❯ ❯
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