst ♥②♥♦ qr ss ♥tr ②

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó ♥á♠ r rst♥

s♦ó♥ s ó♥ s♥♦ r♦ ♥♦s ró♥ s♣rsó♥ ω2(q)

ω2±(q) =

K +G

M±√K2 +G2 + 2KG cos qa

M,

q s r♣rs♥t ♥ r

❯ ó♥ s♣rsó♥ ♣rs ♦

♥ s rs ω(q) s ♠♥ r♠s

ró♥ s♣rsó♥ sr♠♦s ♣sq ♥♦ t♥♠♦s ♦s át♦♠♦s ♦ ♦♥s ♥ s ♣r♥ ♥ s♥ r♠ r♠

ó♣t q s ñ r♠ úst q② í♠♦s ♦t♥♦ ♥ s♦ rst rs

stt②♥♦ st♦s ♦rs ω2 ♥ s♦♥s q ♥♦s r♦♥♥ ǫ1 ② ǫ2 s♦t♥

ǫ1ǫ2

= ∓ K +Geiqa

|K +Geiqa|♣r ♥♦ ♦s N ♦rs q ♣r♠t♦s② ♣♦r ♦ t♥t♦ 2N ♠♦♦s ♣♦ss ♥ rst♦s s♦♦♥s ♣r q r♦ ♦♥ ♦s2N r♦s rt sst♠

❱♠♦s ♦r str s ♦s s♦♦♥s ω+ ② ω− srr♦♥♦ s s♦♦♥s ♣r q → 0 ②♣r q → π/a

• Pr q ≪ π/a s t♥ q cos qa ≃ 1− (qa)2

2 ②

ω+ =

√2(K +G)

M−O(qa)2 ǫ1 = −ǫ2 ♦s ♦♥s stá♥ ss♦s ♦ ♥tr♦

ω− =

√KG

2M(K +G)qa ǫ1 = +ǫ2 ♦s ♦♥s s ♠♥ ♥ s ♥tr♦

❯ ♦♠♥t♦ s ♦ q ≪ π/a

• Pr q = π/a s t♥

ω+ =

√2K

Mǫ1 = −ǫ2 ♦s ♦♥s stá♥ ss♦s ♦ ♥tr♦

ω− =

√2G

Mǫ1 = +ǫ2 ♦s ♦♥s s ♠♥ ♥ s ♥tr♦

♦r ♥ ♣r q = π/a

unun+1

∝ eiqna

eiq(n+1)a= e−i(π/a)a = e−iπ = −1

② ♣♦r t♥t♦ ♦s ♠♦♠♥t♦s ♥ s s ♥s stá♥ ss♦s ♦

s♦ó♥ ω+ s ♠ r♠ ó♣t ② ω−s ♠ r♠ úst

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❯ ❯

❯ ♦♠♥t♦s s ♦ q = π/a ♠♦♦s ♦♥t♥s

♥ s♦ ♣♦r③ó♥ tr♥srs ♠♦♠♥t♦ srí r

❯ ♦♠♥t♦s s ♦ q = π/a ♠♦♦s tr♥srss

♦♠♦ ♠♦s ♥ r♠ ó♣t t♥t♦ ♣r q → 0 ♦♠♦ ♣r q = π/a ♦s ♦♥s r♥ ss♦s♥ ♦ r♥ ♥♦ ♦♥tr ♦tr♦ ♣r♦ ♦♥ s ♥tr♦ ♠ss ♦♥ s♦ rsts ó♥♦s ♦s ♦s ♦♥s t♥♥ rs s♥♦ ♦♣st♦ ② ♣♦r ♦ t♥t♦ st t♣♦ ♠♦♠♥t♦ ♦ ♣♦r♠♦s ①tr ♦♥ ♠♣♦ étr♦ ♥ ② ♣♦r s ♠♦t♦ st r♠r ♥♦♠r r♠ ó♣t st ♥ó♠♥♦ s q stá s♦♦ rt ♦s rstsó♥♦s q ♠♥♦♥á♠♦s ♥ ♥tr♦ó♥ t♠

• ♣♦♥♠♦s ♦r q ♥ s ♦♥st♥ts ♠ ♥ s ♥tr♦♥s s ♠♦♠②♦r q ♦tr K ≫ G ♥ st s♦ srr♦♥♦ ró♥ s♣rsó♥ t♥♠♦s

ω+ =

√2K

M[1 +O(G/K)] ǫ1 ≃ −ǫ2 r♠ ó♣t

ω− =

√2G

M[1 +O(G/K)]

∣∣∣ sin(1

2qa

) ∣∣∣ ǫ1 ≃ ǫ2 r♠ úst

P♦r t♥t♦

r♠ ó♣t ♣♥s ♣rs♥t ró♥ ♦♥ q ② s r♥ s ró♥ ♥♠♦é tó♠ ♦♥ ♦s ♠ss M ♥s ♠♥t ♥ rs♦rt ♦♥st♥t Kst♦ s ♣ ♦♠♣rr ♦♥ ♠ét♦♦ tr♦♥s rt♠♥t ♦s ♦♥ ♥ ♦♣♠♥t♦ é♥tr ♦s ♦rts r ♥s strs qí ♥ ♦♣♠♥t♦ é ♥tr s s ♥ró♥ s♣rsó♥ ♣♥

r♠ úst s ♣rát♠♥t ♥ ♥ át♦♠♦s ♠s 2M ♥♦s♠♥t ♥ rs♦rt é ♦♥st♥t G ♦♠♦ K s ♠② r♥ ǫ1 = ǫ2 ♦r♠ q ♦s ♦♥s s♠♥ ♥ ♦♥ ♠s♠ s

• ♥ ♥r ② sú♥ ♦ ♦ ♠ás rr ♣ ♦rrr q s r③s ♥tr♥s s♠♦é ♦♠♥♥ ♦s ♠♦♦s ó♣t♦s ♦♥ ♦ q s r♠ ó♣t srá str ② s♣r sr♠s ústs ♦ r♣♦ st r♠ s ♣qñ ② q s ♠② ♣♥ ♣♦r ♦ qrá ♥ ♣♣ ♠② ♣qñ♦ ♥ tr♥s♣♦rt ♥rí trés r rst♥

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó ♥á♠ r rst♥

• s♦ tr♠♥s♦♥♥ s♦ tr♠♥s♦♥ s ♠♠♦s p ♥ú♠r♦ ♦♥s s ② 3p ♠♦♦s ♥♦r♠s

♣r ♦r q sts 3p r♠s trs s s♦♥ ústs

(♣r s ♥ í♠t q → 0 ró♥ s♣rsó♥

s ♥ ω(q) ∝ q)② t♦s s 3p− 3 rst♥ts s♦♥ ó♣ts

(♥ s q ω(q = 0) 6= 0

)

óts q st♦ s ♥ ♥r③ó♥ s♦ rst♥♦ s r♦♥s ♥ ♥ ♠♦é ♦♥ pát♦♠♦s ♠♦é t♥ trs r♦s rt trsó♥ ② 3p−3 r♦s rt ró♥

♥ st s♦ ♥♦ s♦♠♥t ② q ♦♥srr s r♦♥s s♣rsó♥ ωs(q) ♥ st♥tsr♦♥s q s♥♦ t♠é♥ str ó♠♦ rí ♣♦r③ó♥ ǫs(q) ♥ ♥ó♥ ró♥ ♣r♦♣ó♥ q

♥ ♥ ♠♦ sótr♦♣♦ s s q s ♣♥ s♦r s s♦♦♥s ♦r♠ q ♣r q♥ s r♠s sté ♣♦r③ ♦ r♦ ró♥ q st♦ s ǫ ‖ q ② s ♦trs ♦s r♠s♣♦r③s ♣r♣♥r♠♥t ǫ ⊥ q ② ♦s r♠s tr♥srss ② ♥ ♦♥t♥

♥♦ s♦tr♦♣í s♣r ② ♥ rst s ♥ ♠♦ r♠♥t ♥sótr♦♣♦ st♦ ♥♦ s rt♦♥ ♥r ♣r♦ s s r♥♦ q ② ♥ r♠ ♦♥t♥ ② ♦s tr♥srss ♣r rtsr♦♥s t s♠trí P❩ s ♣♦r s r③ó♥ q s s ♥♦ r♠s ♦♥t♥s② tr♥srss ♥s♦ ♥ r♦♥s st♥ts ss t s♠trí ♥ st s♥t♦ s ♥trs♥t♦♠♣rr ♣r♦♣ó♥ s ♥ ♠♦s ♥sótr♦♣♦s ♦♥♦ ♥ s s♥trs Ó♣ts ♣r♥♣s t ♦♥ ssó♥ ♥tr♦r

• ♦t s♦r r♠s ústs ② ♦♥st♥ts ástss r♦♥s r ♦♥t ♦♥ ♦rrs♣♦♥♥ts ♥ r♠ úst ♣♥ r♦♥rs

♦♥ s ♦♥st♥ts ásts rst ② q ♣r♦♥ t♥s♦♥s ♥♦r♠s s♦r ♦ú♠♥s r♥s rst s ♥ ♣♥t♦ st ♠r♦só♣♦

❯ ♥ ♦♥t♥

• ♥ ♥ rst ú♦ ♦♥ ♥ ró♥ ♣r♦♣á♥♦s ♦r♦ ❳ q ♦♥ ♦♥ ♥ ú♦ rsts ♦♥ s ♦♥t♥ s t♥ ♥ ♦♥ ♦♠♣rsó♥s♥ q ♣r③♥ t♥s♦♥s trs és r ♠♠♦s ♠ó♦ ① C11 ♠♥t q r♦♥ r③ s♦r ♣♥♦ rst♥♦ ♦♥ ♦♥ó♥ ró♥

Fx = C11∂ux∂x

,

t♥r♠♦s

Fx (x+ δx)− Fx =∂Fx∂x

δx

=⇒ ρ∂2ux∂t2

= C11∂2ux∂x2

,

s♥♦ ρ ♥s ♠♦st ♠♦♠♥t♦ ♠r♦só♣ ♥♦s ♣r♦♣♦r♦♥ ♦ sts ♦♥s ♦♥t♥sásts

√C11/ρ q sr ♦♥♥t ♦♥ í♠t q → 0 s r♦♥s rst♥s q

♠♦s st♦

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

PÓ ❯ P ❯

• P♦r ♦tr ♣rt s s♠♦s ♣♥s♥♦ ♥ ♥ ró♥ ♣r♦♣á♥♦s ♦ r♦ ♠s♠♦ ❳q s ♦♥♥t ♦♥ ♥ ú♦ rst s ♦♥ s trsrs és r ♣r ♥t♥só♥ ③r r♦♥ ♦♥ ♠ó♦ r③ C44 s ♥

Fx = C44∂uy∂x

Fx (x+ δx)− Fx =∂Fx∂x

δx =⇒ ρ∂2uy∂t2

= C44∂2uy∂x2

,

q ♦rrs♣♦♥ ♥ ó♥ ♥ ♦♥ ② ♦ s√C44/ρ

❯ ♥ trsrs

• ♥ ♦trs r♦♥s s♥ ♠r♦ s ♦s ♦♥ s r♦♥♥ ♥ ♦r♠ ♠♦♠ás ♦♠♣ ♦♥ ♦s st♥t♦s ♦♥ts r③ ♠tr

7.5 PÓ ❯ P❯

st ♦r ♠♦s st♦ st♥♦ s ♥ ♣♥t♦ st tór♦ s r♦♥s s♣rsó♥ ♦s ♠♦♦s ♥♦r♠s ♥ rst ❨ ♦ ♠♦s ♥♦ ♥♦ ♥ s ♥s ♦♥st♥ts ásts rst s Dµ,ν K G t q ♥ ♣r♥♣♦ ♥♦s s♦♥ s♦♥♦s ② q s♠♦s q s♦♥ ís r

P♦r s♦ s ♠♣♦rtrs ♣♥trs s s ♣♦s ♠r ①♣r♠♥t♠♥t s r♦♥s s♣rsó♥ ω(q)

♠♦s ♥r ♥ ③ ♥tr♦♥s s♦r ♥ rst s♣rr q ♦s ♥tr♦♥s trs♥s♥ ①s t rst ♥ t♦ ♦s ♥tr♦♥s ♥♦ t♥♥ r étr ② ♣♦r ♦♥s♥t♥♦ ♥tr♦♥♥ ♦♥ ♦s tr♦♥s P♦r ♦♥s♥t ♦s ♥tr♦♥s q trs♥ rst ♥r ♥tr♦♥♦ ♦♥ ♣♦só♥ ♦s ♦♥s ♠s♠♦ q ♦♥t♥♥ ♦s ♥ú♦s st♦ s ♦♥♦s ♦♥♦♥s s ♥tr♣rt ♥t♦♥s q ♥rí ♣r ♦ ♥ ♣♦r ♥ ♥tró♥ trsr rst s ró♥ ♦ s♦ró♥ ♦♥♦♥s ♠r ♦s á♥♦s ② s ♥rís s ♦s ♥tr♦♥srst♥ts s ♣ ①trr ♥♦r♠ó♥ rt s♦r s♣tr♦ ♦s ♦♥♦♥s

♣♦♥♠♦s q p s ♠♦♠♥t♦ ♥ ♥ ♥tró♥ ♥♥t ♥rí E = p2/2Mn qs rst ♦♥ ♠♦♠♥t♦ p′ ② ♥rí E = p′2/2Mn ♣♦♥♠♦s ♠ás q ♥á♠ ♦s♦♥s rst s ♣ srr ♥ ♣r♦①♠ó♥ r♠ó♥ ② q ♦♠♥③♦ ①♣r♠♥t♦ só♦ stá ♥ ♥ st♦ ♥ q ② nq,s ♦♥♦♥s t♣♦ q, s q s♠♣♠♥t r♣rs♥t♠♦s♣♦r nq,s ♥ st♦ só♦ s n′q,s t♥r♠♦s q

E′ − E = −∑

q,s

~ωq,s∆nq,s ∆nq,s = n′q,s − nq,s,

♦r♠ q s ② s♦ró♥ ♦♥♦♥s t♥r♠♦s q E′ > E ② q nq,s > n′q,s

♦s ♥tr♦♥s s♦♦ ♥tr♦♥♥ rt♠♥t s ♦♥ ♦s ♥ú♦s q ♦♣♥ ♦r♥ 10−15 ♦♠♥ ♦ ♠♥t ♦♣♠♥t♦ ♦♥ ú♥ ♣♦s ♠♦♠♥t♦ ♠♥ét♦ ♦ q t♥ ♠♣♦rt♥ ♣r st♦ ①♣r♠♥t

♦s só♦s ♦♥ ♣r♦♣s ♠♥éts

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó ♥á♠ r rst♥

P♦r ♦tr ♣rt ♥tró♥ ♥tró♥ó♥ ♥tr♦ ♥ ♣r♦①♠ó♥ ♥ q s♣♦♥♠♦s qstá srt ♣♦r ♥ ♥tró♥ ♣rs s

Hn−i =∑

R

W(r−R− u(R)

)

s♥♦ r ♦♦r♥ ♣♦só♥ ♥tró♥ st ♥tró♥ ♥♦ rí s ♠♠♦s r ♣♦r r+R0♦♥ R0 ∈ ② u(R) ♣♦r u(R−R0) ② q

∑

R

W(r+R0 −R− u(r−R0)

)=∑

R

W(r− (R−R0)− u(R−R0)

)

=∑

R′

W(r−R′ − u(R′)

)= Hn−i

st s♠trí ♠t♦♥♥♦ ♠♣ ①st♥ ♥ ② ♦♥sró♥ q ♥♦ s ♠strqí q ♣♦♠♦s srr ♦♠♦

p′ − p = −∑

q,s

~q∆nq,s + ~G ♦♥ G ∈ RR.

♥♠♦s ♠♦♠♥t♦ rst♥♦ ♦♥ó♥ ♦♠♦ ~q ♥t♦♥s ② ♦♥sró♥ ♥♦s q ♠♦ ♠♦♠♥t♦ ♥tró♥ s q s r ♥ ♠♦♠♥t♦ rst♥♦ t♦♦s♦s ♦♥♦♥s s♦ ♥ t♦r

② q rr ♣s ①st♥ ♥ ♠♦♠♥t♦ rst♥♦ ♦s ♦♥♦♥s ♥♦ ♠♣ q rst t♥ r♠♥t♠♦♠♥t♦ q ♣r♦♥ ♦s ♦♥♦♥s ♠♥r ♥ q♦s ♣r♦s♦s ♥ ♦s q ♣r③♥ ♦♥♦♥s ② tr♦♥s ♠♦♠♥t♦rst♥♦ sst♠ ♦♥♦♥s tr♦♥s s ♦♥sr s♦ ♥ ♥ ♥t ~G

• sí ♣s t♥♠♦s ♦s ②s ♦♥sró♥ ② s q s ♣ ①trr ♥♦r♠ó♥r s r♦♥s s♣rsó♥ ω(q) Pr r♦ r♠♥t str♠♦s r♦s s♦s

♦ ② ♦só♥ ♦♥ ♦♥♦♥s ♥ st s♦ ♥rí ♥ rst s ♥ ♠♦♦ q t♠♣♦♦ ♠ ♥rí ♥tró♥ ♦só♥ s ást P♦r t♥t♦ p′ − p = ~Gs♥♦ G ∈ ② s♥♦ p′ ② p ♦s ♠♦♠♥t♦s ♥s ♥tró♥ ♥♥t ② s♥t ♠♠♦sp′ = ~k′ ② p = ~p s ♠♣ q ♦ ♥ k = k′ ♦ ♥ k′ = k + G q ♦♥♥ ♦♥ s ♦s♦♥♦♥s ♦♥ ♣r s♣rsó♥ r②♦s ❳ st♦ r s♣rr ② q ♥ ♥tró♥ ♠♦♠♥t♦ p = ~k ♣ ♥tr♣rtrs ♦♠♦ ♥ ♦♥ ♣♥ t♦r ♦♥ k ♠♥r qs s♣rsó♥ s q♥t ♦só♥ ást ♥ ♦♥ tr♦♠♥ét q sr♥ s♦♥♦♥s ♦♥ P♦r t♥t♦ s♣rsó♥ ♥tr♦♥s ♦♥ ♥ r s♥ ró♥ ♥ s♦ró♥ ♦♥♦♥s ♠s♠♥♦r♠ó♥ r strtr rst♥ q s♣rsó♥ r②♦s ❳

♦só♥ ♦♥ ♥ ♦♥ó♥ ♥ st s♦ ♣♦♠♦s st♥r

s♦ró♥ ♥ ♦♥ó♥ E′ = E + ~ωs(q)

p′ = p+ ~q+ ~G

♠só♥ ♥ ♦♥ó♥ E′ = E − ~ωs(q)

p′ = p− ~q+ ~G

♥ ♦s s♦s ♥ ♦s q s s♦r ♦ s ♠t ♥ ♦♥ó♥ r♠ s ♦♥ t♦r ♦♥ q

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

PÓ ❯ P ❯

♦♠♦ rst q ωs(q±G) = ωs(q) ♣r t♦♦ G ∈ ♣♦♠♦s srr ♦♥sró♥ ♥rí ♦♠♦

s s♦r ♥ ♦♥ó♥p′2

2Mn=

p2

2Mn+ ~ ωs

(p′2 − p2

~

)

s ♠t ♥ ♦♥ó♥p′2

2Mn=

p2

2Mn− ~ ωs

(p′2 − p2

~

)

①♣r♠♥t♠♥t s ♦♥♦♥ t♥t♦ p ♦♠♦ E = p2/2Mn P♦r ♦♥s♥t s s♠♦s ♦r♠ ωs(q) s ♦♥s ♥♦s rá♥ ♦r p′ ♦r ♥ ♥ ♥r s t♥♠♦s ♥ ó♥(♥ s

)q ♥♦s r♦♥ s trs ♦♠♣♦♥♥ts p′ qr rs q ó♥ s♣

♥ ♦♥♥t♦ s♣rs ♥ s♣♦ p′ ② q st♠♦s ♥♦ p′ ♥ s s♣♦ ♣r♦ ①♠♦sq rq ♥ s ♦♥s

♥ rs♠♥ s s♦♥♠♦s ♥ ró♥ tr♠♥ ♣r ♦srr ♦s ♥tr♦♥s s♣rs♦s♣♦r rst ♥t♦♥s s♦♦ ♦srr♠♦s ♥ ♦♥♥t♦ srt♦ ♥rís E′ ♦rrs♣♦♥♥ts E′ = p′2/2Mn ♥ sts ♦♥♦♥s ♣♦♠♦s ♠r p′ − p ② E′ − E ♦♥ ♦ q s♠♦s ♣♦r q rst t♥ ♥ ♠♦♦ ♥♦r♠ r♥ (E′ − E)/~ ② ②♦ t♦r ♦♥ s±(p′ − p)/~

❱r♥♦ ♥rí ♥♥t ♦r♥tó♥ rst ② ró♥ ♥ q tt♠♦s♦s ♥tr♦♥s s ♣♦♠♦s ♦♥sr rst♦s ①♣r♠♥ts q ♥♦s ♣r♠t♥ ♦t♥r s♣tr♦ ♦♥♦♥s ♠str st♦ s rt♦ r♦ stá s s ♣♦s st♥r ♦s ♥tr♦♥srt♦s ♣♦r ♥ ♣r♦s♦ ♥ q ♥tr♥ ♥ ú♥♦ ♦tó♥ q♦s ♣r♦s♦s ♥ ♦s q♣rt♣♥ ♦s ♦ ♠ás ♦♥♦♥s

♦só♥ ♦♥ ♦s ♦♥♦♥s ♥ st t♣♦ ♣r♦s♦s ♥tró♥ ♣ s♦rr ♦ ♠tr♦s ♦♥♦♥s t♠♣♦ ♦ s♦rr ♥ ♦♥ó♥ ② ♠tr ♦tr♦ ♥ st út♠♦ s♦ s ♣ ♥tr♣rtrq ♥tró♥ ♦s♦♥ ♦♥ ♥ ♦♥ó♥ ♥♦s ♠♦s ♥ ♥ ♣r♦s♦ ♥ q s s♦r♥ ♦s ♦♥♦♥s s ②s ♦♥sró♥ s♦♥

E′ = E + ~ωs(q) + ~ωs′(q′)

p′ = p+ ~q+ ~q′ + ~G

② ♥♦s q q

E′ = E + ~ωs(q) + ~ ωs′(p′ − p

~− q

)

❯ ♦♥♦ ♦♥t♥♦ ♠t♦♥ó♥♦ ② ♣♦s ♠♦♥♦♦♥ó♥

st ó♥ ♥♦s q s ♦r q str tr♠♥♦ ② s ♠♦s ♥ró♥ tó♥ ♦s ♥tr♦♥s s♣rs♦s s♦♦ ♣rrá♥ ♦♥ ♥rís srts ♠r♦

ísst♦ó♦

❯

❯

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❮ Ó ♥á♠ r rst♥

q s t♦r ♦♥ ♥♦ ♦s ♦♥♦♥s s♦r♦s ② ♥♦ ♦ ♣♦♠♦s r rrá ♦ r♦ t♦ P❩ ② s ♥rís s ♦s ♥tr♦♥s ♥ ró♥ tó♥ rrá♥ ♦♥ stró♥ qst♦ ♥♦s ♥ q ♦s ♥tr♦♥s s♣rs♦s ♣♦r st ♣r♦s♦ ♦ srá♥ tt♦s ♥ ♥ ró♥tr♠♥ ♦♥ ♥ stró♥ ♦♥t♥ ♥rí

❯ s♣rsó♥ ♥tr♦♥s ♥

♥t♠♥t s♦ ♥♦ s♦♦ ♦rr ♥ s♦ ♣r♦s♦s ♦♥ ♦s ♦♥♦♥s s♥♦ ♥ t♦♦s ♦s s♦s ♥ ♦sq ♥tr♥♥ ♠ás ♥ ♦♥ó♥ ♣s t♥r♠♦s ♠ásrs ♥tr♠♥s q ♦♥s ♣r♦♥♥ts s ②s ♦♥sró♥ P♦r ♦ t♥t♦ s♠♣r srá♣♦s st♥r ♦s ♣r♦s♦s ♥ ♦s q ♥tr♥s♦♠♥t ♥ ♦♥ó♥ ② q ♥ s s♦ ♦♥tr②♥♦♥ ♣♦s ♥ ♥♦s ♣r tr♠♥s ♥rísr♥t stró♥ ♦♥t♥ ♦s ♣r♦s♦s♠t♦♥ó♥♦s

P♦r ♦tr ♣rt ♦s rst♦s ①♣r♠♥ts ró♥ ♥tr♦♥s ♥♦ ♥ ♣♦s t♥ r♣t♦s♦♠♦ ♦s q ♠♦s st♦ ♣r ♦s ♣r♦s♦s ♥ s♦♦ ♦♥ó♥ srí♥ ♥♦♥s δ r s♥♦ qstrí♥ s③♦s ♣♦r ♦s ♣r♦s♦s r♦s ♦♥♦♥s r③ó♥ ♦ str ♥ q ♦s rsts rs♥♦ s♦♥ r♠ó♥♦s ♠♥r q ♦s st♦sst♦♥r♦s ♣r♦①♠ó♥ r♠ó♥ ♥♦ s♦♥st♦s st♦♥r♦s ♥ rst r sí q rstr ♥♦ strá ♥ ♥ st♦ ♥♦ ♣♦r ♥ ♦♥♥t♦♥ tr♠♥♦ ♥ú♠r♦s ♦♣ó♥ ♦♥♦♥ss♥♦ ♥ ♥ ♦♠♥ó♥ ♥ ts st♦s

❯ s♣rsó♥ ♥tr♦♥s ♥

♥ t♦♦ s♦ s♥ ♠r♦ ♦s ♣♦s s s♥r♦♥♦r ♦♥ r st♦ s q s ♣r♦①♠ó♥ r♠ó♥ s s♥t♠♥t ♥ ♣rq sr♣ó♥ ♥ tér♠♥♦s ♦♥♦♥s s s♥♦á st♦ s q ♦s st♦s st♦♥r♦s r♠ó♥♦ss♥ ♥ ♥ ♣r♦①♠ó♥ ♦s ①t♦s rstr ♣♦♠♦s sr t③á♥♦♦s ♦♥ ♥ ♠ ♥t τfonones q sr ♥ ♠♥r♣r♦①♠ ♦ó♥ st♦ st♦♥r♦r♠ó♥♦ ♦♠♦ st ♠ ♥t s♦♥ ♥rt♠r ♦r♥ ~/τfonones ♥ ♥rí ♦tó♥ st♦ r s ②s ♦♥sró♥s ♣♦r s q ♥ts t③á♠♦s ② ♣rr ♦s ♣♦s s③♦s ①♣r♠♥ts

ísst♦ó♦

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❯

❯

❯

❯

❯

❯

♣ít♦

PP Ó

8.1 P❮

❨ s ♦♠♥t♦ q r t♠♣rtr ♦r s♣í♦ t♦♦s ♦s só♦s s♠♥②♣♦r ♦ s ♦r ás♦ ts t♠♣rtrs ♠ás ♣r T → 0 s ♦sr q cv → 0 s♥ s st♥♥ ♦s s♦s

s♥ts ♥ ♦s q cv ∝ T 3

ts ♣r ♦s q s ♣♦ss sr♥r ♦s ♦♥tr♦♥s cv ∝ AT +BT 3 ♠♦s q tér♠♥♦ ♥ ♥ T s ♦ ♦♠♣♦rt♠♥t♦ ♦s tr♦♥s ♦♠♦ ② s stó ♥ ♠ ♥ q s trtó ♠♦♦

st ♦♠♣♦rt♠♥t♦ ♥♦ ás♦ t♥♠♦s q ①♣r♦ ♦♥ ♦s ♠ét♦♦s q st ♦r ♠♦s ♦srr♦♥♦ ♦ ①♣ó♥ q s ♦♥só r ♥ sr♣ó♥ á♥t ♦♥stt②ó♥♦ ♦s ♣r♠r♦s tr♥♦s t♦rí á♥t só♦ ② t♦rí á♥t ♥ ♥r ♣s ♠♦♦ ♥st♥ ♥♦ ♦s ♣r♠r♦s é①t♦s r ís tó♠

♥ ♥ sst♠ á♥t♦ ♥s ♥rí tér♠ ♥ sst♠ ís♦ u ♣♦♠♦s ①♣rsr♦♠♦

u(T ) =1

V

∑i εie

−εi/kBT

∑i e−εi/kBT

= − 1

V

∂

∂β

[ln∑

i

e−βεi

],

♦♥ T s t♠♣rtr rst β ≡ 1/(kBT ) s εi s♦♥ s ♥rís ♦s st♦s st♦♥r♦s ♠t♦♥♥♦ ♣r rsts rs

H =∑

R

P2(R)

2M+ U eq +

1

2

∑

R,R′,µ,ν

uµ(R)Dµν(R−R′)uν(R).

② s♠t♦r∑

i stá ①t♥ t♦♦s ♦s st♦s i ♣r♠t♦s sst♠♦♠♦ ♠♦s str ♣♥♥ u(T ) s r♦ q ♣♦♠♦s trtr ♠t♦♥♥♦ ♥tr♦r s♥♥s t♥r ♥ ♥t ♦♥tró♥ U eq ② q ést ♥♦ ♣♥ t♠♣rtr

♠é♥ s ♦♠♥t♦ ♥tr♦r♠♥t q sr♣ó♥ á♥t s ♦s♦♥s ♥rst r♠ó♥♦ rs ♦♥ N ♦♥s ♣ rs ♦♥sr♥♦ rst ♦♠♦ ♦♥stt♦ ♣♦r3N ♦s♦rs ♥♣♥♥ts ♥♦ ♦s s t♥ ♥ ró♥ s♣rsó♥ q ② ♠♦sst♦ ♣r s♦s ♦♥ ♦♠trí s♥ s♦s 3N ♦s♦rs ♥♣♥♥ts s♦♥ ♦s♠♦♦s ♥♦r♠sás♦s rstP♦r t♥t♦ ♦♥tró♥ ♥rí t♦t ♥ ♠♦♦ ♥♦r♠ rtr③♦ ♣♦r t♦r q ② r♠ s s (

nqs +1

2

)~ωs(q)

s♥♦ nqs ♥ú♠r♦ ①tó♥ ♠♦♦ ♥♦r♠ (q s) s ♥ ♥ú♠r♦ ♥tr ♥rí ♥st♦ rst ♥ tr♠♥ ♣♦r s♠ s ♥rís s♦s ♠♦♦s ♥♦r♠s st♦ s

E =∑

qs

(nqs +

1

2

)~ωs(q),

♥ ③ q s s♣♥ ♦s ♥ú♠r♦s ①tó♥ ♥♦ ♦s 3N ♠♦♦s ♠♥r q s♠t♦r

∑q stá ①t♥ P❩ ②

∑s s 3p r♠s s♣tr♦ r♦♥ p s

♥ú♠r♦ át♦♠♦s ♦ ♦♥s s strtr rst♥

ísst♦ó♦

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❮ Ó Pr♦♣s tér♠s ♦s só♦s

♥ t♦rí á♥t ♠♣♦ tr♦♠♥ét♦ s s q ♥rí ♦s st♦s♣r♠t♦s ♥ ♠♦♦ ♥♦r♠ ♠♣♦ ró♥ ♥ ♥ s (n + 1/2)~ω s♥♦ ω r♥ ♥r ♠♦♦ Pr♦ ♥♦ s ♥ú♠r♦ ①tó♥ n ♠♦♦ ② r♥♥r s ω s♥♦ ①st♥ ♥ ♥ú♠r♦ n ♦t♦♥s ② ♥rí s ~ω

♠s♠ ♠♥r r♠♦s q ♥ ♥ st♦ ró♥ rst ② nqs ♦♥♦♥s t♣♦ s② ♦♥ t♦r ♦♥ q st♦ s q♥t r q ♠s♠ ♠♥r q ♦tó♥ s ♣rtí ♦♥ó♥ s ♣rtí ♠♣♦ ♥rí ♠á♥ ró♥ rst

♥♦í ♥♦ s s♥ ♠r♦ ♦♠♣t

♥ú♠r♦ ♠♦♦s ♣r k ♦tó♥ s ♠♥trs q ② 3p r♠s st♥ts ♣r q r♦♥

r♥ ω ♦tó♥ s ♣r♦♣♦r♦♥ |k| ② ♦tó♥ ♥ ♠♦♠♥t♦ ~k ♠s♦ss ♥♦ s♦♥ rts ♣r ♦s ♦♥♦♥s rr ♥♥ t♦r♠ ♦ ♣r ♠♦♠♥t♦ ♥ ♣rtí ♥ ♥ rst tqt ♣♦r ♥ t♦r ♦♥

♥ ♦s ♦♥♦♥s q ♣rt♥ P❩ ♠♥trs q k ♦s ♦t♦♥s s rtrr♦ P♦r s♦ ②♦ q ② 3p ♦♥♦♥s ♣r q ② s♦♠♥t ♦t♦♥s ♣r k s ♥rís tér♠s ♣♦r♥ ♦♠♥ s♦♥

∑

s

∫

PZB

dq

(2π)3~ωs(q)

exp [~ωs(q)/kBT ]− 1♣r ♦♥♦♥s

2

∫

espacio q

dq

(2π)3~ck

exp [~ck/kBT ]− 1♣r ♦t♦♥s

Pr s♦ ♥ ♦♥♥t♦ ♦♥♦♥s rst♦ q s ♦t♥ ♣r ♥rí s

u(T ) =1

V

∑

qs

~ωs(q)

[fs(q) +

1

2

]s♥♦ fs(q) =

1

exp [~ωs(q)/kBT ]− 1.

r♦ ♣♦r ①♣rsó♥ q ♥ó♥ fs(q) ♥♦s ♥ú♠r♦ ①tó♥ ♠♦ ♠♦♦♥♦r♠ (q s) t♠♣rtr T s ♥ú♠r♦ ♦♥♦♥s t♣♦ qs q ② ♥ sst♠♥ qr♦ tér♠♦ ♥♦ t♠♣rtr s T

♦r s♣í♦ s ú rt♠♥t ♣rtr ♥rí sst♠ r♥♦ rs♣t♦ t♠♣rtr

cv =1

V

∑

qs

∂

∂T

[~ωs(q)

exp [~ωs(q)/kBT ]− 1

]

② ♣♦r ♦ t♥t♦ ♣♥ s r♥s ωs(q) ♦s ♠♦♦s ♥♦r♠s sst♠

8.1.1 í♠t t♠♣rtrs ts

s ♠♣ q kBT ≫ ~ωs(q) ♣r t♦s s r♥s ♦s ♦♥♦♥s ♥t♦♥s t♦♦s ♦s ♠♦♦s♥♦r♠s strá♥ ♦♥ t♦s ♥ú♠r♦s ①tó♥ ♥♠♦s x ≡ ~ωs(q)/kBT ② srr♦♠♦s ♣rx≪ 1

1

ex − 1≃ 1

x+ 12x

2 + 16x

3 + . . .≃ 1

x

[1− x

2+x2

12+ . . .

],

❱és ♥ r♣s♦ s♦♠r♦ ♥ ♣r♠r ♣ít♦ t①t♦ ís á♥t t①t♦s s♥tr ♠s♠♦ ♥♦♠r ♠str ♠ás q ♦s ♦♥♦♥s s♦♥ ♦s♦♥s ♦ ♣rtís q ♦♥ stíst ♦s♥st♥ ♦r st♦s t♠s rr ♦s ♦♥t♥♦s s s♥trs r♠♦♥á♠ ② ♥ t♦♦ s♦ ♦s ts ♠ás té♥♦s

srá♥ st♦s ♥ s♥tr á♥ stíst

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

P❮

♥♦s q

cv ≃1

V

∑

qs

∂

∂T

[~ωs(q)

kBT

~ωs(q)

]=kBV

∑

qs

1 =3NkBV

,

t♦♠r s♦♠♥t ♣r♠r tér♠♥♦ ♥ ó♥ st s ② ♦♥ ② Ptt ás

8.1.2 í♠t t♠♣rtrs s

♠♦s q ♣r ♥ rst ♥♥t♦ ♦♥♥t♦ t♦rs q ♣r♠t♦s ♣♦r s ♦♥♦♥s ♦♥t♦r♥♦ ♦r♥♦♥ r♠♥ s ♥s♦ ♥ s ♥ q s♠♥♦ ó♥ rí ♣r♠♥t st♦ ♥♦s ♥ q ♦r s♣í♦ s ♣ srr ♦♠♦

cv =∂

∂T

∑

s

∫

PZB

dq

(2π)3

[~ωs(q)

exp [~ωs(q)/kBT ]− 1

].

t♠♣rtrs ♠② s ♦s ♠♦♦s ó♣t♦s ♣♥s ♦♥tr②♥ ♥tr ② q ♣r ♦s s♠♣ q ~ωs(q) ≫ kBT ♣r♦ sí ♦♥tr②♥ s trs r♠s ústs ♣r s q ~ωs(q) → 0 sq → 0 P♦r t♥t♦ ss t♠♣rtrs s♦♠♥t ♦♥tr②♥ ♦r s♣í♦ s r♥s ωs(q)ústs ♥ í♠t q → 0 ♦♥ r

sí ♣s ♣r r ♦♠♣♦rt♠♥t♦ ♦r s♣í♦ ♥ st í♠t s t♠♣rtrs♠♦s s♣♦♥r q

♥s♦ ♥ rsts ♦♥ s ♣♦tó♠ s♦♦ ♦♥srr♠♦s s r♠s ústs

♠♣③♠♦s ωs(q) ♣♦r ωs(q) = cs(q)q ♣r♦①♠ó♥ á ♣r q → 0 ② q s♦s s♦♥ ♦s♠♦♦s q ♦♥tr②♥ cv

♥tr s♦r P❩ ①t♥♠♦s t♦♦ s♣♦ rí♣r♦♦ ② q s♦♠♥t ②♦♥tró♥ ♥tr ♥♦ ωs(q) → 0

sts trs s♣♦s♦♥s ♥♦s ♣r♠t♥ srr

cv =∂

∂T

∑

s

∫

PZB

dq

(2π)3

[~ωs(q)

exp [~ωs(q)/kBT ]− 1

]=

∂

∂T

∑

s

∫

PZB

dq

(2π)3

[~cs(q)q

exp [~cs(q)q/kBT ]− 1

]

=∂

∂T

((kBT )

4

~3c33

2π2

∫ ∞

0dx

x3

ex − 1

),

♦♥ s ♥♦ ♥ ♦ ♠ c

1

c3≡ 1

3

∑

s

∫dΩ

4π

1

c3s(q),

♣r♦♠♦ ♥r s cs(q) s♦r P❩ ② s♦r s r♠s

P♦r ♦♥s♥t

lımT→0

cv =∂

∂T

(k4BT

4

~3c33

2π2π4

15

)=

2π2

5kB

(kBT

~c

)3

.

érs q st rst♦ ♦ ♠♦s ♦t♥♦ ♣r qr rst ♥s♦ ♦♥ s

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó Pr♦♣s tér♠s ♦s só♦s

♥ rs♠♥ ♠♦s ♥♦♥tr♦ ♥ ♦♠♣♦rt♠♥t♦ ♣r♦♣♦r♦♥ T 3 ♥ ♦r s♣í♦ ♥♦T → 0 st♦ stá r♦ ♦♥ ♦ rst♦s ①♣r♠♥ts

st ♦♠♣♦rt♠♥t♦ ♥ T 3 s á♦ ♥♦ ~ωs(q) ≫ kBT ② st♥♦ ωs(q) ♥ ♣rt ♥ s♣tr♦ ♣♦r ♦ q s ♥t q q str r ♦r♥ ③♦♥ ② q t♠♣rtr sr ①tr♠♠♥t

8.1.3 ♦♦ ② ♣r t♠♣rtrs ♥tr♠s rst rs

♦♠♦ ② ♦♥ ② Ptt ás t♠♣rtrs ♥r♦rs ♠♥t s r♦ q② ♥ r♥♦ ♠♣♦ t♠♣rtrs ♥ s q ♥♥♥♦ ♦s ♦s í♠ts q ♠♦s strs á♦

Pr ♦ s ♣r♦♣♦♥ ♠♦♦ ② ②s rtrísts s♦♥

♠♦♦ ♥♦ t♦♠ ♥ ♦♥sró♥ s♣tr♦ ♦s ♦♥♦♥s ωs(q) ② sstt② t♦s sr♠s s♣tr♦ r♦♥ rst ♣♦r trs r♠s ústs ♦♥ ♥ ♠s♠ ró♥ s♣rsó♥ ♥ ω = cq qí c q t♥ ♠♥s♦♥s ♦ r♣rs♥t ♥ ♦ s♦♥♦

♠ás ♥ ♠♦♦ s sstt② ♥tr ♥ P❩ ♣♦r ♥ ♥tr s♦r ♥ sr

② r♦ kD q ♦♥t♥ ♦s N t♦rs ♣r♠t♦s ♣♦r s ♦r♥ ♦♥r♠♥st♦ qr r q

4

3πk3D =

(2π)3

VN =⇒ k3D = 6π2ni ni = ♥ú♠r♦ ♦♥s ♣♦r ♥ ♦♠♥

á♦ ♦r s♣í♦ s♥♦ ♠♦♦ ② q ♥t♦♥s ♦♠♦ x ≡ ~cq/kBT

cDv =∂

∂T

(3~c

2π2

∫ kD

0

q3dq

exp [~cq/kBT ]− 1

)=

∂

∂T

(3

2π2k4BT

4

~3c3

∫ ΘD/T

0

x3dx

ex − 1

)= cv (ΘD, T ) ,

♦♥ s ♥♦ r♥ ② ωD = ckD t♠♣rtr ② kBΘD = ~ωD = ~ckD

①♣rsó♥ ♥tr♦r ♥♦s ♦r s♣í♦ qr t♠♣rtr T ♥ ♥ó♥ ♥ ú♥♦ ♣rá♠tr♦ ΘD

Pr T ≫ ΘD ♦r s♣í♦ cv (ΘD, T ) s ♦♥rt ♥ ♦♥ ② Ptt cv = 3nikB =24, 943 ♠♦−1 −1 ♠−3 Pr rst♦ s t♠♣rtrs ♦r s♣í♦ ② q ♥♦s ♥ ♦♠♣♦rt♠♥t♦ ♥♦ ♣r T → 0 ② ♣r T ≫ ΘD ♥♦ t♥ ♣♦r qé t♥r ♥ ♦♥♦r♥♦♥ ♦s ①♣r♠♥t♦s ♥ r ♥♦s ♥ ór♠ ♥tr♣♦ó♥ ♣r ♦r s♣í♦ ② ♥♦s♦sq s ró♥ ♦♥ t♠♣rtr

• ó♠♦ r ♣rá♠tr♦ ♠♣ír♦ ΘD

óts q ♣rá♠tr♦ ΘD ♥♦ ♣rá♠tr♦ s♦♥♦♦ c q s t③♦ st ♦r sí

❨ s r♦ s ♥ ♦s♦♥s ♠② ♥♦ P♦r ♠♣♦ s ♠♠♦s s ♦♥st♥ts ásts rst q stá♥ r♦♥srt♠♥t ♦♥ s ♦s s q ♣r♥ ♥ ♥ó♥ c ♣♦♠♦s ♦♥r♦♥tr cv tór♦ ♦♥ ①♣r♠♥t♥ ♦s r♦s ♥♦s s ♥♦♥tr♦ q ♦♠♣rr c s♥♦ ♠♥t ①♣r♠♥t♦s ást♦s ② c ♠♥t ♦♥ ①♣r♠♥t♦s ♦r♠trí sr♦ t♦rí①♣r♠♥t♦ s ♠♥♦r q ♦s rr♦rs ①♣r♠♥ts ∼ 1%② ♣♦r ♦♥s♥t r♦ s ♣rt♦

s ♥trs♥t r ♥♦tr q kD ∼ kF ∼ kPZB ♦♠♦ ♣ ♦♠♣r♦rs s♥ ♠♦ sr③♦

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

P❮

q ♦♥♥ str ó♠♦ r st ♣rá♠tr♦ ♠♣ír♦ q ♥♦s ♣r♠t r ♦r s♣í♦ t♦s s t♠♣rtrs

❯♥ s ♠♥rs ♣♦ss r♦ s r ♦♥r lımT→0 cDv ♦♥ ♦r s♣í♦

①♣r♠♥t s t♠♣rtrs ♥ st í♠t s ♠♣ q ΘD/T → ∞ ② ♣♦♠♦s ♣r♦①♠r ♥tr ∫ ΘD/T

0

x3dx

ex − 1≃∫ ∞

0

x3dx

ex − 1=π4

15.

stt②♥♦ ♥ ♦r s♣í♦

lımT→0

cv =∂

∂T

(π2

10

k4BT4

~3c3

)= t③♥♦ ΘD) =

12

5π4nikB

(T

ΘD

)3

.

st ó♥ ♣r♠t r ♦r ΘD ♠♥t ♦s rst♦s ①♣r♠♥ts ♦r s♣í♦Pr ♦s rsts ♦s r♦s ♥♦s s ♦t♥ st ♠♦♦ ♦s ♦rs s♥t t

t♠♣rtrs ②

tr tr♥t ♣r r ΘD s t③r ♥ó♥ c q s ♦ ♥ts

1

c3=

1

3

∑

s

∫dΩ

4π

1

c3s(q)

② r ΘD ♣rtr s ♦♥st♥ts ásts ①♣r♠♥ts r♥ ♥tr ♦s rst♦s♦t♥♦s ♠♥t st ♠♥r ♣r♦r ♥ ♦♠♣ró♥ ♦♥ ♦s ♠♥r ♣r♦r ♥tr♦r♥♦ s ♠② r♥ ② ♣r♦①♠♠♥t ♥tr♦ rr♦r ①♣r♠♥t ∼ 1− 2%

P♦♠♦s ♠ás s♣♦♥r q ór♠ ② ♣r cv ♥♦ s s♦♦ ♥ ór♠ ♥tr♣♦ó♥s♥♦ q ①t♠♥t ♦s ♦rs cv(T ) ♥ st ♥tr♣rtó♥ ♣rá♠tr♦ ΘD ♣♥ t♠é♥ t♠♣rtr st ♣r♦♠♥t♦ r ♥ ♥ó♥ ΘD(T ) ♥ ③ cv(T ) s t③ s♥ trtr

❯ cv ②

Pr str ♦♠♣♦rt♠♥t♦ ♦rs♣í♦ ♠♦♦ ② ② q r ♥tr q ♣r ♥

cDv (ΘD, T ) =∂

∂T

(3

2π2k4BT

4

~3c3

∫ ΘD/T

0

x3dx

ex − 1

)

= 9nikB

(T

ΘD

)3 ∫ ΘD/T

0

x4ex

(ex − 1)2dx

q ♥♦s ♥ ♦♠♣♦rt♠♥♦ ♣ ♦♥strs♥ t ♠t♠át ♣r st♠r ♦r ♥tr ♦ r③rs ♥♠ér♠♥t ♦♠♦ r

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó Pr♦♣s tér♠s ♦s só♦s

8.1.4 ♦♦ ♥st♥

st ♠♦♦ ♥tr♦r ② ♣r♠r♦ q ①♣ó ♣♦rqé q ♦r s♣í♦ s♥r cv → 0 t♠♣rtrs s q r ♥ ♠♦t♦ s♦r♣rs ♣r ís ás ♠ás ♠♦♦ ♥st♥ sró ♣r ♠♦strr q s ♦s♦♥s ♠á♥s ♦s ♦♥s ♥ rststá♥ ♥t③s ♠s♠ ♠♥r q ♦s stá♥ ♦s ♦s♦rs ró♥ P♥

s s út trtr s r♠s ó♣ts ♦♥ st ♠♦♦ q s♣♦♥ q r♥ ♦s ♠♦♦s♥♦r♠s s ♥ r♥ ♦♥st♥t ωE ♥♣♥♥t q st♦ s ♦ q ♥ ♠♦s s♦s sr♠s ó♣ts t♥♥ ♥ r♥♦ ró♥ r♥s str♦ ♠♥r q ♣♥ ♣r♦♠rs♣r sr ♥ ♦r ♦♥st♥t ωE

♥ r s ♥ ♥ r♠ úst ② s ♣r♦①♠ó♥ s♥♦ ♠♦♦ ② ② ♥ r♠ó♣t ② s ♣r♦①♠ó♥ ♦♥ ♥ ♠♦♦ ♥st♥

❯ ♠ úst ② r♠ ó♣t♣r♦①♠s ♣♦r ♥ ♠♦♦ ② ② ♥ ♠♦♦ ♥st♥

8.1.5 ♦♠♣ró♥ ♦s ♦rs s♣í♦s tró♥♦ ② r♦♥ r

♥ ♠♦♦ r♦♠♠r s♠♦s q ♦r s♣í♦ ♦s tr♦♥s ♥ ♦ qr t♠♣rtr ♣♦r

celecv =π2

2nkB

T

TF.

♠é♥ s♠♦s q t♠♣rtr ♠♥t s t♥ q celecv /cclasv ∼ 10−2 ♠ás t♠♣rtrs♠② s q ♣r♦♥ s r♦♥s r rst♥ t♥ ♦r♠

cvibrv =12

5π4nioneskB

(T

ΘD

)3

.

Pr ♥ ♠t ♣♦r ♦♥s♥t ♦r s♣í♦ t♦t ♥♦ T → 0 srá

cmetal,totalv ∼ AT +BT 3.

♥ú♠r♦ tr♦♥s n ♥ ♠t s n = Zniones s♥♦ Z ♥ ♠t ♥t♦♥s

celecv

cvibrv

=5

24π2Z

Θ3D

T 2 TF,

s ♥tr♦rs ♠♦♦ ② ♥♦ s ♦t♥ q cv ∝ T 3 ♥♦ T → 0

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

Ó ❯

♠♥r q s

T > T0 ≡ 0, 145

(ZΘD

TF

)1/2

ΘD =⇒ cvibrv (T ) > celecv (T )

② ♦♠♥♥ ♦s ♦♥♦♥s st♦ s ♣r♦ ♣♦♦s r♦s ♥ ② q TF ≫ ΘD ② ♦r T0 s♣qñ♦

8.1.6 t♠♣rtr ② ② ♦s ♦rs ♠♦s ♦s ♦♥♦♥s

t♠♣rtr ② ♣rát♠♥t t♦♦ s♣tr♦ r♦♥ só♦ stá ①t♦st r♥ ♠á①♠ ωM ∼ ωD

st♦ s♥ q s T > ΘD ♥ ♥r♠♥t♦ ♥ t♠♣rtr ♥♦ rá ♣rr ♥♦s ♠♦♦s ró♥ s♥♦ q ♠♥trá ♦♣ó♥ ♥ú♠r♦ ♠♦♦s ①t♦s ♦tr ♠♥r r♦s q ♠♥trá ♣♦r t♥t♦ ♥rí t♦t s♦ ♦s ♠♦♦s ♦♣♦s ♣r♦ ♥♦ ♥rí ♠ ♦s ♦♥♦♥s

♥ ♣rtr s s t③ ♠♦♦ ② ♣r st♠r ♦s ♦rs ♠♦s ♥rí 〈εfonon〉 ♣r♦♠r ♥rí ♠♦♦ ♦ ♦♥ó♥ s ♦t♥ q

T ≪ ΘD 〈εfonon〉 =π2

5kBT rí ♥♠♥t ♦♥ T

T > ΘD 〈εfonon〉 =2

3kBΘD ♥♦ rí ♦♥ T

♥ ♦♥s♥ ♥rí tér♠ r(q ♣ ♦♥srrs ♦♠♦ ♣r♦t♦ ♥rí

♠ 〈εfonon〉 ♠t♣ ♣♦r ♥ú♠r♦ ♦♥♦♥s 〈nfonon〉)sr ♣r st♠r ♣♥♥ ♦♥

t♠♣rtr ♥ú♠r♦ ♠♦ ♦♥♦♥s ♥ sst♠

T ≪ ΘD ured ∝ T 4 〈nfonon〉 ∝ T 3

T > ΘD ured ∝ T 〈nfonon〉 ∝ T

8.2 Ó ❯

♥ st ♣rt♦ ♠♦s str ♥ s ♣r♦♣s íss q ♣r♥ ♥tr♦r s♥tr♦♥s rst♥s ♥r♠ó♥s st♦ s srr♦r ♥rí ♣♦t♥ ♥tró♥ ♥tr♦s ♦♥s ♠ás á ♣r♦①♠ó♥ r♠ó♥

❯ ♦só♥ ♦♥♦♥s ♥ ♥s♦

❯♥ ♠♦stró♥ ①st♥ st♦s t♦s♥r♠ó♥♦s s♦♥ ♦s ①♣r♠♥t♦s ♥tró♥♦♥ó♥♦♥ó♥ ♦♥ rst♦ ♦tr♦ ♦♥ó♥ ♥♦s ♦s s♦♥

• ♥ ♥ rst s ♥tr♦♥r ♥ ③ ♦♥♦♥s ♦♥t♥s r♥ ω1 ♦♥ ♦tr♦③ ♣r♦ ② r♥ s ω2 rst♦ ♥tró♥ s ♥ trr ③ ♦♥♦♥s ♦♥t♥s r♥ ω3 = ω1 + ω2 ♦♥ ♥ ♦♥rsó♥ ♥rí ♥ ♥rí ♦s ♦♥♦♥s♥♥ts ♥ ♥ ♦♥t ♥tró♥ ♠

• ♥t ♥ ♠♦♥t ①♣r♠♥t ♦ s♥ ♥tr♦♥r ♦s s trsó♥♦s str♦s

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó Pr♦♣s tér♠s ♦s só♦s

♦♥♦♥s ♥ ③♦♥ ♥tr ♥ s♦ rr és r ♦sr q s ♥r ♥ trr③ ♥ ③♦♥ ♥tró♥ s♦♠♥t ♥ qs r♦♥s q r♥ t♥t♦ ♦♥sró♥ ♥rí ♦♠♦ s♠♦♠♥t♦ ♦s ♦s s ♦♥♦♥s ♦s ♦♥♦♥s ♥♦ ③ s♦♥ rst♥ts ♥tró♥ ♦s ♦s s♥tr♦rs ♥ st s♦ ♥tr♦♥ ♥ ♦♥ ♦♥t♥ ♦♥ ♦trtr♥srs ♣r r r ♦♥♦♥s ♦♥t♥s

♥tró♥ ♥tr ♦♥♦♥s ♣ ①♣rs ♥ ♠♥r s♥ ♣rs♥ ♥ ♦♥ó♥ ♥♥ rst s ♥ ♣rtró♥ ♣ró q ♠♦ ♥ s♣♦ ② ♥ t♠♣♦ s ♦♥st♥tsásts só♦ ♣♦r ♠♦ ♦s tér♠♥♦s ♥r♠ó♥♦s ♠t♦♥♥♦ st ♦♥ó♥ ♥r ♥ qr ♦tr♦ ♦♥ó♥ q s ♣r♦♣ ♥ rst ♠♦♦ q s ♣r♦ ♦♠♥t♥ ♥tró♥ ♥tr ♦s ♦s ♦♥♦♥s ❱ ssó♥ ♥ ♣♦♦ ♠ás t só♥

♠ás ♦s ♦s ①♣r♠♥t♦s rt♦s t♦s ①st♥ ♦tr♦s ♥ó♠♥♦s q rqr♥ ♥só♥ ♦s tér♠♥♦s ♥r♠ó♥♦s ♥ ♠t♦♥♥♦ ♦s ♦♥s ② q s ♥ ♦♥♦♦ q♥s ♦♥s♥s íss ♣r♦①♠ó♥ r♠ó♥ ♥♦ s ♠♣♥ ♦♥ ♣rsó♥ ♥ ♦s rstsrs ♥♠r♠♦s ♦r ♥s s

♦ ①st ①♣♥só♥ tér♠ ♥ rst ② q ♥ ♣r♦①♠ó♥ r♠ó♥ ♣♦só♥♠ ♦s ♦♥s ♥♦ ♣♥ t♠♣rtr ♦♥ ♣r ♥ rst rs ♦♥ ♣♥t♦ ♦rrs♣♦♥♥t

s ♦♥st♥ts íss áts s♦tr♠s s♦♥ s

s ♦♥st♥ts ásts ♥♦ ♣♥♥ ♣rsó♥ ② t♠♣rtr

♣ ♦rí s ♦♥st♥t ts t♠♣rtrs T ≫ ΘD ♣r♦ ①♣r♠♥t♠♥t♥♦ s♠♣r s r ♦r ♦♥ ② Ptt ♦♠♦ ss t♠♣rtrs ② ♣♦ ③ á♦ ás♦ s ♠② ♣♦s q sts r♥s s ♥ q ♣r♦①♠ó♥r♠ó♥ ② ♥♦ s á ss t♠♣rtrs

♦ ①st ♥tró♥ ♥tr s ♦s♦♥s r ♣s ♥ ♣r♦①♠ó♥ r♠ó♥ ♦s ♠♦♦s♥♦r♠s s♦♥ ♥♣♥♥ts s♦ q srí ♥♥t ♦♥t tér♠ rκred

♥ s♣rsó♥ ♥tr♦♥s ♣♦r ♥ rst s ♦t♥♥ ♣♦s ró♥ ♣r♦♦♦s ♣♦r s②s ♦♥sró♥ ♦s ♣r♦s♦s ♥ s♦♦ ♦♥ó♥ ♦♥ ♥ ♥r ♥ ♥ ♦ q s♥ ♠ ♠② rt ♦s tér♠♥♦s ♥r♠ó♥♦s ♠t♦♥♥♦ ♦s ♦♥s

8.2.1 ①♣♥só♥ tér♠ ♦s rsts

♥ st ♣rt♦ ♠♦s str ♠♥r tt ② s♥ ♥trr ♥ ts ①♣♥só♥tér♠ ♥ rst ♦♠♥t s ♣♦s s ①♣♥só♥ s s ♥tr♦♥ tér♠♥♦s ♥r♠ó♥♦s ♥ trt♠♥t♦ rst

Pr ♦ s♣♦♥♠♦s ♥ ♦s♦r ♥r♠ó♥♦ ás♦ st♠♦s ♣s ♥ í♠t tst♠♣rtrs ♥♠♥s♦♥ q sr ♠♦♠♥t♦ ♥ ó♥ ♦♥ rs♣t♦ ♣♦só♥ qr♦ ♠s♠♦ s ♥ ♣♥t♦ st ♣♦t♥ q tú s♦r ♦s♦r ♣ rs♥t♦♥s q ó♥ stá st♦ ♥ ♣♦t♥ q t♥ ♠♥♦s ♥ tér♠♥♦ ♥ x3 q ♦rr ♣rt♣r♠♥t r♠ó♥ ♥ x2 ♥♦s ♦♠♦s ♣s ♦s tér♠♥♦s árt♦s ② s♣r♦rs st♦ s

U(x) ≃ cx2 − gx3.

♦r♠♠♥t s ♥sr♦ q ② ♦s r♠s ♥ ② ♦tr ♣r q ♦rr♥ ♦s ♣r♦s♦s trs ♦♥♦♥s s♦♠♥t ♦♥srár♠♦s ♦s tér♠♥♦s r♠ó♥♦s s♥♦ ♦♥ó♥ ♥♦ rí ♣r♠r ♦♥ó♥ ♣s ♦s ♦♥♦♥s s♦♥

♦♠♣t♠♥t ♥♣♥♥ts ♥ ♣r♦①♠ó♥ r♠ó♥

ísst♦ó♦

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Ó ❯

t♠♣rtr T ♦r ♠♦ s♣ró♥ s ó♥ rs♣t♦ s ♣♦só♥ qr♦♥ ♣♦r

〈x〉 =∫ +∞−∞ dx xe−U(x)/kBT

∫ +∞−∞ dxe−U(x)/kBT

♦s s♣③♠♥t♦s s♦♥ ♣qñ♦s ♦r♠ q s ♠♣ q gx3 ≪ kBT ♣♦r♠♦s srr♦r ①♣♦♥♥

∫ +∞

−∞dx xe−U(x)/kBT ≃

∫ +∞

−∞dxe−cx

2/kBT(x+ gx4/(KBT ) + . . .

)≃ 3

√π

4

g

c5/2(kBT )

3/2

∫ +∞

−∞dx e−U(x)/kBT ≃

√π

c/(kBT

❯ ❱ró♥ ♦♥st♥t r ♦♥ T r

♠♥r q

〈x〉 ≃ 3g

4c2kBT ∝ T.

st ♦r♠ ♥só♥ tér♠♥♦s♥r♠ó♥♦s ♥ st ♠♦♦ ♥♠♥s♦♥ q s♣ró♥ ♥tr ♦s ♦♥s ♥♦s rst sr♣r♦♣♦r♦♥ T ② ♣♦r t♥t♦ sr q ♥ só♦s ①♣♥ ♦♥ t♠♣rtr ♠♥r ♥

st♦ stá r♦ ♦♥ ♦s ①♣r♠♥t♦s ♦s q s ♦♥♦ ♦♥t ①♣♥só♥ ♥♥ r s ♣♥ r ♦s rst♦s ♣r ♦s♦rs ①♣r♠♥ts ♦♥st♥t r rsó♦ ♥ ♥ó♥ t♠♣rtr t♠♣rtr ② r s ΘD = 92

r③ ♣r♦♠♦ q sr ó♥ s♣③rs qr♦ s 〈F 〉 ≃ −2c 〈x〉 + 3g⟨x2⟩② s

♥♦ ② r③s ①tr♥s ♠♣rs q 〈F 〉 = 0 ♥ qr♦ tér♠♦ t♠♣rtr T st♦s♥ q

〈x〉 = 3

2

g

c

⟨x2⟩≃ 3

2

g

c2〈U(x)〉 .

Pr ♥ ♦s♦r r♠ó♥♦ s r q 〈E〉 = 2 〈U(x)〉 ② s ♦s tér♠♥♦s ♥r♠ó♥♦s s♦♥♣qñ♦s ♣♦♠♦s srr q 〈x〉 ≃ 3

4gc2〈E〉

♠♠♦s r0 s st♥ ♥tr ♦s ♦♥s ♠ás ♣ró①♠♦s

〈x〉r0

≃ 3

4

g

r0c2〈E〉

❯ ❱ró♥ α ② cv ♦♥ T

② ♦♥t ①♣♥só♥ ♥ srá

α =1

r0

d 〈x〉dT

≃ 3

4

g

r0c2cv,

♥ ①♣rsó♥ q ♥♦s ♥ ró♥ ♣r♦♣♦r♦♥ ♥tr α ② cv q s♦♦♠♣r♦ ①♣r♠♥t♠♥t

♥♦ ♦rs ①♣rsó♥ ♥tr♦r ♣rts t♠♣rtrs T > ΘD s t♥

α ≃ 3

4

gkBr0c2

∼ 10−4 − 10−5 K−1,

ísst♦ó♦

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❮ Ó Pr♦♣s tér♠s ♦s só♦s

r♦ ♦♥ ♦s rst♦s ①♣r♠♥ts

❯t③♥♦ ♣rá♠tr♦ rü♥s♥ γ s ♣ str ♠♦r ró♥ ♥tr α ② cv

α =γκ

3Vcv,

♦♥ γ s ♣rá♠tr♦ rü♥s♥ q t♦♠ ♦rs ♥tr ♣r ♠ts κ s ♦♠♣rs ② V s ♦♠♥ ♣♦r át♦♠♦

8.2.2 ♠♥s♠♦ ♦só♥ ♥tr ♦♥♦♥s

♦♠♦ ② s ①♣♦ ♥♦ sr♠♦s ♠t♦♥♥♦ ♥tr♦ ♣r♦①♠ó♥ r♠ó♥♣r♥ ♦s ♠♦♦s ♥♦r♠s st♦s ♠♦♦s ♥♦r♠s s♦♥ ♥♣♥♥ts ♥♦s ♦tr♦s ♦r♠ q s ♦♥♦♥s s ♥ s ② ♥♦ t♥ ♦s♦♥s t♥♥♦ ♥ ♠ ♥♥t♠♥t r♥ ♠r♦ s ①st♥ tér♠♥♦s ♥r♠ó♥♦s ♥ ♠t♦♥♥♦ s♦s tér♠♥♦s ♦♣rá♥ st♦s ♠♦♦s♥♦r♠s ♥♦ r ♦♥♦♥s ♦♥ ♥ ♠ ♥t ♠ás t♥t♦ ♦s t♦s rs ♠♣r③s ♦ s s♣rs rst ♠trá♥ ♠ ♦s ♦♥♦♥s s♣♠♥t t♠♣rtrs ♠② s

st♠♦s ♠♥r ♠♥t ♠♥s♠♦ ♦só♥ ♦s ♦♥♦♥s ♦ q ♦ s♦♥♦ ♣♥ ♥s ♦ rst ♣♦♠♦s ♦♥srr q ♥ ♦♥ s♦♥♦r q1 qs ♣r♦♣ ♦ r♦ ♥ rst ♥ q ② s stá tr♥s♠t♥♦ ♦tr ♦♥ q2 ♣rs♥trá ♥♠♦ó♥ ♥ s s♣r♣♦só♥ ♠s ♦♥s ♠♠♦s C ♥ ♦♥st♥tq s♣ ♠♥t s ♠♦ó♥

u(r, t) = exp(i[q1 · r+ C cos (q2 · r− ω2t)− ω1t

]),

♦♥ tér♠♥♦ ♥tr ♦rts r♦s ♥ s s♣rs s ♦♥st♥t ♦♠♦ ♦ s s ♦t♥ r♥♥♦ ♣r s♣r s ♦♥st♥t ró♥ ② s♥♦q vs = dr/dt t♥♠♦s

vs ·q1 − q2C sin (q2 · r− ω2t)

ω1 − ω2C sin (q2 · r− ω2t)= 1.

st♦ r♣rs♥t ♥ ♦ s♦♥♦ ♠♦ ♣♦r ♣rs♥ ♥ ró♥ r♥ω2 ② t♦r ♦♥ q2

C ≪ 1 ♦♣♠♥t♦ ♣qñ♦ ró♥ ♣ srrs ♦♠♦

u(r, t) = ei(q1·r−ω1t)[1 + iC cos (q2 · r− ω2t) + . . .

]=

= ei(q1·r−ω1t) +1

2iCei[(q1+q2)·r−(ω1+ω2)t] +

1

2iCei[(q1−q2)·r−(ω1−ω2)t] +O(C2)

② ♥♦ ♦s trs tér♠♥♦s r♣rs♥t

♣r♠r♦ ♥ ♦♥ó♥ (q1 ω1)

s♥♦ ♥ ♥♦ ♦♥ó♥ (q3 = q1 + q2 ω3 = ω1 + ω2)

trr tér♠♥♦ ♦tr♦ ♥♦ ♦♥ó♥ (q3 = q1 − q2 ω3 = ω1 − ω2)

st♦ ♥♦ s ♠ás q ♦tr ♠♥r ①♣rsr ♣r♦①♠ó♥ t♠♣♦ ró♥ q ♠♦s srr ♠♦♦ r♦♠♠r ♦s tr♦♥s ♥ ♥ ♠t

st♦ s ♣♥ s t♥s♦♥s ② ♦r♠♦♥s q ② ♥ ♠s♠♦ ♦ q ♦ s♦♥♦ ♣♥rá ♥s ♠tr q s ♦ ♠s♠♦ q r ♦r♠ó♥

♠s♠♦ P♦r ♦♥s♥t rá ♥ ♠♦ó♥ ♥ s ② q ♦ s s st♠♥t ♦ s♦♥♦ ♠♥r ♠ás ♥r ♦s ♦s ♠♦♦s q s ♣r♦♣♥ ♥ ♥ ♠♦ ♦♥ ♥s r ♥ ó♣t srí♥ ♠♦s ♦♥ í♥ rró♥ r s t♥rí ①st ♦♣♠♥t♦ ♥tr ♦s ♠♦♦s ♠♦ó♥ ♠♣t ② s st út♠ ♠♦ ♠♦♦s ♥♦ ♦♣♦s ♠♦ó♥ ♥ tr♠♥ ♣♦r ♠♦ s ♥ ♠♦ó♥ s

C = 0 ♦ s♦♥♦ ♥ ♣♦r vs · q1/ω1 = 1

ísst♦ó♦

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❯❱

② t♦♦ ♦ r q ② ♣r♦s♦s st trs ♦♥♦♥s ♥ ♦s tér♠♥♦s ♥ ♦r♥ O(C)

s r♦ q q3 = q1 + q2 ♣ qr r P❩ P♦r ♦ ② q r♦♥ ♦ss♠♦♠♥t♦s s

q3 = q1 + q2 +G G ∈ RR.

♦s ♣r♦s♦s ♥ ♦s q G = 0 s ♠♥ ♥♦r♠s ② q♦s ♥ ♦s q G 6= 0 s ♠♥ ❯♠♣♣❯ st♦s út♠♦s s♦♥ r♦♥s r ♥ ♦♥ó♥ ♠ás ♥ ♠só♥ ♦ s♦ró♥ ♥ ♥♦♦♥ó♥

♦t ♥ ♥ ró♥ ♥tr♦♥s ♦ r②♦s ❳ s ♣rtís tr♥s♠t♥ rst♥②♥♦ ss ♦♥♦♥s ♥ ♠♦♠♥t♦ ~ (k′ − k) ♠♥trs q ♥ qr ♣r♦s♦ ♥tr♦♥♦♥s s ♦ ❯ ♠♦♠♥t♦ t♦t rst ♥♦ ♠ s ♣♦r ♦ q ♣♥sr q~q s ♠♦♠♥t♦ ♦♥ó♥ s rt ♣♦rq ♦♥ó♥ rt♦r ♦♥ ♠♦♠♥t♦~ (q+G) s ♠s♠♦ ♦♥ó♥ q ♦♥ó♥ ~q rérs q ♠♦♠♥t♦ tr♥s♣♦rt♦♣♦r ♥ ♦♥ó♥ s ♥♦

❯ ♣r♦s♦s ② ❯

8.3 ❯❱

♦ ♠♦s qí srr♦r ♥ t♦rí t ♦♥t tér♠ ♦s rsts s♥♦s♦♦ ♦sqr s rtrísts íss ♣r♦♠ t③♥♦ ♥ ♣r♦①♠ó♥ ♠♥t t♠♣♦ ró♥ ♣r ♦s ♦♥♦♥s

♥ ♥r ♦♥t tér♠ t♦t t♥ ♥ ♠á①♠♦ ♣r♦♥♥♦ t♠♣rtrs sT < 80 ② rá♣♠♥t sr t♠♣rtr r♥♦ ♠á①♠♦ stá ♥tr 103 ② 20× 103

❲♠ s♠♥ó♥ ♦♥t tér♠ κ ♦♥ T s ♠ás rá♣ ♥ ♦s s♥ts ♦r♠q Tamb ♦s ♠ts ♦♥♥ ♠♦r ② ①♣♦♥s ♠♥t ♦♥ ♠♦r Tamb qqr ♠t

♥ ♦s s♦③♦s q s ♣rs♥t♥ ♥ s rs s ♣♥ ♦srr s rtríst ♥ ③r♦rt ② ♦r q t♥ ♥ ♦♥t tér♠ ♠②♦r q ③r♦ ♣r t♠♣rtrs tssí ♦♠♦ ♦trs rtrísts q ♣rs♥t♥ κtotal(T ) ♥ ♠ts ② s♥ts

érs q ② s ③♦ ♥ ♣r♦①♠ó♥ s♠r ♣r st♦ ♥á♠ ♦s tr♦♥s ♥ ♦s ♠ts

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❮ Ó Pr♦♣s tér♠s ♦s só♦s

r♠♦s ♥s ♣r♦①♠♦♥s ♣r st ssó♥ tt

❯sr♠♦s ♥ t♠♣♦ ró♥ τ(T )

♦♠♥t ♦♥srr♠♦s r♠s ústs st♦ s str♠♦s rsts rs

st♦ s r③♦♥ ♥ ♠♦s s♦s ♣♦rq ♥ ♦s ♣r♦s♦s ♥ ♦s q ♥tr♥♥ ♦♥♦♥s ②♣♦r t♥t♦ q ♥②♥ ♥ κred(T ) ♣r ♦ r♣♦ ∂ω/∂q q s ♣qñ ♥ sstrs r♠s ó♣ts

óts q

jQ =∑

q,s

~ωs(q)∂ωs(q)

∂qfs(q)

s trs r♠s ústs s srr♠♦s ♠♥t ♥ ♠♦♦ ② ♦♥ ω = cq

①r♠♦s ♥ qr♦ tr♠♦♥á♠♦ ♦ ♠♦♦ q ♥rí s ♥ ♦♥ó♥s♣és ♥ ♦só♥ ♥ ♣♥t♦ x0 s ueq(T (x0))

♣♦♥♠♦s q s♦♠t♠♦s rst ♥ ♣qñ♦r♥t t♠♣rtr ♦ r♦ ❳ ♠♠♦scx ♦ s ♥ ♦♥ó♥ s♣és ♥ ♦só♥ ♦ r♦ ♦ ♣♦♠♦s srr ♦♥tró♥ s ♦♥ó♥ tr♥s♣♦rt tér♠♦ ♦♠♦

cx ×[u(x0 − l cos θ)− u(x0)

]

♦♥ l s ♠♥♦ r ♠♦ ② θ á♥♦ q s ♠str ♥ r

Pr♦♠♥♦ ♥ t♦s s r♦♥s s s q ♣ r ♥ ♦♥ó♥ ♣♥t♦ x0

jQ =⟨cx

[u(x0 − l cos−θ)− u(x0)

]⟩θ=

1

4π

∫dΩ cx

[. . .]

≃ 1

4π

∫ 2π

0dϕ

∫ π

0dθdΩ c cos θ

[− l cos θ

[∂u∂x

]x0

]

≃ −1

2cl[∂u∂x

]x0

∫ 1

−1dµµ2 = −1

3c l

∂u

∂T

(− ∂T

∂x

)x0

② ♦♥t tér♠ srá

κ =1

3cv c l =

1

3cv(T ) τ(T ).

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♣♥♥ κred(T ) ♥ ♣♦r Cv(T ) q s ♦♥♦ ♣♦r ♥♦s♦tr♦s ② ♣♦r τ(T ) ②♦á♦ s ♥ ♣r♦♠ í st ② ♦♠♣♦

ts t♠♣rtrs T ≫ ΘD

sts t♠♣rtrs ♥ú♠r♦ ♦♣ó♥ ♦s ♦♥♦♥s s

fs(q) =1

e~ωs(q)/kBT − 1≃ kBT

~ωs(q)

② ♦♠♦ s ♥♦ ♣♥ rt♠♥t q

st ♠♥t♦ ♥ ♥ú♠r♦ ♦♥♦♥s ♠♥tr T ♦♥ ♥ ♠②♦r ♣r♦ ♦só♥ ♠♥r q ♦♠♣♦rt♠♥t♦ τ srá ♣r♦①♠♠♥t ♣r♦♣♦r♦♥ 1/T sts t♠♣rtrs ♦r s♣í♦ s ♦♥st♥t ② ♦t♥♠♦s rst♦ q κred ∝ 1/T ♣r♦①♠♠♥t

st ssó♥ stá rt♠♥t r♦ ♦♥ rst♦ ①♣r♠♥t q ♥♦s q

κred(T ≫ ΘD) ∝ 1

Tα,

♦♥ α ∈ (1, 2)

s t♠♣rtrs T ≪ ΘD

♥ st s♦ ♥ú♠r♦ ♦♥♦♥s ①t♦s s ♣qñ♦ ② s♦♠♥t ♣rrá♥ ♦r♠♣r ♦s q t♥♥ ♥ ♥rí ωs(q) ≪ ωD ② ♣♦r ♦ t♥t♦ q ♠♣♥ q q ≪ kD

s♦♠♥t s t♥♥ ♥ ♥t ♦s ♦♥♦♥s ♠②♦rtr♦s ♥rí t♦t ♦s ♦♥♦♥s q♦s♦♥ sr ♠♦ ♠♥♦r q ~ωD ② s ♠♦♠♥t♦ t♦t sr ♠♦ ♠♥♦r q ~kD

♥ ♦só♥ ♥rí t♦t s ♦♥sr ♠♥r q ♥rí t♦t ♦s ♦♥♦♥s rst♥tssrá t♠é♥ ♠♦ ♠♥♦r q ~ωD st♦ s s♦♦ s ♣♦s s t♠é♥ ♦s s♠♦♠♥t♦s q ♦s♦♥♦♥s s♦♥ ♠ó♦ ♠♦ ♠♥♦r q kD

♥ rs♠s ♥ts ♦s t♦rs ♦♥ ♦s ♦♥♦♥s ♥♥ts ② s♥ts ♠②♦rí s ♦s♦♥s s♦♥ ♣qñ♦s ♥♦ s ♦♠♣r♥ ♦♥ kD ♦♠♦

q1 + q2 = q′1 + q′2 +G,

♥sr♠♥t ♣r s♦s ♣r♦s♦s ♠②♦rtr♦s ♠♣rs q G = 0

Pr♦s♦s

♦♠♦ ♠♦s r t♠♣rtrs s s♣rr q ♣rát♠♥t t♦♦s ♦s ♣r♦s♦ss♥ ♠♥r q s ♦♥sr ♠♦♠♥t♦ rst♥♦ t♦t

♥ ♦♥s♥ s s♦♦ ♦♥sr♠♦s st♦s ♣r♦s♦s ② s ♥ ú♥ ♥st♥t tr♠♥♦ s♠♣ q ∑

q,s

q fs(q) 6= 0,

rst q st ♦r ♥♦ rí ♥ qr♦ tér♠♦ ♦ jQ ♥♦ s ♥ ♣r♦rs♠♥t ♥qst♠♦s ♥ ♥ sst♠ ♦♥ ♥ r♥t ∇T r♦

♥♠♦s st rst♦ ♦♥ ♠②♦r t ♣♦♥♠♦s ♥ ♦♥♥t♦ ♣rtís ♥ qr♦tér♠♦ sts ♥ ♥s ♥rí εi ② ♦♥ s♠♦♠♥t♦s qi sst♠ stá s♦ ♠♥rq Etotal =

∑i niεi = cte ② q qtotal =

∑i qiεi = cte ♦♥ ni s ♥ú♠r♦ ♣rtís ♥

st♦ i ♠♠♦s Ω ♣r♦ ♥♦♥trr ♥ stró♥ s ♣rtís ♥tr♦s st♥t♦s st♦s st♦ s ♥ ♦♥♥t♦ tr♠♥♦ ♦rs ni ② s ♦♥sr♠♦s ♣rtís♥♣♥♥ts s ♠♣ q Ω =

∏iΩi ♦♥ Ωi s ♥ú♠r♦ ♣♦ss ♦♦r ni

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó Pr♦♣s tér♠s ♦s só♦s

♣rtís ♥ st♦ εi ♠♠♦s gi ♥ú♠r♦ st♦s q ② ♦♥ ♥rí εi st♦ s ♥ró♥ ♥rí εi ♦ ♥s st♦s st ♦r Ωi ♣r ♦s♦♥s s

Ωi =(ni + gi − 1)!

ni!(gi − 1)!

♦ q qr♠♦s s ♠①♠③r Ω ♦♥ s rstr♦♥s Etot = cte ② qtot = cte st♦ s ♣ r ♦ ♦♥ ♠ét♦♦ ♦s ♠t♣♦rs r♥ ♥tr♦♠♦s γ ② β ♣r s rs qtot ② Etot ♣♦♠♦s srr q

∑

i

δni

[∂ lnΩi∂ni

− βεi + ~γ · qi]= 0 ∀β y γ

♦♠♦∂ lnΩi∂ni

= βεi − ~γ · qi

② s♥♦ ♣r♦①♠ó♥ tr♥ ♥♦♥tr♠♦s q ∂ lnΩi/∂ni = ln(gi+ni

ni♣♦♠♦s s♣r ♣r

♦t♥rni =

gieβεi−~γ·qi − 1

t③♠♦s ♥ ♥♦tó♥ s♦♥t♥ ♣r s ♥rís r♣r♠♦s ♥♦tó♥ q ♠♦sst♦ s♥♦ ♥tr♦r♠♥t gi = D(ω) ② ni(ω, T ) = D(ω)fBE(ω, T ) P♦r ♦tr ♣rt t♠♥ts t♥ q ni ≫ gi és ♣r♦♠ ❩♠♥s② ② tt♠♥

Pr♦s♦s

♥ ♦♥s♥ stró♥ qr♦ ♥s ♦♥♦♥s ②♦s s♠♦♠♥t♦s s ♦♥sr♥ ♥ s♦s♦♥s s

fBE,s(q) =1

e~ωs(q)/kBT−~γ·qi − 1,

♦♥ γ s ♥ t♦r ♦♥st♥t q s t q∑

q,s =cte ①♣rsó♥ ♣r fBE,s(q) s ♣ ♥tr♣rtr ♦♠♦q ♦r ♥rí ♦s ♦♥♦♥s s ♠ s ♥ í♥~kBTγ · q és r ♥ ③ q s ♥rís

r♦r♠♦s q

jQ =∑

q,s

~ωs(q)∂ωs(q)

∂qfs(q),

② t♥♥♦ ♥ ♥t s ♥♦♥s q ♣r♥ ♥ s ①♣rsó♥ s r♠♥t q s ♦r♥ ♥rís r ♦r③♦♥t s ♠♣ q jQ = 0 ♣r♦ s ♦r♥ s t♦♠ ♥ í♥ rs♥t♦♥s ♦ tér♠♦ jQ 6= 0 ② s♦r♣r♥♥t♠♥t t♥r♠♦s ♦ tér♠♦ ♥ qr♦ ♣r ♥r♥t t♠♣rtr ♥♦

P♦r ♦ t♥t♦ ♥st♠♦s ♥r ♦s ♣r♦s♦s ❯ ♣r q ♥ s♥ ♥ r♥t t♠♣rtr ♣ ♦♥srs ♥ qr♦ tér♠♦ ♥ sst♠ ♦♥♦♥s st♦ s q s ♦t♥jQ = 0 q ♥st♠♦s ♥r st♦s ♣r♦s♦s ❯ s s♥♠♥t q s♦♥ ♦s ♣r♦s♦sq ♠ás ♠♥ ♦ r♣♦ s ♣rtís s ♦♥♦♥s

② q r♣rtr ni ♦♥♦♥s ♥ gi st♦s ♦♥ ♠s♠ ♥rí εi ♦♦ ② ♣s ni + gi− 1 ♣♦ss r♣rtr ♥ gi− 1st♦s és ❩♠♥s② ② tt♠♥ ♣á ♦♠♦ t♥t♦ ni ♦♠♦ gi s♦♥ ♠♦ ♠②♦rs q 1 s ①♣rs♦♥s s ♣s♣rr

♦ q s q♥t s ♥♣r♥♦ ♣r ♦♥rtr ♣r♦t♦∏

i ♥ ♥ s♠∑

i ♠♦s rr q ♦s ♣r♦s♦s t♠é♥ ♠♥ ♦ r♣♦ ② ♥ ♠ás ♥t♦ ♦ tér♠♦ ♣r♦ ♠♦s

st♦ q stíst♠♥t s ♦t♥rí ♥ ♦ st♦♥r♦ ♥♦ ♥♦ ♥s♦ ♥q ♥♦ ② r♥t t♠♣rtr

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❯❱

Pr♦s♦s ❯

Pr♦s♦s ❯

stró♥ fs(q) q ♠♦s ♥♦♥tr♦ ♥ts ♥♦♣ sr rr ♥ó♥ stró♥ ♦s♦♥♦♥s ♥ ♥ rst ♣♦rq t♦r♠ ♦ ♥♥q ♠♦s ①tr♠♦s P❩ ♥ sr q♥ts sr q γ(q = ±π/a) = 0 ② ♦♠♦ ♦♥s♥ γ(q) ♥♦♣ sr ♦♥st♥t

P♦r ♦tr ♣rt ♠♦s st♦ q ♦s ♣r♦s♦s ♠♣♥q γ s ♥ t♦r ♦♥st♥t ♣♦r ♦ q s♦♦ ♦s ♣r♦s♦s ❯s♦♥ ♦s s♥ts q γ(q) 6= cte Pr ♣♦r ♣r♦rs♦s ♣r♦s♦s ❯ ♥ t♥r q r♥♦s ♦s ♦rs P❩ Pr ♣s r③♦♥ ♣♥sr q ♠♦ ♠t♣♦r r♥ γ = cte ♣r ♣sr γ(q) 6= ctes♦♠♥t srá ♣r r ±π/a

♦♠♦ r♦ ♦♥ ♦ ♦♠♥t♦ ♠♣rsq γ(q = ±π/a) = 0 ♣♦♠♦s s♣♦♥r q r~kBTγ(q) · q t♦♠ ♦r♠ s♥ r

♦♦ st♦ ♦ ♣♦♠♦s rs♠r ♦♠♦ s♦♠♥t ♥♦ ② ♦♥♦♥s t ♥rí ♥ rstst♦ s ♦♥ q r♥♦s P❩ ♣ r rsst♥ tér♠ s ♦♦ ♣♦r t♥t♦ q s t♠♣♦ ró♥ s♦s ♦♥♦♥s r♥ ♥rí q ♦♥tr♦r ♦r♠ ♦♥t tér♠κred(T )

s t♠♣rtrs ♣♥s ② ♦♥♦♥s s t♣♦ ①t♦s ♣♦r ♦ q ② ♥ t♦♥ts tér♠

♦r ♥ ♣r q ② ♣r♦s♦s ❯ ♠♥♦s ♥♦ ♦s ♦♥♦♥s q ♦s♦♥♥ ♥ t♥r ♥s♠♦♠♥t♦ q ∼ kD ② q s♠♦s q kD ∼ G ② ♥ ♥rí ♦r♥ ~ωD P♦r t♥t♦ ♥ó♥ stró♥ ♣r s♦s ♦♥♦♥s srá

fs(q) =1

e~ωs(q)/kBT − 1≃ 1

eΘD/T − 1≃ (ΘD ≫ T ) ≃ e−ΘD/T .

st ♥ú♠r♦ r ①♣♦♥♥♠♥t r t♠♣rtr ② ♣r T ≪ ΘD ♦♥tκred(T ) ♠♥trá ♦♥ r♥ r♣③ ② q τ(T ) ∼ eT0/T ♦♥ T0 ∼ ΘD

Pr♦ s st♠♦s ♥ t♠♣rtr T s♥t♠♥t ② q T ≪ ΘD ♣r q κred(T )♣rs♥t st ♦♠♣♦rt♠♥t♦ ①♣♦♥♥ s r♦ q ♠♥♦ r ♠♦ ♦s ♦♥♦♥s ♣r sr ♠② r♥ ② ♣♦rá sr ♦♠♣r ♦s ♠♥♦s rs ♦s ♦tr♦s ♠♥s♠♦s ♣♦ss ♦só♥ ♦s ♦♥♦♥s ♦♥ s ♠♣r③s ♦ ♠♣r♦♥s r rst♥ ♦♥ s s♣rs trs rst s ♥♦ stá♥ ♣s

P♦r t♥t♦ s r ♥ í♠t ♥ q l(T ) ♥♦ ♣♥ ♠ás q ♠str ♥ stó♥② ♥♦ ♥r♠♦♥ st♦ s t♠♣rtr rst P♦r ♦♥s♥t ♥ í♠t ♥ qT −→ 0 ② T ≪ ΘD ♠♥♦ r ♠♦ l sr ♦♥st♥t ② ♦♥t tér♠ r♦♠♦ κred(T ) ∝ T 3 ♦ ♣♥♥ ♦r s♣í♦ ♦♥ t♠♣rtr

♥ rs♠♥ ♠② s t♠♣rtrs ② ♥ ♦♠♣♦rt♠♥t♦ κred(T ) ∝ T 3 ② ♦♥t tér♠ ♣♥ ♣r③ ② t♠ñ♦ ♠str ♦s ♣r♦s♦s ❯ stá♥ ♦♥♦s

♥ s♦ ♥ ♠str ♣♦rst♥ ♦♥♥t ♣♥rá ♠② rt♠♥t t♠ñ♦ ♦s rstt♦s ♠str

óts q st rst♦ ♣♥r rst♥ ♠str ② ♥ t♦ ♥ ♠trs ♠♦r♦s s st♦ qκred ∝ T

2 ♣r T ≤ 1

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó Pr♦♣s tér♠s ♦s só♦s

r s♥♦ t♠♣rtr ♠♣③♥ ♣rr ♣r♦s♦s ❯ ② ♥ ♦♥s♥ ② ♥ s♠♥ó♥rá♣ ♦♥t tér♠ ♦r♠ ①♣♦♥♥ κred(T ) ∝ eT0/T

♦♦ st♦ s ♦ q s ♦sr tt♠♥t ♥ ♦s rst♦s ①♣r♠♥ts ♣r s♥ts♦♠♦ s ♥ s rs

♦ts s♦r ♦♥t tér♠ κ(T )

• ♦♥t ♦♥ ♦s ♦♥♦♥s ♠②♦rtr♦s t♠♣rtr T

s r♦ q ♥tr③ ♦♥t♦r ♦s ♦♥♦♥s ♥r s♦r ♦♣♠♥t♦ ♦♥qr s♣rtí sst♠ ② ♦ ♦♣♠♥t♦ ♥ ♥r ♣♥rá ♦♥t ♦♥λ ♦s ♦♥♦♥s

t♠♣rtrs s T & ΘD ♦s ♦♥♦♥s t♥♥ ♥ ♥rí r♥ ωmax ♦ q ♥♦s q λ ∼ 2a ♦♥ a s ♣rá♠tr♦ r

Pr ♦trs t♠♣rtrs ♥ st♠ó♥ r③♦♥ ♣♦rí sr

λ ≃ 2a

(ΘD

T

)=⇒ q ≃ π

a

(T

ΘD

),

q ♥♦s λ ≃ 200 a ♣r T ≃ 3 ♠♥trs q ♣r T ≃ 300 t♥r♠♦s λ ≃ 2 a

• ♥s♠♦s ♦só♥ ♦s ♦♥♦♥s

♦t ♦♠♦ ② ♠♦s ♦♠♥t♦ ♥ts s♣♦♥r♠♦s q s ♥s ♥tr♦s ♦só♥s ♦s t♦s ♦s st♥t♦s ♣r♦s♦s ♦só♥ ♥ sr ♥♣♥♥ts ② t♦s P♦r ♦tr♣rt ② s♠♦s q ♦s ♣r♦s♦s ♥♦ ♥tr♥♥ ♥ ♦♥t tér♠ κ(T )

Pr♦s♦s ❯ ② st♦s

♦só♥ ♦♥ t♦s ♣♥ts ♠♣r③s sót♦♣♦s t♥ st s♦ ♦ q ♠♣♦rt s ró♥ ♥tr λ ♦s ♦♥♦♥s ② t♠ñ♦ ♦stá♦ t♠♣rtr s r q λ ≫ a ♦ q q ♣r♦s♦ ♦só♥ s s♠ ♥tró♥ ③ ♦♥ ♣rtís t♠ñ♦ ♠② ♣qñ♦ s♣rsó♥ ② q ♣ srst ♠♥t ♣r♦①♠ó♥ ♦r♥ s♣rsó♥

❱és ♣♦r ♠♣♦ r♦ s♦♥ tr♦♥á♠ ás s♥ ó♥ ♣rt♦ ② ♥ ssó♥ ♣r♥ ♣rt♦

r s♥tr ís á♥t

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❯❱

♣r♦ ♦só♥ rst sr ♣r♦♣♦r♦♥ q4 ♣♦r ♦ q ♥ ♥str ♣r♦①♠ó♥ t♠♣♦ ró♥ ♣♦♠♦s srr q τ(T ) ∝ T−4 ② s t♦ s♦r ♦♥t tér♠ s♣♦r ♦♥s♥t

κimp(T ) ∝ T 3︸︷︷︸cv

T−4︸︷︷︸τ

= T−1.

t♠♣rtrs ts t♥♠♦s s♥ ♠r♦ q λ ≃ a ♦♠♦ ♦r s♣í♦ s ♦♥st♥t♦t♥♠♦s q

κimp(T ) ∼ cte.

♦s♦♥s ♦♥ s s♦♦♥s

♥ s♦ s s♦♦♥s st♦rsó♥ r rst♥ s r♦ ♥ st♦ s ♥♦♣♦♠♦s r q stá ♦③♦♠♦ ♥ú♠r♦ s♦♦♥s s ♣r♦♣♦r♦♥ T−2 s ♦♥tró♥ ♦♥t s

κimp(T ) ∝ T 2.

• Ór♥s ♠♥t ♣r κ(T )

❯♥ ♠tr s♥t ♠② ♣r♦ ♦♥ ♥ t♠♣rtr ② ΘD ♣ ♣rs♥tr ♥♦♥t tér♠ r♥ Tamb

s s t♦ s♦ ♠♥t t♠♣rtr ♠♥t

♠♥t κdiamante ≃ 2000 ❲♠ ΘD = 2000

♦r κCu ≃ 400 ❲♠ ΘD = 340

st♦ s ♣r♦ ♣r ró♥ á♠♥s s♠♦♥t♦rs ♣♦rs s♦r ♠♥t ♦q s ♣r♠t t♥r ♥ rrró♥ ①♥t

♣♦ ♠á①♠♦ ♦♥t κ(T ) s ♥♦r♠♠♥t ♠② t♦ ♦♥ ♦rs tí♣♦s ♥♦s 2× 104 ❲♠ ♣r rsts ♣r♦s ② ♣r ♥ t♠♣rtr ♣r♦①♠ ΘD/20

♥ í♠t s t♠♣rtrs s ♣ st♠r κ(T ) ♦♥

♥ t♦ ♠♦s ♥ s♦ stá♥r

ΘD ≃ 300 ♥s ♣r♦① 104 ♠3 Ps♦ tó♠♦

Cv −−−→T→0

1941

(TΘD

)3

♠♦

460 J/mol−K si T = 4K

3680 J/mol−K si T = 8K

Vsonido ≃ 5000 ♠s dmuestra ≃ 3 ♠♠ ≃ lfonon

κ = 13 cv l vsonido ♠♦

2300 W/m−K si T = 4K

18400 W/m−K si T = 8K

Pr r♦s ② ♣ást♦s s t strtr rst♥ r ♥ r♥ ♥ú♠r♦ ♦s♦♥s ♦s ♦♥♦♥s P♦r ♦♥s♥t t♥♥ ♥ ♦♥t tér♠ κ(Tamb) ∼ 0,1 ❲♠

❨ ♠ás st♦s ♠trs s♦♥ ①♥ts s♥ts s T

❯♥ á♦ ♠ás ♣rs♦ q ♥♦ ♠♦s ts ♥♦s rí κimp(T ) ∝ T−3/2

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó Pr♦♣s tér♠s ♦s só♦s

• ❱ró♥ ♦♥t ♦♥ ♦s sót♦♣♦s

♥ s♦ q ♠str ♥ ♠tr rst♥♦t♥ ♥ stró♥ ③r ♦s sót♦♣♦s q♦♥stt②♥ rst s stró♥ ♣rtr ♣r♦ r

♥♥ ♥ ♦♥t tér♠ κ(T ) ♣sr ♠② ♥♦t sú♥ s ♥ r ♥ q s♣rs♥t♥ s ♦♥ts ♥♦r♠ ② ♥rq♦ ♦♥ 74

• ❱ró♥ ♠á①♠♦ ♦♥ s ♠♣r♦♥s

tr ② ♣♦só♥ ♠á①♠♦ ♦♥t tér♠ ♣♥♥ r♥♠♥t s♠♣r♦♥s r ♦♠♦ s ♦sr ♥ ♦s rst♦s ①♣r♠♥ts

• ❱ró♥ s t♠♣rtrs

♦♠♣♦rt♠♥t♦ ①♣♦♥♥ r♥t κ ♦♥ T ♣r s t♠♣rtrs s ♦sr①♣r♠♥t♠♥t ♦♠♦

κ(T ) ∝ e−ΘD/bT con b ≃ 2− 3

st♦ stá r♦ ♦♥ ♦ q ② s ♦ ♥ts

tr♠♥ó♥ T0 ♦ b s ♠② í ♣r♦ ♦ ♠♣♦rt♥t qí s ♣ró♥ tér♠♥♦①♣♦♥♥ q r ♦♥♠♥t♦ ♦s ♣r♦s♦s ❯ r ♥♦ t♠♣rtr

8.4 ❯❱ ❯ÓÓ

♥ ♦s ♠ts ♦s tr♦♥s ♥ ♦♥trr t♠é♥ tr♥s♣♦rt tér♠♦ ♣ s♣♦♥rq ♦♥tró♥ r ② ♦s tr♦♥s s♦♥ ♥♣♥♥ts ② ♣♦r t♥t♦ q s♦♥ ts

κ = κred + κelec

② s ♣♠♦s ♠s♠♦ ♠♦♦ ♦♥♣t q ♠♦s s♦ ♣r ♦s ♦♥♦♥s s t♥ q

κelec(T ) =1

3lelec vF celecv =

π3

3

Nk2Bm∗vF

lelec(T ) T

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❯❱ ❯Ó Ó

♦♥ s sstt♦ ♦r s♣í♦ tró♥♦ ♣♦r ①♣rsó♥ q ♠♦s ♦t♥♦ ♥ ♠♦♦ ♥ sts r♥st♥s q ♣♦r str ♦♠♣♦rt♠♥t♦ lelec(T ) ♣r tr♠♥r κelec(T )

ts t♠♣rtrs

♥ st í♠t s ♦s♦♥s q ♠t♥ ♠♥♦ r ♠♦ ♦s tr♦♥s lelec s♦♥ s s ♣rs♥ ♦s ♦♥♦♥s

♦♠♦ ② ♠♦s st♦ q ♥ú♠r♦ ♦♥♦♥s s ♣r♦♣♦r♦♥ T ♣♦♠♦s st♠r q lelec(T ) ∝1/T ② q κelec(T ) = cte st♦ s ♦♥sst♥t ♦♥ ♦s rst♦s ①♣r♠♥ts ♥♦ ♦rs

tí♣♦s cv ≃ 0,01 credv ≃ 3 × 104 Joule/m3−K vF ≃ 106 m/s ② λelec ≃ 10−8 m ♦t♥♠♦s qκelec ≃ 102 ♠ ② ♣♦r ♦♥s♥t q

κredκelec

=credv v lfononescelecv vF lelec

≃ 5× 10−2

♦r♠ q κ ♦s ♠ts s ♣rát♠♥t t♦ ♦s tr♦♥s

♥ ♠r♦ ♥ ♠trs ♦♥ ♠s ♠♣r③s ♦ ♥ ♦♥s ♣ ♦rrr q κelec ∼ κred② ♦♠♦ rst♦ ♦♥t tér♠ t♦t rst sr ♠② ♣qñ

Tamb ♦♥st♥tá♥

κ ♠

② s t♠♣rtrs

♥ s ♣r♦①♠s T = 0 ♥ú♠r♦ ♦♥♦♥s s ♠② ♣qñ♦ ② s ♦s♦♥s ♦str♦♥s s♦♥ s ♠②♦r♠♥t s ♠♣r♦♥s ♠♣r③ rst

♥ú♠r♦ ss ♠♣r♦♥s ♠♣r③s ♥♦ rí ♠♥r ♣r ♦♥ T ♥t♦♥st♥r♠♦s q

lelec ∝1

nimp= cte,

♦♥ ♠♦s ♠♦ nimp ♦♥♥tró♥ ♠♣r③s ♠♣r♦♥s

♥ ♦♥s♥ ♦ q celecv ∝ T ♦t♥♠♦s q

κelec ∝ T

♠♣rtrs ♥tr♠s

Pr st r♥♦ t♠♣rtrs ♦♥tró♥ r ♦♥t tér♠ srá♥ ♦♠♣♦rt♠♥t♦ s♠r q ♠♦s st♦ ♣r ♦s s♥ts ♥tr♦r♠♥t ♦♥t r ♦♥ ♥ ♠á①♠♦ ② ♥ ♠♥t♦ ①♣♦♥♥ ♠♥tr t♠♣rtr s♥t♠♥t s♣rr ♣s q ♥ st r♥♦ s ♦♥tró♥ r q s ♠ás ♠♣♦rt♥t

♥ ♠r♦ s s t♦♠ ♥ ♥t q ♣r T ≪ ΘD ♥ú♠r♦ ♦♥♦♥s ①t♦s s ♣r♦♣♦r♦♥ T 3 ♦ q ♦r s♣í♦ r s ♣r♦♣♦r♦♥ T 3 s ♦t♥ q

κelec ∝ T−2

ró♥ q s ♦♥r♠ ♥ ♠♦s s♦s ♣♦r ♦s ①♣r♠♥t♦s

s s r③ó♥ ♣♦r q ♦♥st♥tá♥ t♥ ♥ κ ♠♦ ♠♥♦s q s ♦♥ts tér♠s ♦ ② s ♠♦ q t♠♥t ♥ s s♦ κelec ∼ κred

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❮ Ó Pr♦♣s tér♠s ♦s só♦s

♦♥t ♥ér ♥ ♠t ♦♥t tér♠

♥♦ ♥ rs♠♥ ♥str♦ ♥áss ♦♥tró♥ tró♥ ♦♥t tér♠♥♦s q

κelec ∝ T t♠♣rtrs ♠② s ♦♥ ♦♠♥♥ s ♦s♦♥s ♦♥ ♠♣r③s ② lelecs ♦♥st♥t

κelec ∝ T−2 t♠♣rtrs s T ≪ ΘD ♦♥ s♦♥ s ♦s♦♥s ♦♥ ♦♥♦♥s s ♦♠♥♥ts ② lelec rí ♦r♠ ♥rs♠♥t ♣r♦♣♦r♦♥ T 3

κelec = cte t♠♣rtrs ts T ≫ ΘD ♦♥ ♦♠♥♥ s ♦s♦♥s ♦♥ ♦♥♦♥s ② lelec rí ♦r♠ ♥rs♠♥t ♣r♦♣♦r♦♥ T ♠♣ ② ❲♠♥♥r♥③ st♦ sq κelec/(σT ) = cte

sts ♣r♦♥s ♠♣♥ q κelec ♣rs♥tr ♥ ♠á①♠♦ ♦s rst♦s ①♣r♠♥ts♥ ♠ts ♦♥ ♦♥tró♥ κelec ♦♥t tér♠ s r♥ s♥ ♦♥r♠rsé♥s ♦s rst♦s ♣r s ♥ ♣r s t♠♣rtrs ♥tr♠s s ♠♥ ♦♠♣♦rt♠♥t♦s t♣♦ κelec ∝ T−x ♦♥ 2 < x < 2,5

8.5 ❱ P T

♠♦s q ♦s tr♦♥s ♦ ♥ rst ♣rt♦ ♥ r ♥ ♦♥t ♥♥tst♦ s♥ q ♦♥t étr ♥t ♦s ♠ts s ♥tr♠♥t ss♦♥s r r rs♣t♦ ♣r♦ ♣rt ♥s♦ ♥ ♠str ♠ás ♣r s♥t♦s ♠♣r③s ♦ s♣rs q ♠t♥ ② ♥ ♥t rsst ①st♥ sr♦♥s tér♠s ♦s ♦♥s r ♦♥ ♣ró♥ s ♦s♦♥s tró♥♦♥ó♥ ♦♠♦② s stó ♥ ♥tr♦ó♥ ♠♦♦ sts ♦s♦♥s s♦♥ s s♥ts ♠♥t♦ rsst ♦s ♠ts ♦♥ t♠♣rtr ♠♥trs q rsst s♦ ①st♥ ♠♣r③s ♥ ♠str s s ♦♥st♥t

str ró♥ ρ(T ) s ♣ ♦♥srr ♥tró♥ ♦s tr♦♥s ♦♥ rst♦♠♦ ♥ ♣r♦s♦ s♦ró♥ ♦ ♠só♥ ♦♥♦♥s ♥ q ♥rí ② s♠♦♠♥t♦ ♦str♦♥s ♠♥ r♦ ♦♥

k → k′ εk = εk′ ± ~ωs(k− k′ +G) = εk′ ± ~ωs(k− k′)

♦♥ + ♥ ♠só♥ ♥ ♦♥ó♥ ② − s♦ró♥ ♥ ♦♥ó♥ ② ♦♥ s t♥♦ ♥ ♥t q t♦r ♦♥ ♦♥ó♥ rr q k′ = k+ q−G ♦♥sró♥ ♥rí ♠♣

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❱ P T

q s♦♦ q♦s ♦♥♦♥s ♦♥ ♥rí

~ωs(q) = ±(εk+q − εk)

♣♥ ♣rt♣r ♥ ♦só♥ ♦♥ tró♥ k st ♦♥ó♥ ♥ ♥ s♣r ♥ s♣♦ k

q ás s♦♥ ♦s ♦♥♦♥s q ♣♥ ♦s♦♥r ♦♥ tró♥ ♦r ♥ ~ωs(q) s ♣qñ♥ s ♥rís ♦s tr♦♥s ② ♣♦r ♦ t♥t♦ q sr ♥ t♦r t q εk+g s♣rát♠♥t εk

s s♣r q♥rét εk = cte ♥ ♣rtr εk = εF ② q ♦s tr♦♥sq tr♥s♣♦rt♥ r étr t♥♥ ♣rát♠♥t s ♥rí ♦ q εk′ = εk ♥t♦♥s ♦st♦rs q ♦s ♦♥♦♥s q ♦s♦♥rá♥ ♦♥ ♦s tr♦♥s srá♥ q♦s q ♣rt♥ ①tr♠♦ k ♣r tr♠♥r s♦r ♠s♠ s♣r q♥rét

♠♣rtrs ts T ≫ ΘD

sts t♠♣rtrs ♥ú♠r♦ ♦♥♦♥s ♥ ♥ ♠♦♦ ♥♦r♠ s

f(q) =1

e~ω(q)/kBT − 1≃ kBT

~ω(q)∀q

♦r♠ q ♥ú♠r♦ t♦t ♦♥♦♥s q t♥♥ ♥ s♠♦♠♥t♦ q ♦♠♦ q ♠♦s ♥♦♥ts sr ♣r♦♣♦r♦♥ T ♦ q rsst ♠♥trá ♦♥ ♥ú♠r♦ ♦s♦♥s sr♦ q ésts ♦ rá♥ ♦♥ ♥ú♠r♦ ♦♥♦♥s ② ♣♦r t♥t♦

ρ(T ) ∝ T s T ≫ ΘD

♠♣rtrs s T ≪ ΘD

♥ st s♦ s♦♦ ♦s ♦♥♦♥s ♦♥ ♥rí ~ω(q) . kBT ♣♥ sr s♦r♦s ♦ ♠t♦s ♣♦r ♦str♦♥s ♥ s♦ró♥ s ♥t s♦♥ ♦s ú♥♦s ♦♥♦♥s ♦s q ② ♥ ♥ú♠r♦ ♣r ♦ ♣r♥♣♦ P ② ♦ q ♦s tr♦♥s t♥♥ ♥ ♥rí ♣r♦①♠ εF ± kBT ♦s ♦♥♦♥s q s ♠t♥ s♦♦ ♣♥ t♥r ♥ ♥rí ♦r♥ kBT

P♦r t♥t♦ s T ≪ ΘD s ♠♣ q s♠♦♠♥t♦ q ≪ kD ② st♠♦s ♥ ③♦♥ ♥ ró♥ s♣rsó♥ st♦ s s ♠♣ q ω(q) ≃ csq st♦ qr r q t♦r ♦♥ ♦♥ó♥ srá

q .kBT

~cs

❱♦rs tí♣♦s εF ∼ 10 ❱ ωM ∼ 1012 − 1014 s−1 ~ ≃ 6,6× 10−16 ❱ · s ~ωM ∼ 0,01− 0,1 ❱

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❮ Ó Pr♦♣s tér♠s ♦s só♦s

q ♠♣ ♥ ♦r ♣qñ♦ q ♦♠♦ ♠ás q rr ♦♥ó♥ q ② ♠♦s ♦ω(q) ≃ csq ♥ú♠r♦ ♦♥♦♥s q ♠♣♥ s ♣r♦♣♦r♦♥ q2 ② ♥ ♦♥s♥ ♣r♦♣♦r♦♥ T 2

♥ú♠r♦ ♦s♦♥s ♣♦r ♥ t♠♣♦ ♥tr ♦s tr♦♥s ② ♦s ♦♥♦♥s s ♣r♦♣♦r♦♥ r♦ ♥ ♦♥st♥t ♦♣♠♥t♦ ♥tr ♠s ♣rtís s t♠♣rtrs T ≪ΘD st r♦ s ♣r♦♣♦r♦♥ T ② ♥ rs♠♥ ♥ú♠r♦ ♦s♦♥s ♣♦r ♥ t♠♣♦ sts t♠♣rtrs s ♣r♦♣♦r♦♥ T 3 tr ♠♥r ①♣rsr st rst♦ s s♥t ♥ú♠r♦ ♦s♦♥s s ♣r♦♣♦r♦♥ ♥ú♠r♦ ♦♥♦♥s ♥ rst ② t♠♣rtrs T ≪ ΘD

t♥♠♦s q 〈nfonon〉 ∝ T 3

♦♠♦ ② s ♦♠♥t♦ s t♠♣rtrs ♦s ♣r♦s♦s ♥ ♦s q ♥tr♥ ♥ s♦♦ ♦♥ó♥♠♥ t♦r ♦♥ ♦s tr♦♥s ♥ ♥t ♠② ♣qñ ♦ q s ♠♣ qq ≪ kD ② q q ≪ kF ♦ ♦s tr♦♥s |v(k)| s ♣rát♠♥t ♦♥st♥t s♦r s♣r r♠ s r♦ q s ♦s♦♥s s t♠♣rtrs s ♦♥♥tr♥ ♥t ②♣♦r t♥t♦ s♦♥ ♠♥♦s ts ♣r ♦♣♦♥rs ♦rr♥t étr

♣♦♥♥♦ ♥ s♣r sér sér sótr♦♣ ②q s ♦s♦♥s s♦♥ ásts s T ♥tr♠♦ ♥rí s ♠♦ ♠♥♦r q ~ωD ② s ♦s♦♥s ♥♦ ♣♥rr t♠♣♦♦ ♥rí ♠ ♦s tr♦♥s s ♣♠♦strr q ♥ú♠r♦ t♦ ♦s♦♥s ♣♦r ♥ t♠♣♦ s ♣r♦♣♦r♦♥ ♥♠♥t

T 2 · T 3 = T 5

♦♥ T 2 ♣r♦♥ ♠♦♠♥t♦ ♥r s ♦s♦♥s♥ t♦ s ♥♦s ♠♦s ♥ r ② ♦♠♦ ② ♠♦s ♦♠♥t♦ ♥ ♥ ♦só♥ tró♥ ♣s♦♦ ♥ ♣♦ró♥ ∆k ♠♦♠♥t♦ ♥ ♦♥s♥ ② q r ♥ ♣r♦♠♦ ♥r t♦s s♦s♦♥s

♥ rs♠♥

ρ(T ) ∝ T 5 s T ≪ ΘD

♦♠♦ s ♠♦s ♥♦♥tr♦ ♦♠♣♦rt♠♥t♦ ♥r ρ(T ) ♦♥T ♣r♦ s♥ str ♦s ♦rss♦t♦s ρ st♦ s í r ♣s ♥tró♥tró♥♦♥ó♥ s ♠② í trtr ♥ ♠♥t P♦r ♦s♣ ♦♠♣rr rsst ♣rst♥t♦s ♠ts ♦♥ rsst T = ΘD rst♦ q s ♠str♥ r s ①♥t ♣s tst♠♣rtrs s ♥ ♠♥trs q s t♠♣rtrs t♥ ♥ ②♥ T 5

♦ s ♠♦strrá qí Pr r ♥ st♠ó♥ s ♣r♦♠♦ ♥r ♣r ♦s♦♥s ásts s ♣ r♠♥tr s♥t ♠♥r

r♦ ♦♥ r ∆k = q sin(θ/2) ② sin(θ/2) = q/(2k) ♣♦r ♦ q ∆k = q2/(2k) ♠♦s ♥ tró♥ s♠♦♠♥t♦k ♣r♦ ♠♣♦ étr♦ E ♥ú♠r♦ ♦s♦♥s ν q s♦♥ ♥srs ♣r r♥r tró♥ s♠♦♠♥t♦ k ♣st♠rs ♦♠♦ ν = k/∆k ≃ 2(k/q)2 ∝ q−2 ♦♠♦ r♦ ♦♥ s ♠♣ q q . kBT/(~c) sts t♠♣rtrs ♥♦sq q q ∝ T ② q ν ∝ T−2 ♥ú♠r♦ t♦ ♦s♦♥s q sts t♠♣rtrs r♥♥ tró♥ s ♣r♦♣♦r♦♥ T 2❯♥ ♦s ♠♣♦rt♥t q ♠♦s rr ts t♠♣rtrs q s ♦r♥ k ② q ♠♦s ♦ q kD ∼ kF ② ν ∼ 1P♦r ♦♥s♥t ts t♠♣rtrs ♥♦ s ♥sr♦ r③r ♣r♦♠♦ ♥r ♥ ssó♥ ρ(T ) ② sí ♠♦s ♦ ♥ts

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❱ P T

② ❲♠♥♥r♥③

♦♦ st♦ q st♠♦s ♥♦ ♣♦♠♦s ♥r♦ ♥ st♦ ② ❲♠♥♥r♥③q ♥♦s r③ó♥ κ/(σT ) s ♦♥ts étr ② tér♠ ♥ ♠t

Pr tr♦♥s rs ♠♦♦ r♦♠♠r s t♥ q

κ

σT=π2

3

(kBe

)2

.

ts t♠♣rtrs s ♦♥sr♠♦s q ss t♠♣rtrs s ♠♣ q κtotal ≃ κelectt♥♠♦s q κ t♥ ♥ ♦♥st♥t ♠ás ss t♠♣rtrs σ ∝ 1/T ♠♥r q ♦♥t κ/(σT ) s ♦♥st♥t

♥♦ s t♠♣rtrs s♦♥ ♠② s T → 0 s♠♦s q ♦♠♥♥ s ♦s♦♥s ♦s tr♦♥s ♦♥ s ♠♣r③s sts ♦s♦♥s sí ♣♥ ♠r ♥♦t♠♥t ♦r s♠♦♠♥t♦ k tró♥ q ♦s♦♥ ♦ q stá r♦ ♦♥ s ss ♠♦♦ P♦r ♦♥s♥t ♥♦ ♦♠♥♥ ss ♦s♦♥s ♦♥ s ♠♣r③s t♠é♥ s ♠♣ q r③ó♥κ/(σT ) s ♦♥st♥t

♥ ♠r♦ s st♠♦s s t♠♣rtrs T < ΘD ② t③♠♦s ♥ ♠♦♦ t♠♣♦ ró♥ s ♦s♦♥s ♦♥ ♦s ♦♥♦♥s ♥ ♣srs ♦♥ ♥ t♦r (T/ΘD)

2 ② q tr♥sr♥ s♠♦♠♥t♦ ∆k s ♣qñ ♦♠♦ ② s st♦ ♥ ♦♥s♥

κ

σT∼(kBe

)2( T

ΘD

)2

.

❱ró♥ ♦♥t ❲ ♦♥ t♠♣rtr

♣♦r t♥t♦ ♣♦r r ó♠♦ s ♦♠♣♦rt♠♥t♦ r③ó♥ κ/(σT ) ♣rqs t♠♣rtrs ♥tr♠s ♦♥ T <ΘD ♦♥ ó♥ ♣ r ♦rs♦s ♣r ♦ ♦♥t ①♣r♠♥t♠♥ts ♥♥tr q ♣r ♥ ♠í♥♠♦ és r st ♦ q ♠♦s st♦ st♥♦♥ ♠♥r ♥tr♣rtr s ♠í♥♠♦ s s♥t t♠♣rtr q s ♣rs♥t ♠í♥♠♦ ♦♥r ♦♥ s t♠♣rtrs s q ♠♥s♠♦ ♦♠♥♥t s♣rsó♥ ♦s tr♦♥s ♠♣③ sr ♥tró♥ ♦♥♦♥♦♥s ♥rí

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♣ít♦

P❯❯

9.1 ❯Ó

♥ó♥ ♠ás ♥r ♥ t♦ rst♥♦ ♣ sr ró♥ rst ♦♥ stró♥♠r♦só♣ ♦s ♦♥s s ♠② r♥t stró♥ ♥ ♥ rst ♣rt♦

t♦ ♠ás rtríst♦ q ♣r ♥ t♦♦s ♦s rsts s ♥t♠♥t s s♣rsrst♥s

ró♥ ♠♣ró♥ rst♥ s ♦r♥ ♥ s♣♦ r ♥t♦♥s t♦ s♠ ♣♥t t♦ s ①t♥ ♦ r♦ ♥ í♥ ♥ rst t♦ s ♠ ♥

♦ í♥ ❨ s ①t♥só♥ s ♥ t♦ ♥ s♣rs t♦ s ♠ s♣r

♦♥♥ rs q ♥ó♥ t♦s ♥② t♠é♥ ♦♥♦♥s tr♦♥s ② ♦s①t♦♥s ♠♥♦♥s ♦♥s s♣í♥ ② ♥♦ só♦ ♦s t♦s ♦③♦s ♥ r

9.2 P❯❯

♦s t♥r♠♦s ♣r♠r♠♥t ♥ str ♦s t♦s ♣♥ts ♥♦ ♥♣é ♥ ♦srsts ó♥♦s ①st♥ r♦s t♣♦s ♠♣♦rt♥ts t♦s ♣♥ts

❯ át♦♠♦ ♥trst

t♦♠♦ ♥trst

♣ ♥ ♣♦só♥ ♥ s♣♦ q ♥♦r♠♠♥t ♥♦stá ♦♣ s rs ♠♣s ♣♦♦ ♦♠♣ts ♣♥♦♠♦r át♦♠♦s ♥trsts ♠ás á♠♥t q s q♥♦ ♦ s♦♥ ♠♥r q ♦s át♦♠♦s ♦ ♦♥s ♣qñ♦s♣♥ ♦♣r ♣♦s♦♥s ♥trsts ♦♥ ♠ás q ♦s ♠②♦r ♦♠♥ ♦♠♦ s ♥tr s♠trí r ♦s át♦♠♦s ♥trsts s ♦♦♥ ♥tr♠♥♦s ♣♥t♦s ♥tr ♣♥t♦s ② ♥♦ ♥ qr ♣♥t♦ s♣♦

❱♥t r

s ♥ át♦♠♦ ♦ ó♥ ♦♠t♦ ♥ ♥ ♣♦só♥ ♥♦r♠♠♥t ♦♣ ♥ ♥ rst ♦♥t s♥ ♥ át♦♠♦ étr♠♥t ♥tr♦ ♥♦ ♣r♦ r♦tr ♥tr r rstPr♦ ♥ ♠♦ ♥ ♦s rsts ó♥♦s sí s rí r ② ♣♦r ♦ t♥♥ ♣rs♥tr ♦s t♦s♣♦r ♣rs

sí

❯ t♦s ♦tt② ② r♥

t♦ ♦tt② ♥ ♥t ó♥ ♣♦st♦ ♦rrs♣♦♥ ♦tr ó♥ ♥t♦ ♦♥srá♥♦s s♠♥r ♥tr r t♦ r♥ s ♥♦♠♥ó♥ ♥ ♥t ó♥ ♦♥♥ r ♥trst s♠r ♦ ♥♠♣r③ ♦♥ ♥ ♥ú♠r♦ tr♦♥sr♥t ♦s át♦♠♦s rst

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❮ Ó t♦s rst♥♦s ♣♥ts

♦s t♦s ♦tt② ♣r♥ ♠②♦r♠♥t ♥ ♦s r♦s ♥♦s ② ♦s t♦s r♥ ♥♦s r♦s ♣t ♥ qr rst r ♥ qr♦ tér♠♦ ①st♥ ♠♦s t♣♦s t♦s♦♠♦ ♦♥s♥ tó♥ tér♠ ♦ ♥ ♦trs ♣rs s t♦♥s stísts

t♦♠♦s ①trñ♦s

♦♥ ♦s q s stú♥ ♥trst♠♥t ② s st♦ ♥ ♦s t♦s r♥ ♦ sstt②♥♦ ♦s ♦♥s rst ♠♥♦ s strtr tró♥ s ♠♣♦rt♥t ♥ s♦ ssttó♥② t♠é♥ ♣ sr ♠② ♠♣♦rt♥t s ♦♠♥ s♦ ♦s át♦♠♦s ♥strsts

❨ s ♦♠♥t♦ q s♣rrs ♥ ♣qñ♦ s rá ♠ás ♥t ♥ú♠r♦ t♦s♣♥ts ♥ qr♦ tér♠♦ ♥ ♥ rst q ♠♥t ♦♥ t♠♣rtr ♣♥♥♦ ♥rí ♦r♠ó♥ ♦s t♦s ♣r♦♠♥rá ♥ t♣♦ ♦tr♦ ♦s

♥ú♠r♦ t♦s q ② ♥ ♥ só♦ ♥ qr♦ tr♠♦♥á♠♦ s ♣ ♠♥tr♠♥t r♦s ♠ét♦♦s ❯♥♦ ♦s s t♠♣♦ ♣r♦♠♥t♦ ♣♦r q s rá♣♠♥t t♠♣rtr ♥ ♠tr s t♠♣rtrs s trs ♣♦s s t③r ♦♠r♦ró♥ ♦♥ ♣rtís ♥rs ts ♦♠♦ ♣r♦t♦♥s ♦ ♥tr♦♥s q ♣♥ s♣③r ♦sát♦♠♦s rst ♣♦r ♦só♥ rt ♦ ♣♦r t♦ ♥t♠♥t♦ ♦ ❯♥ trr ♣♦s s ♦r♠ó♥ ♥ rí♦ rst

P♦r ♦tr ♣rt s ♦♠ú♥ s♦ ♣ ró♥♦ ② ♦①í♥♦ ♥ ♠ts q s ♥tr♦③♥ át♦♠♦s①trñ♦s ♥trsts ♣♦r só♥ trés s s♣rs

9.3 P ❱

• ♣♦♥♠♦s ♥ ú♥♦ t♣♦ ♥ts ♥ ♥ rst rs ♥ú♠r♦ ♥ts♥ qr♦ t♠♣rtr T n(T ) s ♥ ♠♥t ①t♥s ♣♦♠♦s r♦ ♠♥t ♥♠♥♠③ó♥ ♥rí r s

G = U − TS + pV p = ♣rsó♥

♦♠♥ rst s ♣ r ♣r♦①♠♠♥t s♣♦♥♥♦ q ♦♣ ♠s♠♦ ♦♠♥q rst ♣rt♦ (N +n) ♦♥s st♦ s ♦♥ N ♦♥s ② n ♥ts q ♦♣♥ ♠s♠♦ rP♦r t♥t♦

V (n) = (N + n)v0

♦♥ v0 s ♦♠♥ ♣♦r ó♥ ♥ rst ♣rt♦

s♦♠♦s ♦r ♥ ♦♥♥t♦ tr♠♥♦ ♣♦s♦♥s s ♥ts ♥♠♦s ♥ó♥F0(n) = U − TS ♣r s ♦♥♥t♦ s n s ♣qñ♦ ♥t♦♥s F0 ♣♥r n ♣r♦ ♥♦ ♣♦só♥ s ♥ts ♥♦ ② ♥tró♥ ♥tr s ♥ rst ①st ♥ r♥ ♥ú♠r♦ ♣♦ss ♦♦r n ♥ts ♥ (N + n) ♣♦s♦♥s ó♥s ♣♦ss s♦ ♥ ♦♥tró♥ ♥tr♦♣í

Sconfig = kB ln

((N + n)!

N !n!

).

P♦r ♦♥s♥tG(n) = F0(n)− TSconfig + p(N + n)v0

♣s F0 ♥② ♥tr♦♣í ♣r♦♣ ♦♥ró♥ s ♥ts r♥♦ rs♣t♦ n s t♥

∂G

∂n=∂F0

∂n+ pv0 − kBT ln

(N

n

)

ísst♦ó♦

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P ❱

♦♥ s t③♦ ♠ ♣r♦①♠ó♥ tr♥ q s lnN ! ≃ N(lnN − 1) ♥♦ N ≫ 1② s s♣st♦ q N ≫ n st út♠ ♦♥ó♥ ♣r♠t ♠ás srr q

∂F0

∂n≃ ∂F0

∂n

∣∣∣∣∣n=0

= ε,

♦♥ ε s ♥rí ♦r♠ó♥ s ♥ts Pr ♠♥♠③r ♠♦s r r♦ r

ε+ pv0 − kBT ln(N/n) = 0 =⇒ n = Ne−(ε+pv0)/kBT .

♣♦♥♠♦s q ♠t♦♥♥♦ s ♣ srr ♦♠♦ ♥ s♠ ♠t♦♥♥♦ qr♦② ♥ ♠t♦♥♥♦ r♠ó♥♦ s♦♦ ①st♥ ♦♥♦♥s ♥ rst P♦r t♥t♦

F = F equil + F fonones(T ),

s♥♦ F fonones ♥rí r ♦s ♦♥♦♥s st ♥rí s ♥♦r♠♠♥t ♠♦ ♠ás ♣qñ qU equil ♠♥r q ♥ ♣r♠r ♣r♦①♠ó♥

ε0 =

(∂U equil

∂n

)

n=0

q s ♥rí rqr ♣r ①trr ♥ ó♥ rst ♥óts q ♥♦ ♣♥ t♠♣rtr♦♠♦ s s♣r q st ♦r s ♣r♦①♠♠♥t ♦r♥ ♥rí ♦só♥ ♣♦r ♣rtí∼❱ s ♣ s♣rr ♦♥tró♥ pv0 ♥ ①♣♦♥♥ ♥ s st♠♦s ♣rsó♥t♠♦sér

♥ rs♠♥n = Ne−ε0/kBT

② s ε0 ≃ 1 ❱ T = 1000 ♥t♦♥s n/N ≃ 10−5 q s ♥ ♥t ♣qñ ♣r♦ ♥♦ ♥

• ② ♠ás ♥ t♣♦ ♥ts ♣♦♠♦s r ♥s nj t♦s ♠♥♠③♥♦ ♥rí ♣r nj s s♣♦♥♠♦s q s♦♥ ♥♣♥♥ts st♠♦s ♥ ♥ rst ó♥♦ ② q①r ♥tr r ∑

j

njqj = 0,

♦♥ qjs r t♣♦ ♥t P♦r ♦♥s♥t ♠ét♦♦ ♦s ♠t♣♦rs r♥ ② q ♠♥♠③r

G+ λ∑

j

njqj

② rst♦ s

nj = Ne−(εj+λqj)/kBT ♦♥ εj =

(∂F

∂nj

)

nj=0

.

♥ ♥r s ♥rís εj♥ str ♠② s♣rs ♥ ♥s kBT ♥s ♦trs ♣s srá♥ ♦r♥ ❱

P♦r ♦♥s♥t s♦♦ rá ♣rát♠♥t ♦s t♣♦s t♦s

♥②ér♠♦s ♦♥tró♥ ♦s ♦♥♦♥s srí ♥ ♦r n ♠②♦r ♣s ♥rí ♦r♠ó♥ s ♥ts srí♠♥♦r

ísst♦ó♦

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❮ Ó t♦s rst♥♦s ♣♥ts

q♦s t♦s ♣♦st♦s ② ♥t♦s ② ♥rí ró♥ s ♠♥♦r t♦♦s ♦s t♣♦s t♦s

n+ = N+e−(ε++λe)/kBT (qj = e)

n− = N−e−(ε−−λe)/kBT (qj = −e)

② ♦s ♠ás t♦s ♣♥s ♣rrá♥ ♣s t♥rá♥ ♦rs nj ≪ n+ ② nj ≪ n−

♥tr r ♦ q s ♠♣ q n− = n+♦♥ r♥ ♣r♦①♠ó♥ P♦r ♦ t♥t♦s ♣ ♠♥r λ ♣r ♦t♥r

n+ = n− =√N+N−e

−(ε++ε−)/2kBT .

9.4 ❯❱ Ó

♦s rsts ó♥♦s rí♥ sr s♥ts ♣♦r ①♥ ♣s ①tó♥ tér♠ ♦str♦♥s s ♣rát♠♥t ♥ ♥ ♠r♦ ♣rs♥t♥ ♥ ♦♥t étr st♥t r♦ ♥♦ ♠♦♠♥t♦ ♦s tr♦♥s s♥♦ ♦s ♦♥s trés r rst♥s♦ s ♦♠♣r♦♦ ①♣r♠♥t♠♥t ♦♠♣r♥♦ tr♥s♣♦rt r r③♦ ♥tr ♦str♦♦s ♦♥ ♠s ♣♦st ♥ ♦s tr♦♦s q s ♣♦♥♥ ♥ ♦♥tt♦ ♦♥ rst

❯ ♦♠♥t♦ ♥t

rsst ♦s rsts ó♥♦s ♣♥r♥♠♥t t♠♣rtr ② ♣r③ ♠str ♥ ♦s r♦s ♥♦s rí 102 108 Ω♠♦♥♥ r♦rr q ♦s ♠ts t♥♥ rssts ♦r♥ µΩ♠ s r♦ q s ♥ts ♥ ♠♥tr ♦♥t ② q ♥rí ♣r ♠♦r ♥ ♥t trés rst és r s ♠♦ ♠♥♦r q♣r ♠♦r ♥ ó♥ ♥ strtr rst♥ ♣rt

♦s rst♦s ①♣r♠♥ts s ♣♥ ①♣rsr ♥♥ó♥ ♦s ♥ú♠r♦s tr♥s♣♦rt t♣♦ ó♥ ♥♦s ♦♠♦ ró♥ ♦rr♥t♦♥ ♣♦r ♦s ♦♥s ♣♦st♦s ♦ ♥t♦s

t± =σ±

σ+ + σ−

② ♣♦r t♥t♦ t+ + t− = 1

♥♦♥tr♦ q ♦♥ ♥ ♦s r♦s ♥♦s s ♠②♦r♠♥t ♦s ♦♥s♣♦st♦s ♥q st♦ ♣♥ t♠♣rtr P♦r ♠♣♦ ♥ s t♥ q t+(525 ♦C) =0,75± 0,1 t+(700 ♦C) = 0,55± 0,1 q t♥ q r ♦♥ ♦r ♦s r♦s ó♥♦s r(Cl−) = 1,62② r(K+) = 1,54

♥ ♦s r♦s ♣t t+ ≃ 1 ♣r t♠♣rtrs ♦r♥ T ≃ 300 ② s♣r♦rs ♥ sts♦ r(Ag+) = 1,13

❱♠♦s ♥ ♠♣♦ ♦s rst♦s ①♣r♠♥ts ♦♥t ó♥ ♥ trs ♠strs r ♥ ♠② ♣r ② ♦s ♦♥ ♠♣r③ s r2 s ♦t♥♥ ♦s rst♦s r

ísst♦ó♦

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❯ ♦♥t ó♥ r

❯ ♦♥t ó♥ r

♦s rst♦s s trs ♠strs ♦♥♥ T s ② ♦♥t ♠♥t①♣♦♥♥♠♥t r ♣♥t♦ só♥ ♦ P♦r ♥♠ st t♠♣rtr s ♥ íq♦ ó♥♦ ♦♥t♦r ♦♥ σ = 3 (Ω− cm)−1

ró♥ t♠♣rtr s ♠ ♥trí♥s ② q t♠♣rtrs ♠ás s ② ♥r ♣♥♥ ♦♥ s ♠♣r③s ♦♠♦ s ♥ r ② s ró♥ s ♠ ①trí♥s

♥ ró♥ ♥trí♥s ♦♥t s ♣ srr ♦♠♦

σ = Ae−E/kBT =⇒ lnσ ∝ 1/T,

♦♥ A ② E s♦♥ ♦♥st♥ts q ♣♥♥ rst ó♥♦ ♥ ♣rtr

ísst♦ó♦

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❮ Ó t♦s rst♥♦s ♣♥ts

t♦s ② ♣r ♦♥t ó♥

♥ ró♥ ①trí♥s s ♦t♥ ♥ ♣♥♥t q ♣♥s ♣♥ ♠♣r③ q strt s♥♦ ♣r♦①♠♠♥t ♠t ♣♥♥t ♦t♥ ts t♠♣rtrs ♦r σsí ♣♥ ♥t ♠♣r③s ♦♠♦ s ♥ r ② s ♦sr♦ q ♥ ♦♥ ♠♣r③s st ♣♥♥ s ♣r♦♣♦r♦♥ ♦♥♥tró♥ P♦r ♦ t♥t♦ t♠♣rtr s♦♦ ♦♥tr② ♦♥t ♥ ♣qñ ró♥ ♦♥s rst ♠♥rq s ♠②♦r♠♥t ♣rs♥ s ♠♣r③s

♦♠♦ s ♥ t ♥tr♦r ♦s r♦s ♥♦s t♥♥ ♥ ♦♥t ♠♥♦r q ♦s ♣t st♦ s ♦ q ♥ ♦s ♣r♦♠♥♥ ♦s t♦s ♦tt② ♠♥trs q ♥ ♦s ♣t ♣r♦♠♥♥ ♦s t♦s r♥

tr ♠♥r ♦♥♥rs q ♦♥t s ♥ t♦ ♥ rtríst ó♥ ♦♥sst♥ tr ♠s t♦ ♦s ♦♥s s ♠♥ ♦ s ①t♦♥s ♦ t♦♥stér♠s ② ♥♦ r♠♥t ♦♥ ♦ q t♦ ♥ ♠♣♦ ♠♥ét♦ s s♣r ♠ás ♠♦ ó♥ ♠♥t ♦♥ t♠♣rtr ♠♥trs q ♦s tr♦♥s s♠♥②

9.5

♥r♠♥t q♦s rsts q t♠♣rtr ♠♥t s♦♥ s♥ts étr♦s t♠é♥ s♦♥tr♥s♣r♥ts ③ ❯♥ rst r♦s♦r ♠ s ♣rr r♦ st tr♥s♣r♥t Prq st♦ ♦rr ♥♦ r tr♥s♦♥s ♣♦ss tró♥s r♦♥s ♦ qr ♦tr♦t♣♦ ♥ s♣t♦ s ❱ ❱ ♥ s♦ q ♥♦ s sí ♣ rs♦r♦♥s ③ ② rst sr tr♥s♣r♥t ♥q ♥♦ ♥♦ ♣rs♥t ♥ s♦ró♥s ♦♦r rst s ③ tr♥s♠t trés ♠s♠♦

Prs♥t♠♦s ♥ r rs♠♥ r ♦♦r ♥♦s rsts

♦s ♠ts r♥ ③ ♠♥r q ♥♦ trs♠t♥ ② t♥♥ r♦

♥tr♦ ♣r♦♦ ♥rís ♠♥t r♦♥♦ ♣r♦ s ❱ ② ♠♥t ♣r♦s tr♥s♣r♥t ♣ s♥ ♠r♦ ♦♦rrs ♠♥t rró♥ ♦ q ♣r♦ t♦s ♥ rrst♥

sr♦ ♠♦ t♥ ♥ r ♥rís ❱ ② ♣♦r ♦ t♥t♦ s♦r ró♥ ③ s♣tr♦

s♦ t♥ ♥ r ❱ ② t♦s s r♥s ss ①t♥ tr♦♥s ❱ ♦♥ ♦ q ♣rs♥t r♦ ♠tá♦ st r♦ s♥ ♠r♦ ♥♦ s ♦ ♣rs♥ ♣♦rt♦rs rs s♥♦ ①tó♥ ♦s ♠s♠♦s ♥ ♥ ♦trPr♦ s s r ♥ ♣í r♦s♦r ♠♥♦r q ♠ ♦♦r ③ tr♥s♠t

ísst♦ó♦

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s r♦♦ r③ó♥ s q r rt s ❱ ♣r♦ s♦ró♥ ♥rt ❱ ♠♣♦r ♥ ♦♥ó♥ ♥♦ s ♠② ♥t♥s ② ♦♦r rst s ③ tr♥s♠t

ó①♦ ♠♥♦ ú♠♥ ♣r♦ ♥♦ t♥ ♦♦r ♥ ♠r♦ rí rsstt♦♥ s r♦♦ ② ③r♦ ♦♥ ♠② ♣♦s ♠♣r③s s ③

♥♦s ♦♠♣st♦s ♦♥ ♠♣r③s ♠♥t♦s tr♥só♥ ♣rs♥t♥ ♦♦r ♥q srs ♥rí ♦s ♠trs ♥♦ sté♥ ♥ s♣tr♦ s st♦ s ♦ q ♠♦s ♦♥s ♠♥t♦s tr♥só♥ t♥♥ st♦s ①t♦s ♥ ró♥ s

♥♦s rsts ♣♥ ♦♦rrs ♣♦r rró♥ ♥r r②♦s ❳ ♦ γ ♦ qs ♣r♦♥ t♦s q ♣♥ r q tr♦♥s ♦ ♦s s ♦♥ ② ♣r③♥ tr♥s♦♥s st♦s ①t♦s ♥ ró♥ s

9.5.1 ♥tr♦s ♦♦r

♦s r♦s ♥♦s ♣r♦s s♦♥ tr♥s♣r♥ts ♥ ró♥ s s♣tr♦ ♥ ♠r♦ s♣♦s ♦♦rr ss rsts rss ♠♥rs

♥tr♦ó♥ ♠♣r③s qí♠s

♥tr♦ó♥ ♥ ①s♦ ♠t ♥♦ ♥t♥♦ rst ♥ ♣♦r ♠t ② ♥r♥♦ rst ♦♥ r♣③ ♣♦str♦r♠♥t ♦sr q ♥s rst s♠♥② q s ♣♥tr♣rtr ♦♠♦ q ♦s át♦♠♦s ♠t s ♦♥③♥ ♥ ♣♦r ② s ♥♥ ♥ rst ♣r♦♥♦♥ts ó♥s ♣r ♥tr③r r rst♦ ♥t♦ s ♥ s♠♥ó♥ ♥s ♣♦r s♣rs♥ ♥ts sí s ♠r♦ ♥ ♣rs♥ ♣♦r ② s ♦♦r ♠♥t

♣♦r ♦♠r♦ tr♦♥s r②♦s ❳ r②♦s γ

♣♦r tróss

❨ ♠♦s ♥♦ q ♥tr r rqr q ♣ró♥ ♥ts s qr♦♥ ♦♥s ♥trsts ♠s♠♦ ♦♥stt②♥t q ♥t s♦ r♥ ♦ ♦♥ ♥ts ♦tr♦♦♥stt②♥t s♦ ♦tt②

♥ ♠r♦ ①st ♦tr♦ ♣r♦♠♥t♦ ♣r ♦ r ♣r ♣♦r ♥t ó♥ ♥t♦s ♥tr③ ♦♥ ♥ tró♥ ♦ r♦s ♦③♦s ♥ s r♥ís ♥t P ♦♥srrsq tró♥ ♦ ♦s tr♦♥s s♠♣♠♦s ♥♦ ♥ tró♥ stá ♦ ♣♦r ♥ ♣♦t♥t♦ ♣♦st♦ ② q t♥ ♥ s♣tr♦ srt♦ ♥rís ás ssó♥ r ♦s st♦s♦③♦s ♥ ♠ s♦r t♦r♠ ♦ ①st ♣♦r t♥t♦ ♣♦s tr♥só♥ ♥trst♦s ♥s ss tr♥s♦♥s ♣r♦rí♥ í♥s s♦ró♥ ó♣t s♠rs s ♦s át♦♠♦ss♦s ② ♣rrá♥ ♣♦s st♦s ♥ s♣tr♦ s♦ró♥ ♥ ♥r sts s♦r♦♥s ♥♦stá♥ ♥ s♣tr♦ s ♥ ♦s r♦s ♥♦s s♥ str ♥ tr♦t

• ♠♥ ♥tr♦s ♦♦r ♦ ♥tr♦s ♠á♥ r♥ ♦♦r ♦s t♦s r qs♦r♥ ③ ♥ s♣tr♦ ó♣t♦ q s ♠ás ♥r q s ♣s ③ s s ♣rts s♣tr♦ ó♣t♦ ♦♠♦ ② s ♦♠♥t♦ ♣♥ str s♦♦s tr♦♥s ♦s st♦s t♦s

rs♦♥♥ s♣í♥ tró♥♦ ♣r♠t♦ ♥tr ♥ ♥tr♦ ♦♠♦ ♥ tró♥ ♦ ♥ ♥t ♥ ó♥ ♥t♦ st♦ s ♣♦②♦ t♠é♥ ♣♦r ♥ sr rtrísts①r♠♥ts

s♦ró♥ ó♣t

♥áss qí♠♦

♦s rsts ♦♦r♦s s♦♥ ♠♥♦s ♥s♦s

ísst♦ó♦

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❮ Ó t♦s rst♥♦s ♣♥ts

•② q ♥♦tr ♦s ♦ss rs♣t♦ ♦s ♥tr♦s

tró♥ ♦ s ♠ ♥ ♥ ♠♣♦ s♠trí ú ② ♦ q s ♥rís rs♦♥ ♦r♥ ❱ tró♥ stá r ♥t ♥♦ s ♣s ♠♣♦ ♦s ♥s ♠♣r③ ♦s

s♦ró♥ ♣r♦ ♣♦r ♥tr♦ ♥♦ s t♥ ♣rs ♦♠♦ ♥ át♦♠♦ s♦ s♠ás ♥ ♥ ♥ s♦ró♥ st♦ s ♦ q ♥r í♥ s♦ró♥ ♥♥rs♠♥t ♣r♦♣♦r♦♥ t♠♣♦ st♦ ①t♦ ♠♥trs q ♦s át♦♠♦s s♦ss♦♦ ♣♥ ①trs rt♠♥t tró♥ ①t♦ ♥tr♦ ♥tr♦♥ rt♠♥t ♦♥ rst♦ só♦ ② ♣ ♣rr ♥rí ♣ ♠♥t ♠só♥ ♦♥♦♥s

• ①st♥ ♦tr♦s ♥tr♦s ♦♦r ♥ r♦s ♥♦s ♦♥ ♣r♦♣s s♣tr♦só♣s st♥ts ②♠② ♥trs♥ts

P♦r ♠♣♦ ♥ ♥tr♦ A ♦♥sst ♥ ♥ ♥tr♦ ♥ q ♥♦ ♦s ss ♦♥s ♣♦st♦s qr♦♥ ♥t s ♠♦ ♣♦r ♦tr♦ ó♥ ♥♦ ♥ s ♠ ♥ + ♣ ♣♦r ♥+ ♦s ♥tr♦s ②♥ts ♦♥stt②♥ ♥ ♥tr♦ s ♣r♦♥ trs ♥tr♦s ②♥ts♥ ♣♥♦ ❬❪ strtr ♣ r♥ ♥♦♠r ♥tr♦

st♥ó♥ ♥tr ♦s st♥t♦s ♥tr♦s s ♥ s ss st♥ts ♣r♦♣s ó♣ts

❯ ♥s r♦s ♥♦s

ísst♦ó♦

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♣ít♦

❮ ❯ ❯ ❨

10.1 ❯Ó

10.2 ❯

♠♦s q ♥tr♦ ♣r♦①♠ó♥ tr♦♥s ♥♣♥♥ts ♦s s♥ts s ♣♥rtr③r ♣♦r ♥ ♥tr♦ ♥rís ♣r♦s ♦ ♣ ② ♠♥t s Eg ♥ ♦♠♦ ♥rí q s♣r ♦r s♣r♦r s ♥s ♥s ② ♦r ♥r♦r s ♥s ís ss♦r♥t♥♦ q T = 0 P♦r ♦ t♥t♦ s s ♦♥♦♥s s♦♥ ts q ♠♦♦ s♠ás♦ q♠♦s st♦ ♣r trtr ♥á♠ ♦s tr♦♥s s á♦ ♥ só♦ s♥t ♥♦ ♦♥ t♠♣rtr

♦r ♥ s T 6= 0 ② ♥ rt ♣r♦ q ♥♦s tr♦♥s s♥ ①t♦stér♠♠♥t trés ♣ ② st♥ s s ♥s ♥ s ♦♥ó♥ ♦r♠q s ♣ ♣r♦r ♦♥ó♥ étr ♣♦r tr♦♥s ♥ ② ♣♦r ♦s ♥ ❱

❱r♠♦s ♠ás ♥t q ró♥ tr♦♥s q s ①t♥ trés ♣ t♠♣rtrT s ♦r♥ t♦r ♦t③♠♥♥ E−Eg/2kBt ♦r♠ q t♠♣rtr ♠♥t kBT ≃0, 025eV = 1/40eV t♥r♠♦s ♥ ♣r♦ e−80 ≃ 10−35 s Eg = 4 ❱ ② e−5 ≃ 10−2 sEg = 0,25 ❱ s ♥t q ♥ s♥♦ ♦s s♦s s ♦srrá ♦♥t

♦s só♦s ②♦s ♣s ♥rí s♦♥ s♥t♠♥t ♣qñ♦s ♣r q ♦♥③♥ t♠♣rtrs ♥r♦rs ♣♥t♦ só♥ s ♠♥ s♠♦♥t♦rs P♦♠♦s r q♦s s♠♦♥t♦rs ♠ás ♠♣♦rt♥ts s♦♥ ♦s q t♥♥ Eg ≤ 2 ❱ ② s rsst tí♣ rí♥s ♦r♥ 10−3 Ω♠ st 109 Ω♠ ♥ ♦s ♠ts ρ ≃ 10−6 Ω♠ ② ♥ s♥ts ρ ≃ 1022

Ω♠

♦♥♥tró♥ ♣♦rt♦rs ♥ rs♦s ♠trs tr♦♥s♠3

ts 9×1022 5×1022 2× 1022 ≃ 1022 − 1023

♠♠ts s 4× 1020 8× 1019 rt♦ 5× 1018 5× 1017 ≃ 1017 − 1021

↑s♠♦♥t♦rs♦♥ ♠♣r③s

♠♦♥t♦rsTamb ≃ 300

♥ 2 ×1016

♣r♦6× 1013

♣r♦ 1010

♦♠♦ ♥ú♠r♦ tr♦♥s ①t♦s rí ①♣♦♥♥♠♥t ♦♠♦ e−1/T ♦♥tétr ♥ s♠♦♥t♦r ♠♥trá rá♣♠♥t ♦♥ T ♥ ♦♥trst ♦♥ ♦s ♠ts ♣r♦s q ♥ú♠r♦ ♣♦rt♦rs s ♦♥st♥t ② s ♦♥t σ = ne2τ/m ♣♥ s♦♠♥t t♠♣♦ ró♥ τ q s♠♥② ♦♥ t♠♣rtr ♦ s ♦s♦♥s tró♥ ♦♥ó♥♥ ♥ s♠♦♥t♦r τ t♠é♥ s♠♥② ♦♥ T ♣r♦ ♥ú♠r♦ ♣♦rt♦rs ♠♥t ♠♦ ♠ásrá♣♠♥t ② ♦♠♣♥s s♦r♠♥t q s♠♥ó♥

❯♥ ♣r♦♠ q sr str ♦s s♠♦♥t♦rs s ♥♥ s ♠♣r③s ♠str ♥ ss ♣r♦♣s íss P♦r ♠♣♦ rsst ♥ t♠♣rtr ♣rr ♥ 1012 s ♠♠♦s ♦♥♥tró♥ ♠♣r③s ♥ ♥ t♦r 103 s s t♠é♥ q

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó rtrísts ♥♠♥ts ♦s ♠trs s♠♦♥t♦rs ② ♠♥ét♦s

rsst ♣♥r ♦♥♥tró♥ ♠♣r③s ♥♦ t♠♣rtr s♠② t st rsst s ♠ rsst ♥trí♥s ② q s ♥ ♠str ♣rt♠♥t♣r ♥ ♦s ♦tr♦s s♦s s ♠ ♦♥t ①trí♥s

♥ ♥r ♥ s♠♦♥t♦r s ♥trí♥s♦ s ss ♣r♦♣s tró♥s stá♥ ♦♠♥s ♣♦r♦s tr♦♥s ①t♦s tér♠♠♥t s ❱ ♠♥trs q s ♠ ①trí♥s♦ s stá♥♦♠♥s ♣♦r ♦s tr♦♥s q ♠♥ s ♥rí ♦ ♣rs♥ ♠♣r③s st♦ s qs ♠♣r③s ♥ tr♦♥s ♥ ♦♥ó♥ ♦ ♣t♥ tr♦♥s ♥ ♥

10.2.1 ♠♣♦s s♠♦♥t♦rs

♦s s♠♦♥t♦rs s♦♥ ♥ ♥r s♥ts ♦♥ts ♦♥ ♥ stró♥ tró♥ s♣q ♥tr ♦s s♥ts s ♠ás ♣r ♥ ♠t

♠♥t♦s s♠♦♥t♦rs

♦♠♥ ❱ sst♠ ♣ró♦ ♦♥ strtrs ♠♥t

P ós♦r♦ r♦♦ ♦r♦ s♥♦ ② q t♥♥ strtrs ♦♠♣s ♣r♦ ♦♥ ♥♦♥t

♦♠♣st♦s s♠♦♥t♦rs

♠♦♥t♦rs ❱ rs♥r♦ ♦ s ♣♦r ♠♣♦ ♦♥ strtr ③♥♥ ♥ s ♣r♦♠♥♥t♠♥t ♦♥t

♠♦♥t♦rs ❱ s s♥ ♠r s♠♦♥t♦rs ♣♦rs ♦ q t♥♥ ♥ ♥♠② ó♥♦ ♦♥ strtrs ③♥♥ ♦

♦tr♦s ♠♦s s♠♦♥t♦rs ♦♠♣♦s

10.2.2 ♣s ♥rí

♦s ♣s ♥rí ♦s s♠♦♥t♦rs ♣♥♥ t♠♣rtr ♠str st♦ s♦ ♣r♥♣♠♥t ♦s r③♦♥s

①♣♥só♥ tér♠ q ♣♦t♥ ♣ró♦ q sr♥ ♦s tr♦♥s rí ♦♥ t♠♣rtr ♠♥r q t♠é♥ ♦ ♥ s ♥s ② ♣ ♥rís

t♦ s r♦♥s r s♦r strtr ♥s ♥rí ♥tró♥tró♥♦♥ó♥ ♣ rr ♦♥ t♠♣rtr r♥♦ ♣♥♥ stró♥ ♦♥♦♥s ♦♥ ♠s♠

st♦s ♦s t♦s s♦♥ ♦♠♣rs ② ♥ r ♥ ♣ ♥ ♥ T ♣r t♠♣rtrs tsTamb ② rát ♥ T ♥♦ t♠♣rtr t♥ r♦

❯ ❱ró♥ ♣ ♦♥ t♠♣rtr

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❯

• ó♠♦ s ♠ ♣ ♥rís

❯♥ s ♠♥rs ♠r♦ s t③♥♦ s ♣r♦♣s ó♣ts rst s♣rr q ② ♥ rt ♥r♠♥t♦ ♥ s♦ró♥ ró♥ ♥♥t ♥ ♠♦♠♥t♦ ♥ q~ω > Eg

st♦ ♥♦s ♣ ó♣t♦ q s r♥ ♠í♥♠ ♥rí ♥tr s ♥s ♣rtr♥s♦♥s ♥tr st♦s rtr③♦s ♣♦r ♥ ♠s♠♦ ♣♥t♦ s♣♦ k ♥♦ k rr t♦ ♣r♠r ③♦♥ r♦♥ sts tr♥s♦♥s s ♠♥ rts ♥ st s♦ s♦ró♥ ♠♥ts r♦ ♠♥r r♣t ♥♦ s ~ω > Eg

Pr♦ s ♠á①♠♦ ② ♠í♥♠♦ s ♥s stá♥ ♥ ♣♥t♦s st♥t♦s k ♣r ♦♥srr ♠♦♠♥t♦rst♥♦ ♥tr♥ ♥ ♦♥ó♥ ② tr♥só♥ s ♠ ♥rt ♦ q ♥rí ♦♥ó♥ srá♠♥♦r q ~ωD ≃ 10−2 ❱ ♥r ♣♦♦ ♥ tr♠♥ó♥ ♦r Eg Pr st s♦ s♦ró♥ ♠♥t s ♠♦♠♥t♦ ♥ q ~ω > ~ωg + ~ωf (q) ≃ Eg ♣r sr r♣t♠♥t♥♦ ~ω > Evert > Eg

❯ s ♣

tr ♠♥r ♠r Eg s ♠♥t ♦s rst♦s ①♣r♠♥ts ♦♥t

♥trí♥s s♠♦♥t♦r sú♥ s rí t♠♣rtr st ♣♥♥ s ♣r♥♣♠♥t ró♥ ♥ú♠r♦ ♣♦rt♦rs r q ♦♠♦ exp−Eg/(2kBT ) ②q ♣♥♥ t♠♣♦ ró♥ ♦♥ t♠♣rtr τ(T ) s ♠♦ ♠ás s ♣s s♥ ♣♦t♥ T r♣rs♥tr − lnσ r♥t 1/(2kBt) ♣♥♥t ♥♦s rá ♠♥r ♣r♦①♠ ♦r Eg♦t ♥ r ♦t♥rí♠♦s E0 ♥♦ Eg(0)

10.2.3 strtr ♥s s♠♦♥t♦rs

♠♦s q s ♣r♦♣s tró♥s ♦s s♠♦♥t♦rs stá♥ tr♠♥s ♣♦r ♣qñ♦ ♥ú♠r♦ ♥ ♦♠♣ró♥ ♦♥ ♦s ♠ts tr♦♥s ①t♦s ♥ ② ♦s♦rrs♣♦♥♥ts ♥ ❱

P♦r s♦ ♦s tr♦♥s ♥ t♥r ♥ ♥rí ♠② ♣ró①♠ ♠í♥♠♦ ❱ ② ♦s ♦s ♠②♣ró①♠ ♠á①♠♦

r q ♠♦♠♥t♦ ♦tó♥ s ♠♦ ♠♥♦r q ♦s tr♦♥s ♦ ♦♥♦♥s

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó rtrísts ♥♠♥ts ♦s ♠trs s♠♦♥t♦rs ② ♠♥ét♦s

P♦r ♦♥s♥t ♣♦♠♦s r♠r q s ♥s tr♦♥s ② ♦s ♥ ♣rt sq ♥♦s ♥trs ♣♥ ♣r♦①♠rs rát♠♥t

ε(k) = εc +~2

2

∑

µν

kµ

(M−1e

)µνkν

ε(k) = εv −~2

2

∑

µν

kµ

(M−1h

)µνkν

♦♥ εc s ♦r ♠í♥♠♦ ② εv ♦r ♥rí ♥ ♠á①♠♦ ❱ qí♦s sí♥s e ② h s rr♥ tr♦♥s ② ♦s rs♣t♠♥t óts q ♥ ♥ sór♠s ♥tr♦rs ♠♦s ♣st♦ ♦r♥ s♣♦ k ♥ ♠í♥♠♦ ② ♥ ♠á①♠♦ ② ♦♠♦ s ó♦ s ② ♠ás ♥ ♠á①♠♦ ♦ ♠í♥♠♦ ♥ s r♥ís sts ♥rís ② q♥r tér♠♥♦s ♦♠♦ ♦s ♥tr♦rs q ♦rrs♣♦♥♥ ♠á①♠♦ ♦ ♠í♥♠♦

t♥s♦r M s r ② s♠étr♦ ② s ♣ ♥♦♥trr ♥ ♦♥♥t♦ s ♣r♥♣s ♦rt♦♦♥sq ♥♦ s♦♥ ♦s ♠s♠♦s ♣r tr♦♥s ② ♦s ♠♥r q s ♥rís t♥ ♦r♠ ♦♥

ε(k) = εc +~2

2

(k21me

1

+k22me

2

+k23me

3

)

ε(k) = εv −~2

2

(k21mh

1

+k22mh

2

+k23mh

3

).

P♦r ♦ t♥t♦ s s♣rs ♥rí ♦♥st♥t r♥ ♦s ♠á①♠♦s ② ♠í♥♠♦s t♥♥ ♦r♠♣s♦ ♦r♠ ② ♣♦só♥ ♦s ♣s♦s ♥ s♣♦ k s s♣♥ ♠♥t ♦s s♦rt♦♦♥s ② s ♠ss ts m1 m2 ② m3

• ♦

strtr ♥s s♦ s ♣r♦①♠♠♥t r

❯ ♥s

❯ ♣s♦s tr♦♥s

s ♥s ♥ s ♥♥tr♥ ♥rs ♥ ♣♥t♦ Γ k = 0 ② ♥ r ♦s ♦s♣s♦s ② r♦s ♦♥ m = 0,49me ② m = 0,16me

trr ♥ s♣r ♣♦r ♥tró♥ s♣í♥órt t♠é♥ r ♣♦rt♦rs t♠♣rtr ♠♥t ② q kBTamb ≃ 0,025 ❱

s ♥s t♥♥ s♠trí sér ② ♣♦r t♥t♦ s ♠ss ts s♦♥ srs

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❯

s ♥s ♦♥ó♥ t♥♥ ss ♠í♥♠♦s ♥ s ss r♦♥s 〈100〉 s♣♦ rí♣r♦♦r ♦r ③♦♥ ró♥ Γ❳ ♥ r P♦r s♠trí st♦ r ss ♣s♦s r♦ó♥ r♦r ♦♦r♥♦ s♣♦ rí♣r♦♦ ♠s t ♦♥t♥ ♦ r♦ s mL ≃ me ② tr♥srs s mT ≃ 0,2me

• r♠♥♦

♥ st s♦ ♦s ♠í♥♠♦s ♥ ♦♥ó♥ stá♥ ♥ s r♦♥s 〈111〉 s♣♦rí♣r♦♦ ♠♥r q ② tr♦ ♠í♥♠♦s st♦ ♥ ♦s ♦rs ③♦♥ r♦ stá q ♦s ♦tr♦str♦ st♦s st♦s ♥ s rs ♦♣sts P❩ s♦♥ ís♠♥t ♦s ♠s♠♦s q st♦s ♦sq ♠♦s s s♣rs q♥réts s♦♥ ♣s♦s r♦ó♥ r♦r s r♦♥s〈111〉 ♦♥ ♠ss ts mL ≃ 1,6me ② mT ≃ 0,08me

♥ ♥ ♥ ② ♦s ♥s ♥rs ♥ k = 0 q s ♣♥ str ♦♥ ♦s♠ss ts 0,28me ② 0,044me rs♣t♠♥t ♥ st s♦ s♦♠♥t♦ s♣í♥órts 0,3 ❱

❯ ♥s ❯ t s ♥s

10.2.4 s♦♥♥ ♦tró♥

ó♠♦ s ♠♥ s ♠ss ts s q st♠♦s ♥♦

♦♥sr♠♦s ♥ tró♥ r♥♦ ♠á①♠♦ ♦ ♠í♥♠♦ s ♥s ♠♥r q ♥rís rát ♥ k ♥ st í♠t ♣r♦①♠ó♥ ♠s t s á ②

Mv = p = ~k

♦♥ ♦ q ó♥ ♠♦♠♥t♦ ♥ tró♥ ♦ ó♥ ♥ ♠♣♦ ♠♥ét♦ ♥♦r♠B s♥ t♥r ♥ ♥t s ♦s♦♥s

~dk

dt= −e

cv ×B,

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

❮ Ó rtrísts ♥♠♥ts ♦s ♠trs s♠♦♥t♦rs ② ♠♥ét♦s

q ♦♥rt ♥M

v

dt= ∓e

cv ×B

t♦♠♠♦s B = Buz ó♥ ♥tr♦r t♥ ♥ s♦ó♥ t♣♦

v = Re(v0e−iωct)

s♥♦ ωc = eB/(m∗cc)

st m∗c s ♠ ♠s t ♦tró♥ ② ♥ ♣♦r ①♣rsó♥

m∗c =

√detM

Mzz

② ♥ sst♠ ♦♦r♥s ♥ q M s ♦♥

m∗c =

√m1m2m3

u21m1 + u22m2 + u23m3

♦♥ ui s♦♥ s ♦♠♣♦♥♥ts uz ♦ r♦ ♦s s ♣r♥♣s M

♥♠♦s ♣s q ♣r ♥ ♣s♦ tr♠♥♦ r♥ ♦tró♥ ωc s♦♦ ♣♥ ♦r♥tó♥ rt ♠♣♦ ♠♥ét♦ B rs♣t♦ ♣s♦ ② q ♥♦ ♣♥ t♦r ♦♥ ♥ ♥rí tró♥P♦r ♦♥s♥t s s ♦r♥t♦♥s rst ② ♠♣♦ ♠♥ét♦ s ♣s sr♦♥s B ② ♦r♥tó♥ ♣s♦ ♦s tr♦♥s ♦♥ó♥ ♥ tr♠♥♦ ♣s♦t♥rá♥ ♥ r♥ ♣r t♦♦s ♦s q ♥rá tr♠♥ ♣♦r t♥s♦r M ♣s♦

♦♠♦ ♦♥s♥ ② ♥ ♥ú♠r♦ ♣qñ♦ r♥s ♦tró♥s st♥ts ♥ s ♦rrs♣♦♥♥t ó♥ ωc = eB/(m∗cc) ♣r s st♥ts m∗c s♦s ♦s ♣s♦s♦♥ ② ♣♦rt♦rs r ♠♦s r♥♦ ♦r♥tó♥ B ♠♥♦ ♦r m∗c ② í ♣♦♠♦s ①trr♥♦r♠ó♥ r ♦s ♦rs mi

①♣r♠♥t♦ rs♦♥♥ ♦tró♥ ♦♥sst ♥ ♥r ♥ ♠♣♦r étr♦ E ró♥ s r♦r♥s q s ♣r♣♥r ♠♣♦ ♠♥ét♦ stát♦ B q♣r♦ ♠♦♠♥t♦ ♦s tr♦♥s ♥♦ r♥ ♠♣♦ étr♦ s ωc = eB/(m∗cc) ② ♥ s♦ró♥ rs♦♥♥

♦♠♥tr ♦ r s tr②t♦rs ♥ s♣♦ r

Pr ♥ ♦r♥tó♥ rst ② ♠♣♦ ♠♥ét♦ ♦s tr♦♥s ② ♦s r♥ ♥ r♥ q stá s♦♠♥t ♣♦r ts♥♦r ♠s t ♣s♦ t♥♠♦s ró♥ ♠♣♦ étr♦ ♦sró♥ rrá ♦♥ ♥t♥s ♠♣♦ B ♦♠♦ s ♠str ♥ ♦s rst♦s ①♣r♠♥ts

st♦ s ♦♥s ♠♥t ♥ ♠♣♦ tr♦♠♥ét♦ ♥ s ró♥ r♥s

ísst♦ó♦

❯

❯

❯

❯

❯

❯

❯

P❯❩ ❯ ❱ ❱ P❯❩

Pr q rs♦♥♥ ♦tró♥ s ♦sr r♥ ωc = eB/(m∗cc) sr ♠②♦rq ♦ ♠♥♦s ♦♠♣r r♥ ♦só♥ ♦s tr♦♥s ♦ q s♥ ♥ ♣r♦♦♦tró♥♦ ♠♥♦r q t♠♣♦ ró♥ ♦s tr♦♥s st♦ s ωcτ > 1

P♦r t♥t♦ ② q trr ♦♥ ♠strs ♠② ♣rs ② ♠② s t♠♣rtrs st♦ q s ♠② ♣qñ♦ ♥ú♠r♦ ♣♦rt♦rs ♦♥t ② q ♥ ♦♥s♥ ♥♦ ②♣♥s t♦ ♣r ♣r♠t♥♦ ♣♥tró♥ ♠♣♦ tr♦♠♥ét♦ r♦r♥ ② ♣ró♥ s rs♦♥♥s

Pr ♣♦r r s ♠s s♥ ♠r♦ ② q ♥rr ♣♦rt♦rs ♣♦r ♠♣♦ ♠♥t♦t♦①tó♥ ♣r q ② ♥ ♥ú♠r♦ s♥t ♣r q rs♦♥♥ s ♠

10.3 P❯❩ ❯ ❱ ❱ P❯❩

♥♦s t♣♦s ♠♣r③s ♦ ♠♣r♦♥s r t♥ ♠♦ s ♣r♦♣s étrs ♥ s♠♦♥t♦r sí ñ♥♦ ♦r♦ s♦ ♥ ♥ ♣rt ♥ s ♥r♠♥t ♦♥t rst ♣r♦ t♠♣rtr ♠♥t ♥ ♥ t♦r 103

♥ ♥ ♠tr ♣r♦ ♥ú♠r♦ tr♦♥s ② ♦s s ♠tr s ♥trí♥s♦ ♦srsts ♦♥ ♦♥♥tr♦♥s ♠② ♣qñs ♠♣r③s q s ♥ ♣♦♦ ♦t♥r ♥ ♦rt♦r♦t♥♥ ♦r♥ 1010 ♠−3 ♠♣r③s st♦ r q ♥ ♦s rsts ♣r③♥ ♣♦rt♦rs r t♠♣rtrs s ② q s ♥ú♠r♦ r③ rá♣♠♥t

P♦r ♦tr ♣rt ♥s ♣♦rt♦rs ♥trí♥s t♠♣rtr ♠♥t Tamb ≃ 300 s ni = 6× 1013 ♠−3 ♥ ni = 7× 109 ♠−3 ♥ ♣♦r ♦ q ♥♦ s♣rr ♦t♥r ♥trí♥s♦ Tamb ♥q sí s ♦t♥♦ ♥trí♥s♦ st♠♣rtr

❱♠♦s str s♣♠♥t s ♠♣r③s s♠♦♥t♦rs ♦♥ts ♦♠♦ ② ♦♠♦t♥♥ strtr ♠♥t át♦♠♦ stá r♦♦ ♦tr♦s tr♦ st♦s ♥ s♣♦só♥ttrér

s ♠♣r③s tr♥ts ♥ ② s ♣♥t♥ts P s sstt②♥ ♦s át♦♠♦s ♥♦r♠s ② ♥♦ s stú♥ ♥ ♣♦s♦♥s ♥trsts sú♥ s ♣♦♦ sr ♠♥t♠s ♦♥st♥t r ♦s s♠♦♥t♦rs ♦♥ ♠♣r③s

qs ♠♣r③s q ♦♥tr②♥ ♠♥tr ♥s tr♦♥s ♥ ♥ ♦♥ó♥ s ♠♥ ♦♥♦rs ♦s át♦♠♦s t♥♥ ♥ tró♥ ♠ás q ♦s át♦♠♦s s♠♦♥t♦rés♣ ② s q ♠♥t♥ ♥s ♦s ♥ ❱ s ♠♥ ♣t♦rs ♦s át♦♠♦s t♥♥♥ tró♥ ♠♥♦s q ♦s át♦♠♦s s♠♦♥t♦r és♣st♦s ♥♦♠rs s ①♣♥ ♣♦rq ♥s ♠♣r③s ♥ tr♦♥s rst ② ♦trs r♥ ♥sq ♣t♥ tr♦♥s ❱ r♥♦ ♣♦r ♦♥s♥t ♦s ♥ ❱

• ❱♠♦s str ♣r♦♠ ♥ s♠♦♥t♦r ♦♥t♠♥♦ ♦♥ ♠♣r③s ♦♥♦rsát♦♠♦s s ♥ ♣♦r ♠♣♦ ♥♦s ♠♦s ♥ ♥ ss ♠♣r③s ♥ ♣r♠r ♣r♦①♠ó♥ ♣♦♠♦s♣♥sr q át♦♠♦ s s stú ♥ ♥ s ♣♦s♦♥s ② t③ tr♦ ss ♥♦tr♦♥s ♣r ♦r♠r ♦s tr♦ ♥s ♦♥ts strtr ♠♥t ♣s ♥ r ♣♦st ♥ r ♥ q s st♦ ♠♣r③ ② ♥ tró♥ ♦♥

❨ ♦ s ♥♣♥♥t♠♥t t♠♣rtr ♣s ♦♠♦ ② s ♦♠♥t♦ s s q s T ♠♥t s♥t♠♥t s s♠♣r ♥ ♦♠♣♦rt♠♥t♦ ♥trí♥s♦

rá ♠ás ♥t q ♥rí ♥♦♥③ó♥ ss ♠♣r③s s ♦r♥ 0,01 ❱ s r♥ ♥♦♠r ♠♣r③s ♦♥♦rs ♠é♥ s s ♠ ♠♣r③s ♣t♦rs

ísst♦ó♦

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❮ Ó rtrísts ♥♠♥ts ♦s ♠trs s♠♦♥t♦rs ② ♠♥ét♦s

s q ♥rí ♦♥③ó♥ s s 9, 81 ❱ ❨ ♣♦r ♦tr ♣rt ♥ rst ♦♥♠♣r③s s s ♠♦ ♥ ♥rí ♦♥③ó♥ ♦r♥ 0,01 ❱ á s stó♥ís ♥t♦♥s tró♥ s♦r♥t q ♣r♦ ♠♣r③

s ♣♥t♦ st ♦③ó♥ ② ♦s ♣♦ss s ♥ tró♥ ♦③♦ r♦r r ♣♦st ♠♣r③ ♥rí ♣ srqr s♥ rstr♦♥s s ♥ tró♥ ①t♥♦ t♦♦ rst ♦♠♦ ❱ stá ♥ ② s ♥rí str♥sr♠♥t r r ♥rís ♦ ♣ tró♥ t♥r ♥ ♥rí ♥ ♥ st s♦ ② q rst ♣r♦ ♣r♦ ♣rt tró♥ srá ♥ ♦♠♥ó♥♥ st♦s ♦ ♣r♦ t♥r ♠♥♦r ♥rí ♣♦s ♥ ♦♥s♥ tró♥ stár ♠í♥♠♦ ② s ♥á♠ s ♣ srr ♠♥t ♥ ♠s t m∗

P♦r ♦tr ♣rt s s q ♥ ♥ ♣♦t♥ ♥tr ①st ♥ ró♥ ♥r ♥tr ♥rí r ♥ st♦ ② r♦ tr②t♦r tró♥ ♥ ♦ st♦sí ♥ ♥ ♣♦t♥ ♦♠♥♦ ♥ ♥♥t♦ s s q 〈r〉 s ♣r♦①♠♠♥t ♣r♦♣♦r♦♥ n2 ② q s ♥rí s ♣r♦♣♦r♦♥ −1/n2 s st♦ stá ♦P♦r ♦tr♦ ♦ ♥ ♥ ♣♦t♥ ♥ ♥t♦ ♦♠♦ ♣♦r ♠♣♦

U(x) = − U0

cosh2(x/r0),

♦♥ r0 s ♥ ♣♦t♥ s t♥ ♣r♦①♠♠♥t q E ∝ −1/r20

♦♠♦ st♠♦s ♥ ♥ strtr t♣♦ ♠♥t ♣♦t♥ q sr tró♥ s♦r♥t t♥r ♣r♦①♠♠♥t ♥ rt s♠trí sér P♦r ♦ ♣♦♠♦s t③r ♦ q ♠♦s ♦♠♥tr r s ♥rís r ② ♦s r♦s ♦s st♦s ♦s ♥ ♣♦t♥s ♥trs② r♠♥tr q ♥rí ♦♥③ó♥ t ♥ts ♣ str r♦♥ ♦♥ ♥ st♦ ♦② tr②t♦r t♥ ♥ r♦ ♠② r♥

P♦r t♥t♦ s♣rr q st st♦ r ♣♦st ♠♣r③ ♦♠♦ ♥ r♣♥t ♣♦r ♦♥st♥t étr ♠r♦só♣ s♠♦♥t♦r ❱♠♦s ♣s ♦♥srrq tró♥ s♦r♥t ♠♣r③ ♦♠♦ ♥ ♣rtí ♠s m∗ q sr ♥ ♣♦t♥ +e/ǫrst s ♥ ♣r♦♠ é♥t♦ át♦♠♦ ró♥♦ ♠♥♦ m ♣♦r m∗ ② −e2 ♣♦r −e2/ǫ ♥rí r rs♣t♦ ♦s st♦s ♦♥t♥♦ t♠é♥ ♠♦s st♦s s♣rsó♥ ♥st s♦ s♦♥ ♦s st♦s ♦♥♦ q r♣rs♥t♥ ♦s st♦s ♥♦ ♦s s

ε =m∗

m

1

ǫ2× 13,6 eV r0 =

m

m∗ǫ a0,

q ♥♦s ♥ ♥rí r ♦r♥ 0,001 ❱ ② ♥ r♦ r0 ♦r♥ 50 100

♦♠♦ ♣r s♦ ♥ ♣ str♦ t♥r♠♦s ♥ ♦♥st♥t étr r♥ ♥t♦♥s ♥♥r st ♥rí r s s♠♣r ♣qñ ♦♠♣r ♦♥ Eg

• st♦s r♠♥t♦s s ♣♥ ♥ ♠ s♦ ♠♣r③s ♣t♦rs q ♥ r ♥s tró♥♦s εa ♥rí r♠♥t s♣r♦r ♥rí ♠á①♠ ♥ ♥εv ② s♠♣r ♣qñ ♦♠♣r ♦♥ Eg

♥ r s ♥ r♣rs♥t♦ ♦s ♦s s♦s ♠♣r③s ♦♥♦rs ♦♥ ♥ ♠♣r③ εd ♠♣r③s ♣t♦rs ♦♥ ♥ ♠♣r③ εa

♥ ② st③ ➓ ♠♣♦ Pr st ♣♦t♥ U(0) = U0 ② U(2r0) =7

100U0

Pr r♠♥♦ ǫGe = 15,8 ♣r s♦ ǫSi = 11,7 ♠♥t ♦♥st♥t étr ♠ ♥ ♠♥r ♦r♠ó♥ s ♥♦♥s ♦♥ tró♥s r♥t ♥♠♣♦ ①tr♥♦ é ♥ ♥ ♠t ǫ s ♠② r♥ ② q ♦s tr♦♥s ♣♥ ♠♦rs r♠♥t ② ♣♥t♥ ♥ st♥s♣qñs s t③ ♥ t♦rí ♣rtr♦♥s ♣r sr♣ó♥ ♦s ♠♦s ♥♦s ♣♦r s ♠♣♦ ①tr♥♦ s♠♦sq ♥ ①♣rsó♥ ♣r s ♥♦♥s ♦♥ ♣rtrs ♣r♥ s r♥s ♥rís ∆E ♥tr ♦s st♥t♦s st♦s sst♠ ♥ ♦♥s♥ ♣r ♥ ♣ ♣qñ♦ s ♦t♥r ♥ ǫ r♥ ❨ st♦ s ♦ q ♥ ♠♥r s t♥ ♣r ② ♥ ♣ s ♠♥♦r q ♥ ② s ǫ s ♠②♦r

ísst♦ó♦

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P❯❩ ❯ ❱ ❱ P❯❩

♦t ♠♣♦rt♥t st♦s st♦s ♦③♦s ♦ ♥s ♠♣r③ ♥♦ s♦♥ s s ♣♥t♦ st ♦s tr♦♥s

♥ Ed s ♥ ♥ ♥ q ♣ r s♦♠♥t ♥ tró♥ ♣♦rq s r ♦s ♥tró♥ ♦♠♥ ♥tr ♦s rí ♠r ♥rí ♥ ♦♥ rst♦ q ♥♦ ♣r ♥ st♦ ♦③♦ ♦♥ ♦♣ó♥ ♦

♥ ♥ ♦s ♣s ♦ ♠s♠♦ ♥ st s♦ ♣s s♦♠♥t ♣ r ♥ ú♥♦ ♦ ②♣♦r t♥t♦ ♥ tró♥ ♦ ♥♥♥♦ ♣♦r rs ♦♥③♦ ♦ ♥ ♦s ♦ q ♥♦s q rí ♦s tr♦♥s ♥ ♥ εa

• s♠s ①♣r♠♥ts ♦♥③ó♥ st♦s ♥s ♠♣r③ ♦t♥s ♠♥t①♣r♠♥t♦s s♦ró♥ ♥rrr♦ ② ♦♥t t♠♣rtr s♦♥ s q s ①♣♦♥♥♥ t

❱ Eg ♥ P s

r♣♦ r♣♦ ❱

♦♠♦ s ♣r s♠♦♥t♦r ♥rí ♦♥③ó♥ s ♠② ♣r ♣sr sr♠♣r③s r♥ts t♥t♦ ♥ s♦ tr♦♥s ♦♠♦ ♦s P♦r ♦tr ♣rt ♥rí ♦♥③ó♥ ♦s ♥s ♦③♦s q ♥tr♦♥ sts ♠♣r③s s ♦♠♣r kBT t♠♣rtr♠♥t st♦ s♥ q ♦♥③ó♥ tér♠ s ♥♠♥t ♣r ①♣r t♣♦ ♦♥tétr q ♣rs♥t♥ ② ♣♦r ♠♣♦ ♦♥♥tró♥ ♦s ♦♥♦rs s ♠♦ ♠②♦rq ♦s ♣t♦rs ♦♥③ó♥ tér♠ rá q ② ♠ás tr♦♥s ♥ q ♦s ♥ ❱ ♣♦r ♦ q ♦s ♣♦rt♦rs ♠②♦rtr♦s s♦♥ ♥t♦s ② ♠tr r ♥♦♠r ♠tr t♣♦ n ♥ s♦ ♦♥trr♦ ♦s ♣♦rt♦rs ♠②♦rtr♦s s♦♥ r ♣♦st ② ♠tr s t♣♦p és ♠ás ♥t

ró♥ r ♦s ♣♦rt♦rs s ♠♥t ♠s ♦♥t ② ♣r♦♣s tr♠♦étrs ♦s s♠♦♥t♦rs ♦♥t♠♥♦s ♣♦r ♠♣r③s

ísst♦ó♦

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❮ Ó rtrísts ♥♠♥ts ♦s ♠trs s♠♦♥t♦rs ② ♠♥ét♦s

10.4 Ú P ❯

♦s ♥ú♠r♦s q ♦♥tr♦♥ ♦♠♣♦rt♠♥t♦ ♦♥t ♥ s♠♦♥t♦r t♠♣rtr T s♦♥ ♥ú♠r♦ tr♦♥s ♣♦r ♥ ♦♠♥ ♥ nc(T ) ♥ú♠r♦ ♦s ♣♦r ♥ ♦♠♥ ♥ ❱ pv(T )② ♦♠♦ s rá ♠ás ♥t st♦s ♥ú♠r♦s ♣♥♥ ♥ ♠♥r rít ①st♥ ♠♣r③ ♥ s♠♦♥t♦r

s q ♣rs♥ ♠♣r③ s♥ s♠♦♥t♦r ♣rr ♥s ♥rí ♥ r ♥rí s♦♦s st♦s ♦③♦s ♦ ♠♣r③ ② q st♦s ♥s ♥♦ tr♥ ♦r♠ ♣r ♦r♠ s ♥s ♥ ♣♦r t♥t♦ s ♥ss st♦s ♥ s

♥ ♠r♦ ② q st♦s ♥s ♦③♦s ♣♥ ♦♥③rs ♦♥ rt ♣r r r ♣♦rt♦rs r ♥ s ♥s sí q ♥②♥ ♥ ♣♦t♥ qí♠♦ µ(T ) s ♠♦♥ r♠ q stá ♥♦ ♣♦r stíst r♠r

P♦r t♥t♦ s ♥s ♣♦rt♦rs tr♦♥s ♦ ♦s ♣♦r ♥ ♦♠♥ s ♣r ♠♥t s ♦♥s

nc(T ) =

∫ ∞

εc

dε Dc(ε)1

e(ε−µ(T ))/kBT + 1

pv(T ) =

∫ εv

−∞dε Dv(ε)

[1− 1

e(ε−µ(T ))/kBT + 1

]=

∫ εv

−∞dε Dv(ε)

1

e(µ(T )−ε)/kBT + 1

♦♥ s s♣st♦ q ú♥♠♥t ② ♥ ❱ ② ♥

Pr tr♠♥r ♦r ♣♦t♥ qí♠♦ µ ♠♦s ♦♥♦r ú♥ t r ♦s♥s ♠♣r③ q ①st♥ ♥ rst Pr s♠♣r s ♦♥s ② ♣♦♠♦s

♣♦t♥ qí♠♦ s ♥rí r ♥sr ♣r ñr ♥ ♥ ♣rtí ♥ sst♠ Pr ♠ás ts s♦r s stró♥ ♦♥súts s♥tr á♥ stíst ❱é♥s t♠é♥ ♦s ♦♥t♥♦s

s♥tr ♠♦♥á♠ ♦♠♦ s ①♣♦♥♥s r♥ ♠② rá♣♠♥t st♦ ♥♦ t♥ ♥ ♠♣♦rt♥ ①s

ísst♦ó♦

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Ú P ❯

s♣♦♥r q s r♥ s ♦♥♦♥s

εc − µ(T ) ≫ kBT µ(T )− εv ≫ kBT

sts ♦♥♦♥s ♥♥ q ♣♦t♥ qí♠♦ stá ♠② ♥♦ ♥ ♥s kBT s♥s ♥rí

♦♠♦ µ(T ) s ♥rí r ♥sr ♣r ♣ró♥ ♥ ♥ ♣rtí ♥ sst♠ ② ♥ ♥ s♠♦♥t♦r ♣r♦ ♥ í♠t T → 0 s r♥ ♦s ♣rtís tró♥ ② ♦ s s ñ ♥ ♥rí Eg sst♠ s♣rr q ♥ ♥r s µ ≃ 1

2Eg ❨ ♥ t♦♦♠♦ s rá ♠ás ♥t s ♠♣ q µ(T ) = 1

2Eg ♣r T → 0 s s r♥ s ♦♥♦♥s ♦ q ♦♥r ♦♥ st st♠ó♥ µ(T ) q ♠♦s ♦

P♦r ♦ t♥t♦ s Eg ' 0,1 ❱ ② kBTamb ≃ 0,025 ❱ ♣r r♦ q ② ♥ r♥♦ ♦rs µ(T ) ♣r ♦s q s ♦♥♦♥s s sts♥ ♥ st s♦ s♠♦♥t♦r s q s♥♦ ♥r♦ ♦ q ♥♦ ①st ♥ró♥

s ♦♥s s♦♥ rts s ♣ srr

1

e(ε−µ)/kBT + 1≃ e−(ε−µ)/kBT ε > εc

1

e(µ−ε)/kBT + 1≃ e−(µ−ε)/kBT ε < εv

q ♥♦s r♣r stíst ①♦t③♠♥ ♦s q♥ ♣s s ♥ss ♣♦rt♦rs

nc(T ) = Nc(T )e−(ε−µ(T ))/kBT

pv(T ) = Pv(T )e−(µ(T )−ε)/kBT

♦♥ s ♥♦♥s

Nc(T ) =

∫ ∞

εc

dε Dc(ε)1

e(ε−εc)/kBT + 1

Pv(T ) =

∫ εv

−∞dε Dv(ε)

1

e(εv−ε)/kBT + 1

♥ rr ♦♥ t♠♣rtr T ♥ ♠♥r ♠♦ ♠ás s q s ①♣♦♥♥s q stá♥①♣íts ♥ s s

♥ t♦ ♦ q ♦s ♥tr♥♦ ♥ s ♥♥ ♥ ♥t♦ q ε s εc ♦ εv ♥s♥ts ♥s kBT ♥t♦♥s ♣♦r♠♦s ♥trr s♥♦ ♣r♦①♠ó♥ rát ♥rí ♦s ♥s tr♦♥s ② ♦s ♥t♦♥s ♥♥♦ ♠s t

m3c,v = me,h

1 me,h2 me,h

3

s t♥ ♥s st♦s

Dc,v(ε) =√

2|ε− εc,v|

√m3c,v

~3π2

♥ s♦ ♦♥trr♦ s q s ♥r♦ ② s♠♣ó♥ ♣r ② q qí ♠♦s ♣r♦♣♦♥r ♥♦ s ♣r

s s ♠♦t♦ q s♥ sst♠s ♥♦ ♥r♦s ♣s ♥s ♣rtís s s♥t♠♥t ♦♠♦ ♣rq s ♣ stíst ás ♣sr sr r♠♦♥s ♥s ♣♦rt♦rs nc(T ) ② pv(T ) q sst♠s ♥♦ ♥r♦s ② s ♣ ♣r stíst

ísst♦ó♦

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❮ Ó rtrísts ♥♠♥ts ♦s ♠trs s♠♦♥t♦rs ② ♠♥ét♦s

② ♠♦ rs x =√ε− εc,v ♥♦s ♦♥rt ss ♥trs ♥ ♦trs ♦r♠∫∞

0 x2e−a2x2dx =

√π/(4a3) sí

Nc(T ) =1

4

(2mckBT

π~2

)3/2

= 2,5×(mc

m

)3/2(

T

300 K

)3/2

× 1019 cm−3

Pv(T ) =1

4

(2mvkBT

π~2

)3/2= 2,5×

(mv

m

)3/2(

T

300 K

)3/2

× 1019 cm−3

st rst♦ á♦ ♣r ♥ s♠♦♥t♦r ♥♦ ♥r♦ ♥♦s ♥ q s ♥ss ♣♦rt♦rs t♥t♦ nc ♦♠♦ pv Tamb q ♥♦s ♣r♦♣♦r♦♥♥ s ♦♥s ♥ sr ♦♠♦♠á①♠♦ ♦r♥ 1018 ♦ 1019 ♠−3

♣r♦t♦ ♠s ♥ss ♣♦rt♦rs rst sr ♥t♦♥s

nc(T ) pv(T ) = Nc Pv e(εc−εv)/kBT ∝ T 3 e−Eg/kBT

q s ♥ ó♥ ♥r ♣r ♥ s♠♦♥t♦r ♥♦ ♥r♦ ♣s ♣r r s♦♠♥t♠♦s t③♦ ♦ q s ♥♦ ♥r♦ ② q s ♠♣ q ♥s st♦sD(ε) s ♣r♦♣♦r♦♥

√ε st ó♥ s s ♠r ② ó♥ ♠ss ② ♥♦s q

♣r ♥ t♠♣rtr tr♠♥ s s♥t ♦♥♦r ♥ú♠r♦ tr♦♥s ♣r ♦♥♦r ♦s ② rs

qí s ♦♥ rst sr s ♣rs♥ s ♠♣r③s ♣s s♠♦s q s ♣♥♦♥③r ♦♥ ♦♥♥♦ s♠♦♥t♦r tr♠♥♦ t♣♦ ♣♦rt♦rs ❱♠♦s st♥r♣s ♦s s♦s í♠t

s♦ ♥trí♥s♦ s ♠♣r③s ♦♥tr②♥ ♥ ♠♥r s♣r ♥s ♣♦rt♦rs ♠tr

s ♠② ♣r♦ ♦ ♥♦s ♥trs♠♦s ♥ s ts t♠♣rtrs st♠♦s ♥ s♦ ♥ s♠♦♥t♦r

♥trí♥s♦

♥ st s♦ ♥ú♠r♦ tr♦♥s ♥ s ♥ú♠r♦ ♦s ♥ ❱ ♦r♠q

ni(T ) = nc(T ) = pv(T )

② ♠♦s

ni(T ) =1

4

(2kBT

π~2

)3/2 (mcmv

)3/4e−Eg/2kBT ∝ T 3/2e−Eg/2kBT

= 2,5×(mc

m

)3/4(mv

m

)3/4( T

300 K

)3/2e−Eg/2kBT × 1019 cm−3

ni(T ) =√Nc(T )Pv(T ) e

−Eg/2kBT

♥♦ st ①♣rsó♥ pv(T ) ♦ nc(T ) ♥ ♣♦♠♦s ♦t♥r ♣♦t♥ qí♠♦ µ(T )♣r s♦ ♥trí♥s♦

µi(T ) = εv +1

2Eg +

1

2kBT ln(Pv/Nc)

= εv +1

2Eg +

3

4kBT ln(mv/mc)

❨ q ♠♦s ♣♦ s ♦♥♦♥s ó♥ st♦ á♥t♦ s♠♦♥t♦r ←→ tr♦♥s ♦s srí q♥t ♥ ó♥ qr♦

qí♠♦

ísst♦ó♦

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Ú P ❯

♦r♠ q ♣r t♠♣rtrs ♠② s s t♥ q

µi(T ) = εv +1

2Eg

② ♣♦t♥ qí♠♦ s stú ♥ ♠♦ r ♥rís

T 6= 0 ♥t♦♥s

µi(T ) ≃ εv +1

2Eg ± skBT,

s♥♦ s ♥ ♥ú♠r♦ ♣qñ♦ ♣♦t♥ qí♠♦ stá r♥♦ ♥tr♦ r ♥rís

♦♠♦ rs♠♥ s s ♠♣ q Eg ≫ kBT ♥ s♦ ♥trí♥s♦ s♠♦♥t♦r s ♥♦ ♥r♦② s r♥ s ♦♥♦♥s

s♦ ①trí♥s♦♥ st s♦ ♦♠♦ s ♥t s ♠♣r③s sí ♥②♥ ♥ ♥ú♠r♦ ♣♦rt♦rs q ② ♥

s♠♦♥t♦r s♠♦♥t♦r s ♥♦ ♥r♦ s s r♥♦ ② ó♥ ♠ss ♠♥r q s ♥♠♦s

nc − pv = ∆n 6= 0

② s t♥rá qnc pv = n2i

s t♦s s ♥ss ♣♦rt♦rs s ú♥ ♠s♠ t♠♣rtr T sts ♦s ♦♥s♦t♥♠♦s

nc =1

2

√(∆n)2 + 4n2i +

1

2(∆n)

pv =1

2

√(∆n)2 + 4n2i −

1

2(∆n)

♥ s♦ ♥trí♥s♦ s ♦♥s q♥ rs

e−εc/kBT =niNce−µi(T )/kBT

♦♥ µi s ♣♦t♥ qí♠♦ ♥ s♦ ♥trí♥s♦ ② ♣♦♠♦s rsrr s ♦♥s ♦♠♦

nc = Nce−εc/kBT eµ(T )/kBT =

Nc

Ncnie−µi(T )/kBT eµ(T )/kBT = nie

(µ(T )−µi(T )

)/kBT

pv = nie−(µ(T )−µi(T )

)/kBT

sts ♦♥s ♥♦s ♥ q s ♣♦t♥ qí♠♦ ①trí♥s♦ s ♠②♦r q ♥trí♥s♦ µ > µis ♠♣ q nc > ni st♦ s q ② ♠ás tr♦♥s q ♦s ♥ s♦ ♥trí♥s♦ ♠♥trsq s µ < µi s t♥ q ② ♠ás ♦s q tr♦♥s ♥ s♦ ♥trí♥s♦ pv > ni st rst♦r ♦♠♣♦rt♠♥t♦ ♥r µ sú♥ t♣♦ ♠♣r③s

② s ♦t♥ rt♠♥t q

∆n(T )

ni(T )= 2 sinh

(µ(T )− µi(T )

kBT

)

st ó♥ ♦♠♦ ② s ♦♠♥t♦ s á ♣r s♠♦♥t♦rs ♥♦ ♥r♦s ♣♦r ♦ q ♦r ∆n/ni sr t q µ− µi ♥♦ s ♦♠♣r Eg

Pr sr ♠ás ♥ttt♦s sr♠♦s

∆n(T )

ni(T )= 10r µ(T )− µi(T ) = m kBT.

ísst♦ó♦

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❮ Ó rtrísts ♥♠♥ts ♦s ♠trs s♠♦♥t♦rs ② ♠♥ét♦s

② ♣♦♠♦s r t ♦rs

r ≃ 1 =⇒ m ≃ 2,5

r ≃ 2 =⇒ m ≃ 5

r ≃ 3 =⇒ m ≃ 7.

♦♠♦ ♦♥só♥ sú♥ s ♦r Eg sí srá ♠á①♠♦ ♦r q ♣ t♦♠r ∆n ♣rq s♠♦♥t♦r s ♥♦ ♥r♦ ♥ t♦♦ s♦ ♠♦s q ♦s ♦rs m s♥ s♥♦s♥t♠♥t ♣qñ♦s sú♥ ♠♥t♥ ♣♦t♥♠♥t ♦s ∆n(T )/ni(T )

st♠♦s ♦♥sr♥♦ s♦ ♥ s♠♦♥t♦r ①trí♥s♦ ♣r♦ ♥♦ t♠♥t ①trí♥s♦♣♦♠♦s r q s♠♦♥t♦r s s♥♦ ♥♦ ♥r♦ s s ♠♣ q Eg ≫ kBT ♠♥r q ♣r s♦ ♥trí♥s♦

Pr♦ s s t♥ q ∆n ≫ ni ♠♣r③s ♦♥♦rs ② q Eg ≫ kBT ♥q ♥tr♦ ♦s í♠ts ♦s q st♠♦s ♥♦

nc ≃ ∆n≫ ni

pv ≃( ni∆n

)2∆n≪ ni.

P♦r ♦♥s♥t ♥♦ s ♠♣r③s ♦♥♦rs s♦♥ s q ♣♦rt♥ ♠②♦r ♥ú♠r♦ ♣♦rt♦rss♦ ①trí♥s♦ ♦s tr♦♥s s ♦♥rt♥ ♥ ♦s ♣♦rt♦rs ♠②♦rtr♦s ♠♦♦ q s♠♦♥t♦r s ♠ t♣♦ ♥ ♠②♦rtr♦s ♦s tr♦♥s

s ♠♣r③s s♦♥ ♣t♦rs ② ♣♦rt♥ ♠②♦r ♥ú♠r♦ ♣♦rt♦rs s♠♦♥t♦r s♦①trí♥s♦ s♦♥ ♦s ♦s ♦s ♣♦rt♦rs ♠②♦rtr♦s ② s♠♦♥t♦r r ♥♦♠r t♣♦ ♣

♠②♦rtr♦s ♦s ♦s

♦♠♦ ♥♦t ♥ ♣♦♠♦s ♦♠♥tr q s ED s ♥rí ♥ ♠♣r③ ♥ ♦♥♦r ②EA s ♥ ♣t♦r ró♥ ♥ r♠ µ ♦♥ t♠♣rtr ♣ rs ♥ r♥óts q ♦♠♦ ② s ♦♠♥t♦ ♠②♦r T s♥ str ♠ás ♥tr♦ ró♥ ♥trí♥s

P♦r ♠♣♦ s t♥♠♦s ♦s ♦rs Eg = 1 ❱ kBT ≃ 140

❱ ♥t♦♥s ∆n ≃ 103ni

ísst♦ó♦

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PP Ó

10.5 PP Ó

• ♦♥♥ r ♥ts ♥ ♥ r r♣s♦ ♥s s ♣r♦♣s áss s q s ♣ró♥ ♦s ♥ó♠♥♦s ♠♥ét♦s

♣r♠r ♣♥t♦ ♠♣♦rt♥t s rs ♥t q ♦s ♥ó♠♥♦s ♠♥ét♦s ♥ ♥ ♦t♦ át♦♠♦s ♦ ♦♥s ② ♣♦r ♦ t♥t♦ ♥ ♥ só♦ s♦♥ ♥ t♦ á♥t♦

♥ ♥♥ ①♣ó ♣r♠♥ts♠♦ ♦♠♦ ♦ ♥♠♥t♦ ♣♦♦s ♠♥ét♦s♣r♠♥♥ts ♦ ó♥ ♥ ♠♣♦ ♠♥ét♦ s ♣ ♠♦strr q ás♠♥t ♥ ♦♥♥t♦ ♣rtís rs ♥♦ ♣♥ rr ♠♦♠♥tt♦s ♠♥ét♦s ♣r♠♥♥ts

P♦r ♦tr ♣rt ♣rs♥ ♠♥ts♠♦ q s ♥ s♥ t♥♥ ♦s ♠trs ①♣sr ♠♣♦ ♠♥ét♦ st♦ ② ♥③ s ♣ ①♣r s♦♠♥t s s t♦♠♥♥ ♥t st♦s ♥rét♦s srt♦s ♦♥ rt st r♥t ♦s ♠♣♦s ①tr♥♦s

♦♦ st♦ s ♣ str ♣♥s♥♦ q stíst ás ♥♦s q stró♥ s ♣rtís ♥ ♦s ♣♥ s♦♠♥t t♦r ♦t③♠♥♥ e−E/kBT ♦♥ E s ♥rí t♦t sst♠ ♦tr ♥ E s ♠s♠ ♦♥ ♠♣♦ q s♥ ♠♣♦ ② q ♥ ♠♣♦♠♥ét♦ ♥♦ tú tr♦ s♦r s rs P♦r ♦ t♥t♦ s ♥t s q s ♣r ♠♣♦ ♥♦í ♦rr♥ts ♥ ♠♦♠♥t♦s ♠♥ét♦s t♠♣♦♦ ♦s rá s♣és ♣r♦ ♥trs s ♦♥t ♠♣♦ ♣rrá♥ ♦rr♥ts ② ss ♠♦♠♥t♦s ♠♥ét♦s s♦♦s ♣r♦ srá♥ ♥♦s ♥ qr♦tér♠♦

• ♠é♥ ② q r ♥♦tr q s ♣r♦♣s ♠♥éts ♦s s♥ts s ♠♦s str♥ s s ♣r♦♣s ♠♥éts ♦s át♦♠♦s ♦ ♦♥s ♦♥stt②♥ts ♦♥ ♥ ♠♦ó♥ ♥t♦r♥♦ rst♥♦ ♥ q s ♥♥tr♥ st♦s ♣♦♥r♠♦s q s ♥tr♦♥stró♥tró♥ stá♥ ♥s ♥ s ♣r♦♣s ♦s át♦♠♦s ♦ ♦♥s ♥ st♦s s♥ts ② ♥♦r♠♦s st ♠ás ♥t s ♥tr♦♥s tró♥tró♥ r♦ ♥

♥ ♦s ♠ts t③r♠♦s ♥ sró♥ tr♦♥s ♥♣♥♥ts

• ❱♠♦s ♥r ♦r ♦ q s ♥t♥ ♣♦r ♠♥t③ó♥ ♥ts ♥ ♦♥♥ ♦♠♥tr q♥ s sst♥s ♦ ♣r♠♥éts ♥♦r♠s s ♦rr♦♥s ♦s ♠♣♦ ♠♥ét♦ ①tr♥♦s♦♥ ♠② ♣qñs T = 0 s ♥ ♥s ♠♥t③ó♥ M ♥ sst♠ á♥t♦ ♦♠♥ V ♥ ♥ ♠♣♦ ♠♥ét♦ ♥♦r♠ H ♥ q s♣♦♥♠♦s q M s ♣r♦ H ♦♠♦

M(H) = − 1

V

∂E0(H)

∂HT = 0 K

♦♥ E0 s ♥rí st♦ ♥♠♥t sst♠ ♥ ♣rs♥

t♠♣rtr T 6= 0 ② q ♣r♦♠r s♦r t♦♦s ♦s st♦s q ♣♥ str ♦♣♦s

M(H,T ) = −∑

nMn(H)e−En/kBT

∑n e−En/kBT

T 6= 0 K

s♥♦

Mn(H) = − 1

V

∂En(H)

∂H

❱és ♣♦r ♠♣♦ ♥áss ♣r♠♥ts♠♦ ♥ s♥tr á♥ stíst ♦ ♥ ♥ ♥áss ás♦ ♦s ♠♦♠♥t♦s ♠♥ét♦s r♦s ♣♦r rs q s ♠♥ ♦ ó♥ ♥ ♠♣♦

♠♥ét♦ ♥ r ♥ ss♣t ♥ s ♣r♠r ③ q ♠♦s s♥ts ♦ ó♥ ♥ ♠♣♦ ♠♥ét♦ ♣s ♥tr♦r♠♥t s♦♦ ♠♦s ♦

♦♠♣♦rt♠♥t♦ ♦s tr♦♥s ♦♥ó♥ ♥ ♠t ♥ ♥ ♠♣♦ ♠♥ét♦ s♦ s ♦ q ♥ ♠♦s s♦s ♦r s♦t♦ s♣t s ♥r♠♥t ♠② ♣qñ♦ ♦r♥ 10−4 ♦

10−5 r♥t s♦ ♣♦r ♠♣♦ ss♣t χ = −1 ♥ ♠♥ét ♣rt♦ ♥ s♣r♦♥t♦r

ísst♦ó♦

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❮ Ó rtrísts ♥♠♥ts ♦s ♠trs s♠♦♥t♦rs ② ♠♥ét♦s

①♣rsó♥ s ♣ srr ♦♠♦

M(H,T ) = − 1

V

∂F

∂H

♦♥ F = −kBT ln(∑

n e−En/kBT

)

ss♣t ♥ sst♥ s ♥ ♦♠♦ ♠♥t ♠♥s♦♥

χ = −∂M∂H

= − 1

V

∂2F

∂H2

♥ s♦ ♥ q rs♣st sst♠ ♠♣♦ ♠♥ét♦ s ♥ M = χH

♠♥t③ó♥ ♥ ♠str ♣ ♠rs T ♦♥st♥t ♠♥t r③ q r ♥♠♣♦ ♥♦ ♦♠♦é♥♦ s♦r s♣st ♥ ró♥ ♥t ♠♣♦ ♥ t♠ñ♦ ♠str♥ t♦ s s♣♦♥♠♦s ♥ ♠♣♦ H ♣r♦ ❩ ♠♦r ♠str x x+ dx s ♠ ♥rí r

dF = F(H(x+ dx)

)− F

(H(x)

)=∂F

∂H

∂H

∂xdx = −VM ∂H

∂xdx

② ♣♦r t♥t♦ r③ f ♣♦r ♥ ♦♠♥ q s r s♦r ♠str s

f = − 1

V

dF

dx=M

∂H

∂x

② ♠ f ♥♦s rt♠♥t ♠♥t③ó♥ M

♦t s ♣r♦♣s tr♠♦♥á♠s ♦s ♠trs ♠♥t③♦s s ♣♥♦t♥r ♠♥t s ①♣rs♦♥s tr♠♦♥á♠s ♦♠♥s ♥ s q s sstt② ♣rsó♥ p ♣♦r ♠♥t③ó♥ M ② ♦♠♥ V ♣♦r ♠♣♦ H sí s r♠♦s 0 H ♠♥t ♠♣♦ ♠♥ét♦ q tú s♦r ♥ sst♥ ♥rí ♣♦r♥ ♦♠♥ ♠s♠ ♠ ♥ ♥ ♥t

∫dU = −

∫ H

0M dH = (lineal) = −

∫ H

0χH dH = −1

2χH2.

10.6 ❮ Ó

• ♦♦s ♦s ♠trs ♣♥♥♦ ♦r ② s♥♦ ss♣t ♠♥ét χ ♣♥srs ♦♠♦

♠♥ét♦s χ < 0 ♥ ♥r s t♥♥ ♦rs |χ| ≪ 1 ② ♦ q s ♠♥t③♥ ♥ró♥ ♦♣st ♠♣♦ ♠♥ét♦ ①tr♥♦ s♦♥ ♠♣♦s ③♦♥s ♥t♥s ♠♣♦ ♠♥ét♦

ísst♦ó♦

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❮ Ó

Pr♠♥ét♦s χ > 0 ♠é♥ s♥ sr ss ♦rs |χ| ≪ 1 Pr♦ χ ♥♦ s ♦♥st♥t② ♠♥t③ó♥ s str ♣r ♠♣♦s ♠♥ét♦s rts ② s t♠♣rtrs ♦r♠♥♦ ♥ M(H) ♣r t♠♣rtrs s ♦ ♠♣♦s ♠♥ét♦s t♦s

♠♥t③rs ♥ ró♥ ♠♣♦ ♠♥ét♦ ①tr♥♦ s♦♥ ♠♣♦s qs ③♦♥s♦♥ ♠♣♦ ♠♥ét♦ t♥ ♠á①♠ ♥t♥s ♠ás ss♣t χ ♣♥ t♠♣rtr ♦♠♦ χC/T rst♦ ♠♦ ② r

rr♦ ♥trr♦ ② rr♠♥ét♦s ♦♥ ♠♥t③ó♥ s♣♦♥tá♥ ♥♥ ♦rs ♠②♦s χ ♦r♥ ♥s 105 s ♥ ♠tr ♣r♠♥ét♦ tí♣♦ ♠é♥ ♣rs♥t♥stró♥ ♥q ♦♠♣♦rt♠♥t♦ χ(H) s st♥t♦ ♥ ♣r♠♥ét♦ ♣rs♥t♥♦ ♥ró♥ s♦♥t♥ ♠♥t③ó♥ ♦♥ H

♠♠♦s Θc t♠♣rtr ♥ rr♦♠♥ét♦ s tr♥s♦r♠ ♥ ♣r♠♥ét♦ ♦q s♠♣r ♦rr s ♦sr♦ q ró♥ χ ♦♥ T ♣r T > Θc ♥ ♣♦r ② r❲ss

χ =C

T −ΘP

♦♥ C ♦♥st♥t r ② ΘP ♠♦ ♣♥t♦ ♣r♠♥ét♦ r ♥ ♣♦♦ ♠②♦r q Θc

♦♦ q♦s rsts ②♦s ♦♥stt②♥ts s♦♥ ♣r♠♥ét♦s ♥♠♥t st♦ s q t♥♥♠♦♠♥t♦s ♠♥ét♦s ♣r♠♥♥ts s♦♥ ♣s ♣rs♥tr t♣♦ ♦♠♣♦rt♠♥t♦ ♦♦♣rt♦q ♦♥ rr♦♠♥ts♠♦

♦ q ♠♥t③ó♥ ♠♦♥♦rsts rr♦♠♥ét♦s s ♥sótr♦♣ s r♦♥s ♥ sq stró♥ ♠♥ét s ♦t♥ ♣r ♦rs rt♠♥t ♣qñ♦s H s ♠♥ á♠♥t③ó♥ í ♠♥t③ó♥ ♥ ♦s s♦s ♦♥trr♦s

tró♥ s♥ q ss♣t s ♥ ♥♦ ♣♦r ♠♣♦ H s ♠② r♥ ú♥ ♦s rst♦s①♣r♠♥ts r ♥ó♥ χ(H) s ♥ ♥ó♥ ♠♦♥ót♦♥ r♥t

♦♥st♥t C s ♠ ♦♥st♥t r ás ♥t s rá♥ s ♦♥♦♥s ♣r q s ♠♣ s ② r

ísst♦ó♦

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❮ Ó rtrísts ♥♠♥ts ♦s ♠trs s♠♦♥t♦rs ② ♠♥ét♦s

♥♠♥t ♦r♠ ② s ♠♥s♦♥s ♥ rr♦♠♥t♦ rí♥ ♦♥ s ♠♥t③ó♥♠♥t♦stró♥ st♦ s♥rá sú♥ ♣r♥♣♦ tr q ♣ó♥ sr③♦s t♥só♥ ♦ ♦♠♣rsó♥ ♥ ♠tr ♥rá ♥ ♠♦r ♦ ♣♦r ♠♥t③ó♥ ♠str t♦♠♥t♦ást♦

• ♥ ♥r str ♥ sst♥ ♥ ♥ ♠♣♦ ♠♥ét♦ ♣r ♥ ♥ ♠♦♠♥t♦ ♠♥ét♦♣♦r ♥ ♦♠♥ ♠♦ ♠♥t③ó♥ ♥ ♠trs sótr♦♣♦s M ② H s♦♥ ♣r♦s ②♥t♦♥s ró♥

M = χH

♦♥ ss♣t χ stá ♥ ♣♦r ♥ ♦♠♥ ② ♥ st s♦ s ♥ sr ♠tr s ♥sótr♦ ♥t♦♥s χ s ♥ t♥s♦r

♠♥ét♦s Pr♠♥ét♦s rr♦♠♥ét♦s

−18× 10−5 (cm3/mol) 580× 10−5 1000 tr♥só♥

−0,9× 10−5 2 360×10−5 ♦ 240 tr♥só♥

−0,8× 10−5 4 120× 10−5 150 tr♥só♥

−0,3× 10−5 Pt 26× 10−5 ♥tá♥♦

|χ| ≃ 10−5 χ = 1 s ♠♥ét♦ ♣rt♦ |χ| ≃ 10−5 ② ♥tá♥♦

ísst♦ó♦

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❮ Ó

♦♠♦ ② s ♦♠♥t♦ ♥ts ② ♥③ ♥♦s ♥ q ♠♥ts♠♦ sr ♦♥trí♥s♦ ♥ t♦♦s ♦s ♠trs ♥ ♦s át♦♠♦s ♦♥ s♣s ♥s ♦s ♠♦♠♥t♦s ♠♥ét♦s s♣í♥ ② ♦s ♠♦♠♥t♦s ♠♥ét♦s ♦rts s ♥♥ s♣r♠♥t st♦ s♥ q ♦s rsts ss ♥♦s ② ♠♦s rsts ó♥♦s ♥♦ ♣rs♥t♥ ♣r♠♥ts♠♦ ② s♦♥ ①♥ts ♣r st♦♠♥ét♦s

♥ ♦s rsts ♦♥ts ② ♥ ♠♦s rsts ó♥♦s ①st ♥ ♦♣♠♥t♦ ♦s tr♦♥st q ♦s ♠♦♠♥t♦s ♣♦rs ♠♥ét♦s s ①st♥ ♥♦ ♣♥ ♠r s ró♥ s♥ ♠r ♦♥ró♥ tró♥ ss ♦♥stt②♥ts st♦ q st t♣♦ rsts t♠♣♦♦ s♥♣r♠♥ét♦s

P♦r ♦tr ♣rt ♠♦s ♦s ♠♥t♦s tr♥só♥ ② t♦s s trrs rrs ♣rs♥t♥♦♠♦ só♦s ♦♠♣♦rt♠♥t♦s ♣r♠♥ét♦s t♥♥ s♣s ♥tr♥s ♥♦ ♥s ② s♠♥t♦rqr♦ ♦s ♠♦♠♥t♦s ♠♥ét♦s ♥s ♣r♦♥ ♣♥t♠♥t♦ ♣r♦♦ ♣♦r ss♣s ♠ás ①tr♥s ♦s ♣r♦♣♦s át♦♠♦s ♦ ♦♥s

❨ ♣r♦♣ ①st♥ st♦s ♥rét♦s srt♦s ♥ ♦s sst♠s ís♦s