❯ ❯

P ❩ ❯ P ❯

❯

ssrtçã♦ ♣rs♥t à ❯♥rsr ❱ç♦s ♦♠♦ ♣rt s ①ê♥s ♦ Pr♦r♠ Pósrçã♦ ♠ís ♣ ♣r ♦t♥çã♦ ♦ tít♦ str ♥t

❱

Ficha catalográfica preparada pela Seção de Catalogação e Classificação da Biblioteca Central da UFV

T Rodrigues, João Henrique, 1987- R696p Processo de desmagnetização e histerese de um gelo de spin 2013 artificial em uma geometria triangular / João Henrique Rodrigues. – Viçosa, MG, 2013. xiii, 69f. : il. (algumas color.) ; 29cm. Orientador: Lucas Alvares de Silva Mól Dissertação (mestrado) - Universidade Federal de Viçosa. Referências bibliográficas: f. 67-69 1. Magnetismo. 2. Monopoles magnéticos. 3. Materiais nanoestruturados. 4. Física do estado sólido. I. Universidade Federal de Viçosa. Departamento de Física. Programa de Pós- Graduação em Física Aplicada. II. Título. CDD 22. ed. 538

♦ st ssrtçã♦ ♠str♦ ♦ ♥ç♦ ê♥ qqr ♣ss♦ q

♦♥s rt♦s trés

á três ♠ét♦♦s ♣r ♥r s♦r ♣r♠r♦ ♣♦r r①ã♦ q é ♦ ♠s ♥♦r

s♥♦ ♣♦r ♠tçã♦ q é ♦ ♠s á trr♦ ♣♦r ①♣rê♥ q é ♦ ♠s

♠r♦

♦♥ú♦

rç♦ ♣r♠r♠♥t ♠♥ ♠í ♠ ♣ ♦ã♦ ♦sé ♦rs ♣♦r

s♠♣r ♠ r ♠ ❱ç♦s ①♥♦ ♥rt♠♥t r♦ q s ♠♥s ú♥s ♣r♦

♣çõs q r♠ ♦s st♦s ♠♥ ♠ã ♥ ♦s ♣♦r s♠♣r s ♣r♦♣r

♦♠♦ r ♠ ♠♠ ♥s ♠♥s s ♥ó♣♦s ♠♥ r♠ã ♦s♥ s

♦rs ♣ ♣r♦♣çã♦ ♦♠ ♠♥ ê♠ ♣♦r ♣♥r

♦ q ♦ ③r ♥♦s ♠s ♣ró①♠♦s ♥♦s ♦t♦r♦ ♠ r♠ã♦ ú♦ ésr

♦rs ♣ ♣ê♥ ♦♥srçã♦

rç♦ ♦s ♠s t♦s sté♦ ♦♠r♦ ♥é ♥tô♥♦ r♥②

♥tô♥♦ ♥♥ ♦s ♣ ót♠ r♣çã♦ q♥♦ ♦s st♦ ♦s ♠s ♣r♠♦s

♥ é♦r ❱♥ís ❱t♦r ①♠♥♦ ♦♥ ♥í♥ ❲s♦♥ ♥

ss♦♥ ♣ ♠③ ♠s ♦rt q ♦ ♥♦ss♦ ♣r♥ts♦ ♣r♥♣♠♥t ♠ ♣r♠♦

P♦ ♣♦r sr ♠ r♥ ♠♦ ♠ tr♥♦ r ♦♦s

rç♦ à ♣♦s ót♠♦s ♥♦s ♦♥ê♥ ♣s ♠ts rs

q ♠ tr♦① ♥ss ♣q♥♦ t♠♣♦ ♠ q ♥♦s ♦♥♠♦s

♦s ♠s ♠♦s ís ♦r♦♦r ♥♥② ♦② r

s♠t♦ ♦s♦♥ rt♦ró t♦♥♥♥ r♥r r

③ ♥ré♦♦tó r♥♦str♦ r♦rã♦ ♦♥r♦♦♠♦ P

r♦s♠ st♦ç s♦♥ r♥♥♦s ♥ t♥

t ♠♦♥ ♦♦Ptr ②♥ ♦tr♦s ♣ ót♠ ♦♥ê♥ ♠③

♣r♥③♠ ♥st ♣rí♦♦

♦s ♠s ♠♦s ❱ç♦s ô♥s♠♦ r♠rr á♦ ①

ss ♦ss♥r♦♥str♦ ♥ r♦♥r♦♦ t♥t♥ ②

t♦♥ rr♥ ♠♥♥ r♥♥r♥s r♥

♥ ♥♥t ♠♥♠♥♥ ③③♥ r♦sót♦

r♥r♥ê r♥ ③ ♥ ❱♥ís ♥ árá ♦ ♠

t♦s ♦tr♦s ♣♦r ♠ ③r♠ ♠ ót♠ ♦♠♣♥ ♠ ❱ç♦s ♥s tórs ♥s rr♦ts

té q é♠ s rrt á ♠♦r

♦s ♠s ♠♦s ♥ó♣♦s rã♦ r♦ ♥r♦ ②

♥ ♦❱r♠♦ ♥r♦ r♥♥♦♣t ♦♦ts

③r♦ ♥ ♥②♥② r ♦♦♦ s tt

♠rst árr♥ ♥ré❱♦ô ♦ ♦♦sr

s r♥ ②♥r ♣♦s ót♠♦s rt♦s ♠♦♠♥t♦s ♠♥ st

♠ ♥ó♣♦s

❯♠ r♠♥t♦ s♣ ♠ ♦r♥t♦r s ó ♣ ót♠ ♦r♥tçã♦

♣ ♣ê♥ ♣s sssõs ♦♥strts q t♠♦s ♠ ♠ ♠str♦

♦s ♣r♦ss♦rs ♠ s♣ ♦ ♠♦s♦ ♥tô♥♦ r♦s r♥ ♠♦ ❱♦

♦♥♥♦ r♦ ①r q ♠ ♥s♥♦ sr ♠ ♣r♦ss♦rr ♥♦♥ár♦s

♦ P

❯❱

P ♦ P

t♦♦s ③r♠ ♣rt ♠♥ r ♠♥ trst③ ♦ ♠ s♥r♦

♠t♦ ♦r♦

❯

❯ ①

❯ ①

①

♥tr♦çã♦

t♦ st tr♦

strtr ssrtçã♦

♥♠♥tçã♦ tór

♥ts♠♦

❯♠ ♣♦♦ stór ♦ ♠♥ts♠♦

♥trçã♦ ♣♦r ♠♥ét

Pr♠ ss♣t ♠♥ét

♥s♦tr♦♣ ♠♥ét

♦♠í♥♦ ♠♥ét♦ ♠t s♣r♣r♠♥ét♦

♦s s♣♥s

♦s s♣♥s rts ♠ s♠çõs ♦♠♣t♦♥s

s ♦ s♣♥ ♠ ♦trs ♦♠trs

Pr♦t♦♦♦s s♠♥t③çã♦ ♦s s♣♥s rts

Pr♦t♦♦♦s ①♣r♠♥ts

Pr♦t♦♦♦s ♦♠♣t♦♥s

♦tçõs ♠t♦♦♦

♦tçõs

st

Pr♦ss♦s s♠♥t③çã♦

♠♣♦ ♠♥ét♦ r♦t♦♥

♠♣♦ ♠♥ét♦ ♠ ♠ ú♥ rçã♦

♦♥srçõs ♥s

st♦s sssõs

Pr♦ss♦ s♠♥t③çã♦ trés ♠♣♦ ♠♥ét♦ ①tr♥♦ ♠

♠ ú♥ rçã♦

Pr♠r ♥ás ♦♠♥t s♦r♠ ♥♦s sít♦s r

♥ ♥ás s♥♠♥t♦ r♠♣♦ ♠♥ét♦ ①tr♥♦

s♦r♠ ♥♦s sít♦s r

Pr♦ss♦ s♠♥t③çã♦ trés ♠♣♦ ♠♥ét♦ ①tr♥♦ r♦t

♦♥

♦♥sõs ♣rs♣ts

♦♥sõs ♣rs♣ts

❯

r strs

tr♥r s♣♥s ♣r♦♣♦st ♣♦r ❲♥♥r

♦♥rçã♦ ♦ ♦ á

♦♥♦♣♦♦s ♠♥ét♦s

♣r♦♣♦st ♣♦r ❲♥

s ♦♥rçã♦ ért r qr

♠♠ r r ♣r♦③ ♠ ♦rát♦r♦

r strs ♠ ♥♥♦

♥trçã♦ ♥tr ♦s ♠♦♥♦♣♦♦s ♠♥ét♦s ♥♦ ♦ s♣♥

r♦r

♣♦ ért r r♦r

♦♥②♦♠

♦♠é

♦♥②♦♠ ♦♠é

s t♦♣♦♦s ♣rs♥ts ♥ r tr♥r

tr♥s ♥ r tr♥r

Pr♦ss♦s s♠♥t③çã♦ ♣r♦♣♦st♦s ♣♦r ❲♥

st ♥st tr♦

strs Mag ♣♦r ~Bext ♣r s♦r♥s

strsV ♣♦r ~Bext ♣r s♦r♠ ♥

strsV ♣♦r ~Bext ♣r s♦r♠ b′i = ±10%

①

strs ❱ért V 7

strs V ♣♦r ~Bext ♣r s♦r♠ b′i = 20%

strs Mag ♣r ~Bext ♦♠ s♥♠♥t♦ s♠ s♦r♠

strs strçã♦ ♥â♠ ♠ ért

strs V ♣♦r ~Bext ♣r θ = 10♦ s♦r♠ ♥

strs Mag ♣r ~Bext ♦♠ s♥♠♥t♦ b′i = 10%

strs V ♣♦r ~Bext ♦♠ b′i = ±10% θ = 10♦

strs Mag ♣♦r ~Bext ♦♠ s♥♠♥t♦ b′i = 20%

strs V ♣♦r ~Bext ♣r b′i = 20% θ = 10♦

strs V ♣♦r ~Bext ♦♠ s♦r♠ θ = 10♦

♦t♦♥ Mag ♣♦r θ ♣r s♦r♠ | ~Biext| = bc

♦t♦♥ V ♣♦r theta ♣r s♦r♠ ♥ ♦♠ | ~Biext| = bc

♦t♦♥ V ♣♦r θ ♣r s♦r♠ 10% ♦♠ | ~Biext| = bc

♦t♦♥ V ♣♦r energia ♣r s♦r♠ 10% ♦♠ | ~Biext| = bc

♦t♦♥ V ♣♦r θ ♣r s♦r♠ 20% ♦♠ | ~Biext| = bc

♦t♦♥♠♥s ♦s érts r ♣r ♦s t♣♦s s♦r♠

①

❯

❯ ♦ã♦ ♥rq ❯♥rs r ❱ç♦s rr♦2013 Pr♦ss♦ s♠♥t③çã♦ strs ♠ ♦ s♣♥ rt ♠ ♠ ♦♠tr tr♥r r♥t♦r s rs ó♦♦r♥t♦rs r♥♦ ♦rs Prr ❲♥r ①♥r ♦r ♦

st tr♦ st♠♦s ♥s ♣r♦ss♦s s♠♥t③çã♦ ♠ ♦ s♣♥

rt ♠ ♦♠tr tr♥r trés ♦s ♣r♦ss♦s t③♥♦ ♠♣♦ ♠♥é

t♦ ①tr♥♦ ♣r♠r♦ ♣r♦ss♦ s♠r ♦ ♣r♦ss♦ strs ♠♣♦ ♠♥ét♦

♠ ♠ ú♥ rçã♦ ♦ s♥♦ trés ♠ ♠♣♦ ♠♥ét♦ r♦t♦♥

rs♥t Pr st♦ t③♠♦s s♠çõs ♦♠♣t♦♥s ♦♥t r♦ ♦♥sr♥♦

q ♦s s♣♥s r r♠ t♣♦ s♥ q sts sr♠ ♥rt♦s s♦♠♥t q♥♦

♦♥çã♦ −~si · ~Bi ≤ bi ♦ss stst ♦♠♦ t♥tt ♣r♦①♠r ♥♦sss s♠çõs

♦s ①♣r♠♥t♦s t♦s ♠ ♦rtór♦ ♣t♠♦s ♦s ♦rs ♥♦sss s♠çõs ♣r

sr♠ ♣r♦♣♦r♦♥s ♦s ♦rs ♦t♦s ♠ ♦rtór♦ ♦♥sr♠♦s ♠ s♦r♠

♥♦s s♣♥s r ③♥♦ ♦♠ q ♦ ♦r ♦ ♠♣♦ ♠♥ét♦ ♠í♥♠♦ ♣r ♥rsã♦

bi s ♠ strçã♦ ss♥ bi → bc ± b′i st út♠ ♦♥srçã♦ sr r

①♦ ♦♥strr ♦s ♦s s♣♥s ♠ ♦rtór♦ r♥♦ árs s

r♥ts ♠ ♠ ♠s♠ r é♠ st♦ ♦♥sr♠♦s ♠ ♦tr s♦r♠ ♥♦

♣r♦ss♦ s♠r ♦ strs ♦ s♥♠♥t♦ ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♦♠

r ♦ ss ♣q♥s ♠♥sõs s rst♦s ♥♦ss♦ tr♦ ♦r♠ ♦t♦s

trés r qrs ♦♠ 30 s♣ç♠♥t♦s r s ♠és ♦r♠ t♦♠s

50× 10 ♠♦strs rs r♥ts ♠♦strs r sts rst♦s

♣r♠♦s q ♥♦ ♣r♦ss♦ s♠r ♦ strs s rs ♦r♠ s♠♥t③s

①

♦♥sr♠ ♥çr ♦ st♦ ♥♠♥t ♠s♠♦ ♦♠ r♥s s♦r♥s ♥q♥t♦

♥♦ ♣r♦ss♦ ♠♣♦ ♠♥ét♦ r♦t♦♥ ♣r♠♦s q s rs r♠ s♠♥t

③s ♠s ♥♥♠ ♥ç ♦ st♦ ♥♠♥t ♦ ♣r♦ss♦ r♦t♦♥ ♦ st♦

♥♠♥t r s♦♠♥t ♥ç♦ ♥♦ ♥tr♦r s rs s♠ s♦r♠

①

❯ ♦ã♦ ♥rq ❯♥rs r ❱ç♦s rr②2013 ♠♥t③t♦♥ ♥ ②strss ♣r♦ss ♦ ♥ rt s♣♥ ♥ tr♥r ♦♠tr② sr s rs ó ♦srs r♥♦♦rs Prr ♥ ❲♥r ①♥r ♦r ♦

♥ ts ♦r st s♦♠ ♠♥t③t♦♥ ♣r♦t♦♦s ♦r ♥ rt s♣♥ ♥

tr♥r tt tr♦ t♦ ♣r♦sss s♥ ①tr♥ ♠♥t s rst

♣r♦ss s s♠r t♦ t ♣r♦ss ♦ ②strss s♥ ♠♥t ♥ ♦♥ rt♦♥

♥ t s♦♥ tr♦ r♦tt♦♥ ♥ rs♥ ♠♥t ♦r ts s

♦♥t r♦ ♦♠♣tr s♠t♦♥s ♦♥sr♥ tt t tt s♣♥s r ♣♣ ♦♥②

♥ t ♦♥t♦♥ −~si · ~Bi ≤ bi s sts ♥ ♥ tt♠♣t t♦ rt ♦r s♠t♦♥s

t♦ ♦rt♦r② ①♣r♠♥ts ♣t ♦r s♠t♦♥s s t♦ ♣r♦♣♦rt♦♥ t♦

t s ♦t♥ ♥ ♦rt♦r② ♥ ♦♥sr s♦rr ♥ t tts

s♣♥s ♥ s ② tt t ♠♥♠♠ rrs♥ ♠♥t bi ♦♦s ss♥

strt♦♥ bi → bc±b′i s st ♦♥srt♦♥ ♦ rt♦♥ ♦ t t②

♦ ♦♥strt♥ t s♣♥ s ♥ t ♦rt♦r② ♥rt♥ sr r♥t s♥s ♥

tt ♦r♦r ♦♥sr ♥♦tr ♥ ♦ s♦rr ♥ t ②strss ♣r♦

ss t ♠s♥♠♥t ♦ t ①tr♥ ♠♥t t t tt t♦ ts s♠

♠♥s♦♥s rsts ♦ ♦r ♦r r ♦t♥ tr♦ sqr tts t

tt s♣♥s ♥ rs r t♥ r♦♠ 50 × 10 s♠♣s r♥ts tts

♥ s♠♣ t♦ tt ♥ ts rsts r③ tt t ♣r♦ss s♠r t♦

①

②strss t tt r ♠♥t③ ♥ ♠♥ t♦ r t r♦♥ stt ♥

t ♠♦r s♦rrs ❲ ♥ t ♣r♦ss ♦ r♦tt♦♥ ♠♥t r③ tt

t tt r ♠♥t③ t ♥♦♥ r t r♦♥ stt ♥ ts r♦tt♦♥

♣r♦ss ♦♥② t r♦♥ stt s ♥ s♠ r♦♥ ♦ t tt t♦t t

s♦rr

①

♣ít♦

♥tr♦çã♦

♥trss ♦ ♦♠♠ s♦r ♦ ♠♥ts♠♦ ♦s ♠trs ♠♥ét♦s t♠ ♦r

♥s ♠t♦ ♥ts s ♥t ré ♥t ♥ ♠♥ts♠♦ t r♥

♠♣♦rtâ♥ ♥ stór ♦♠♦ ♣♦r ①♠♣♦ ①♥♦ ♦ ♦♠♠ s r ♣♦s ♠rs

♥ é♣♦ s r♥s ♥çõs t♠♥t ♦ ♠♥ts♠♦ é s♦ ♥♦ ♥♦♥

♠♥t♦ ♣r♦s s♠♣s ♦♠♦ ♠ ♠♣♥ ♦ ♠ ♣r♦s ♠s ♦♠♣①♦s

♦♠♦ rçã♦ ♠♥ét ♦s ♠ s♦ rí♦ ♦♠♣t♦r ♦ s♥♦r

stór ár♦s ♣sqs♦rs s♦rr♠ s rtrísts ♦s ♠trs ♠♥ét♦s

♥ê♥ q sts s♠ ♥♦ ♠♦ ♠ q s ♥♦♥tr♠ ❯♠ rtrísts

♠♣♦rt♥t ♦s ♠trs ♠♥ét♦s é q sts sã♦ ♦♠♣♦st♦s ♣♦r ♣q♥♦s rsts

♦♠ ♦♠í♥♦ ♠♥ét♦ s♠♣s q s ♦♠♣♦rt♠ ♣♦ ♠♥♦s ♦♠♦ ♠ ♣r♠r

♣r♦①♠çã♦ ♦♠♦ ♣♦♦s ♠♥ét♦s ♥trçã♦ ♦♥♦ ♥ ♥tr sss ♦♠í

♥♦s ♣♦ rr rtrísts ♥trss♥ts ♣r sts ♠trs P♦r ①♠♣♦ ♥s

♥trr♦♠♥t♦s ♣rs♥t♠ st♦s ♥♠♥ts ♥r♦s ♦ rstrçõs

♦r♥s ♦♠étr

❯♠ s rtrísts ♦srs ♠ t♦♦s ♦s ♠trs ♠♥ét♦s é q sts

s♠♣r ♣rs♥t♠ ♥♦ ♠í♥♠♦ ♦s ♣♦♦s ♠♥ét♦s té ♦ ♥ã♦ ♦ ♦sr ♥

♥tr③ ♥♥♠ ♣rtí q s ♦♥tss ♣r♦♣r s♠r à ♠ ♠♦♥♦♣♦♦

s r♥t♠♥t s♦rs ♠ ss ♠trs q ♣♦r♠ ①r ♥♦s s

t♦s ♦s ♠♦♥♦♣♦♦s ♠♥ét♦s ♦s ♦s s♣♥s sts ♠trs ♦ ♦sr♦

q s ①tçõs ♦ s st♦ ♥♠♥t sã♦ qs♣rtís q ♦♠♣♦rt♠s

♥tr♦çã♦

♦♠♦ ♠♦♥♦♣♦♦s ♠♥ét♦ s ♠♥s♦♥s ♦s s♣♥s rts ♦r♠

♠♦♥ts ♦♠ ♦ ♥tt♦ str ♠♦r sss ①tçõs ♠s ♥③♠♥t ♥ã♦ ♥

♦♥trr♠ ♥♥♠ ♠ét♦♦ ♥t q ♦♥s ♦tr ♦s st♦s ♥♠♥ts sts

rs

♠ ♥♦ss♦ tr♦ ♠♦s ①♣♦rr ♦s ♣r♦ss♦s s♠♥t③çã♦ ♣r

t♥tr s♠♥t③r ♦tr ♦ st♦ ♥♠♥t ♠ t♣♦ s♣ ♦ s♣♥

rt ♣r♦♣♦st♦ r♥t♠♥t

♥tr♦çã♦

t♦ st tr♦

♦t♦ st tr♦ é str ♦ ♣r♦ss♦ s♠♥t③çã♦ s♥♦

♦tr ♦ st♦ ♥♠♥t ♠ r ♦ s♣♥ rt ♠ ♠ ♦♠tr

tr♥r trés ♠ ♠♣♦ ♠♥ét♦ ①tr♥♦ Pr st♦ srá ♥s ♦çã♦

♠♥t③çã♦ ♦s ♣♦ssís t♣♦s érts ♣rs♥ts ♥♦ sst♠ trés ♦s

r♥ts ♣r♦ss♦s s♠♥t③çã♦ ♦ ♣r♠r♦ trés ♠ ♠♣♦ ♠♥ét♦

r♦t♦♥ ♦ s♥♦ trés ♠ ♠♣♦ ①tr♥♦ ♠ ♠ ú♥ rçã♦ ♦♠♦ ♦

♣r♦ss♦ strs

strtr ssrtçã♦

s♥♦ ♣ít♦ st ssrtçã♦ ♠str♦ ♣rs♥t ♠ ♣q♥

♥tr♦çã♦ s♦r ♠♥ts♠♦ s♠çõs ♦♠♣t♦♥s ♦s s♣♥ st ♥tr♦

çã♦ ♦♥té♠ s ♥♦r♠çõs ♠í♥♠s ♥ssárs ♣r q ♦ t♦r ♦♥s ♦♠♣♥r

♣r♥♣ ♣♦r trás st tr♦ é♠ st♦ ♥st ♣ít♦ ♥♦♥tr♠s s ♠♦

tçõs st tr♦ ♦s tr♦s ♠s ♠♣♦rt♥ts q ♥♦s r♠ ♥ r③çã♦

st s r♦s rt♦s ♦s qs ♥♠♥tçã♦ tór ♦ ①trí ♥♦♥tr♠s

♥♦ ♥ st ssrtçã♦ ♠ rê♥s ♦rás

trr♦ ♣ít♦ ♠♦str ♦s ♣r♦♠♥t♦s r③♦s ♥ ①çã♦ st

tr♦ trés st s♣rs q ♦ t♦r ♦♥s ♥t♥r ♦s ♣r♦♠♥t♦s

r③♦s té s ♦r ♥trss st ♦♥s r♣r♦③r ♦s rst♦s ♥♦♥tr♦s

é♠ st♦ ♥ ①♣♠♦s ♦ ♠♦t♦ ♠s ♦♥srçõs ♠♣♦rt♥ts q ♦r♠

t♦♠s r♥t r③çã♦ st tr♦

qrt♦ ♣t♦ ♣rs♥t ♦s rst♦s ♦t♦s rs ①♣çõs ♣r

♦s ss ♥t♥♠♥t♦s sts rst♦s sã♦ ①♣rss♦s ♠ rá♦s strçõs ts

s ①♣çõs sã♦ s ♦♠ ♥tt♦ srr ♠♦strr s ♠♣çõs íss

rst♦

♦ q♥t♦ ♣t♦ sã♦ ♣rs♥ts s ♦♥sõs ♥♦ss♦ tr♦ st

♣ít♦ srá ♠♦str♦ s ♦ ♥♦ss♦ ♦t♦ ♦ ♥ç♦ ♦s ♣♦♥t♦s ♣♦st♦s ♥t♦s

♥tr♦çã♦

♥♦ss ♠t♦♦♦ ♥♦sss ♣rs♣ts ♣r tr♦s tr♦s

♣ít♦

♥♠♥tçã♦ tór

♥ts♠♦

❯♠ ♣♦♦ stór ♦ ♠♥ts♠♦

s ♣r♠r♦s st♦s ♥ô♠♥♦s ♠♥ét♦s ♦♥tr♠ ♠t♦ t♠♣♦ ♥

ré ♥t ❬❪ ♦ sé♦ ❱ s t♦ ♦sr♦ ♠ ♠ ss

♥s ♣rt♦ ♥és q ♣q♥s ♣rs ♦♥s ♦♠♦ ♠♥tt t♥♠

♣ trr t♥t♦ ♦t♦s rr♦ q♥t♦ s trír♠ s t♥t♦

①♣r st ♥ô♠♥♦ ③♥♦ q sss ♣rs ♣♦ssí♠ ♠ s♣é ♠ q

tr ♣♦r trçã♦ ♦♠ ♦ rr♦ ♥rt s s ♥ã♦ ♦ ♣r♠r

♣ss♦ ♦srr st ♥ô♠♥♦ ♥♠♥t s ♠ s♦ ♦sr♦s ♣♦r ♠

♣st♦r ♦s ♥s q t♥ r♣r♦ q ♣q♥s ♣rs ♠ ♣rss ♥s

s rr♦ s s♥á ♥ ♣♦♥t rr♦ s ♦ á t♠é♠ ♥í♦s

q s♦rt ♦ ♠♥ts♠♦ ♦ t ♠t♦ t♠♣♦ ♥ts ♥ ♥t ♥

sss ♥ô♠♥♦s s♣rtr♠ ♠s ♥trss ♥♦ s♦ ❱ q♥♦ ♦♠çr♠

srr s ♣r♠rs ♦srçõs tr♦s ♠s ♦r♦s s♦r ♦ ♠♥ts♠♦

tr ♦s ♣♦s ♠ Prr r♦rt st♦ ♣r♦♥♠♥t ♦s

♥ô♠♥♦s ♠♥ét♦s trés ár♦s ①♣r♠♥t♦s ♦sr♦ ár♦s ♥ô♠♥♦s

♥trss♥ts s♦r ♦ ♠♥ts♠♦ trés sts r♦rt ♦sr♦ q t♦♦s ♦s

♠♥t♦s s♠♣r tr♠ ♦s r♥ts ♣♦♦s ♥♦rt s ♦s ♦♣♦st♦s s♠♣r s tr♠

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

♥♠♥tçã♦ tór

♠ ♦tr♦s ♦s ♠♥t♦s ♠♥♦rs q ♥♦ ♦ qr s ♥♦♥tr♠ ♦s ♣♦♦s

♦♣♦st♦s trés sts ♦tr♦s st♦s ♦s ♥tsts q é♣♦ ♦♥ír♠ q

♦s ♥ô♠♥♦s étr♦s ♠♥ét♦s sr♠ ♦♠♣t♠♥t st♥t♦s s q st

t♦ ♥ã♦ é r ♣♦s ♠ ♥s rst♥ rst s♦r q ♦s ♥ô♠♥♦s

étr♦s ♠♥ét♦s stã♦ ♦rt♠♥t ♦s st t♦ ♦ ♦sr♦ r♥t ♠

ss ①♣rê♥s ♠ s rst ♣r q ♦rr♥t étr ♦♥s

♥trrr ♥ ♦r♥tçã♦ ♠ úss♦ ♣ró①♠♦ ♦♥♥♦ q ♦rr♥t étr

♣r♦♦ t♦s ♠♥ét♦s ♠ ss ③♥♥çs

s ♣ss♦s s♥ts ♥♦ ♥ç♦ ♦♠♣r♥sã♦ ♦ ♠♥ts♠♦ ♦r♠ ♦s ♣♦r

r② ♦s♣ ♥r② ♠♣r ♥r ♥③ ♦♠ s♦rt ♥çã♦

♠♥ét r② ♥r② ♦srr♠ q r ♣♦ssí rr ♠ ♦rç tr♦♠♦tr③

trés ♠ ♠♣♦ ♠♥ét♦ rá ♥③ ♦sr♦ q ♥③

r r ♣ ♠♥ç ①♦ ♦ ♠♣♦ ♠♥ét♦ P♦r ♠ ♠s r ①

♦♥s rs♠r t♦♦ ♦ ♦♥♠♥t♦ s♦r ♠♥ts♠♦ tr ♠ qtr♦

qçõs ♦♥s ♦♠♦ qçõs ①

~∇ · ~E =ρ

ε0;

~∇ · ~B = 0;

~∇× ~E = −∂ ~B

∂t;

~∇× ~B = +µ0ε0∂ ~E

∂t+ µ0

~J ;

♠ q ~E é ♦ ♠♣♦ étr♦ ~B é ♦ ♠♣♦ ♠♥ét♦ ρ é ♥s ♦♠étr s

rs étrs ~J é ♥s s♣r ♦rr♥t étr ǫ0 é ♣r♠

étr ♥♦ á♦ µ0 é ♣r♠ ♠♥ét ♥♦ á♦

é♠ st♦ sts qçõs st♠ ss tórs sós q♥t♦ à ①stê♥

s ♦♥s tr♦♠♥éts s qs ss ♦s sã♦ s ♦ ♦r ①♣r♠♥t

♦ ③

♥♠♥tçã♦ tór

♥trçã♦ ♣♦r ♠♥ét

sssã♦ ♥tr♦r ♠♦s q ♦s ♠♥t♦s s♠♣r t♠ ♦s ♣♦♦s st♥t♦s

♥♦rt s ♦ s♦ ♠s s♠♣s sts ♠♥t♦s t♠é♠ ♣♦♠ sr ♥tr♣rt♦s

♦♠♦ s rs ♣♦♥ts ♠♥éts ♠s♠ ♥t♥s s♥s ♦♣♦st♦s s♣

rs ♣♦r ♠ stâ♥ ~d ♦ ♥sr♠♦s ♦ ♠♣♦ ♠♥ét♦ ♠ ♣♦♦ ♠♥ét♦

~m1 r♥s stâ♥s ~r ♠ rçã♦ ~d ♠♦s q ♦ ♠♣♦ ♠♥ét♦ r♦ ♣♦r

st ♣♦♦ ♥ ♦r♠ ♦ sst♠ ♦♦r♥s é ♦ ♣♦r ❬❪

~B1 =3 ( ~m1 · ~r1) ~r1 − ~m1

~r3,

♠ q ~m1 é ♦ ♠♦♠♥t♦ ♠♥ét♦ ♦ ♣♦♦ ~r1 é ♠ t♦r ♥tár♦ r

♣rs♥ç ♦tr♦ ♣♦♦ ♠♥ét♦ ~m2 sr ♠ ♥trçã♦ trés ss

♠♣♦s ♠♥ét♦s ①♣rssã♦ q sr ss ♥trçã♦ é ♣ ♥r

♥trçã♦ E12 = − ~m1. ~B2

E12 = −~m1. ~B2 =µ0

4π

[

~m1 · ~m2 − 3(~m1 · r12) · (~m2.r12)

r312

]

,

♠ q r12 é ♦ t♦r ♥tár♦ stâ♥ ♥trs ♦s ♣♦♦s ~m1 ~m2

ss qçã♦ ♣♦ sr st♥ ♣r ♦ s♦ N ♣♦♦s ♠♥ét♦s ♥tr

♥ts ♠♥r q ♥r t♦t srá srt ♣♦r

Edip =∑

i 6=j

−~mi · ~Bj =∑

i 6=j

µ0

4π

[

~mi · ~mj − 3(~mi · rij) · (~mj · rij)

r3ij

]

.

st s♦♠tór♦ t♦♦s ♦s ♣♦♦s ♠♥ét♦s ♥tr♠ ♦♠ t♦♦s ♦s ♦tr♦s ♣♦♦s

♠♥ét♦s ♦♠ ①çã♦ s ♠s♠♦s i 6= j ❱ q st qçã♦ sr ♣r ♦

st♦ qqr sst♠ ís♦ q ♦♥t♥ ♣♦♦s ♠♥ét♦s ♠ ♠ sst♠

♦♠♣♦st♦ ♣♦r ár♦s ♣♦♦s ♠♥ét♦s ê♥t♦s ~m ♥ q ♠♥♦r stâ♥ ♥tr

sts é ♦ ♣♦r a st qçã♦ ♣♦ sr rsrt ♦♠♦

Edip =∑

i 6=j

µ0

4π

a3

a3

[

~mi · ~mj − 3(~mi · rij) · (~mj · rij)

r3ij

]

♥♠♥tçã♦ tór

Edip =∑

i 6=j

µ0µ2

4πa3

[

~si · ~sj − 3(~si · rij) · (~sj · rij)

(rij/a)3

]

Edip =∑

i 6=j

D

[

~si · ~sj − 3(~si · rij) · (~sj · rij)

R3ij

]

,

♠ q ~m = µ ·~s µ é ♦ ♠♦♦ ♦ ♠♦♠♥t♦ ♠♥ét♦ ~s é rçã♦ ♦ s♥t♦ st

♠♦♠♥t♦ D = µ0µ2/4πa3 Rij = rij/a é ♦ ♦r ♠ ♥s s♣ç♠♥t♦

r st ♥♦♦ s♦♠tór♦ ♣♦s ♦srr q D ♦♥té♠ t♦s s ♥s ♦

♥t♦ s♦♠tór♦ ①♥♦ s ♦trs rás si Rij ♠♥s♦♥s é♠ st♦ ♦

♦r D ♦r♥ ♦r♠ r♥③ ♥r ♥trçã♦ ♥tr ♦s ♣♦♦s

❯♠ t♦ r♦s♦ é q ♥ã♦ ♦ ♦sr ①♣r♠♥t♠♥t ①stê♥ ♠

♣rtí q ♦♥té♠ s rtrísts ♠ r ♠♥ét s♦ ♠ ♦♥tr

♣rt ♠ ís ♠tér ♦♥♥s ♦ ♣r♦♣♦st♦ q ♦s ♠♦♥♦♣♦♦s ♠♥ét♦s

♠r♠ ♠ ♠ ss ♠tr ①ót ♦♥ ♦♠♦ ♦s s♣♥s st

rá r ♠s r ♥♦ ♦rrr s ♣ró①♠s sssõs

Pr♠ ss♣t ♠♥ét

s ♠trs ♣♦♠ ♣rs♥tr r♥ts ♦♠♣♦rt♠♥t♦s q♥♦ sã♦ ①♣♦st♦s

♠ ♠♣♦ ♠♥ét♦ ①tr♥♦ ❬❪ P♦r ①♠♣♦ q♥♦ sts ♠trs sã♦ ♦♦♦s

♣rt♦ ♠ ♦s ♣♦♦s ♠ ♠ã s ♣♦♠ sr trí♦s r♣♦s ♦ ♥ã♦ rr♠

♥♦ ♦s ♠trs sã♦ r♠♥t r♣♦s s sã♦ ss♦s ♦♠♦ ♠♥ét

♦s ♥♦ ♦s ♠trs sã♦ r♠♥t trí♦s sts ♣♦♠ sr ss♦s ♦♠♦

♣r♠♥ét♦s q♥♦ s sã♦ ♦rt♠♥t trí♦s ♦♠♦ rr♦♠♥ét♦s ♣r♦

♣r q rtr③ st ♦♠♣♦rt♠♥t♦ é ♣r♠ ♠♥ét µ(B/H)

♠ ♥s r♦s át♦s ♣rs♥ts t♠é♠ ♦tr♦ t♣♦ ♣r♠ ♠

♠♥s♦♥ st ♣r♠ é ♦♥ ♦♠♦ ♣r♠ rt é

♣ r③ã♦ ♣r♠ ♦ ♠tr s♦r ♣r♠ ♦ á♦ ♦ s

µr = µ/µ0 ♠ q µ0 é ♣r♠ ♦ á♦ µ0 = 4π ·10−7B/H st ♠♥r

t♠♦s q s ♣r♠s rts ♦s ♠♥t♦s sã♦ r♠♥t ♠♥♦rs ♦

q 1 ♦s ♣r♠♥t♦s sã♦ r♠♥t ♠♦rs ♦ q 1 ♦s rr♦♠♥t♦s sã♦

♠t♦ ♠♦rs ♦ q 1

♥♠♥tçã♦ tór

s ♠térs ♠rs♦s ♠ ♠ ♠♣♦ ♠♥ét♦ ♣♦♠ ♣rs♥tr ♠ ♠♥t

③çã♦ ~M ss ♠♥t③çã♦ é srt ♠t♠t♠♥t ♣♦r

~M = (µr − 1) · ~B = xm · ~B,

♠ q xm é ss♣t ♠♥ét ♦♠♦ ♣♦s ♣rr ss♣t

♠♥ét stá ♥t♠♠♥t à ♣r♠ ♠♥ét st ♦r

♥t♥s ♦♠ q ♠ ♠tr ♣♦ sr ♠♥t③♦ ❱ q ♥s ♠tr

s ♦♠♦ ♦s ♣r♠♥t♦s rr♦♠♥t♦s ♣r♦③♠ ♠ ♠♥t③çã♦ ♦r ♦

♠♣♦ ♠♥ét♦ ♥q♥t♦ ♦tr♦s ♠trs ♣r♦③♠ ♠ ♠♥t③çã♦ ♦♥trr

♦ ♠♣♦ ♠♥ét♦ ♦♠♦ ♦s ♠♥ét♦s t 2.1 ♠♦str ssçã♦ ♦s

♠trs t♦s ♥st sssã♦ trés ss ♣r♠s ss ss♣ts

♠♥éts

ssçõs ♦s ♠trs ♠♥ét♦s

trs Pr♠ ♠♥ét s♣t ♠♥ét♠♥t♦s > 1 > 0Pr♠♥t♦s < 1 < 0rr♦♠♥t♦s << 1 << 0

strs ♠♥ét

❯♠ t♦ ♥trss♥t s♦r ♥s ♠trs rr♦♠♥ét♦s é q q♥♦ s

♣ ♠ ♠♣♦ ♠♥ét♦ ①tr♥♦ rs♥t ♥sts té q s ♠♥t③çã♦ s str

♣♦s ♠♥s st ♠♣♦ té q st s ♥♦ ♦srs q ♠♥t③çã♦

sts ♥ã♦ ♠♥ tã♦ r♣♠♥t q♥t♦ ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ q♥♦ ♦

♠♣♦ ①tr♥♦ ♦r ♥♦ rá ♥ ♠ ♠♥t③çã♦ r♠♥s♥t ♥st ♠trs

Pr q ♠♥t③çã♦ sts ♠trs s ♥ é ♥ssár♦ ♣r ♠ ♠♣♦

♠♥ét♦ ♥ rçã♦ ♦♥trár ♦ ♣♦ ♥♠♥t st ♠♣♦ é ♦♥♦

♦♠♦ ♦rç♠♣♦ ♦r♦ ♦♥t♥r ♠♥tr ♦ ♠♣♦ ♥ rçã♦ ♦♣♦st

♠♥t③çã♦ ♠♥t③çã♦ st ♠tr rá ♥rtr ♦♠♣t♠♥t st

♣r♦ss♦ ♣♦ sr ♥♦♠♥t r♣t♦ té q ♠♥t③çã♦ s ♥rt ♥♦♠♥t

♥♠♥tçã♦ tór

r 2.1 ♠♦str ♠♥t③çã♦ ♠ ♠tr rr♦♠♥ét♦ ♥♦ ♣r♦ss♦ srt♦

♠

r r strs ♠ ♠tr rr♦♠♥ét♦

st ♥ô♠♥♦ é ♦♥♦ ♦♠♦ strs ♠♥ét ♦ ♦ q sr

r ♠♥t③çã♦ ♦♠♦ ♠♦str♦ ♥ r 2.1 é ♦♥♦ ♦♠♦ r

strs

♥s♦tr♦♣ ♠♥ét

③♠♦s q ♠ ♦r♣♦ é ♥s♦tró♣♦ q♥♦ ♣♦ ♠♥♦s ♠ ss ♣r♦

♣rs íss t♠ ♠ t♥ê♥ r♦♥ ❬❪ P♦r ①♠♣♦ q♥♦ ♠s

sst ♠♥ét ♠ ♠tr ês q ♠ rçã♦ st é ♣r

③♠♦s q st ♠tr t♠ ♠ ♥s♦tr♦♣ sst ♠♥ét ❱ q

♦ ♠♥ts♠♦ ♥♦s ♠trs ♠♥ét♦s ♣♥ ár♦s t♦rs ♥tr s t♠♦s

♥t♥s ♦s ♠♦♠♥t♦s ♠♥ét♦s stâ♥ ♥tr sts ♠♦♠♥t♦s

s♠tr r s ♥trçõs ♣rs♥ts P♦s ♣rr q s ♣r♦♣r

s sts ♠térs sã♦ ♦rt♠♥t ♣♥♥ts ♦ s t♠♥♦ s ♦r♠ ♦ q

♣♦ rr ♠ ♥s♦tr♦♣ ♠♥ét ①st♠ três t♣♦s ♣r♥♣s ♥s♦tr♦♣s

♠♥éts ♥s♦tr♦♣ ♠♥t♦rst♥ ♥s♦tr♦♣ ♠♥t♦strt t♠é♠ ♦

♥ ♠♥t♦ást ♥s♦tr♦♣ ♠♥t♦stát ♦ ♦r♠

♥s♦tr♦♣ ♠♥t♦rst♥ ♦♦rr ♣r♥♣♠♥t ♦ ♦ ♦♣♠♥t♦

s♣♥órt st ♦♣♠♥t♦ ♥③ ♠ rçã♦ ♣rr♥ ♦s s♣♥s q t♥t

♥♠♥tçã♦ tór

♦r♥tá♦s ♠ ♠ ♠s♠ rçã♦ ♦ à s♠tr r ♥trçã♦ tr♦

♥tr ♦s s♣♥s é t ③♥♦ ♦♠ q ①st♠ ①♦s ♣rr♥s ♠♥t③

çã♦ t♥♦ ss♠ ♠ ♥s♦tr♦♣ ♠♥t♦rst♥ st ♣rrê♥ ①♦s stá

ss♦ ♠ ♥r ♥s♦tr♦♣ q é ♠♥♠③ q♥♦ ♦s ♠♦♠♥t♦s

♠♥ét♦s s ♦r♥t♠ ♠ ts ①♦s ♥♦♠♥♦s ①♦s ás ♠♥t③çã♦

♥s♦tr♦♣ ♠♥t♦strt ♦♦rr ♦ ♠♥ç s ♠♥sõs ♠

rst q♥♦ st é ①♣♦st♦ ♠ ♠♣♦ ♠♥ét♦ ①tr♥♦ s q ♥r

♥trçã♦ ♥tr ♦s ♠♦♠♥t♦s ♠♥ét♦s ♣♥ stâ♥ ♥trs sts ss

rs♣ts ♦r♥tçõs ♥♦ ♣s ♠ ♠♣♦ ♠♥ét♦ ♥ r ♦rç♥♦ ♦s

♠♦♠♥t♦s ♠♥ét♦s ♦tr♠ ♠ ♥♦ ♦r♥tçã♦ ♥r ♥trçã♦ ♠♥t

s stâ♥s ♥tr ♦s ♠♦♠♥t♦s ♠♥ét♦s s ♠♦♠ ♣r q ♥r t♦t

♦ sst♠ s r③ st♦ ♣r♦♦ t♥sõs ♥ r q rst♠ ♠ ♠♥çs ♥

♦r♠ st st ♥ô♠♥♦ é ♦♥♦ ♦♠♦ ♥s♦tr♦♣ ♠♥t♦strt s♥♦

st ♣♦st♦ q♥♦ r s ♦♥ ♥ rçã♦ ♠♥t③çã♦ ♦ ♥t q♥♦

st ♠♥ ♥ rçã♦ ♠♥t③çã♦

❯♠ t♦r ♠♣♦rt♥t ♥s♦tr♦♣ ♠♥ét sã♦ s ♠♥sõs ♦s ♠trs

s rã♦s ♠♥ét♦s ♦♠ ♠♥sõs ♠t♦ ♣q♥s ♠♥♦rs q ♠ µm s ♠

♥t③♠ ♦ ♦♥♦s ♦s ①♦s ás ♠♥t③çã♦ s♥♦ ♣rtr♦s s♦♠♥t ♣

tçã♦ tér♠ ♦ st♦ ♦s ♠♦♠♥t♦s ♠♥ét♦s ♦s át♦♠♦s sts rã♦s s

♥♠ ♠ ♠ ú♥ rçã♦ ♣rs♥t♠ ♠ ♠♥t③çã♦ ♥♦r♠ ♣♦♥♦s

③r q s ♣♦ss♠ ♠♦♥♦♦♠í♥♦s s♦ ♦♦rr ♠ ♠♥ç ♥ rçã♦ ♦ ♠♦

♠♥t♦ ♠♥ét♦ t♦♦s ♦s át♦♠♦s q ♦♠♣õ stá ♣rtí rrã♦ s♠t♥♠♥t

♣♦s stã♦ r♠♥t ♥♦s ♦ à ♠ ♥trçã♦ tr♦ s rã♦s ♠♥ét♦s

♦♠ ♠♦rs ♠♥sõs ♣rs♥t♠ ♦s ♦ ♠s ♦♠í♥♦s ♠♥ét♦s ♥tr♦

♦♠í♥♦ ♦s ♠♦♠♥t♦s ♠♥ét♦s ♣♦♥t♠ ♠ ♠ ♠s♠ rçã♦ ♠s ♥ã♦ ♥ss

r♠♥t r ♠ ♥♠♥t♦ ♥tr ♦ ♠♦♠♥t♦ ♠♥ét♦ ♠ ♦♠í♥♦ ♦♠ ♦

♦tr♦ ③♥♦ ♦♠ q ♥s rã♦s ♦♠♦ t♦♦ str♠ s♠♥t③♦s ♠ ♠

s ♠r♦só♣ st t♦ ♦♦rr ♦ ♥trçã♦ ♣♦r ♦♥♦ ♥ ♥

tr ♦s ♠♦♠♥t♦s ♠♥ét♦s ♦♠í♥♦ ♣r♦③r♠ ♠ ♥r ♠♥t♦stát

♦ s♠♥t③♥t q ♣♦r s♣rr s ♥r ♠♥t♦rst♥ ♠♥t♦str

♥♠♥tçã♦ tór

t st ♠♥r ♠♥t③çã♦ ♠ ♠tr ♠♥ét♦ t♠é♠ ♣♥rá ♦

rr♥♦ ♦s rã♦s ♠♥ét♦s s ♣ró♣r ♦r♠ st ♥ô♠♥♦ é ♦♥♦ ♦♠♦

♥s♦tr♦♣ ♠♥t♦stát ♦ ♥s♦tr♦♣ ♦r♠

♦♠í♥♦ ♠♥ét♦ ♠t s♣r♣r♠♥ét♦

s sssõs ♥tr♦rs ♦ ♣rs♥t♦ q ♠♥t③çã♦ ♠

♣rtí ♣♥ árs rtrísts ♦♠♦ s ♦r♠ ♦ s t♠♥♦ s ♣r♦

♣rs ♦s ♠♥t♦s qí♠♦s q ♦ ♦♠♣õ P♦r ①♠♣♦ ♦s rr♦♠♥t♦s ♣♦♠

♣rs♥tr ♠ ♠♥t③çã♦ s♣♦♥tâ♥ ♠s♠♦ ♥ sê♥ ♠♣♦ ♠♥ét♦ ①

tr♥♦ ❱♠♦s q ♣♥♥♦ s ♠♥sõs sts rr♦♠♥t♦s st ♠♥t③çã♦

♣♦ sr ♣♦r ♠ ú♥♦ ♦♠í♥♦ ♠♥ét♦ ♦ ♣♦r ár♦s ♦♠í♥♦s ♠♥ét♦s

st sssã♦ ①♣♦r ♥ê♥ ♠♥sã♦ ♠ ♣rtí ♥ ♠♥t③çã♦ st

♥♠♥t ♠♦s ♦♥srr ♠ ♣rtí ♥ q ♦s ♠♦♠♥t♦s ♠♥ét♦s

♦s át♦♠♦s q ♦♠♣õ s ♠♦♠ ♦r♥t♠♥t st ♠♦♦ ♦ ♠♦♠♥t♦ ♠♥ét♦

t♦t ♣rtí srá ♦ ♣♦r µ = µat ·N ♠ q µat é ♦ ♠♦♠♥t♦ ♠♥ét♦ tô♠♦

N é ♦ ♥♠r♦ át♦♠♦s ♠♥ét♦s st ♣rtí ❬❪ ♦ s♦ ♠s s♠♣s ♦

♠♦♠♥t♦ ♠♥ét♦ st ♣rtí é tr♠♥♦ ♣ ♥s♦tr♦♣ ♣rtí ♣♦

♠♣♦ ♠♥ét♦ ①tr♥♦ ❯♠ t♦ ♥trss♥t s♦r s ♣rtís ♠♥éts é q

sts ♣♦♠ ♠r s ♠♥t③çã♦ ♥tr♥s♠♥t ♦ s ♦ ♠ t♠♣♦

r①çã♦ rtríst♦ τ ♦ ♠♦♠♥t♦ ♠♥ét♦ sts ♣rtís ♠♠ ♠

st♦ qír♦ ♣r ♦tr♦ t♠♣♦ r①çã♦ τ ♣♥♥t rqê♥

t♥tts ♠♥çs ♦r♠ ν0 = 1010Hz ♦ t♦r ♦t③♠♥♥ e+E

kbT ♦

s

τ = τ0 · e+E

kbT ,

♠ q τ0 = 1/ν0 E é rrr ♥r q s♣r ♦s ♦s st♦s qír♦

q é ♦ ♣♦ ♦♠ ♣rtís V ③s ♥s ♥r ♥s♦tr♦♣

ka ♣rtí T é t♠♣rtr ♣rtí kb é ♦♥st♥t ♦t③♠♥♥

rqê♥ st♦s ♠ st♦ qír♦ ♦tr♦ ♣♦ sr ♠♥t

♥♠♥tçã♦ tór

trés ♦ ♥rs♦ ♦ t♠♣♦ r①çã♦ τ

ν = ν0 · e−E

kbT .

trés s qçõs 2.12 2.13 ♣♦s ♦♥r q ♣r ①s t♠♣rtrs

♦ r♥s ♦♠s kbT ≪ E ♦ t♠♣♦ r①çã♦ τ é ♠t♦ r♥ ♠♥r

q ♣rtí ♠♥té♠ ♦ s ♠♦♠♥t♦ ♠♥ét♦ ♠ ♠ st♦ qír♦ ♣r

ts t♠♣rtrs ♦ ♣r ♣q♥♦s ♦♠s kbT ≫ E ♦ t♠♣♦ r①çã♦ τ

t♥ sr ♠t♦ ♣q♥♦ ♦ ♠♦♠♥t♦ ♠♥ét♦ st ♣rtí ♠ r♣♠♥t

♠♥ç ♦ ♠♦♠♥t♦ ♠♥ét♦ ♠ ♣rtí ♣♦ sr ♦sr♦ ♣♦r r♥ts

♥str♠♥t♦s ♠ ♦ t♠♣♦ q ♣r♦ ♣r ♦tr ss ♠ é

♦♥♦ ♦♠♦ t♠♣♦ rtríst♦ tm ♥♦ ♦ t♠♣♦ rtríst♦ ♠

çã♦ ♠ ♣r♦ tm é ♠t♦ ♠♦r ♦ q ♦ t♠♣♦ r①çã♦ τ ♥ã♦ é ♣♦ssí

♠r ♦ ♠♦♠♥t♦ ♠♥ét♦ ♠ ♠ ú♥♦ st♦ qír♦ st ♣rtí é ss

♥tã♦ ♦♠♦ s♣r♣r♠♥ét ♥♦ ♦ t♠♣♦ rtríst♦ ♦ ♣r♦ tm

é ♠t♦ ♠♥♦r ♦ q ♦ t♠♣♦ r①çã♦ τ ♣r♥t♥♦ q ♦ ♠♦♠♥t♦ ♠♥ét♦

♣rtí stá ♣rs♦ ♠ ♠ ú♥♦ st♦ qír♦ é t♦ q ♣rtí é ♦

q ♠♣♦rt♥t ♥t③r q rtríst s♣r♣r♠♥ts♠♦ ♠

♣rtí ♣♥ ①s♠♥t ♦ ♣r♦ ♦srçã♦ P♦r ①♠♣♦ ♥q♥t♦

♠ ♣rtí ♣♦ sr s♣r♣r♠♥ét ♠ ♠ ♠ ♠♥ét ♦♥♥♦♥

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

s♣tr♦s♦♣ ♦ssr q ♠♦r ♣♦r ♦t tm = 10−8s

é♠ ♦ t♠♣♦ r①çã♦ τ ♠♥sã♦ s ♣rtís ♠♥éts tr♠♥

♦ ♥ú♠r♦ ♦♠í♥♦s ♠♥ét♦s q st ♣♦ tr ❱♠♦s q ♣rtís ♦♠ r♥s

♠♥sõs ♣♦ss♠ ár♦s ó♠♥♦s ♠♥ét♦s ♠t♦♠í♥♦s q ♥ã♦ ♥ssr

♠♥t ♣♦♥tr♠ ♠ ♠ ♠s♠ rçã♦ s ♣rtís ♣q♥s ♠♥sõs ♣♦ss♠

♠ ú♥♦ ♦♠í♥♦ ♠♦♥♦♦♠í♥♦ ♣♦♥♦ st sr ♦q♦ ♦ s♣r♣r♠♥é

t♦ é♠ sts três r♥ts st♦s ♠♥ét♦s q ♣♥ ♣r♥♣♠♥t

ss ♠♥sõs s ♦r♠ ♦s át♦♠♦s s ♦♠♣♦sçã♦ ①st ♠ ♦tr♦ st♦

♠♥ét♦ ♥tr ♦s st♦s ♠♦♥♦♦♠í♥♦ ♦q♦ ♦ ♠t♦♠í♥♦ ♦ órt st

st♦ ♥tr♠ár♦ ♦s ♠♦♠♥t♦s ♠♥ét♦s ♦s át♦♠♦s s ♦ ♦r♠ rr ♦r

♥♠♥tçã♦ tór

♠♥♦ ♠ órt ♠♥ét♦ ①♠♣♦ ♠s s♠♣s ♣r st st♦ sr ♠

♣q♥♦ s♦ ♠♥ét♦ st s♦ ♠♥ét♦ ♠♥t③çã♦ st sr rr

str ♦ ♦♥♦ ♣♥♦ ♦ s♦ ♠ s ♥tr♦ ♦s ♠♦♠♥t♦s ♠♥ét♦s ♦s át♦♠♦s

♣♦♥t♠ ♣r ♦r ♦ ♣♥♦ ♦ s♦ st t♦ ♣♦ sr ♦sr♦ ♠ ♠ sst♠

s♣♥s q ♥tr♠ ♦♠ ♠ ♥trçã♦ tr♦ q t♠ ss ♦r♥tçõs rs

♦♠♦ ♦sr♦ ♣♦r t ❬❪

♦s s♣♥s

❯♠ rtríst ♥trss♥t ♥s sst♠s ♦♠ ♥trçõs ♦♠♣tts

é q ♦s ss st♦s ♥♠♥ts ♠ r sã♦ ♥r♦s ♣rs♥t♥♦ ♠ rs

trçã♦ ♦♠étr s♠♥t ♠ ♠tr é ♦♠tr♠♥t rstr♦ q♥♦

♣♦ss Ω0 st♦s ♥♠♥ts ♥r♦s r♥♦ ♠ ♥tr♦♣ rs ♦ st♦

♥♠♥t S = kBln(Ω0) st♦ ♦♦rr ♣♦s ♦ rr♥♦ s♣♥s ♠ ♠ r

♠♣ q t♦s s ♥trçõs s♠ ststs ♦ ♠s♠♦ t♠♣♦ ①♠♣♦ ♠s

♦♥♦ rstrçã♦ ♦♠étr é ♠ r tr♥r s♣♥s t♣♦ s♥ ♦

♣♦s ♥trr♦♠♥t♠♥t ♣r♦♣♦st♦ ♣♦r ❲♥♥r t ❬❪ ♠ r

♠ sst♠s ♥trr♦♠♥ét♦s ♥trçã♦ ♠♥♦r ♥r é ♣r ♦s s♣♥s

q ♥♠s ♥t♣r♠♥t s ♦ str r ♣r♦♣♦st ♣♦r ❲♥♥r t

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

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

st t♣♦ sst♠ rr ss r♥ts ♦♥rçõs ♠♥♦r ♥r ♦ q r

♠ ♥tr♦♣ ♣♦r st♦ ♣♦r S ≈ 0.366kB

r tr♥r s♣♥s t♣♦ s♥ ♦♣♦s ♥trr♦♠♥t♠♥t ❱q ①st♠ ♦♥rçõs r♥ts q ♣♦ss♠ ♠♥♦r ♥r ♦ sst♠

♥♠♥tçã♦ tór

❯♠ t♦ ♥trss♥t é q st ♥ã♦ ♦ ♦ ♣r♠r♦ sst♠ st♦ ♣rs♥tr

rstrçã♦ ♦♠étr ♠ ❲♠ ♥q t ❬❪ ❬❪ ♦srr♠ q ♦ ♦

á ♣rs♥t ♠ ♥tr♦♣ rs st rst♦ ♦ ♦♦ sr♦ ♣♦r ♥s

P♥ ❬❪ t P♥ t ♦srr♠ q ♥♦ ♦ á í♦♥ O2− st

♥♦ ♥tr♦ ♠ ttrr♦ st r♦ ♣♦r ♦tr♦s qtr♦ í♦♥s O2− ♦③♦s

♥♦s érts ♦ ttrr♦ ♣♦r qtr♦ ♣rót♦♥s H+ ♦s ♣ró①♠♦s ♦s st♦s

♦ ♦ ♣♦s♦♥♠♥t♦ ♦s ♣rót♦♥s H+ ♦s ♣rt♦ ♦s st♦s ♦ át♦♠♦ O2−

♥tr st ♦♠tr ♣♦ sr r♦♥ ♦♠ ♠♦♠♥t♦s ♣♦♦s ♠♥ét♦s t♣♦

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

♣r ♦r ttrr♦ st ♦♥rçã♦ ♥♦♥tr ♥♦ ♦ á s♠♣s

♣♦ sr srt ♣ rr ♦ ♦ q st q rá s♠♣r ♦s ♠♦♠♥t♦s

♠♥ét♦s ♣♦♥t♥♦ ♣r ♥tr♦ ♦s ♣♦♥t♥♦ ♣r ♦r r 2.3 é ♠

♠♦♦ strt♦ ♦♥rçã♦ ♦ ♦ á s ♥tr♣rtçã♦ ♦♠ ♠♦♠♥t♦s

♣♦♦

r s r♥s srs r♥s sã♦ ♦s í♦♥s O2− s ♣q♥s srs ♣rts sã♦♦s ♣rót♦♥s H+

♥t♠♥t rrs t ❬❪ s♦rr♠ ♠ ss ♠trs ♠♥ét

♦s ♦♠ strtr rst♥ s♠♥t ♦ ♦ á sts ♠trs ♣♦r ①♠♣♦

Ho2T i2O7 Dy2T i2O7 ♣rs♥t♠ ♠ strtr t♣♦ ♣r♦♦r♦ ♠ r tr♠♥

s♦♥ ♦♠♣♦st ♣♦r ttrr♦s ♦♠ érts ♠ ♦♠♠ st r ♦s ♠♦♠♥t♦s

♠♥ét♦s stã♦ ♦③♦s ♥♦s érts ttrr♦s ♠ ♠ rt ① t♠

♣rtr ♦s ♠♦♠♥t♦s ♠♥ét♦s s ♥♠ ♦♠ ♦ ♥tr♦ sts ♦ st t♣♦

strtr sts ♠trs ♣rs♥t♠ ♠ s♦r♠ ♦♥rçã♦ ♦r♥tçõs

♦s ♠♦♠♥t♦s ♠♥ét♦s s♠r ♦s ♥♦♥tr♦s ♥♦ ♦ á rr ♦ ♦

♦s ♠♦♠♥t♦s ♠♥ét♦s ♣♦♥t♥♦ ♣r ♥tr♦ ♦s ♣r ♦r ttrr♦

♥♠♥tçã♦ tór

s ♠♥st ♦♠ ♠ ♠♥♠③çã♦ ♥trçã♦ ♥r s♣♥s♣♥ ♦ st

r♥ s♠♥ç sts ♠térs ♦r♠ ♥♦♠♦s ♦♠♦ ♦s s♣♥ t♦s ♣s

qs♦rs ❬❪ ❬❪ ❬❪ ❬❪ ❬❪ ❬❪ ❬❪ ❬❪ ❬❪ t♥t♦ tór♦s q♥t♦ ①♣r♠♥ts

①♣♦rr♠ ís sts ♠trs ♠ rs♦s tr♦s ♥tr s ♦s tr♠

r♥ r♣rssã♦

♣r♠r♦ tr♦ sr t♦ ♦ t♦ ♣♦r st♥♦♦ ♦♦r♦rs ❬❪

sts t♦rs ♥tr♣rtr♠ q s ①tçõs ♦ st♦ ♥♠♥t ♦s ♦s s♣♥s

♣♦r♠ sr st♦ ♦♠♦ qs♣rtírs ♠r♥ts q s ♦♠♣♦rt♠ ♦♠♦ ♠♦♥♦♣♦

♦s ♠♥ét♦s s ♦srr♠ q ♣rs ♠♦♥♦♣♦♦s ♠♥ét♦s rs ♦♣♦sts

♣rr♠ ♣ ♦çã♦ rr ♦ ♦ ♠ ①s t♠♣rtrs ♥trçã♦ ♥tr

s rs ♠♥éts ♥ã♦ r tr♠♥ ♣♦ st♦ ♥rét♦ ♥rsã♦ ♦s s♣♥s

♠s s♠ ♣ ♥trçã♦ ♦♦♠♥ ♠♥ét ♠ ás ♠♦♥♦♣♦♦s ♥tr♥

ts ♠ ts t♠♣rtrs ♦s ♦s s♣♥s s ♦♠♣♦rt♠ ♦♠♦ ♠ ♣r♠♥t♦

♦♥♥♦♥ srçã♦ s ①tçõs ♦♠♦ ♠♦♥♦♣♦♦s ♥ã♦ ♣r♦ ♦s ♦s

s♣♥s ♦s ♠♦♥♦♣♦♦s rs ♦♣♦sts sã♦ ♦s ♣♦r ♠ str♥ s♠ t♥sã♦

♦♠♣♦st ♣♦s s♣♥s ♥rt♦s q ♠ sts ♦s ♠♦♥♦♣♦♦s ❯♠ t♦ ♥trss♥t

é q sss ♠♦♥♦♣♦♦s ♠♥ét♦s ♥ã♦ sr♠ qs ♣rst♦s ♣♦r r ❬❪ ❬❪

♣♦s ♥ t♦r r str♥ q ♦s ♦s ♠♦♥♦♣♦♦s é ♥♥ts♠♠♥t ♥

♠ t♦ s♦♥♦ ♥ã♦♦srá q ♥s ①♦ ♠♣♦ ♠♥ét♦

♠ r ♦tr ♦s ♦s s♣♥s str♥ é r ♦srá ♣♦ ♠r

s ♦♠♣r♠♥t♦ s ♦r♠ s♠ ♥♥♠ st♦ ♥r q ♥ã♦ s ♥trçã♦

♦♠♥ ♠♥ét s rs ♠ ss ①tr♠s ♦ st t♦ r

♠♥ét ♦ ♦ s♣♥s ♥ã♦ st r♦♥ ♦♠ r étr ♣ qçã♦

qD = h/µ0e ♦♠♦ ♣rst♦ ♣♦r r ♠ q qD é r ♠♥ét e é r

étr r 2.4 str s qs♣rtírs ♠r♥ts ♥♦ sst♠ s♥♦ s

♣ str♥

♥♠♥tçã♦ tór

r ♦♥♦♣♦♦s ♠♥ét♦s s srs r♠ ③ ♠r♥ts ♥ r ♦s s♣♥ ♦t q ♥ r q ♦s ♦s ♠♦♥♦♣♦♦s é str♥ ❬❪

s♥♦ tr♦ sr t♦ ♦ t♦ ♣♦r ❲♥ t ❬❪ trés r

çã♦ t♦rá sts t♦rs rr♠ ♠ ♦rtór♦ rs ♠♥s♦♥s ♥♥♦♠

♥t♦s ♦♠ ♣r♦♣rs s♠rs ♦s ♦s s♣♥ sts rs r♠ ♦♠♣♦sts ♣♦r

♥♥♦s ♦♥s ♣r♠♦② s Fe20Ni80 ♥♦ q ss ①♦s ♠s ♦♥♦s

tr♠ ss ♦r♥tçõs ♦ ♦♥♦ ♦s ♦s ♣r♥♣s ①♦s r s♣ç♠♥t♦

♥tr s s ♠s ♣ró①♠s t♠é♠ ♦♥♦ ♦♠♦ s♣ç♠♥t♦ r r♠

♥tr 320nm 880nm s ♠♥sõs sts s r♠ ①s 80nm×220nm×25nm

♦ sts ♠♥sõs s ♦♥rçõs ♠♥éts s s r♠ stás ♠

t♠♣rtr 300K s ♦♠♣♦rt♠s ♦♠♦ ♣♦♦s t♣♦ s♥ ♦ s s s

t♥♠ ♠ ♦♠í♥♦ ♠♥ét♦ s♠♣s ♦ ♦ ♠♦♠♥t♦ ♠♥ét♦ ♣♦r ♦tr ♦s

♦rs st♥t♦s ♠s♠♦ ♠♦♦ ♦ ♦♥♦ ♦ ♠♦r ①♦ sts ♠♦♠♥t♦s

♠♥ét♦s r♠ ♣r♦①♠♠♥t 3 × 107 ♠♥t♦♥s ♦r ♦ ♠♣♦ ♠♥ét♦

r♦ ♣♦r s ♥♦ ♥tr♦ s s ♠s ♣ró①♠s r♠ ♦r♠ 10Oe r♥♦

♠ ♥r ♥trçã♦ ♥tr sss s ♦r♠ 10−19J q♥t 104K

r 2.5 ♠♦str ♠ ♠♠ strt r ♣r♦♣♦st ♣♦r ❲♥ ♠♥s

t♦♠ ♦r r♦s♦♣② ♥t ♦r r♦s♦♣② r ♦♥strí

♣♦r

♥♠♥tçã♦ tór

r ♠♠ ♠♦str r rstr ♣r♦♣♦st ♣♦r ❲♥ ♦♠ s ♣♦sçõss s ♣♦ssís ♦r♥tçõs ♦s ♠♦♠♥t♦s ♠♥ét♦s sts s ♠♥s sã♦rs♣t♠♥t s ♠♥s r rstr r ♣♦r ❲♥ ♠♠ s rõs ♣rts r♥s r♣rs♥t♠ ♦s ♣♦♦s ♥♦rt s ❬❪

trés sts rs ♦s t♦rs str♠ ♠ r♥ t rstrçã♦ ♣r

s♥t ♥♦ sst♠ s♣rr♠ s ♦♥rçõs érts ♠ r♥ts r♣♦s t♦♣♦

ó♦s r 2.6

r st r ♠♦str s ③sss ♣♦ssís ♦♥rçõs érts ♣rs♥ts ♥♦sst♠ ♣r♦ ♥♦♥trr ♠ s♣ ♦♥rçã♦ ♠ ♠ r ♦♥rçã♦ tór é♠ st♦ sts ♦♥rçõs stã♦ ♦r♥s ♥rt♠♥t s értsTypeI sã♦ s ♦♥rçõs ♠♥♦r ♥r s ♦♥rçõs TypeIV sã♦ s ♠♦r♥r ❬❪

sr ♥ r 2.6 q s♦♠♥t ♦s érts TypeI TypeII ♦♠ rr

♦ ♦ ♦s s♣♥s ♣♦♥t♥♦ ♣r ♥tr♦ ♦s s♣♥s ♣♦♥t♥♦ ♣r ♦r ♦ ért

trés st r ♦♥s q ♠ ♠ r ♦s s♣♥s ♣♦♥t♠ ♠ rçõs

tórs ♣r♦ ♥♦♥trr érts q ♦r♠ rr ♦ ♦ sr

s♦♠♥t 37.5% s ❲♥ t ♦srr♠ ♠ ss ①♣r♠♥t♦s q ♣r♦♣♦rçã♦

érts q ♦♠ rr ♦ ♦ r ♣r♦①♠♠♥t 70% ♠s 30% ♦

q ♦ ♦r s♣r♦ sr♦ ♥ q st ♦r rs ♠♦♥♦t♦♥♠♥t ♦♠ ♦

♠♥t♦ ♦ s♣ç♠♥t♦ r st♦ ♥ ♣rs♥ç ♥trçõs q ♦r♠ ♦

♥♠♥tçã♦ tór

♣r♠♥t♦ rr ♦ ♦ ③♥♦ ♦♠ q st sst♠ s ss♠ ♠

♦r♠ ♦s ♦s s♣♥s rst♥♦s

❯♠ t♦r ♥trss♥t ♥ ♣r♦çã♦ ♦s s♣♥s rts ♠ ♦rtór♦

é q s s r sã♦ ♠♣rts ❯♠ st♦ ♠s t♦ t♦ ♣♦r ♦ t

❬❪ ♠♦str♦ q s s ♣♦♠ tr rss ♠♣rçõs ♦♠♦ ♦r♠ ♠

ss strtrs ♠♣rsã♦ ♠ ss ♣♦s♦♥♠♥t♦s ♠ ss ♥♠♥t♦s r

♠♦str s ①♣tts r s rs ♦s s♣♥s ♣r♦③s ♠

♦rtór♦

r ♠♠ ♠ r r ♣r♦③ ♠ ♦rtór♦ ♠ r ❬❪

♦ sts ♦r♠s s ♣r♦♣rs íss sã♦ ♠ ♣♦♦

r♥ts ♠s ♠ rçã♦ s ♦trs P♦r ①♠♣♦ P♦r t ❬❪ ♦srr♠

q ♥t♥s ♦ ♠♣♦ ♠♥ét♦ ♥ssár♦ ♣r ♥rtr ♦ ♠♦♠♥t♦ ♠♥ét♦

♣♦r ♦s sít♦s r ♥ã♦ r ♣r t♦s s s s t♦rs ♦srr♠

①♣r♠♥t♠♥t q ♥t♥s ♠é ♦ ♠♣♦ ♠♥ét♦ ♣r ♦♥sr ♥

rtr ♦ ♠♦♠♥t♦ ♠♥ét♦ s s é b = 320Oe ♦♠ ♠ s♦r♠ ss♥

b = ±60Oe s ♣s ♦r♠s s s

♦s s♣♥s rts ♠ s♠çõs ♦♠♣t♦♥s

s ♦s s♣♥s rts trír♠ t♥çã♦ ár♦s ♣sqs♦rs ♦

à ①stê♥ ①tçõs ♦ts q s ♦♠♣♦rt♠ ♦♠♦ qs♣rtís s♠

♥ts ♠♦♥♦♣♦♦s ♠♥ét♦s ♠ s♠çõs ♦♠♣t♦♥s é rt♠♥t á

r♣r♦③r ♠ r rt ♦s s♣♥s ♠ qsqr ♦♥rçã♦ ♦♥çã♦

s t♥♦ ss♠ ♦s st♦s sts rs

♥♠♥tçã♦ tór

❯♠ st♦ ♥trss♥t t♦ ♣♦r ❲②s♥ t ❬❪ ♠♦strr♠ q ♥rsã♦ ♦

♠♦♠♥t♦ ♠♥ét♦ ♠ ♥♥♦♠♥t♦ ♣s♦ trés ♠ ♠♣♦ ♠♥ét♦

♣♥ ♦rt♠♥t ♥t♥s ♦ ♠♣♦ r♠♥t rçã♦ s♥t♦ st

♠♣♦ s t♦rs ♦srr♠ q ♦s ♦rs ♥t♥s ♦ ♠♣♦ ♠♥ét♦ ♣r

♥rtr ♦ ♠♦♠♥t♦ ♠♥ét♦ ♦ ♥♥♦♠♥t♦ r♠ ♠t♦s ♣ró①♠♦s ♣rt♠♥t

♥♣♥♥t♠♥t ♦ â♥♦ ♥♦ q r ♣♦ s q st ♦ss ♦♥trár♦

♦r♥tçã♦ ♦ ♠♣♦ r♦ ♣♦ ♥♥♦♠♥t♦ r 2.8 ♠♦str ♦ rá♦

árs rs strs ss ♥♥♦♠♥t♦ Pr q s rs strs ♣r

r♥ts ♦r♥tçõs ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ s ♠ ♣r♦①♠♠♥t ♣r

♠ ♠s♠♦ ♦r ♦ ♠♣♦

r ♠♠ ♠♦str r strs ♣r r♥ts â♥♦s ♠ rçã♦ ♦①♦ x ♥♥♦ r ❬❪

❯♠ ♦tr♦ st♦ s♦r ♦s ♦s s♣♥s rts s♥♦ s♠çã♦ ♦♠♣t

♦♥ ♥s♦ ♥r ♥trçã♦ ①tçõs s♦r ♦ st♦ ♥♠♥t ♠

♦ s♣♥ rt ♥ r qr ó t ❬❪ ♦srr♠ q s ①tçõs

♠s ① ♥r ♣♦♠ sr sts ♦♠♦ ♠ ♣r ♠♦♥♦♣♦♦s ♠♥ét♦s ♦s

♣♦r ♠ str♥ ♥rét♦ st tr♦ ♦s t♦rs ♦♥ír♠ q qçã♦ q

♦r♥ ss ♥trçã♦ sr ♣♦r

V (R) =a

R+ b ·X(R) + C.

st qçã♦ ♦ ♣r♠r♦ tr♠♦ r♣rs♥t ♥trçã♦ ♦♦♠♥ ♥tr ♦s

♠♦♥♦♣♦♦s ♠♥ét♦s ♦ s♥♦ tr♠♦ stá r♦♥♦ ♦♠ ♥r ♦ str♥ q

♥♠♥tçã♦ tór

♣♥ s ♦r♠ ♦ s ♦♠♣r♠♥t♦ X(R) ♦ út♠♦ tr♠♦ stá r♦

♥♦ à ♥r rçã♦ s rs s ♦rs ♥♦♥tr♦s ♣r ♦♥st♥t

♦r♠ a = −4.0Da b = 10.0D/a c = 23.4D s ♠♦♥♦♣♦♦s ♣rs♥ts ♥st r s

♦♠♣♦rt♠ ♦♠♦ ♦s ♠♦♥♦♣♦♦s ♠ ❬❪ ♣♦s s str♥s q ♠ sss sã♦

♥réts ♣♦rt♥t ♦srás sã♦ ♦r♥ts ♦ s t♠ ♠ s♥t♦ ♣♦

r③çã♦ ♥trí♥s♦ ♦♠♦ ♠ í♠ã s t♦rs ♦srr♠ ♠ ♠ ♣r♠r ♥s q

♥r ♥trçã♦ V (r) ♠ q r = R/a t♥ ♠ ♦♠♣♦rt♠♥t♦ ♣r♥t♠♥t

♥r ♦♠♦ ♠♦str♦ ♥♦ ♥st r 2.9 ♦ ♥t♥t♦ ♦ ③r♠ ♠ rrssã♦

♥ã♦ ♥r ♦♠ ♥çã♦ fq(R) = q/R+ b′R+ c ♥ tr ③ strí

rrssã♦ ♥r ♥ ♦♥t♥ r♠ s ♣rr♠ q s rs ♠♥éts

♥tr♠ ♣♦ ♣♦t♥ srt♦ ♣ qçã♦ 2.14

r st rá♦ ♦s ♣♦♥t♦s sã♦ ♦s ♦rs ♦t♦s ♣♦s t♦rs ♥ r♠é ♥çã♦ ♦♥sr♥♦ s♦♠♥t ♦♥trçã♦ ♥r ③ stá ♦r♦ ♦♠ qçã♦2.14 ❬❪

s ♦s tr♦s ♣rs♥t♦s ♥st ssssã♦ sã♦ rr♠♥ts ♣♦r♦ss ♣r

str ♦s ♦s s♣♥s s♦ s♠ ♦♥r♠s ①♣r♠♥t♠♥t ♣♦s trés s

sr ♣♦ssí ③r ♣rsõs s♦r ♦ ♦♠♣♦rt♠♥t♦ ♦s ♦s s♣♥s rts ♦♠♦

♦♠♣♦rt♠♥t♦ s rs ♠ ♦trs ♦♠trs ♦ ♦♠♦ s ♦♠♣♦rtr ♠ r

♠♥ét ♥ ♣rs♥ç ♠ ♠♣♦ ♠♥ét♦

♥♠♥tçã♦ tór

s ♦ s♣♥ ♠ ♦trs ♦♠trs

r ♦ s♣♥ ♠s st é r qr ♠♦str ♥ r

t♦s st♦s á ♦r♠ t♦s ①♣♦r♥♦ ♦♠♣r♥♥♦ ♦s ss ♠stér♦s

♦ st♦ ♥s ♣sqs♦rs ♦♠çr♠ ①♣♦rr ♦s ♦s s♣♥ ♠ ♦trs

♦♠trs s ①♠♣♦s ♠s ♦♥♦s sã♦ s rs r♦r ❬❪ ♦♠é ❬❪

♦♥②♦♠ ❬❪ tr♥r ❬❪ s ♣ró①♠s ssçõs srá ♠♦str♦ ♠ ♣♦♦

♠ sss ♦♠trs ♠ s♣♦ ♦str q ♦ t♦r ss ♦♠ ♠s

t♥çã♦ ♣rt r♦♥ à r tr♥r ♣♦s ♥♦ss♦ tr♦ ♦ r③♦ ♦♠

s ♥st r

r♦r

r r♦r é ♠ r s♠♥t ♠ ♣r t♦♦s ❯♠ s

♠♥rs ♦tr st r sr trés rtr ♥s s♣♥s r qr

st♦s ♠s t♦s st r ♦r♠ t♦s ♣♦r ❨♥ ♥❳♥ ❲♥ ❬❪

sts t♦rs str♠ st r ♠ r♥ts ♦♥rçõs r ♦s ss

st♦s ♥♠♥ts ss st♦s qs♦s

r s ♠♥s ♠♦str♠ s três r♥ts ♦♠trs ♣r r r♦r❬❪

r♥t r qr st r ♣rs♥t érts ♦♠ ♦s três s♣♥s

é♠ ♦s tr♦♥s érts ♦♠ qtr♦ s♣♥s sts r♥ts érts sã♦ ♣rs♥

t♦s ♥ r sã♦ s♣r♦s ♠ ♦s r♣♦s r♥ts t♦♣♦♦s ♦♠♦ ♥

r ♦s érts r♣♦ stã♦ ♦r♥♦s ♣♦r ♦r♠ ♥r ♦ ♠♥♦r

♣r ♦ ♠♦r

♥♠♥tçã♦ tór

r s ♠♥s ♠♦str♠ rs♣t♠♥t s r♥ts t♦♣♦♦s ♣rr r♦r ♣r ♦s três s♣♥s ♣♦r érts ❬❪

s ♦♥②♦♠ ♦♠é

r ♦♥②♦♠ é ♠ r ♦♠ ♦♠tr ①♦♥ s ♥♦♠ s ♦r♥

♦ r♥ s♠♥ç ♦♠ ♦♠ ♦ ♥ês ♦♥②♦♠ st

r ♦s s♣♥s s ♦③♠ ♥s rsts r ss ♠♦♠♥t♦s ♠♥ét♦s ♣♦♥t♠ ♦

♦♥♦s sts r♥♦ ♠ ♦♥rçã♦ s♣♥s ♣♦r ért r ♠♦str

r ♦♥②♦♠ ♦s ♣♦ssís t♣♦s érts ♣rs♥ts ♥st t♣♦ sst♠

r ♠♠ ♠♦str ♠ ♦♥rçã♦ r ♦♥②♦♠ ♦♠ s♣♥s ♣♦s♦♥♦s ♦r♥t♦s ♥s rsts ♦s ①á♦♥♦s ♠♠ ♠♦str ♦s t♣♦s érts♣♦ssís ♥♦♥trr ♥st r

st t♣♦ sst♠ ♣♦s ♦srr q ♦s érts ②♣ ♥ã♦ ♣r♦③♠

♠ ♠♥t③çã♦ rst♥t ♥q♥t♦ ♦s érts ②♣ s♠♣r ♣r♦③♠ ♠ ♠

♥t③çã♦ s♣♦♥tâ♥ ♦ ♦♥♦ ♦ ♣♥♦ ♦s s♣♥s

r ♦♠é é ♠ r ♦♠ ♠ ♦♠tr s♠r ♠ str

♦ s ♥♦♠ s ♦r♥ ♦ s r♥ s♠♥ç ♦♠ ♠ tr♦♥ st

♠ ♣♦♥s r ♠♦str ♠ r ♦♠é ♦s s♣♥s s

♥♠♥tçã♦ tór

s♠♥ç ♦♠ st ♠ ♣♦♥s

r ♠♠ ♠♦str ♠ ♦♥rçã♦ r ♦♠é ♦♠ ♦s s♣♥s ♣♦s♦♥♦s ♠ ss érts ♠♠ ♠♦str r♥ s♠♥ç st ♣♦♥s ♦♠ str Pr ♠♦r s③çã♦ ♦ s♥♦ ♣rt♦ ♠ ♣q♥ str ♥♦ ♥tr♦ st st

♦ ♥sr ♦s t♣♦s érts st r ♣♦s ♦srr q sts sã♦ ①t

♠♥t ♦s ♠s♠♦s érts ♣rs♥t ♥ r ♦♥②♦♠ Pr sr ♠s ①t♦ ♣♦s

♦srr q s rs ♦♥②♦♠ ♠é sã♦ ♥ r ♠s♠ r r

♠♦str ♦♠♦ é ♣♦ssí ♦tr ♠ r ♦♥②♦♠ trés r ♦♠é

r s ♠♥s ♠ ♠♦str♠ ♦♠♦ é ♣♦ssí ♦tr ♠ r ♦♥②♦♠trés ♠ r ♦♠é

♥♠♥tçã♦ tór

r♥r st ♣♦r ó

♥t♠♥t ó t ❬❪ ♣r♦♣sr♠ ♠ ♥♦ ♦♥rçã♦ ♣r r

♦ s♣♥ ♦♠ ♦♠tr tr♥r st ♥♦ ♦♠tr é s♠♥t r ♣r♦

♣♦st ♣♦r ❲♥♥r t ♠s ♦ ♥és ♦s s♣♥s r♠ ♥♦s érts ♦s trâ♥♦s sts

♠ ♥♦ ♥tr♦ s rsts ss ♠♦♠♥t♦s ♣♦♦ ♠♥ét♦ ♣♦♥t♠ ♦ ♦♥♦

sts ♥♦ s♣♥s ♣♦r ért s t♦rs st tr♦ ♦sr♦ ①stê♥

t♣♦s érts q ♣♦♠ ♣rr ♥♦ sst♠ ♦s ss♦ ♠ r♣♦s

t♦♣♦ó♦s

r s t♦♣♦♦s ♣rs♥ts ♥ r s♣♥ ♦♠ ♦♠tr tr♥r stsr♣♦s stã♦ s♣r♦s ♣♦r t♣♦s ♦♥rçõs ♥r ♣♦r s♣♥ ❬❪

st r rr ♦ ♦é ♣♦r s♣♥s ♣♦♥t♥♦ ♣r ♥tr♦ s♣♥s

♣♦♥t♥♦ ♣r ♦r ♦ ♥tr♦ ♦ ért trés r 2.15 ♣♦s ♦srr

q s♦♠♥t s s três ♦♥rçõs érts q ♦♠ rr ♦ ♦ sã♦

s ♠♥♦rs ♥rs ♦t q ♦trs s ♦♥rçõs q ♦♠ rr ♦

♦ t♠ ♥r ♠♥♦r q ♠ s ♦♥rçõs q ♦♠ rr ♦ ♦

st r♣♠♥t♦ é r♥t ♦ q ♦ ♦sr♦ ♥ r qr ♠ q t♦s s

♦♥rçõs q ♦♠ rr ♦ ♦ sã♦ s ♠♥♦rs ♥r é♠ st♦

♥♠♥tçã♦ tór

♣♦s ♦srr ①stê♥ t♣♦s rs ♠♥éts s♠♣s r ♣♦r

s♣♥s ♣♦♥t♥♦ ♣r ♥tr♦ ♣r ♦r ♦ rs ♣ r ♣♦r s♣♥s

♣♦♥t♥♦ ♣r ♥tr♦ ♦ ért ♣r ♦r ♦ rs r tr♣ r

♣♦r s♣♥s ♣♦♥t♥♦ ♦ ♣r ♥tr♦ ♦ ♣r ♦r ♦ ért

st ♠s♠♦ tr♦ ♦s t♦rs ♠♦str♠ q s ①tçõs ♠s ①s

♥rs ♣♦♠ sr ♥tr♣rts ♦♠♦ ♠♦♥♦♣♦♦s ♠♥ét♦s ♠ ❬❪ sts

r♠ ♦s ♣♦r ♠ str♥ ♥rét st ♣r♦ss♦ ♦♦rr ♦ ♦çã♦ rr

♦ ♦ ♠ r ♠ s st♦ ♥♠♥t ♦♠♣♦st s♦♠♥t ♣♦r érts t♣♦

V 1 ♦çã♦ rr ♦ ♦ ♦♦rr ♦ ♥rsã♦ ♦ ♠♦♠♥t♦ ♠♥ét♦

♠ sít♦ r r♥♦ ♦s érts t♣♦ V 4 sss érts ♣♦♠ sr ♥tr♣rt♦s

♦♠♦ ♠♦♥♦♣♦♦s rs ♦♣♦sts ♦ s♣♥ ♥rt♦ ♦♠♦ str♥ rçã♦ sts

♦s ♠♦♥♦♣♦♦s s♦♠♥t ♦♦rr ♦ ♦çã♦ rr ♦ ♦ ♦s érts

s♣rçã♦ sts rs ♣♦ ♦♦rrr s♠ q ♠s ♥♥♠ ♦tr ♦çã♦

rr ♦ ♦ ♥st r Pr st♦ é ♥ssár♦ ♥rtr ♠ s♣♥ q st ♥t

♥♦ ♦♠ ♦ ♣r♠r♦ ♥rt♦ st ♣r♦ss♦ ♠♦s q ♦ ♦♠♣r♠♥t♦ str♥

é ♣r♦♣♦r♦♥ stâ♥ ♥tr s rs ♦s érts r♦s ♥st ♣r♦ss♦ sã♦ ♦

t♣♦ V 5 ♦s qs ♦♠ rr ♦ ♦ P♦s ♠r st t♣♦ s♣rçã♦

s♣rçã♦ ♣♦r str♥ ♥r ♣♦s ♦r♠ str♥ é s♠♣s♠♥t ♠ ♥ rt

①st ♦tr ♠♥r s♣rr s rs ♠♥éts s♠ q ♦çã♦ rr

♦ ♦ st s♥♦ ♠ét♦♦ é ♥ssár♦ q s ♥rsõs s♣♥ ♣r

q s rs s♠ s♣rs ♠ ♠ s♣ç♠♥t♦ r st ♠♥r r

♠♥ét é st ③♥♦ ♠♦♠♥t♦ ③③ r♥♦ érts t♣♦ V 2 st

t♣♦ ♥rsã♦ ♣♦ sr ♥♦♠♦ ♦♠♦ s♣rçã♦ ♣♦r str♥ ♥tsrr ♣♦s

♦r♠ str♥ é s♠r s ♥ts ♠ srr st ♣r♦ss♦ str♥ t♠ ♦

♦♠♣r♠♥t♦ ♦ ♦r♦ s♣rçã♦ s rs ♠♥éts sts s str♥s

♣♦♠ sr ♠♦r s③s trés r 2.16

♥♠♥tçã♦ tór

r st r ♠♦strs ♦s ♦s t♣♦s str♥s q s♣r♠ s rs ♠♥étss♠ q ♦çã♦ rr ♦ ♦ s♣rçã♦ ♣♦r ♠ str♥ ♥r s♣rçã♦ ♣♦r♠ str♥ st♦♦t st ♠♥s ♦s s♣♥s ♥③s sã♦ ♦s s♣♥s ♥rt♦s q ♦r♠♠ sstr♥s ♦s r♥s ♣♦♥t♦s ③s r♠♦s sã♦ ♦s ♠♦♥♦♣♦♦s ♠♥ét♦s rs ♦♣♦sts ♦s ♣q♥♦s ♣♦♥t♦s rs ♠r♦s ♥♠ rs♣t♠♥t ♦s érts t♣♦ II t♣♦V ❬❪

❯♠ t♦ ♥trss♥t é q ♦s t♦rs srr♠ q sr ♣♦ssí s♠♥

t③r ♦ st♦ ♠♥t③♦ r ♦♠♣♦st♦ ♣♦r érts t♣♦ V 5 ♦tr ♦ st♦

♥♠♥t ♦ sst♠ s♠♣s♠♥t ♦♠ ♥rsã♦ ♦s s♣♥s ♠ ú♥ rçã♦

st ♠♥r ♥â♠ érts ♣r ♥çr ♦ st♦ ♥♠♥t sr

♣♦r V 5 → V 4 → V 1

Pr♦t♦♦♦s s♠♥t③çã♦ ♦s s♣♥s

rts

Pr♦t♦♦♦s ①♣r♠♥ts

sssã♦ ♥tr♦r ♦ st♦ q ♠♦♥♦♣♦♦s ♠♥ét♦s ♠r♠ trés

①tçõs ♦ st♦ ♥♠♥t ♦s ♦s s♣♥s ♥tã♦ ♠ ♦s ♥trsss ♦s ♣s

qs♦rs sts ♠trs sr ♥♦♥trr ♠ ♣r♦ss♦ q ♦♥sss s♠♥t③r

r ♦ sst♠ ♦ s st♦ ♥♠♥t á q ❲♥ t ❬❪ ♦srr♠ q

sss rs ♥ã♦ s ♥♦♥tr♠ ♠ ss st♦s ♥♠♥ts ♣ós sr♠ ♦♥strís

sts ♠s♠♦s t♦rs ♥ t♥tt ♦tr ♦ st♦ ♥♠♥t ♦s ♦s s♣♥

♣r♦♣sr♠ ♣r♦t♦♦♦s s♠♥t③çã♦ ♠ t♦♦s ♦s ♣r♦t♦♦♦s ♦s t♦rs t

♥♠♥tçã♦ tór

③r♠ ♠ ♠♣♦ ♠♥ét♦ ①tr♥♦ q r ♠ t♦r♥♦ r ♦s s♣♥s ♦♠

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

♣r♦t♦♦♦ ♣r ♣r♦t♦♦♦ ♦ ♣r♠r♦ ♣r♦t♦♦♦ ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♦♠ç

♦♠ ♠ ♥t♥s ♥ Hi st r ♠♥í♦ ♠ SH ♣r ♣ss♦ t♠♣♦

Ts Pr ♠♥ç ♦r ♥♦ ♠♣♦ ♠♥ét♦ ♣♦r st ♠♣♦ t♠

é♠ r ♥rt ♦ s♥♦ ♣r♦t♦♦♦ ♦ ♠♣♦ ♦♠ç ♦♠ ♠ ♥t♥s ♥

Hi ♠♥í ♥r♠♥t ♠ r③ã♦ RH té zero trr♦ ♣r♦t♦♦♦ ♦♠ç ♦♠

♠ ♦r ♥ Hi ♠♥í ♠ ♦rs srt♦s SH ♣r ♣ss♦ t♠♣♦ Ts

♠s r♥t ♦ ♣r♠r♦ ♣r♦t♦♦♦ ♠♥t♥♦ s ♣♦r③çã♦ r 2.17 ♠♦str

♠ rá♦ ♦♠♦ ♥t♥s sts ♠♣♦s r♠ ♠ rçã♦ ♦ t♠♣♦

r st rá♦ é ♠ strçã♦ ♦s ♣r♦ss♦s r③♦s ♣♦r ❲♥ ❬❪

♠ ♠ r ♦s s♣♥s ♠♥t③ ♦♠ s♣ç♠♥t♦ r 320nm

❲♥ t ♦srr♠ q ♦ ♣r♠r♦ ♣r♦t♦♦♦ ♦♥s ♥rtr 51.8% s

s q ♦♥stt♠ r st ♦r é ♠ ♣ró①♠♦ ♦ 50% ♥♥♦ sr

♠ ♦♠ ♣r♦t♦♦♦ s♠♥t③çã♦ tr♦ t♦ ♥trss♥t é q ♠♥t③çã♦

♠é r mtot = 0.056 ♣♦r sr ♠♥t té ♠ ♦r mtot = 0.152

s ♦ s♣ç♠♥t♦ r ♦ss ♣r 720nm s ♦tr♦s ♦s ♣r♦t♦♦♦s ♥ã♦ ♦r♠ tã♦

♥t ♦♠♦ ♦ ♣r♠r♦ ♣♦s ♠♥t③çã♦ ♠é r ♠♦s ♣r♦t♦♦♦s ♦

mtot ∼ 0.5 ♠t ♦♠♣t ♠♥t③çã♦ r mtot = 1.0

♥♠♥tçã♦ tór

Pr♦t♦♦♦s ♦♠♣t♦♥s

♥trss ♦tr ♦ st♦ ♦ ♦s s♣♥s é tã♦ r♥ q ♠t♦s ♣sq

s♦rs ♦♠çr♠ str ♥â♠ sts ♠ s♠çõs ♦♠♣t♦♥s ❯♠

r♣♦ ♣sqs♦rs q tr♠ ♠ r♥ ♥trss ♥ ♥â♠ ♦s ♣r♦ss♦s

s♠♥t③çã♦ ♦ rs t ❬❪ ❬❪ rs t ♣r♦♣sr♠ q ♥rsã♦

♠♥t③çã♦ ♠ ♠ s♣♥ t♣♦ s♥ ♦♦rrr q♥♦ s

− ~si · ~Bi ≤ hc,

♦ss stst st s t♠♦s q~si é ♦ ♠♦♠♥t♦ ♠♥ét♦ ♠

i ~Bi é ♦ ♠♣♦ ♠♥ét♦ t♥♦ ♥st hc sr ♦ ♠ó♦ ♦ ♠♣♦ ♠♥ét♦

♠í♥♠♦ tr♦ ♣r ♥rsã♦ ♦ ♠♦♠♥t♦ ♠♥ét♦ i ♠ ♠ ss

tr♦s ❬❪ s t♥tr♠ ♦tr ♦ st♦ ♥♠♥t ♦s ♦s s♣♥s trés

♠ ♠♣♦ ♠♥ét♦ r♦t♦♥ s♥♦ s ♦♥çã♦ ♥rsã♦ ♠♦♠♥t♦

♠♥ét♦ s ♦♥ír♠ q st ♣r♦t♦♦♦ s♠♥t③çã♦ ♥ã♦ r ♥t

♣♦s s♦♠♥t ♠s árs rs ♦♥sr♠ ♥çr ♦ st♦ ♥♠♥t

♠ ♦tr♦ tr♦ ❬❪ s ♣r♦♣sr♠ ♠ r ♠♣rt ♥ q s s s♦r♠

s♦r♠ ♥ ♣♦sçã♦ ♥ ♦r♥tçã♦ ss ♠♦♠♥t♦s ♠♥ét♦s ♥♦ ♠♣♦ ♠♥ét♦

♠í♥♠♦ tr♦ ♥ ♥trçã♦ ♦♣♠♥t♦ ♥tr ♦s s♣♥s sts rs s

♦srr♠ q é ♠♣♦ssí ♦tr ♦ st♦ ♥♠♥t trés ♦ ♣r♦t♦♦♦

s♠♥t③çã♦ trés ♠ ♠♣♦ ♠♥ét♦ ①tr♥♦ r♦t♦♥

♣ít♦

♦tçõs ♠t♦♦♦

♦tçõs

s st♦s ♦s ♦s s♣♥ sã♦ r♥ts sts ♣♦♠ ♦rr ♥í♦s t♥t♦

t♥♦ó♦s ♦♠♦ ♥♦s ê♥s P♦r ①♠♣♦ trés ♦ ♦ s♣♥ ♣♦rí♠♦s

♦♥sr ♥♦s t♥♦♦s r♠③♥♠♥t♦ ♦s ♠ ♦r♠t♦ ♥ár♦ ♣♦s ♦s

s♣♥s r só ♣♦♠ ♦tr ♦s ♦rs st♥t♦s ♦ ♣♦♥t♦ st ê♥

ás ♦ st♦ ♦s s♣♥s ♣♦ r ♠ ♠♦r ♥t♥♠♥t♦ r s

♣r♦♣rs ①tçõs ♦ts ♣s♦♣rtís ♠ sst♠s ♠tér ♦♥

♥s

♦ ♣ít♦ ♥tr♦r ♠♦s q qs♣rtís ♠r♥ts ♥♦s ♦s s♣♥s

s ♦♠♣♦rt♠ ♦♠♦ ♠♦♥♦♣♦♦s ♠♥ét♦s ♦s ♣♦r ♠ str♥ ♥rét sts

qs♣rtís sr♠ ♦ à ♦çã♦ rr ♦ ♦ ♦ s sã♦ ①tçõs s♦r

♦ st♦ ♥♠♥t t♦s ♣sqs♦rs r♣r♦③r♠ ♦s ♦s s♣♥s ♠ rs

rts ♦♠ ♥trss str s ♣r♦♣rs sts ♠trs ♣r♥♣♠♥t

qs s ♦s ♠♦♥♦♣♦♦s ♠♥ét♦s ♥t♦ ♠ rs rs ♦♠♦ ♠ s♠çõs

♦♠♣t♦♥s ♠♦s q ♥ã♦ ♦ ♥♦♥tr♦ ♠ ♠ét♦♦ ③ q ♦♥s r ♦s

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

♠s t♦s ♦s ♠♦♥♦♣♦♦s ♠♥ét♦s ♣♦s sts sr♠ ♦♠♦ ①tçõs s♦r

♦ st♦ ♥♠♥t

ó ♦♦r♦rs ♠ ♠ ss tr♦s ❬❪ ♣r♦♣sr♠ ♠ ♦♥r

çã♦ tr♥r ♣r ♦s ♦s s♣♥s rts st ♠s♠♦ tr♦ ♦♠♥t♦

♦tçõs ♠t♦♦♦

q sr ♣♦ssí r st r ♠ st♦ ♦♠♣t♠♥t ♠♥t③♦ ♣r ♦

st♦ ♥♠♥t s♠♣s♠♥t ♥rt♥♦ ♠ ♥r s♣♥s ss♦ r

♠♥t ♦r ♣♦ssí st r ♣♦r ♥♦s ♦rr ♠♦r ♥t♥♠♥t♦ ♦s ♠♦♥♦♣♦♦s

♠♥ét♦s s ♥â♠ ♦♠♦ ♣♦r ①♠♣♦ ♦s st♦s ♠♥t♦stát ♥♦s ♦s

s♣♥s rts ♥tã♦ ♦ ♦t♦ ♥♦ss♦ tr♦ sr rr ♣óts sts

t♦rs trés ♠ ♣r♦t♦♦♦ s♠♥t③çã♦ trés ♠ ♠♣♦ ♠♥ét♦

①tr♥♦ s ts ♠t♦♦♦ s ♠ ♥♦ss♦ tr♦ srá srt ♦♠ ♠s

ts ♥♦ ♣ró①♠♦ ♣ít♦

st

r st ♥st tr♦ ♦ r ♣r♦♣♦st ♣♦r ó t ❬❪ ♦

s♣♥ rt ♥ q ♦s s♣♥s r sã♦ t♣♦ s♥ ♣♦♥♦ ♦tr s♦♠♥t ♦s

♦rs srt♦s ♠ ♠ ♦♠tr tr♥r strtr♠♥t sts s♣♥s

stã♦ ♦♥♥♦s ♥♦ ♣♥♦ r xy stã♦ ♦③♦s ♥♦ ♥tr♦ s rsts

trâ♥♦s qátr♦s ♥♦s qs ss ♠♦♠♥t♦s ♠♥ét♦s stã♦ ♦ ♦♥♦s sts

r♥♦ ♠ ♦♥rçã♦ ss s♣♥s ♠ ért r st é ♠♦str ♥

Figura3.1 r♥♥♦s s♦♠♥t ♣ q♥t s♣♥s ♦t q ss ♦rs

sã♦ ss s s s♠♣r ♦♥s♠ r trâ♥♦s qátr♦s ♦ ♦r♠t♦ r

♦♠♦ t♦♦ é ♠ ♣ró①♠♦ ♠ qr♦ ①st ♥♦ ♠í♥♠♦ s♣♥s ♣♦r ért

r s♠r r st ♥st tr♦

♦♠♦ ó t st♠♦s r trés s♠çõs ♦♠♣t♦♥s ♠s

♦tçõs ♠t♦♦♦

r♥t s tr♦ t♥t♠♦s ♣r♦①♠r ♦ ♥♦ss♦ ♠♦♦ s rs rs r

s ♠ ♦rtór♦ ♣♦r t♦r Pr st♦ ♥s♠♦s ♦ tr♦ r③♦ ♣♦r

Pt ♦♦r♦rs ❬❪ ♣t♠♦s ♦s ♣râ♠tr♦s st tr♦ ♦♠ s ♥♦s

ss s♠çõs st rt♦ ♦s t♦rs rr♠ ♠ r ♦s s♣♥s

rts s♠r ♦♥strí ♣♦r ❲♥ t ♠s ♦♠ r♥ts ♠♥sõs

s s ♦s ♥♥♦♠♥t♦s t♠é♠ sã♦ ♦♥sttí♦s Pr♠♦② ♠s tê♠ ♠♥

sõs 290nm ♦♠♣r♠♥t♦ 130nm rr 20nm s♣ssr ♦s ♥tr♦s

s s ♠s ♣ró①♠s stã♦ s♣r♦s ♣♦r ♠ stâ♥ a = 500nm ♦ ♦

sr♦ q sts s sã♦ s♥t♠♥t ♣q♥s ♣r q t♥♠ ♠ ú♥♦ ♠♦

♥♦♦♠♥♦ ♠♥ét♦ sã♦ s♥t♠♥t r♥s ♣r q ♦ t♠♣♦ r①çã♦

sts s ♠t♦ ♠♦r ♦ q ♦ t♠♣♦ s ♠s ①♣r♠♥ts ♥ã♦ ♥trr♥♦

♥s ♥ss st r trés sts ♦s ♦♥í♠♦s q ♦ ♠♦♠♥t♦ ♠♥é

t♦ sts s é µ = M0V ∼ 6.0 · 10−16Am2 ♦ q ♦ ♦♠ s s é

V = 290nm · 130nm · 20nm = 7.5 · 105(nm)3 = 7.5 · 10−27m3 ♠♥t③çã♦

strçã♦ ♦ Pr♠♦② é M0 = 7.958 · 10−4A/nm ss♠ ♣r♠♦s q

♠♥t ♦ ♠♣♦ ♠♥ét♦ ♥♦ ♥tr♦ ♦ ♥♥♦♠♥t♦ ③♥♦ ♠s ♣ró①♠♦ é

[ ~B] ≡D

µ∼ 0.5mT,

♠ q

D =µ0µ

2

4πa3∼ 3.0 · 10−19J,

J = Am2T µ0 = 4π · 10−7N/A2

st ♠s♠♦ tr♦ ♦ ♥♦♥tr♦ q ♦ ♠♦♦ ♦ ♠♣♦ ♠♥ét♦ b

♥ssár♦ ♣r ♥rtr ♦ ♠♦♠♥t♦ ♠♥ét♦ ♠ ♥♥♦♠♥t♦ é ♦ ♣♦r hc =

(550± 55)Oe sr♥♦ st ♠♣♦ ♠ ♥s ♥tr♥♦♥s t♠s q

b = (500.0± 50.0)Oe = (500.0± 50.0) · 10−4T = (50.0± 5.0)mT.

♦tçõs ♠t♦♦♦

s é ♥ssár♦ ♠ ♠♣♦ ♠♥ét♦ ♣r♦①♠♠♥t ③s ♠♦r

♦ q ♦ ♠♣♦ ♠♥ét♦ ♦ ♥♥♦♠♥t♦ ③♥♦ ♠s ♣ró①♠♦

♠ ♥♦ss♦ tr♦ s s ♥tr♠ ♥tr s s♦♠♥t ♣♦r ♥trçã♦ ♣♦r

ss♠ ♦♠♦ ♠♦s ♥tr♦r♠♥t ♣r srr ♦ ♥♦ss♦ sst♠ t③♠♦s ♠

t♦♥♥ ♣rs♥t ♥ qçã♦ 2.10

Edip =∑

i 6=j

D

[

~si · ~sj − 3(~si · rij)(~sj · rij)

R3ij

]

.

st qçã♦ st♠♦s ♦s ♦rs D D/µ ♠♦♦ q sts ♦rs

♠♥tss♠ s ♣r♦♣♦rçã♦ ♦♠ ♦s ♦rs ♥♦♥tr♦s ♣♦r Pt ♠ ♥♦sss s♠

çõs ♥♦♥tr♠♦s q ♦ ♠♦♦ ♦ ♠♣♦ ♠♥ét♦ ♠ ♠ s ③♥♦

♠s ♣ró①♠♦ é | ~B| = 2.0D/µ ♦ tr♦ Pt ♠♦s q ♦ ♦r ♠♦

①♣r♠♥t♠♥t ♣r st ♠♣♦ é 0.5mT st ♠♥r ♣♦♠♦s ♦♥r q

2.0D/µ = 0.5mT −→ 4.00D/µ = 1.0mT.

trés st t♠♦s ♠ ♦♥①ã♦ ♠s r ♥tr ♦s ♦rs ①♣

r♠♥ts ♦s ♦rs s♦s ♠ ♥♦sss s♠çõs ♠ ♥♦sss s♠çõs s♦

♠♦s ♦ ♦r D/µ = 0.5u.r.u.r. ≡ ♥s r③s ♣♦s ss♠ tr♠♦s q

2u.r. = 1mT ♥♦ q ♦ ♠♦♦ ♦ ♠♣♦ ♠♥ét♦ ♥♦ ③♥♦ ♠s ♣ró①♠♦ sr

| ~B| = 1.0u.r. ♠ ♦r ♥tár♦ trés st ♦♥rsã♦ rst♠♦s ♦s ♦rs

♥♦sss s♠çõs ♠♥r q sts ♦rrs♣♦♥ss♠ ♦s ♦rs ♦t♦s ①♣r

♠♥t♠♥t P♦r ①♠♣♦ ♦ ♦r ♦ ♠♦♦ ♦ ♠♣♦ ♠♥ét♦ b ♣r ♥rtr ♦

♠♦♠♥t♦ ♠♥ét♦ s s ♠ ♥♦sss s♠çõs sr ♦ ♣♦r

b = (50.0± 5.0)mT −→ (100.0± 10.0)u.r.

qçã♦ 3.6 ♠♦s q b é ♦ ♣♦r ♠ strçã♦ ss♥ ♥tr

♠ r ♦♠ ♠ s♦ r st qçã♦ t♠é♠ ♣♦ sr rsrt ♦♠♦

b = (100.0± 10.0)u.r. = bc +b,

♦tçõs ♠t♦♦♦

♠ q bc = 100u.r. b = 10u.r. = 10%bc b ♦♠♦ ♦ st♦ ♥ sssã♦ str

r♣rs♥t♥♦ s♦r♠ s ♣♦r ♠♣rçõs ♥s s ♠ ♥♦ss♦ tr♦ ♣r

♠♦r ♥t♥♠♥t♦ st r ♠♥♣♠♦s ♦s ♦rs b ♣r ♦srr q sr

♦ t♦ ss s♦r♠ ♥♦ ♦♠♣♦rt♠♥t♦ r Pr st♦ ♠♦♠♦s ♦ ♦r

ss s♦r♠ 0% bc ♠ r ♦♠ s ♦♠♣t♠♥t ♦♠♦ê♥s té 20%

bc ♠ r ♦♠ s ♦♠ ♦ ♦r♦ s♦r♠ ♥♦♥tr ①♣r♠♥t♠♥t

q sr♠ ss ♣♦r r♥s ♠♣rçõs ♥s s

♠ qqr s♠çã♦ ♦ ♥♦ss♦ tr♦ s s r tê♠ ss ♣♦sçõs

①s ♠♥té♠ ss rtrísts ♦♠♦ strtr rçã♦ ♦ ♠♦♠♥t♦ ♠♥ét♦

♥♣♥♥t ♦s t♦rs ♠♣♦st♦s ♥♦ ♣r♦r♠ ú♥ ♣r♦♣r s s q

♣♦ ♠r é ♦ s♥t♦ ♦ ♠♦♠♥t♦ ♠♥ét♦ ♥rsã♦ ♦s ♠♦♠♥t♦s

♠♥ét♦s s s s♦♠♥t ♦♦rr♠ trés ♠♣♦ ♠♥ét♦ ♠s ♥♥♠ ♦tr♦

t♦r ♦♥çã♦ ♣r st ♥rsã♦ q s♠♦s ♦ ♠s♠ ♣r♦♣♦st ♣♦r rs t

❬❪ ❬❪

− ~Btoti · ~si ≤ bi,

♠ q ~Btoti é ♦ ♠♣♦ ♠♥ét♦ t♦t ♦ ♣ s♦♠ ♦ ♠♣♦ ①tr♥♦ ~Bext ♠s ♦

♠♣♦ ~Bi q é ♦ ♠♣♦ r♦ ♣♦s ♦tr♦s s♣♥s r ♥ i ~si é ♦ ♠♦♠♥t♦

♣♦♦ ♠♥ét♦ i bi é ♦ ♠ó♦ ♦ ♠♣♦ ♠♥ét♦ ♠í♥♠♦ ♣r ♥rtr

♦ ♠♦♠♥t♦ ♣♦♦ st ♦♠♦ t♥tt ♣r♦①♠r st ♠♦♦ ♦s sst♠s

rs ♠ q s s sã♦ rr♠♥t ♠s s s ♦trs ♦s ♦rs ♦s ♠♣♦s ♠

♥ét♦s ♥ssár♦s ♣r ♥rsã♦ ♦ ♠♦♠♥t♦ ♠♥ét♦ srã♦ r♦s

t♦r♠♥t rs♣t♥♦ ♦s ♦rs st♦s ♣ qçã♦ st s♣rs

q ♦s ♦rs bi s s ♥ã♦ s♠ ♦rr♦♥♦s ♥s ♦♠ ♦s ♦tr♦s

Pr♦ss♦s s♠♥t③çã♦

st tr♦ ♦r♠ s♦s ♦s r♥ts ♣r♦ss♦s s♠♥t③çã♦ ♦

♣r♠r♦ ♦ trés ♠ ♠♣♦ ♠♥ét♦ ①tr♥♦ r♦t♦♥ ♦ s♥♦ ♦ trés

♠ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♠ ♠ ú♥ rçã♦ s♠r ♦ ♣r♦ss♦ strs

♦tçõs ♠t♦♦♦

♥ts ♦♠çr ♣r♦♥r ♥ srçã♦ ♣r♦ss♦ srã♦ ♥♠r♦s ♦s

s♣t♦s ♦♠♥s q sã♦ ♦♥sr♦s ♠ ♣r♦ss♦

1 ♥rsã♦ ♦ ♠♦♠♥t♦ ♠♥ét♦ s s sr ①t♠♥t s ♦♥çõs

ts ♥ sssã♦ ♥tr♦r s ♠♦♠♥t♦s srã♦ ♥rt♦s s♦♠♥t ♣♦r ♠♣♦

♠♥ét♦ ♥ã♦ á t♦rs ♥♦ ♣r♦r♠s q ♦♥s♠ ♦r♠r strtr

r

2 ♠♣♦ ♠♥ét♦ ①tr♥♦ s♠♣r é ♣r♦ à r st é ♥♦r♠ ♠ t♦

st

3 ♥ts ♦♠çr qqr s♠çã♦ é ♣♦ ♠ ♠♣♦ ♠♥ét♦ ①tr♥♦

①tr♠♠♥t r♥ ♥ rçã♦ ♥ ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♠♦♦

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

r ♦♠♣♦st s♦♠♥t ♣♦r érts t♣♦ V 5 ♠♦str♦ ♥ r 2.15 ♣ós st♦

ss ♠♣♦ é ♠♥í♦ té ♦ ♦r s♦ ♣r ♦ ♥♦ s♠çã♦

4 ♠♥ç ♦ ♦r ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♥s s r t♥t♦ ♣♦r

♥t♥s q♥t♦ ♣♦r rçã♦ â♥♦ ♦ ♠♣♦ ♠ rçã♦ às s rá

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

sts sts③♠ ♦♥çã♦ qçã♦ s srr q q♥♦ ♠

s♣♥ é ♥rt♦ é r♦ s t♦♦s ♦s ♦tr♦s s♣♥s r t♠é♠ srã♦

♥rt♦s ♠s♠♦ q sts á t♥♠ s♦ st♦s ♥tr♦r♠♥t

5 s rçõs ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ t♥t♦ ♠ ♥t♥s q♥t♦ ♠ r

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

♠♦♠♥t♦ ♠♥ét♦ srá q q stá ♠s ♣t♦ st♦ trés st♦ s♣r

s q ♦ sst♠ ♦♠♦ t♦♦ ç ♠♥çs áts ♥ã♦ st♥♦ ♣♦ssís

st♦s qír♦ ♣r ♠♥♠③r s ♥r

♠♣♦ ♠♥ét♦ r♦t♦♥

st ♣r♦ss♦ é ♠ ♦♥♦ ♦♠♦ ♠ ♣r♦ss♦ s♠♥t③çã♦ sst

♠s ♠♥t③♦s st r str ① ♥q♥t♦ ♠ ♠♣♦ ♠♥ét♦ ①tr♥♦

♦tçõs ♠t♦♦♦

sr ♣♦ ♥st str r♥♦ ♥t♠♥t ♠ t♦r♥♦ ♦ ①♦ ♥tr r

♠ ♥♦sss s♠çõs r stát ♠♥té♠ ♦s ♦rs s ♣♦sçõs ♦s s♣♥s ♦s

♦rs ss ♠♦♠♥t♦s ♣♦rs ♠♥ét♦s ♦♥st♥ts ♦ ♠♣♦ ♠♥ét♦ tr

s ♠ó♦ ♦♥st♥t ♠s ss ♦♠♣♦♥♥ts ♦ ♦♥♦ ♦s ♦s ①♦s ♣r♥♣ r

x y s♠♣r str♠ ♠♥♦ ♦r

♠♣♦ ♠♥ét♦ ♥ ♣♦♥t ♥ rçã♦ ♦ ①♦ +x st r ♥t♠♥t

♥♦ s♥t♦ ♥t♦rár♦ +x ♣r +y ♠♦♦ ♥ st ♠♣♦ ♦ |Bext| ∼ bc

♣♦s ♦sr♠♦s q ♣r ♠♣♦s ♠t♦ ♠♥♦rs ♦ q ♦ ♠♣♦ ♠í♥♠♦ ♥rsã♦

bc ♥♥♠ ♠♦♠♥t♦ ♠♥ét♦ r ♥rt♦ ♣r ♠♣♦s ♠t♦s ♠♦rs ♦ q

bc ♦ ♠♣♦ r ♠t♦ ♥t♥s♦ ♦s ♠♦♠♥t♦s ♠♥ét♦s r s♠♣r s ♥♠

♦♠ ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ é♠ st♦ ♥t♥s st ♠♣♦ ♠♥í ♠t♦

♥t♠♥t ♠♦♦ q st ♠♥í ♠ 1u.r. ♣r três ♦t ♦♠♣ts ♥ r

rçã♦ ♥t♥s ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♦ ♦t ♦♠ ♥

♦tr ♠ st♦ st♦♥ár♦ ♣r r ♣♦s s♦ ss ♥ã♦ ♦ss r③♦ ♥s

♠♦♠♥t♦s ♠♥ét♦s ♠s s r ♥♥ r♠ ♣rr sr ♥rt♦s

♣r♥♣♠♥t qs ♠s ♣ró①♠♦s ♦r r ♣q♥♦s ♦rs ♦ ♠♣♦

♥trí♥s♦ ♥rsã♦

♠♣♦ ♠♥ét♦ ♠ ♠ ú♥ rçã♦

s♥♦ ♣r♦ss♦ ♦♠♦ ♦ t♦ ♥tr♦r♠♥t é ♠ ♣r♦ss♦ s♠♥t ♦

strs ♥♠♥t ♦♥srs q ♠ r q stá ♠♥t③ ♦

♠ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♣♦ ♥ rçã♦ +x r ♣ós ♥çr st

st♦ ss ♠♣♦ ♠♥ét♦ s♥♦ ♥t♠♥t ♥rt♦ té q ♠♥t③çã♦

r s t♦t♠♥t ♥rt ♣♦r ♠ ♦ ♠♣♦ ♠♥ét♦ é ♥♦♠♥t ♥rt♦

té q r ♦t tr ♠♥t③çã♦ ♦ s st♦ ♥

st ♣r♦ss♦ ♦ s♦♦ ♦ ♠ sstã♦ ♣♦r ó t ❬❪ ♦

♠s♠♦ ♥♦ q ♦s t♦rs sr♠ ♥♦ r st rt♦ ♦s t♦rs ♦♠♥t♠

q ♦ st♦ ♠♥t③♦ ♦♠♣♦st♦ s♦♠♥t ♣♦r érts t♣♦ V 5 ♣♦ ♠♥t sr

♦ ♦ st♦ ♥♠♥t trés ♣♦s ♥rsõs ♠♦♠♥t♦s ♠♥ét♦s

♠♥r q ♦s érts V 5 −→ V 4 s V 4 −→ V 1 ♦ strtr r

♦tçõs ♠t♦♦♦

s♣♦♠♦s q ♠ ♣r♦ss♦ s♠r ♦ strs ♦♥sr ♥çr st ♦t♦

♥ ♥st ♠s♠♦ ♣r♦ss♦ ♣♦s ①♣♦rr ♥ ♠s ♦s ♣r♦♠s ①♣r

♠♥ts q ♣♦♠ ♦♦rrr ♥s ♠s ♠ ♦rtór♦ ❯♠ sts é ♦ ♥♠♥t♦ ♦

♠♣♦ ♠♥ét♦ ①tr♥♦ ♦♠ r s♣♥ ♦r♠♠♥t r ♦s s♣♥s

t♠ ♠♥sõs ♥♥ô♠tr♦s t♥♦ ♦ ①t♦ ♥♠♥t♦ ♠♣♦r ♥tã♦

♦ st t♦ r♠♦s t♠é♠ s♣♦r ♠ ♥♦sss s♠çõs ♠ s♥♠♥t♦

r ♦♠ ♦ ♠♣♦ st ♠♥r ♦ ♠♣♦ q t♥ ♠ ú♥ ♦♠♣♦♥♥t ♦r

♣♦rá tr s ♦♠♣♦♥♥ts ♠ q ♦♥t♥rá ♠t♦ ♥t♥s ♥ rçã♦ x ♦tr

♠♥♦r ♥ rçã♦ y q trá ♥t♥s s♥t ♣r tr s ♠s ♦s ①♣

r♠♥t♦s ♥♦ s♥♠♥t♦ θ rr 0 ♠ ♣rt♦ ♥♠♥t♦

10 ♠ s♥♠♥t♦ ♠t♦ ♠♦r ♦ q ♣♦s s♣rr ①♣r♠♥t♠♥t

♦♥srçõs ♥s

st sssã♦ srã♦ r♠♥t ♣rs♥ts ♠s ♦♥srçõs ♥s s♦r

♠♥r ♦♠♦ ♦r♠ ♦t♦s ♦s ♦s ♣rs♥t♦s ♥♦ ♣t♦ st♦s ♥

♠♥t t♦♦s ♦s ♦s ♣rs♥t♦s ♥st tr♦ ♦r♠ ①trí♦s ♠ r ♦♠

♠♥sõs 30a ① ∼ 30a ♠ q a é ♦ s♣ç♠♥t♦ r st ♠♥r r

trá ♦♥♥t♦s s ♣rt♠♥t ♥s ♥ rçã♦ x ♠ ♦tr♦ ♦♥♥t♦

♣r♦①♠♠♥t s ♥s ♠ ③③ ♥ rçã♦ y ♠♦s ♣r♦ss♦s t♦s

♥ sssã♦ ♥tr♦r ♦r♠ ♥s♦s trés s r♥③s r♥ts ♣r♠r

♦ ♠♥t③çã♦ ♠é r ♥ rçã♦ xMag s♥ ♦ ♣♦♣çã♦ ♦s

r♥ts t♣♦s érts ♣rs♥ts ♥♦ sst♠ r 2.15 r♥③ ♦ ♦t

trés ♠é 50 × 10 ♠♦strs ♠ q sss ♠♦strs ♦r♠ ♦ts 50

rs r♥ts ♣ ♦ ♥s 10 ③s ♥s ♠ ♠s♠ ♣ ♦

♥ssár ♦ ♦ ♠ét♦♦ s♦ ♣r ♥rtr ♦s ♠♦♠♥t♦s ♠♥ét♦s s s

♠ ♥♦sss s♠çõs ♦ ♥♦ss♦ ♣ss♦ ♦♥t r♦ ♦ t♦ ♣♦r ♥♦ ♠í♥♠♦ 4 × Nspins

Nspins ≡ ♥ú♠r♦ s♣♥s r rçõs ♠ q ♠ rçã♦ ♠

r r s♦rt t♦r♠♥t rs s ss ♣♦r tr ♦ s ♠♦♠♥t♦

♠♥ét♦ ♥rt♦

s rs sts sã♦ ♥s ♣♦s ♣r♦ss♦s s♠♥t③çã♦ ♣♦r q

♦tçõs ♠t♦♦♦

♠ ♣ss ♣♦r ♠s rtrísts ♦♥srs ♠ ♣r♦ss♦ s r q

♣ssr♠ ♣♦ ♣r♦ss♦ s♠♥t③çã♦ trés ♠ ♠♣♦ r♦t♦♥ sã♦ s

ss s♦♠♥t ♣ s♦r♠ q ss s s♦r♠ s s rs q ♣ssr♠

♣♦ ♣r♦ss♦ s♠♥t③çã♦ trés ♠ ♠♣♦ ♠♥ét♦ ♠ ♠ ú♥

rçã♦ sã♦ sss ♣ s♦r♠ s s ♣♦ ♥♦ s♥♠♥t♦ ♥tr

♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ r

♣ít♦

st♦s sssõs

Pr♦ss♦ s♠♥t③çã♦ trés ♠♣♦

♠♥ét♦ ①tr♥♦ ♠ ♠ ú♥ rçã♦

♣r♠r♦ ♣r♦ss♦ st♦ ♦ ♦ s♠♥t③çã♦ trés ♠ ♠♣♦

♠♥ét♦ ①tr♥♦ ♠ ♠ ú♥ rçã♦ s♠r ♦ ♣r♦ss♦ strs st

♣r♦ss♦ ♠ ♠♣♦ ♠♥ét♦ ①tr♥♦ s♥t♠♥t ♦rt é ♣♦ ♥ rçã♦ +x

♠♥t③♥♦ r ♥st rçã♦ ③♥♦ ♦♠ q ♣♥s érts t♣♦ V 5 st♠

♣rs♥t ♠ s st ♠♣♦ srá ♥t♠♥t ♥rt♦ té q ♠♥t③çã♦

r s ♦♠♣t♠♥t ♥rt ♣♦r ♠ ♦ ♠♣♦ srá ♥♦♠♥t ♥t♠♥t

♥rt♦ té ♥çr ♦ st♦ ♥ r

Pr♠r ♥ás ♦♠♥t s♦r♠ ♥♦s sít♦s r

♥♠♥t ♥s♠♦s ♦♠♦ sr ♦ ♦♠♣♦rt♠♥t♦ ♠♥t③çã♦ ♠é

r t♠♥♦ L = 30 ♣r r♥ts t♣♦s s♦r♠ r 4.1 ♠♦str ♦

♦♠♣♦rt♠♥t♦ ♠♥t③çã♦ ♠ ♥çã♦ ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♣r ♥♦

♦rs r♥ts s♦r♠

st♦s sssõs

r ♥t③çã♦ ♠é Mag ♣♦r ♠♣♦ ♠♥ét♦ ①tr♥♦ ~Bext ♣r r♥ts♦rs s♦r♠ t♠♥♦ L = 30

st r ♣rs q t♦s s rs sã♦ ♠t♦ ♣ró①♠s ♠ ♠ ♣r

♠r ♥ás ♣♦s ③r q s♦r♠ ♣r s♦♠♥t s③r s rs ♠

♥t③çã♦ q♥t♦ ♠♦r s♦r♠ ♠s s é r ♠♥t③çã♦ sr

q s rs ♦♠ ♠♦rs s♦r♠ sã♦ s ♣r♠rs ♦♠çr♠ s s♠♥t

③r♠ ♦ ♦s sít♦s ♠s ss♣tís ♥rsã♦ ss ♠♦♠♥t♦s ♠♥ét♦s

t♠é♠ sã♦ s út♠s tr♠ ss ♠♥t③çõs ♦♠♣t♠♥t ♥rts

♦ ♦s sít♦s ♠♥♦s ss♣tís ♥rsã♦ ss ♠♦♠♥t♦s ♠♥ét♦s ♠ ♠

s♥ ♥s ♦sr♠♦s q ♥st r ①st♠ ♠ ♣tôs ♦♠ Mag = +1.0

♠ ♣tô ♦♠ Mag = −1.0 ♠ ♣tô ♦♠ Mag ≈ 0.0 sr q ♦s t♠♥♦s

sts ♣tôs sã♦ ♥rs♠♥t ♣r♦♣♦r♦♥s à s♦r♠ r q♥t♦ ♠♦r

s♦r♠ ♠♥♦r sã♦ ♦s t♠♥♦s ♦s ♣tôs s ♣tôs ♠♥t③çã♦ Mag = ±1.0

sã♦ rr♥ts às rs ♦♠♣t♠♥t ♠♥t③s ♦♠♣♦sts s♦♠♥t ♣♦r érts

t♣♦ V 5 ♦ ♣tô ♠♥t③çã♦ Mag ≈ 0.0 é rr♥t à r ♦♠♣t♠♥t s

♠♥t③ ♦♠♣♦st s♦♠♥t ♣♦r érts t♣♦ V 1 st t♦ ♠s ♥t

♦ ♥sr♠♦s rçã♦ ♣♦♣çã♦ érts ♠ rçã♦ ♥t♥s ♦ ♠♣♦

st♦s sssõs

♠♥ét♦ ①tr♥♦ ♣rs♥t ♥ r 4.2

r rçã♦ ♣♦♣çã♦ érts ♠ ♥çã♦ ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ ~Bext

♣r s♦r♠ ♥ ♠♠ a) ♠♦str ♦ ♣r♦ss♦ ♥tr ♦s ♦rs ♠♣♦s 0u.r. 300u.r. ♠♠ ♠♦str ♦♠ ♠s ♥çã♦ s ♣ss♥s tr♥sçõs q sã♦ ís s③r ♥ ♠♠ s ♦♥rçõs ♦s érts ♣tô é r♣rs♥t♦ ♣♦♥rçã♦ ért ①♦ sts ♣q♥ ♠♠ ♥sr ♥ ♠♠ é ♠♠♣çã♦ ♠s t s♥ tr♥sçã♦ ♠♦str♥♦ ①stê♥ érts V 4 V 6♥ s♥ tr♥sçã♦ s♥♦ ♠t♦ ♥r♦rs s ♠s ♣rs♥ts t♠♥♦ L = 30

st♦s sssõs

♦ à s♠tr ♦ ♣r♦ss♦ s♠♥t③çã♦ ♥st r ♥sr♠♦s s♦

♠♥t ♣rt ~Bext ≥ 0.0 ♥♠♥t s ♥t③r q ♦s ♣tôs ♦sr♦s ♥

r 4.1 sã♦ r♠♥t ♦♠♣♦st♦s s♦♠♥t ♣♦r ért t♣♦ V 1 ♦ st♦ ♥♠♥t

r V 5 ♦ st♦ ♦♠♣t♠♥t ♠♥t③♦ sss ♣tôs sã♦ s♣r♦s ♣♦r

s tr♥sçõs ♣r♠r tr♥sçã♦ q r st♦ ♦♠♣t♠♥t ♠♥

t③♦ ♣r ♦ st♦ ♥♠♥t ♠ Bext ∼ 85.0 s♥ tr♥sçã♦ q r

♦ st♦ ♥♠♥t ♣r ♦ st♦ ♦♠♣t♠♥t ♠♥t③♦ ♠ Bext ∼ 220.0

♣r♠r tr♥sçã♦ ♦s érts t♣♦ V 5 sã♦ ♥q♦s ♣r rçã♦ érts

t♣♦ V 4 s ss ♠♥ç é r♣♠♥t ♦♠♣♥ ♥qçã♦ ♦s érts

t♣♦ V 4 rçã♦ érts t♣♦ V 1 ③♥♦ ss♠ ♣♦♣çã♦ érts t♣♦ V 4

sr ♠t♦ ♣q♥ ❯♠ t♦ ♥trss♥t é q s♦♠♥t ♦s s♣♥s ♥♦s ♦♠ ♦ ①♦

x tê♠ ss ♠♦♠♥t♦s ♠♥ét♦s ♥rt♦s ♦ s s♦♠♥t s♣♥s ért

♦r♠ ♥rt♦s s ♣ss♥s st tr♥sçã♦ sã♦ srts ♣♦r V 5 → V 4 → V 1

s♥ tr♥sçã♦ é s♠r ♦ ♥rs♦ ♣r♠r ♠s ♦♠ ♦ ♣r♠♥t♦

♠s ♦s t♣♦s érts t♣♦ V 2 t♣♦ V 6 st tr♥sçã♦ érts t♣♦ V 1 sã♦

♥q♦s ♣r rr érts t♣♦ V 4 sts sã♦ r♣♠♥t ♥q♦s ♣r rr

érts t♣♦ V 2 ♦t q ♦s érts t♣♦ V 2 sã♦ ♠s stás q ♦s érts t♣♦

V 4 ♣♦s s rçã♦ ♥ã♦ é s r♣♠♥t ♣♦r s ♥qçã♦ ♠ s

sts érts sã♦ strí♦s ♣r rçã♦ érts t♣♦ V 6 sts sã♦ r♣♠♥t

strí♦s ♣r rçã♦ érts t♣♦ V 5 s ♣ss♥s st tr♥sçã♦ sã♦ srts

♣♦r V 1 → V 4 → V 2 → V 6 → V 5 st tr♥sçã♦ ♦sr♠♦s q ♦s s♣♥s q ♥ã♦

♦r♠ ♥rt♦s ♥ ♣r♠r tr♥sçã♦ qs q ♥ã♦ stã♦ ♥♦s ♦♠ ♦ ①♦

x tê♠ ss ♠♦♠♥t♦s ♠♥ét♦s ♥rt♦s ♥st tr♥sçã♦ ♦ ♥ sts s

tr♥sçõs ♠♦s q ♠♥t③çã♦ r ♦ ♦♠♣t♠♥t ♥rt ♣ss♥♦

♠ sst♠ ♦♠♣♦st♦ s♦♠♥t ♣♦r érts V 5−x érts V 5 ♦♠ ♠♥t③çã♦ ♥

rçã♦ −x ♣r ♠ sst♠ ♦♠♣♦st♦ s♦♠♥t ♣♦r érts V 5+x érts V 5 ♦♠

♠♥t③çã♦ ♥ rçã♦ +x

♦ ♦♥srr s♦r♠ r♠♦s q st tr ♠t♦ ♥â♠ ♥ r

♦ q ♥ã♦ ♣♦ sr ♥♦t♦ ♥ r 4.1 r 4.3 ♠♦str ♦ rá♦ rçã♦

♣♦♣çã♦ érts ♠ rçã♦ ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♦♠ s ♦♠ s♦r♠

st♦s sssõs

b′i = 10.0%

r rçã♦ ♣♦♣çã♦ érts ♣♦r ♠♣♦ ♠♥ét♦ ①tr♥♦ ~Bext ♣r s♦r♠ b′i = 10.0% t♠♥♦ L = 30

r 4.3 ♣rs q ♦ ♦♠♣♦rt♠♥t♦ ♣♦♣çã♦ érts é s♠r

♦ r 4.2 ♠ r s♠ s♦r♠ ♠s ♥st ♣rs♥t ♠ r♥ rs♠♥t♦

♣♦♣çõs érts t♣♦ V 4 V 6 rs♠♥t♦ sts érts stá rt♠♥t

♦ ♦♠ ♠♥t s♦r♠ ♦s sít♦s ♠♥r q q♥t♦ ♠♦r ♦r

s♦r♠ ♠♦r srá rçã♦ sts t♣♦s érts r③ã♦ st♦ stá à

s♦r♠ ♥♦s sít♦s ♦s qs ♣♦♠ tr ss ♠♦♠♥t♦s ♠♥ét♦s ♠s ss♣tís

b′i < 0.0 ♦ ♠♥♦s ss♣tís b′i > 0.0 ♦ ♥♠♥t♦ ♦♠ ♦ ♠♣♦ ♠♥ét♦

①tr♥♦ ♣r♠r tr♥sçã♦ ár♦s sít♦s ♦♠ b′i < 0.0 trã♦ ss ♠♦♠♥t♦s ♠♥é

t♦s ♥rt♦s ♣r ♥t♥ss ♠♣♦ ♠♥ét♦ ①tr♥♦ ♠♥♦rs ♦ q s rs

s♠ s♦r♠ r♥♦ ♠ ♣♦♣çã♦ érts t♣♦ V 4 ♣ tr♥sçã♦ V 5 → V 4

♥s sts érts srã♦ ♠s stás ♠s♠♦ ♦♠ ♦ ♠♥t♦ ♥t♥s ♦

♠♣♦ ♠♥ét♦ ①tr♥♦ ♦ ♣rs♥ç sít♦s ♦♠ b′i > 0.0 ss♠ ♣♦♣

çã♦ ért V 4 srá rs♥t té ♠ ♦r ♠t ♠♣♦ ♥♦ q ♦s sít♦s ♦♠

st♦s sssõs

b′i > 0.0 t♠é♠ trã♦ ♦s ss ♠♦♠♥t♦s ♠♥ét♦s ♥rt♦s s♠♣r ♠♥♥♦

♣♦♣çã♦ ért V 4 ♣r ♥t♥ss ♠♦rs ♦ q ♠t ❯♠ ♣r♦ss♦

s♠r ♦♦rr ♥ s♥ tr♥sçã♦ ♠♥t♥♦ ♣♦♣çã♦ érts V 4 V 6

r③♥♦ ♣♦♣çã♦ ért V 2 ♦ ♦ rs♠♥t♦ sts érts ♦ t♠

♥♦ ♦ ♣tô ♦ ért V 1 é r③♦ r♥♦ ♠ ♠ s ♦tr ♦ st♦

♥♠♥t ♦ sst♠

tr♦ ért q ♣r ♥st tr♥sçã♦ ♦♠ rqê♥ ♠t♦ ♠♥♦r ♠

rçã♦ ♦s ♠s ♦sr♦s é ♦ ért t♣♦ V 7 r♥t ♦s ♦tr♦s érts ♦

ért t♣♦ V 7 ♣r ♦ s♦ ♣♦♥♦ ♣rr ♦ ♥ã♦ r♥t ♦ ♣r♦ss♦

s♠♥t③çã♦ ♠ r rts q ♦ s ♣r♠♥t♦ ♦♥tç ♦

érts q t♥♠ ♦s sít♦s b′i < 0.0 ♦s sít♦s b′i > 0.0 é♠ ♦s ♦tr♦s ♦s

sít♦s q á tr♠ s ♠♥t③çã♦ ♥rt ♥ ♣r♠r tr♥sçã♦ r 4.4

♠♦str♠♦s ♠ ♣q♥ strçã♦ st ♦♥rçã♦ ♦♠♦ ♦♥tr rçã♦

sts érts

r r ♠ ♠♦str ♣ss♠ ♠ ért ♥ ♦♥rçã♦ ♦ st♦♥♠♥t V 1 ♣r ♦ st♦ ♠♥t③♦ V 5 st ♣ss♠ ♦ ért ♥tr♠ár♦ ♥tr sss ♦s é ♦ V 7 s s♣♥s ♣rt♦ sã♦ s♣♥s ♥tr♦s s♦r♠ ♥ã♦é ♠♣♦rt♥t ♦s s♣♥s ③s t♠ b′i < 0.0 ♦s s♣♥s r♠♦s t♠ b′i > 0.0

♦ ♣r♦ss♦ srt♦ ♣♦r st r ♠♦s q ♦s s♣♥s ③s b′i < 0.0 sã♦

♦s ♣r♠r♦s tr♠ ♦s ♠♦♠♥t♦s ♥rt♦s s♥♦ ♦♥rçã♦ V 1 ♣r ♦♥

rçã♦ V 7 ♦♦ ♠ s ♦♠ ♦ ♠♥t♦ ♥t♥s ♦ ♠♣♦ ♠♥ét♦

①tr♥♦ ♦s s♣♥s r♠♦s b′i > 0.0 trã♦ ♦s ss ♠♦♠♥t♦s ♠♥ét♦s ♥rt♦s

r♥♦ ♦♥rçã♦ ért V 5 sttst♠♥t ♣♦s ♣rr q ♣r♦

st♦s sssõs

♦♥rçã♦ r 4.4 ♣rr ♥ r ♥ã♦ sr tã♦ ♣q♥ 1/16

s♦ ♦♥sráss♠♦s q ♦s s♣♥s r♠♦s ③s tê♠ s ♣r♦s 1/2

♠ ♦tr♠ ♦rs b′i 6= 0.0 s ♦ ♥sr ♥r ♦♥rçã♦ r

2.15 ♠♦s q ♥r ♦♥rçã♦ ♦ ért V 7 27D é é ♠s ♦ q ♦

♦r♦ ♦ q ♦ ért V 6 13D ♦ st♦ s♣rs q ♣ss♠ r

4.4 s♦♠♥t ♦♥tç s♦ t♦♦s ♦s ♦rs b′i ♦tss♠ ♦rs r♥s q sr

sttst♠♥t ♠s í ♦♦rrr st ♦♥sã♦ stá ♦r♦ ♦♠ ♦s ♦s

♦t♦s r 4.3 ♠s t③ ♥ã♦ s ú♥ ①♣çã♦

Pr ♥③r st sçã♦ ♠♦s ♥sr ♠♦r qs ♦s t♦s s♦r♠ ♥

♥â♠ érts r♥t st ♣r♦ss♦ s♠♥t③çã♦ r 4.5 ♠♦str ♦

rá♦ ♥â♠ ♣♦♣çã♦ érts ♠ rçã♦ ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦

♦♠ s ♦♠ s♦r♠ b′i = 20.0%

r rçã♦ ♣♦♣çã♦ érts ♣♦r ♠♣♦ ♠♥ét♦ ①tr♥♦ ~Bext ♣r s♦r♠ b′i = 20.0% t♠♥♦ L = 30

st r ♣♦♠♦s ♥♦tr q ♣r s♦r♥s ♠t♦ r♥s ♦ ♣tô ér

t V 1 é r③♦ ♠ ♦♠♣rçã♦ ♦♠ ♦s rst♦s ♣rs♥t♦s ♥tr♦r♠♥t t♥t♦

st♦s sssõs

♠ s tr q♥t♦ ♠ s rr s♠♦ ss♠ ♠♦s q st ♣r♦ss♦ ♦r

♦♥s rst♦s ♣♦s ♣♦♣çã♦ érts V 1 sr s♣r♦r 95%

♠♥çã♦ ♦ ♣tô ért V 1 ♦♦rr ♦ ♦s sít♦s ♦♠ b′i ≫ 0.0 q tê♠ ss

♠♦♠♥t♦s ♠♥ét♦s s♦♠♥t ♥ rçã♦ x ♦s sít♦s ♦♠ b′i ≪ 0.0 q tê♠ ♦s ss

♠♦♠♥t♦s ♠♥ét♦s ♦ ♦♥♦ ♠♦s ♦s ①♦s ♦ ♣♥♦ r♥t ♦ ♣r♦ss♦

s♠♥t③çã♦ ♦s sít♦s ♦♠ b′i ≫ 0.0 trã♦ ♦s ss ♠♦♠♥t♦s ♠♥ét♦s ♥rt♦s

s♦♠♥t ♣r r♥s ♥t♥ss ♠♣♦ ♠♥ét♦ t♥♦ ♦r♠çã♦ ♦

♣tô V 1 ♦s sít♦s ♦♠ b′i ≪ 0.0 trã♦ ♦s ss ♠♦♠♥t♦s ♠♥ét♦s ♥rt♦s ♣r

♣q♥s ♥t♥ss ♠♣♦ ♣♦♥♦ sr ♥rt♦s té ♠s♠♦ ♥ts ♦s sít♦s

♦♠ ♠♥t③çã♦ s♦♠♥t ♠ x ♣r♥♣♠♥t qs ♦♠ b′i ≫ 0.0 st ♠s♠♦

rá♦ ♣♦♠♦s ♦srr q ♦s érts t♣♦ V 4 sã♦ ♦s ♠s ♣rs♥ts r♥t t♦♦

♦ ♣r♦ss♦ ♠s♠♦ q ♠ ♣q♥s q♥ts

st ssçã♦ ♠♦s q ♦ ♣r♦ss♦ s♠r ♦ strs é ♠ ♦♠ ♣r♦ss♦

s♠♥t③çã♦ ♦s ♦s s♣♥s tr♥rs rts s♠ s♦r♠ ❱♠♦s q

s rs ♦♠ ♣q♥s s♦r♥s ♦♥sr♠ ♥çr ♦s ss st♦s ♥♠♥ts

s rs ♦♠ r♥s s♦r♠ ♦♥sr♠ ♦tr ♠ ♣♦♣çã♦ ért V 1

s♣r♦r 95% ♦s ♠s♠♦s q ♦♠♣õ♠ ♦ st♦ ♥♠♥t

♥ ♥ás s♥♠♥t♦ r♠♣♦ ♠♥ét♦

①tr♥♦ s♦r♠ ♥♦s sít♦s r

❯♠ ♦s ♣r♦♠s q ♣♦r ♣rr ♥ ①çã♦ st ♣r♦♠♥t♦

s♠♥t③çã♦ sr ♠♣rsã♦ ♥r ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♦♠ r

♦s s♣♥s ♦ às ♣q♥s ♠♥sõs r st ♣r♦ss♦ ♦ ♦sr

♦ q ♦ s♥♠♥t♦ ♦ ♠♣♦ ♦♠ ♦ ①♦ x r ♦s s♣♥s ♣r♦♦

♠♥çs ♥♦s rst♦s ♦t♦s ♥tr♦r♠♥t sss ♠♥çs srã♦ ♣rs♥ts

♠ três ♣rts ♥ ♣r♠r ♣rt ss ♥s srá t ♣r ♦ st♦ ♦ s♥

♠♥t♦ ♥s rs s♠ s♦r♠ ♥ s♥ srá t♦ ♦ st♦ ♦ s♥♠♥t♦ ♠

rs ♦♠ s♦r♠ ♥ út♠ srá t♦ ♦ st♦ ♦s t♦s s♦r♠ ♠ ♠

r q ♥ã♦ stá ♥ ♦♠ ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♦ à s♠tr ♦ ss

t♠ ♦s rst♦s st ssçã♦ srã♦ ♣rs♥t♦s ♣r â♥♦s s♥♠♥t♦s

st♦s sssõs

θ ≥ 0.0♦

st♦s ♦ s♥♠♥t♦ ♦ ♠♣♦ ♠ r ♦♠ sít♦s s♠ s♦r♠

r 4.6 ♠♦str ♦♠♦ ♦s r♥ts â♥♦s ♠♣rsã♦ ♦♠ ♦ ①♦ +x

♣♦♠ tr ♠♥t③çã♦ ♠é ♥s rs s♠ s♦r♠

r ♥t③çã♦ ♠é ♦ sst♠ Mag ♠ rçã♦ ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦Bext s r♥ts rs r♣rs♥t♠ r♥ts â♥♦s ♠ r t♠♥♦ 30 s♠s♦r♠

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

tr♠ ♦ ♦♠♣♦rt♠♥t♦ ♠♥t③çã♦ ♠é r♥t ♦ ♣r♦ss♦ s♠♥t

③çã♦ P♦r ♦tr♦ ♦ sts ♠♥♠ ♦ t♠♥♦ ♦ ♣tô Mag ≈ 0.0 ♦♠♣♦st♦

s♦♠♥t ♣♦r érts V 1 r♠ t♠é♠ ♠ ♥♦♦ ♣tô st ♥♦♦ ♣tô r♣rs♥t

♠ ♥♦♦ st♦ ♠♥t③♦ Magxy ♥ q ♠♥t③çã♦ stá ♦ ♦♥♦ ♦s ①♦s

x y ♠♥çã♦ ♦ ♣tô Mag ≈ 0.0 ♦ ♣r♠♥t♦ ♦ t♠♥♦ ♦ ♥♦♦ ♣tô

stã♦ ♥t♠♠♥t ♦s ♦♠ ♦♠♣♦♥♥t ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♥ rçã♦

−y Pr ♠♦r ♥t♥♠♥t♦ st ♣r♦ss♦ srã♦ ♣rs♥ts s strçõs s

♦♥rçõs érts ♦s qtr♦ ♣tôs r 4.10 q♥♦ rs ♦ ♠♣♦

st♦s sssõs

♠♥ét♦ ①tr♥♦ Bext = +200.0 té Bext = −200.0

r strçã♦ ♥â♠ ♠ ért r s ♠♥s sã♦rs♣t♠♥t s ♦♥rçõs ♦ ♣r♠r♦ ♦ s♥♦ ♦ trr♦ ♦ qrt♦ ♣tôs ♣rs♥ts ♥ r 4.6 ♠♠ sr ♣r ♥♦♠r r♣♦ s♣♥s r ①♣çõs♦♥s ♥♦♥tr♠s ♥♦ t①t♦

r 4.7 ♣ss♠ 4.7.a) → 4.7.b) r♣rs♥t s♠♥t③çã♦ ♣

♦ st♦ ♦♠♣t♠♥t ♠♥t③♦ ♣r ♦ st♦ ♥♠♥t ♥â♠ st

♣r♦ss♦ ♦♠♦ ♦ ♦sr♦ ♥ r 4.6 ♥ã♦ ♦ t ♣♦ s♥♠♥t♦ ♦

♠♣♦ ♦♠ ♣ ♣♦r ss ♠♦t♦ s♦♠♥t ♦s ít♦s tr♠ ss ♠♦♠♥t♦s

♠♥ét♦s ♥rt♦s r 4.7.e) s ♦ t♥tr ♠♥t③r r sr

♦ st♦ ♥♠♥t 4.7.b) r ♣r ♦ st♦ ♠♥t③♦ 4.7.d) ♠♦s q ①st

♠ ♥♦ ♦♥rçã♦ ♥tr♠r ♠ ♦♠ ♠♥t③çã♦ ♥s rçõs −x −y

st♦s sssõs

st ♥♦ ♦♥rçã♦ é r rçs à ♦♠♣♦♥♥t ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♥

rçã♦ −y ♦r♦ ♦♠ qçã♦ 3.8 ♦♠♣♦♥♥t ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦

♥ rçã♦ −y ♣r ♥rsã♦ ♦s ♠♦♠♥t♦s ♠♥ét♦s ♦s ít♦s sr

q ♦ ♣r♦ss♦ 4.7.b) → 4.7.c) é ①t♠♥t ♦ ♣r♦ss♦ ♥rs♦ ♦ s♠♥t③çã♦

4.7.a) → 4.7.b) ♠s ♦♠ ♠♥t③çã♦ r ♥s rçõs x y r 4.6

♠♦s q ♠♥çã♦ ♦ ♣tô Mag ≈ 0.0 ♦ ♣r♠♥t♦ ♦ ♥♦♦ ♣tô stã♦

r♦♥♦s ♦♠ ♦ â♥♦ s♥♠♥t♦ ♣♦r ♦♥sqê♥ ♦♠ ♥t♥s

♦ ♠♣♦ ♥ rçã♦ −y s q ♥t♥s ♦ ♠♣♦ ♠♥ét♦ ♥ rçã♦

−y ~By é rt♠♥t ♣r♦♣♦r♦♥ | ~Bext| ·sin θ ♠♥r q q♥t♦ ♠♦r ♦ ♦r

θ ♠♦r srá ♦ ♦r ~By ♥tã♦ ♣♦s ♦♥r q ♣r r♥s ♦rs θ ♦

♠♣♦ ♠♥ét♦ ♥ rçã♦ −y é r♥ ♦ s♥t ♣r ♥rtr r♣♠♥t t♦♦s

ít♦s ♠♥♥♦ ♦ ♣tô Mag ≈ 0.0 r♥♦ ♠ ♥♦♦ ♣tô V 5xy ♦♠♣♦st♦ ♣♦r

érts V 5 ♠s ♦♠ ♠♥t③çã♦ ♥♦s ①♦s x y ❱ q st ♠s♠♦ ♠♣♦

t♥t ♠♥tr ♦s ♠♦♠♥t♦s ♠♥ét♦s ♦s ít♦s ♣♦♥t♥♦ ♥ rçã♦ −y ♣♦r ♠s

t♠♣♦ ♠s♠♦ ♥ ♣rs♥ç ♦tr ♦♠♣♦♥♥t ♦ ♠♣♦ ♥ rçã♦ −x t♥t♥♦

♥rtr sts ♠♦♠♥t♦s ♠ ♦tr rçã♦ ♦ st♦ ♣r♠♦s q ♦ t♠♥♦

♦ ♣tô V 5xy é ♣r♦♣♦r♦♥ θ ~By ♣♦r ♠ ♦sr♠♦s ♠s♠♦ ♣r ♦ ♠♦r

♦r θ ♦ ♠♣♦ ♠♥ét♦ ♥ rçã♦ −x qr ♦rs r♥s ♦ s♥t

♣r ♦♥sr ♥rtr ♦s ♠♦♠♥t♦s ♠♥ét♦s ♦s ít♦s st t♦ stá str♦

♣ ♣ss♠ 4.7.c) 4.7.d) ♠ q ♦s érts V 5xy ♣ss♠ ♣r érts V 5x

érts ♦♠ ♠♦♠♥t♦ ♠♥ét♦ s♦♠♥t ♥♦ ①♦ x r ♥♠♥t t♠ s

♠♥t③çã♦ ♠é ♦♠♣t♠♥t ♥rt

♥â♠ srt ♠ ♣♦ sr ♠♦r ♦♠♣r♥ ♦♠ ①í♦ r

4.8 st r ♠♦str ♥â♠ ♣♦♣çã♦ érts ♠ rçã♦ ♦ ♠♣♦

♠♥ét♦ ①tr♥♦ ♣r â♥♦ 10.0♦

st♦s sssõs

r rçã♦ ♣♦♣çã♦ érts ♣♦r ♠♣♦ ♠♥ét♦ ①tr♥♦ ~Bext ♠ ♠â♥♦ 10.0♦ ♦♠ sít♦s s♦r♠ ♥ ♠♠ a) ♠♦str ♦ ♣r♦ss♦ ♥tr ♦s ♦rs ♠♣♦s 0u.r. 360u.r. ♠♠ ♠♦str ♦♠ ♠s ♥çã♦ s ♣ss♥s tr♥sçõsq sã♦ ís s③r ♥ ♠♠ s ♦♥rçõs ♦s érts ♣tô ér♣rs♥t♦ ♣ ♦♥rçã♦ ért ①♦ sts t♠♥♦ L = 30

♦ ♥sr s rs 4.2 4.8 ♣♦s ♥♦tr q ①st♠ três r♥çs s♥

ts ♣r♠r stá ♥ r♥ ♠♥çã♦ qs ①t♥çã♦ ♦s érts t♣♦ V 2 ♥

st♦s sssõs

s♥ tr♥sçã♦ ♣♦♣çã♦ érts s♥ r♥ç é ♦ ♣r♠♥t♦

♠ trr tr♥sçã♦ t ♣ ♠♥çs érts V 5xy → V 6 → V 5x ❱ q

♦ ért t♣♦ V 6 é ♠ ért tr♥sçã♦ ♠ t♦ á ♦sr♦ ♥ts ♥ r 4.2

út♠ r♥ç é ♠♥ç ♣♦sçã♦ s♥ tr♥sçã♦ ♦ s♦ ♦ ♠♣♦

♣rt♠♥t ♥♦ r 4.2 s♥ tr♥sçã♦ ♦♦rr ♠ ♣r♦①♠♠♥t

| ~Bext| = 220D/µ ♣r ♦ s♦ ♦♠ s♥♠♥t♦ θ = 10.0♦ r 4.7 s♥

tr♥sçã♦ ♦♦rr ♠ ♣r♦①♠♠♥t ♠ | ~Bext| = 180D/µ ♥â♠ ♣♦♣çã♦

érts s rs s♥s ♦♠ ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♣♦ sr srts

♣♦r V 5x → V 4 → V 1 → V 4 → V 5xy → V 6 → V 5x

st ssçã♦ ♦sr♠♦s q ♦ s♥♠♥t♦ ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦

♦♠ ♣ t ♦t♥çã♦ ♦ st♦ ♥♠♥t r r♥t ♦s s♦s

♦♠ s♦r♠ ♦ st♦ ♥♠♥t r stá ♠s ♠ ♥♦ ♦♠♦ ♣♦s

♦♠♣rr ♥s rs 4.2 4.8 ss♠ ♣♦s ♦♥r q ♦ ♣r♦ss♦ s♠r ♦

strs é ♠ ♦♠ ♣r♦ss♦ ♣r s♠♥t③r ♦tr ♦ st♦ ♥♠♥t r

♦s s♣♥s tr♥rs rts ♠s♠♦ s st r stá s♥ ♦♠ ♦

♠♣♦ ♠♥ét♦ ①tr♥♦

st♦s ♦ s♥♠♥t♦ ♦ ♠♣♦ ♠ r ♦♠ sít♦s ♦♠ s♦r♠

st ssçã♦ ♥sr♠♦s ♦♠♦ ♦ s♥♠♥t♦ ♦ ♠♣♦ ♠♥ét♦ ①

tr♥♦ t ♦ ♦♠♣♦rt♠♥t♦ ♠ r ♦♠ s♦r♠ ♥♠♥t ♠♦strr♠♦s

♦♠♦ st t s rs ♦♠ b′i = 10.0% s♦r♠ ♠ s s rs ♦♠

b′i = 20.0% s♦r♠ r 4.9 ♠♦str ♦ ♦♠♣♦rt♠♥t♦ ♠♥t③çã♦ ♠é

♠ r ♦♠ s♦r♠ b′i = 10.0% ♠ rçã♦ ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦

♣r r♥ts â♥♦s s♥♠♥t♦

st♦s sssõs

r ♥t③çã♦ ♠é r Mag ♣♦r ♠♣♦ ♠♥ét♦ ①tr♥♦ ~Bext ♠ r♥ts â♥♦s s♥♠♥t♦ st r ♦s sít♦s tê♠ s♦r♠ b′i = 10.0% t♠♥♦ L = 30

♦♠♦ ♥s ssçõs ♥tr♦rs ♠♦s q s♦r♠ s③ s rs

♠♥t③çã♦ ❱♠♦s t♠é♠ q s♦r♠ ♠♥ ♠s ♦ ♣tô Mag ≈ 0.0

t♥♦ ♦t♥çã♦ ♠ st♦ s♠♥t③♦ ♥ t ♦r♠çã♦ ♦

♣tô V 5xy ♦tr ♦ st♦ s♠♥t③♦ é ♦♥sqê♥ ♦s ♠s♠♦s

♣r♦♠s st♦s ♥s s ssçõs ♥tr♦rs s rs ♦♠ s♦♠♥t s♦r♠

s rs s♦♠♥t s♥s r 4.10 ♠♦str ♥â♠ ♣♦♣çã♦ ért

s ♠ rçã♦ ♥t♥s ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♠ r ♦♠ s♦r♠

b′i = 10.0% ♦♠ ♠ s♥♠♥t♦ θ = 10.0♦ q rá ♥t♥r ♠♦r st

♣r♦ss♦

st♦s sssõs

r rçã♦ ♣♦♣çã♦ érts ♣♦r ♠♣♦ ♠♥ét♦ ①tr♥♦ ~Bext ♠ ♠â♥♦ 10.0♦ st r ♦s sít♦s tê♠ s♦r♠ b′i = 10.0% t♠♥♦L = 30

trés st r ♦srs q ♦ ♣tô V 1 é ♦r♠♦ t♠ s t♠♥♦

♠ ♣♦♦ r③♦ ♦♠♦ ♣♦ sr ♦sr♦ ♥♦s rá♦s ♠♦str♦s ♥tr♦r♠♥t

rs 4.3 4.8 é♠ st♦ ♠♦s q ♦ ♣tô V 5xy t♠ ♦ s t♠♥♦ st♥t

r③♦ ♣r♥♦ ♠s ♠ st♦ tr♥sçã♦ ♥tr ♦s st♦s ♥♠♥t ♦

♦♠♣t♠♥t ♠♥t③♦ ❱ q ♦s érts t♣♦ V 4 V 6 sã♦ ♦s ú♥♦s érts

♦♠ ♠ rs♠♥t♦ ♣rá st é ♠ t♦ ♥trss♥t ♣♦s ♦ ♦♠♣rr ♦♠

r ♦♠ s♦r♠ s♦♠♥t r 4.3 ♠♦s q ♦s t♣♦s érts V 2 V 7 sã♦

♦♠♣t♠♥t ♠♥♦s ♦ ♦ s♥♠♥t♦ ♦ ♠♣♦ ♦♠ r

♦r ♦ ♦♥sr♠♦s s r ♦♠ ♦ ♦r♦ s♦r♠ b′i = 20% r♠♦s

q st ♦r ♦ s♦r♠ t ♦t♥çã♦ ♦ st♦ ♥♠♥t r

♦♠♦ ♦ st♦ ♥ r 4.5 r 4.11 ♠♦str ♦ ♦♠♣♦rt♠♥t♦ ♠♥t③çã♦

♠é ♠ r ♦♠ s♦r♠ b′i = 20.0% ♠ rçã♦ ♦ ♠♣♦ ♠♥ét♦

①tr♥♦ ♣r r♥ts â♥♦s s♥♠♥t♦

st♦s sssõs

r ♥t③çã♦ ♠é r Mag ♣♦r ♠♣♦ ♠♥ét♦ ①tr♥♦ ~Bext ♠r♥ts â♥♦s s♥♠♥t♦ st r ♦s sít♦s tê♠ s♦r♠ b′i = 20.0% t♠♥♦ L = 30

st r ♣♦s ♦srr q ♦ ♣tô Mag ≈ 0.0 ♥ã♦ stá ♠ ♥♦

♣r r♥s â♥♦s ss♠ s♣rs q ♦ st♦ ♥♠♥t ♥ã♦ s ♥

ç♦ sr q ♣r r♥s â♥♦s s♥♠♥t♦ ♠♥t③çã♦ r

s♦r ♠s ♦sçõs r♥t ♥rsã♦ ♠♥t③çã♦ sss ♦sçõs sr♠

♣ró①♠s ♦s ♣♦♥t♦s ♦♥ ♦s ♣tôs r♠ ♦r♠♦s ♥s rs s♠ s♦r♠ Pr ♦♠

♣r♥r♠♦s ♠♦r st ♦♠♣♦rt♠♥t♦ ♠♦s ♥sr s rçõs érts ♣r

s rs ♦♠ s♦r♠ b′i = 20.0% ♦♠ s♥♠♥t♦ θ = 10♦

st♦s sssõs

r rçã♦ ♣♦♣çã♦ érts ♣♦r ♠♣♦ ♠♥ét♦ ①tr♥♦ ~Bext ♠ ♠â♥♦ 10.0♦ st r ♦s sít♦s tê♠ s♦r♠ b′i = 20.0% t♠♥♦L = 30

♦ ♦♠♣rr s rs 4.10 4.12 ♥♦ts q ♦ ♠♥t♦ s♦r♠ ♥ r

♥ã♦ ♠♥t s♥♥t♠♥t ♦s ♦rs ♠á①♠♦s ♣♦♣çã♦ ♦s érts V 4

V 6 s ♣♦r ♦tr♦ ♦ ♠♦s q st ♣♦♣çã♦ rs ♠♥t ♥♦ ♥tr♦

~Bext = 100u.r. ~Bext = 200u.r. st ♠♥t♦ é ♦♠♣♥♦ ♠♥çã♦

♣♦♣çã♦ ♦s érts t♣♦ V 1 ♥st ♠s♠♦ ♥tr♦ é♠ ss♦ ♣♦s ♦srr

q ♦s ♣tôs ♦s érts V 1 V 5xy sã♦ ssttí♦s ♣♦r ♠á①♠♦s ♦s

st ssçã♦ ♠♦s ♦s rst♦s ♣r s rs ♦♠ s♦r♠ ♥♦s sít♦s ♦♠

s♥♦ ♥tr ♦ ♠♣♦ ♣ s♠♦ ♣r ♦s ♠♦rs ♦rs st ♦ ♣r♦ss♦

s♠r ♦ strs ♦♥s ♦tr ♠ ♠é ♠ ♣♦♣çã♦ érts t♣♦ V 1

s♣r♦r 85% ♣r♦①♠♥♦ ♠t♦ ♦ st♦ ♥♠♥t t♦t r ♦

st t♦ ♣♦s ♦♥r q ♦ ♣r♦ss♦ s♠♥t③çã♦ s♠r ♦ ♣r♦ss♦

strs é ♠ ♦♠ ♠ét♦♦ ♣r s♠♥t③r ♦tr ♦ st♦ ♥♠♥t ♦s ♦s

s♣♥s tr♥rs rts

st♦s sssõs

st♦s s♦r♠ ♣r rs s♥

st sçã♦ r♠♦s ♦♠♦ ♥t♥s s♦r♠ ♥♥ ♥ ♠♥t

③çã♦ r ♣r♠♥t s♥ r 4.13 ♠♦str ♠♥t③çã♦ ♠é

r ♠ ♥çã♦ ♥t♥s ♦ ♠♣♦ ♠♥ét♦ ♣r ♠ r s♥

♠ θ = 10.0♦ ♠ rçã♦ ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♣r r♥ts ♥t♥ss

s♦r♠

r rçã♦ ♣♦♣çã♦ érts ♣♦r ♠♣♦ ♠♥ét♦ ①tr♥♦ ~Bext ♠ ♠â♥♦ 10.0♦ ♣r r♥ts ♦rs s♦r♠ t♠♥♦ L = 30

trés st r ♦♥í♠♦s q ♦s t♦s s♦r♠ sã♦ s ♠ rs

♥s s♥s Prs q s rs ♦♠ ♠♦rs s♦r♥s sã♦ s ♣r

♠rs ♥trr♠ ♥♦ ♣r♦ss♦ s♠♥t③çã♦ t♠é♠ sã♦ s út♠s tr♠

s ss ♠♥t③çõs ♦♠♣t♠♥t ♥rts ♣tô ♦♠♣♦st♦ ♣♦r érts t♣♦

V 5xy é rst♠♥t t♦ ♣ s♦r♠ ♣♦s ♦ ♠♥t♦ s♦r♠ s♠♣r

♠♥ ♦ t♠♥♦ st ♣tô s ♥â♠s ♣♦♣çã♦ ♣r s r s♥s

♣r r♥ts ♥t♥ss s♦r♠ á ♦r♠ ♣rs♥ts ♥tr♦r♠♥t tr

és s ♦♥í♠♦s q s♦r♠ t ♦t♥çã♦ ♦ st♦ ♥♠♥t s

st♦s sssõs

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

♣tô ♦r♠♦ ♣♦s érts V 5xy é ssttí♦ ♣♦r ♠ ♠á①♠♦

Pr♦ss♦ s♠♥t③çã♦ trés ♠♣♦

♠♥ét♦ ①tr♥♦ r♦t♦♥

s♥♦ ♣r♦ss♦ st♦ ♦ ♦ s♠♥t③çã♦ trés ♠ ♠♣♦

♠♥ét♦ ①tr♥♦ r♦t♦♥ st ♣r♦t♦♦♦ ♦ ♠♣♦ ♠♥ét♦ é r③♦ ♠

♥ ♦t ♦♠♣t q á ♠ t♦r♥♦ ♣ ♥♠♥t t♥t♠♦s

s♠♥t③r ♣ trés ♦ ♣r♦ss♦ ♣rã♦ st ♣r♦t♦♦♦ ♣r ♠ ♠♣♦

♠♥ét♦ ①tr♥♦ ♠t♦ r♥ ♠♥í♦ ♥t♠♥t st ♣r♦ss♦ ♣rã♦

♠♦s q ♠ ♠é ♣ ♦ ♦♠♣t♠♥t s♠♥t③ ♠s ♥③♠♥t s

♣♦♣çã♦ ért t♣♦ V 1 ♦ ♠t♦ ♣q♥ ♥ã♦ ♦♥s♥♦ ♥çr ♦ st♦

♥♠♥t r ♦ st ♠♦t♦ srt♠♦s ♦ ♣r♦ss♦ ♣rã♦ s♦r

♠♦s ♠ ♥t♥s ♠♣♦ ♠♥ét♦ ♠♥♦r ♣r s♠♥t③r ♣ ♥♦ss

s♦ ♠♣♦ ♠♥ét♦ ♥ ♦ | ~Bext| = bc ♠s♠ ♥t♥s ♦ ♠♣♦ ♠

♥ét♦ ♠í♥♠♦ ♥rsã♦ ♦s ♠♦♠♥t♦s ♠♥ét♦s ♦s sít♦s r ♥♠♥t

t♥t♠♦s s♠♥t③r ♠ ♣ ♦♠♣♦st s♦♠♥t ♣♦r ért V 5 ♠♥t③ ♦

♦♥♦ rçã♦ +x

♥♠♥t ♠♦s ♥sr ♠♥t③çã♦ ♠é r ♠ ♥çã♦ ♦ â♥♦

♣r r♥ts s♦r♥s

st♦s sssõs

r ♥t③çã♦ ♠é Mag ♣♦r â♥♦ θ ♣r ♦rs s♦r♠ ♦♠ ♦

♠♣♦ ♥ | ~Biext| = bc t♠♥♦ L = 30

trés r 4.14 ♣♦♠♦s ♥♦tr q s♦r♠ é ♠ t♦r r♥

♥ê♥ ♣r st ♣r♦ss♦ s♠♥t③çã♦ Prs q st stá rt♠♥t

♦♠ ♦ t♠♣♦ s♠♥t③çã♦ ♣ ♣♦s q♥t♦ ♠♦r s♦r♠

♠♦r ♦ t♠♣♦ ♣r s♠♥t③r ♣

❯♠ t♦ ♥trss♥t é q s rs ♦♠ s♦r♠ ♦♥s♠ rr ♦♥rçõs

érts q ♥ã♦ sr♠ ssís s r s♠ s♦r♠ ♦ ♦s sít♦s ♦♠

b′i < 0.0 ♦♠ b′i > 0.0 st t♦ ♠s ♥t ♦ ♥sr ♦s rá♦s rçã♦

♣♦♣çã♦ érts ♣♦r â♥♦ ♣r s♦r♠ ♥ r 4.15 ♣r s♦r♠

b′i = 10.0% r 4.16 ♣r s♦r♠ b′i = 20.0% r 4.18

st♦s sssõs

r rçã♦ ♣♦♣çã♦ érts ♣♦r â♥♦ θ ♣r s♦r♠ ♥ t♠♥♦ L = 30

r 4.15 ♣♦s ♥♦tr q ♠ ♠ tr♠♥♦ ♥tr♦ rçã♦

érts t♣♦ V 5 ♠♥ r♣♠♥t rçã♦ érts t♣♦ V 1 rs qs ♥

♠s♠ ♣r♦♣♦rçã♦ ♠s ♠ ♣♦♦ ♠♥♦s ❱♠♦s q ♥st ♥tr♦ s rçõs

érts t♣♦ V 4 V 6 V 7 rs♠ ♠t♦ ♣♦♦ ♠s ♦♦ ♠ s ♦t♠ ♥ã♦

①str ♥ r ért t♣♦ V 2 t♠é♠ rs ♣♦♦ ♥st ♥tr♦ ♠s r♥t

♦s ♦tr♦s três ♦♥t♥ rsr ♥t♠♥t té ♥çr ♠ ♦r st♦♥ár♦

st r ♠♦s q ♥♦ ♥ ♦ ♣r♦ss♦ s♠♥t③çã♦ ♠ r s♠

s♦r♠ ①st♠ s♥♠♥t s♦♠♥t ♦s t♣♦s érts t♣♦ V 1 q

♦♠♣õ ♣r♦①♠♠♥t 95% ♦s érts r ♦ ért t♣♦ V 2 q ♦♠♣õ

♦s ♦tr♦s 5% sts érts sã♦ ①t♠♥t ♦s ♦s érts ♠♥♦r ♥r ♠♦s

rs♣t♠ rr ♦ ♦ três s♣♥s ♣♦♥t♥♦ ♣r ♥tr♦ três s♣♥s ♣♦♥t♥♦

♣r ♦r ♦ ért

st♦s sssõs

r rçã♦ ♣♦♣çã♦ érts ♣♦r â♥♦ θ ♣r s♦r♠ 10.0% t♠♥♦ L = 30

r 4.16 ♣♦s ♣rr q ♦ t♠♣♦ s♠♥t③çã♦ r ♦♠

s♦r♠ b′i = 10.0% ♦ ♣r♦①♠♠♥t ③s ♠♦r q ♦ r s♠

s♦r♠ st r ♣rs q t♦♦s ♦s ♣♦ssís érts q ♣♦♠ ♣rr

♥♦ sst♠ ♦♠ ①çã♦ ♦ t♣♦ V 8 tê♠ ♦rs s♥t♦s ♥♦ ♥ ♦ ♣r♦ss♦

s♠♥t③çã♦ ≈ 45% sã♦ V 1 ≈ 33% sã♦ V 2 ≈ 10% sã♦ V 3 ≈ 10% sã♦ V 4 ♠♥♦s

2% sã♦ V 5 V 6 V 7 0% sã♦ V 8 ss rst♦ é ♠ r♥t ♦ rst♦

♥♦♥tr♦ ♠ ♠ r s♠ s♦r♠ ♠ q ♦s ú♥♦s érts q ♣r♠ ♥♦

♥ s♠♥t③çã♦ r♠ ♦s érts t♣♦ V 1 V 2 sr q q♥t

érts ♣rs♥ts ♥s rs ♣♥ ♥r ♦♥rçã♦ st érts ♣♦s ♦s

ért ♠♥♦s ♥rét♦s ♦♠♦ V 1 V 2 t♠ s ♠♦rs ♣♦♣çõs ♥q♥t♦ q

♦s érts ♠s ♥rét♦s ♦♠♦ V 7 V 8 t♠ s ♠♥♦rs ♣♦♣çõs Pr ♠♦r

s③r st rst♦ srá ♠♦str♦ ♥ r 4.17 ♣♦♣çã♦ érts ♥♦ ♥

♦ ♣r♦ss♦ s♠♥t③çã♦ r♣rs♥t♦ ♥ r 4.16 ♣♦r ♥r ♦♥r

çã♦ ért st r ♦ ♦r ♥r ♦ ♠♥t♦ ♠ 51u.r. ♣r q ♥ã♦

♦rs ♥t♦s ♥r ♣♦♣çã♦ érts V 8 ♦ ①í ♦ rá♦

st♦s sssõs

♣♦s s ♣rs♥ç ♥ã♦ ♣rs♥t ♥♥♠ ♥♦r♠çã♦ r♥t ♣♦♣çã♦ ♥

♣r ♦♠♣r♥sã♦ st rá♦

r rçã♦ ♣♦♣çã♦ érts ♣♦r ♥r ♣r s♦r♠ 10.0% t♠♥♦ L = 30

st♦s sssõs

r rçã♦ ♣♦♣çã♦ érts ♣♦r â♥♦ θ ♣r s♦r♠ 20.0% t♠♥♦ L = 30

r 4.18 ♣rs♥t♠♦s ♦s rst♦s ♣r s rs ♦♠ s♦r♠ b′i = 20%

q ♠♦s q ♦ t♠♣♦ s♠♥t③çã♦ é ♠♦r qs ③s ♠ rçã♦ s

rs s♠ s♦r♠ qs ③s ♠ rçã♦ às rs ♦♠ b′i = 10% s ♠♣ts

s ♦sçõs s rçõs érts sã♦ ♥ ♠♦rs ♠ rçã♦ s r ♦♠ b′i =

10% st r ♣r♠♦s q s rçõs érts ♥♦ ♥ ♦ ♣r♦ss♦

s♠♥t③çã♦ ♥ã♦ stã♦ ♠s ♦r♥s ♣s ♥rs ♦♥rçõs érts

♣sr st♦ ♦s érts ♠♥♦rs ♥rs ♦♥t♥♠ s♥♦ ♠s ♥♠r♦s♦s ♦

q ♦s ♦tr♦s érts ♠s ♥rét♦s P♦s rst♦s ss r ♣♦s ♦♥r

q ♠ rs r♥s s♦r♥s st ♠ét♦♦ s♠♥t③çã♦ ♥ã♦ é ♥t

♣r s♠♥t③r ♥♠ ♦tr ♠ r♥ ♣♦♣çã♦ érts V 1

♦ ♥sr s qtr♦ út♠s rs ♦♥s q sr ♠♣♦ssí ♥çr ♦

st♦ ♥♠♥t r trés st ♣r♦ss♦ st ♦♥sã♦ stá ♣r♠♥t

♦rrt ♦ ♥sr♠♦s s rs s♠ s♦r♠ ♠♦s q r♠♥t é ♠♣♦ssí

♥çr ♦ st♦ ♥♠♥t ♣r t♦ r ♠s ③♠♥t ♦sr♠♦s ♠ ♦♠í

♥♦ érts ♦ st♦ ♥♠♥t ♥ rã♦ ♥tr r st rst♦ ♦♦rr

st♦s sssõs

♣♦rq ♦s sít♦s ♦r r ♥tr♠ r♠♥t ♦♠ ♦s ♠s sít♦s P♦r ♦tr♦

♦ ♦s sít♦s ♥trs stã♦ r♦♦s ♣♦r ár♦s ♦tr♦s sít♦s ♣rsr♥♦ ♥t

♦ ♦♥♥t♦ ♦ st t♦ ♦s sít♦s ♦r ♥ã♦ ♦♥s♠ ♥çr ♦ st♦

♥♠♥t ♥t♠♥t ♦♠ ♣rt ♥tr ♣♦s sts t♥♠ s ♥r ♠s

♠♥t ♦♠ ♠♣♦ ♠♥ét♦ ①tr♥♦ s♦r♠ ♦ sst♠ ♠♥t ♦ ♥♠r♦

sít♦s r q ♣r♠ s ♥t ♠♣♥♦ ♦t♥çã♦ ♠ rã♦ ♠

♥ ♦♠♣♦st s♦♠♥t ♣♦ st♦ ♥♠♥t r 4.19 ♣♦s ♥♦tr q

①st♠ r♥s rõs ♦♠♣♦sts s♦♠♥t ♣♦r érts ♥♦ st♦ ♥♠♥t ♠s

sts ♥ã♦ t♠ ♠ ♣rã♦ ♠ ♥♦ ♣r♥♦ t♦r♠♥t ♥♦ sst♠

r ♠♥s ♦s érts r ♣r ♦s t♣♦s s♦r♠ s ír♦s ♣rt♦sr♠♦s rs ③s sã♦ rs♣t♠♥t ♦s érts t♣♦ V 1 V 2 V 3 V 4 ❯♠r ♥t s♠ s♦r♠ sr q rã♦ ♥tr é ♦♠♣♦st s♦♠♥t ♣♦r értst♣♦ V 1 ♠ r ♥t ♦♠ s♦r♠ b′i = 10.0% ❱ ①stê♥ árs rõs♦♠♣♦st ♣♦r érts t♣♦ V 1 ♠s sts stã♦ s♣s t♦r♠♥t ♥ r

st sçã♦ ♠♦s q s♠♥t③çã♦ ♠ r ♦♠♣♦st ♣♦r ért

s V 5 trés ♦ ♣r♦ss♦ ♦♠ ♠♣♦ ♠♥ét♦ r♦t♦♥ ♥t♥s ♥

| ~Bext| = bc ♦r rst♦s ♥trss♥ts s rs s♠ s♦r♠ ♠♦s q sts

sã♦ ♦♠♣♦sts s♦♠♥t ♣♦r érts q ♦♠ rr ♦ ♦ ♦ st♦ ♥

♠♥t é ♥ç♦ ♥♦ ♥tr♦r r s rs ♦♠ b′i = 10% s♦r♠ ♠♦s

q s ♣♦♣çõs érts sã♦ ♦r♥s ♣s ♥rs ♦♥rçã♦ ♦s érts

st♦s sssõs

s♥♦ q ♦s érts ♠♥♦rs ♥rs sã♦ qs ♦♠ ♠♦rs ♣♦♣çõs ♥s

rs ♦♠ b′i = 20% ♠♦s q s♦r♠ é ♠t♦ r♥ ③♥♦ ♦♠ s rs ♥ã♦

♦ss♠ ♥♠ tss♠ r♥ ♣♦♣çõs ért V 1

♣ít♦

♦♥sõs ♣rs♣ts

♦♥sõs ♣rs♣ts

st tr♦ st♠♦s ♦s ♣r♦ss♦s s♠♥t③çã♦ ♣r ♠ ♦

s♣♥ rt ♥♠ r tr♥r ♦ ♣r♠r♦ s♠r ♦ ♣r♦ss♦ strs ♦

♦tr♦ trés ♠ ♠♣♦ ♠♥ét♦ r♦t♦♥

♦ ♣r♦ss♦ s♠r ♦ strs ♦sr♠♦s q é ♣♦ssí s♠♥t③r

♦tr ♦ st♦ ♥♠♥t s rs ♦♠ s♦r♥s ♠♥♦rs ♦ s b′i = 10% s

ts r ♥â♠ ♣♦♣çã♦ é srt ♣♦r V 5 → V 4 → V 1 s♥♠♥t♦

♦ ♠♣♦ ♠♥ét♦ ♦♠ sts rs ♥ã♦ ♥trr ♦t♥çã♦ ♦ st♦ ♥♠♥t

♠s ♣♦r ♦tr♦ ♦ r ♠ st♦ ♥tr♠ár♦ ♦♠♣♦st♦ ♣♦r érts V 5xy ♥

tr ♦ st♦ ♥♠♥t ♦ st♦ ♦♠ ♠♥t③çã♦ ♦♣♦st à ♠♥t③çã♦ ♥

r Pr s rs ♦♠ s♦r♥s s♣r♦rs b′i > 10% ♦sr♠♦s q ♥

♣♥♥t♠♥t ♦ s♥♠♥t♦ ♦ ♠♣♦ ♦♠ r sts sã♦ s♠♥t③s

ss ♣♦♣çõs érts V 1 sã♦ s♣r♦rs 85% ♣r♦①♠♥♦s st♥t ♦

♥♦ss♦ ♦t♦

♦ ♣r♦ss♦ s♠♥t③çã♦ trés ♠ ♠♣♦ ♠♥ét♦ ①tr♥♦ r♦

t♦♥ ♠♦s q ♦ ♣r♦t♦♦♦ ♣rã♦ ♦♠çr s♠♥t③r ♦♠ ♠♣♦

r♥ ♥t♥s ♥ã♦ ♦ ♠ ♠ét♦♦ ♥t ♣♦s s rs r♠ s♠♥t③s

♠s ♥ã♦ ♦♥sí♠ tr r♥s ♣♦♣çõs ért V 1 Pr t♥tr ♦♥t♦r♥r st

♣r♦♠ sr♠♦s q ♥t♥s ♥ ♦ ♠♣♦ ♦ss bc ♦♥rçã♦ ♥

r sr ♦♠♣♦st s♦♠♥t ♣♦r érts t♣♦ V 5 st ♣r♦ss♦ ♦sr♠♦s q

♦♥sõs ♣rs♣ts

♥â♠ ♦s érts stá ♦rt♠♥t ♦♠ s♦r♠ ♥♦s sít♦s Pr s rs

s♠ s♦r♠ ♦sr♠♦s q ♦s st♦s s♠♥t③♦s r♠ ♦♠♣♦st♦s s♦♠♥t

♣♦r érts V 1 V 2 ♦s ♦s érts ♠♥♦rs ♥rs q ♦♠ rr ♦

♦ Pr s rs ♦♠ b′i = 10% ♠♦s q ♦s érts ♦♠ ♠♥♦rs ♥rs ♦r♠

qs q ♦tr♠ s ♠♦rs ♣♦♣çõs Pr s rs ♦♠ b′i = 20% ♠♦s q

s♦r♠ ♥♦ sst♠ é ♠t♦ ♥t♥s ♥ã♦ ♦♥s♠♦s s♠♥t③r ♥♠ ♦tr

♠ r♥ ♣♦♣çã♦ ért V 1

P♦r ♠ ♣♦♠♦s ♦♥r q ♦ ♣r♦t♦♦♦ s♠♥t③çã♦ s♠r ♦ s

trs é ♦ ♠s ♥♦ ♣r s♠♥t③r ♦tr ♦ st♦ ♥♠♥t ♦s ♦s

s♣♥ tr♥rs rts ♣♦s ♠s♠♦ ♣r r♥s s♦r♥s ♦♠ s♥♠♥t♦

♥tr ♦ ♠♣♦ ♠♥ét♦ r ♦♥s♠♦s ♦tr ♠ ♣♦♣çã♦ érts V 1

s♣r♦r 85%

♦♠♦ tr♦ tr♦ ♣rt♥♠♦s str ♦ ♦♠♣♦rt♠♥t♦ s rs ♠

♥éts q ♣r♠ ♥ r ♦s s♣♥s rts tr♥rs ♥♦ ♣r♦ss♦

s♠♥t③çã♦ s♠r ♦ strs ♦♠ ♦ ①í♦ ♥s rst♦s ♦t♦s

♥st tr♦ ♣rt♥♠♦s str ♦ ♣r♠♥t♦ ♥â♠ sts rs ♥

♣rs♥ç ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ trés ♥â♠ ♣♦♣çã♦ ♦s t♣♦s

ért V 1 V 5 ♦s ért s♠ r ♠♥ét V 4 ♦ ért ♦♠ r ♠♥ét

s♠♣s st♦s ♠ ♥♠♥t♦ ♦ ♣r♦ss♦ s♠r st ♠ ♠ r s♠ s♦r♠

♥♠ q s rs s♠♣s rs sã♦ s ♣♦r ♠ str♥ t♥sã♦ ♥t

♦ ♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♣♦ ♥♦ sst♠ s rs ♥ã♦ s♥s

str♥ é ♦ t♣♦ ♥r r♥t s♠♥t③çã♦ é ♦ t♣♦ ♥tsrr r♥t

♠♥t③çã♦ r s rs ♦♠ s♥♠♥t♦ str♥ é s♠♣r ♥r

rê♥s ♦rás

❬❪ rt ❲ ②r st♦r② ♦ trt② ♥ ♥ts♠ Psr r♥②

rr②

❬❪ rts ♥tr♦t♦♥ t♦ tr♦②♥♠s t t♦♥

❬❪ ♦ ♣♥ ♥t trs ♥♠♥ts ♥ ♣♣t♦♥s ♥

t♦♥

❬❪ Ptr ❲♥rr ♥t ♥s♦tr♦♣s ♥ ♥♦strtr ttr

❬❪ ❩♥ P②s

❬❪ Prr ♥ ❲ ♦r♦ P❨

❱❲

❬❪ ❲♥♥r P②s rrt♠ P②s

❬❪ ❲ ♥q s② P②s

❬❪ ❲ ♥q ❲ t♦t ♠ ♠ ♦

❬❪ P♥ ♠ ♠ ♦

❬❪ rrs r♠ ♦rr♦ ❩s ♥ ❲ ♦r② P②s

tt

❬❪ ❲②s♥ ❲ ♦r♦ ó ♥ Prr P②s

♦♥♥s ttr

❬❪ ó ❲ ♦r♦ ♥ Prr P②s

❬❪ s♠♥t♦ ó ❲ ♦r♦ ♥ Prr

P②s

❬❪ Prr ♣♣ P②s

❬❪ ❲②s♥ ❲ ♦r♦ ó ♥ Prr r❳

❬❪ s♠♥t♦ ó ❲ ♦r♦ ♥ Prr

P②s

❬❪ ♦♣s ó ❲ ♦r♦ ❲②s♥ ♥

Prr P❨ ❱❲

❬❪ ó ♥ ♦st ❱ r❳ ❬♦♥♠t❪

❬❪ ó Prr ♥ ♦r♦ ❲ P②s tt

❬❪ st♥♦♦ ♦ss♥r ♥ ♦♥ tr

❬❪ r P ♦r② ♦ tr♦♥s ♥ P♦str♦♥s ♦

♦♥t♦♥ tr

❬❪ ♥rs♦♥ r P♦st tr♦♥ P②s

❬❪ ❲♥ s♦ rts ❲ ♦♥ ♦♦② ♥

♠rt t♦♥ ❱ rs♣ ♥ P r tr

❬❪ ♦ ♥r ♥ ❩♥ ♥ rt P ♠♠rt

❱♥♥t rs♣ Ptr r ♥ t♥ ♠rt P②s

❬❪ P♦r ❱ ❱♦♦ ♥ ❨ ❩ P②s

❬❪ ❲②s♥ ❲ ♦r♦ ó Prr P②s ♦♥♥s

ttr

❬❪ ó Prr ❲ ♦r♦ ♥ ❱

♦st ❯ PP P❨

❬❪ ♠ ❨ P②s

❬❪ ❨♥ ♥ ♥❳♥ ❲♥ P②s

❬❪ ❱ ♦♥r ♥♦tt ②r♠♥ r

♦r③ ♦t♥ ♥ r♥ ❯ PP P❨

❬❪ ♠♥♥ ♦t♠♥♥ P ③r② ♥ ❩ PP P❨

❬❪ ó Prr ♥ ❲ ♦r♦ P②s

❬❪ ❲♥ ❲ ♦♥ s♦ ❳ ❲ r♥ ❱ ♦s

r♠sts P ♠♠rt ❱ rs♣ P r r❳♦♥♠t

❬❪ ❩♦ rs P♦♦ P♦t ♥ t♠♣s P②s tt

❬❪ ❩♦ rs P♦♦ P♦t ♥ t♠♣s r❳

❬❪ Pt Pt♦r♦♥ ♥♦♥♥ ♥s ♥ r P❨

❱❲

❬❪ tr♦♣♦s ❲ ♦s♥t ♦s♥t r ♥ r

♠ P②s

❬❪ ♥t♦s P ❩ ♦r ♥ts P ❱ r♦♥ s♦ rsr

strtr trô♥ rr ís