❋❊▲■P❊ ❆◆❉❘❆❉❊ ❆P❖▲Ô◆■❖

❯▼ ▼❖❉❊▲❖ P❆❘❆ ❉❊❋❊■❚❖❙ ❊❙❚❘❯❚❯❘❆■❙ ❊▼

◆❆◆❖▼❆●◆❊❚❖❙

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ à❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❝♦♠♦♣❛rt❡ ❞❛s ❡①✐❣ê♥❝✐❛s ❞♦ Pr♦❣r❛♠❛ ❞❡Pós✲●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛ ❆♣❧✐❝❛❞❛✱♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❛❣✐st❡r

❙❝✐❡♥t✐❛❡✳

❱■➬❖❙❆▼■◆❆❙ ●❊❘❆■❙ ✲ ❇❘❆❙■▲

✷✵✶✵

FELIPE ANDRADE APOLÔNIO

UM MODELO PARA DEFEITOS ESTRUTURAIS EM NANOMAGNETOS

Dissertação apresentada à Universidade Federal de Viçosa, como parte das exigências do Programa de Pós-Graduação em Física Aplicada, para obtenção do título de Magister Scientiae.

APROVADA: 07 de julho de 2010.

Prof. Tiago José de Oliveira

Prof. Sidiney de Andrade Leonel

Prof. Alexandre Tadeu Gomes de Carvalho

Prof. Lucas Álvares da Silva Mol

Prof. Winder Alexander de Moura Melo (Orientador)

❆♦s ♠❡✉s ♣❛✐s✱ ▲✉③✐❛ ❡ ❏♦sé✱ ❞❡❞✐❝♦✳

✐✐

✧❆ ♠❡♥t❡ q✉❡ s❡ ❛❜r❡ ❛ ✉♠❛ ♥♦✈❛ ✐❞❡✐❛ ❥❛♠❛✐s ✈♦❧t❛ ❛♦ s❡✉ t❛♠❛♥❤♦ ♦r✐❣✐♥❛❧✳✧

❆❧❜❡rt ❊✐♥st❡✐♥

✐✐✐

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ❛ ❉❊❯❙✱ ♠❛♥t❡♥❡❞♦r ❞♦ ❯♥✐✈❡rs♦✱ ♣♦r ❡st❡ ✐♠♣♦rt❛♥t❡ tr❛❜❛❧❤♦✱ ♦♣♦r✲

t✉♥✐❞❛❞❡ ❡ ❝✉✐❞❛❞♦✳ ❆♦s ♠❡✉s ♣❛✐s ▲✉③✐❛ ❡ ❏♦sé ♣♦r✱ ❞❡s❞❡ ❝❡❞♦✱ ♥❛ é♣♦❝❛ ♠❛✐s ❞✐❢í❝✐❧✱

t❡r ♠❡ ❞❛❞♦ t♦t❛❧ ❛♣♦✐♦ ❡ s❡♠♣r❡ ❛❝r❡❞✐t❛♥❞♦ ❡♠ ♠✐♠✳

➚s ♠✐♥❤❛s ✐r♠ãs P♦❧❧②❛♥♥❛ ❡ ▼❛r✐❛✱ q✉❡ s❡♠♣r❡ s♦✉❜❡r❛♠ ♠❡ ❛♥✐♠❛r ❡ ❛❧❡❣r❛r✱

t♦r♥❛♥❞♦ ♠✐♥❤❛ tr❛❥❡tór✐❛ ❡♠ ❱✐ç♦s❛ ♠❛✐s ❢❡❧✐③ ❡ ♠❡♥♦s ❞♦❧♦r♦s❛✳ ◆ã♦ ♣♦ss♦ ❡sq✉❡❝❡r

❞♦ r❡st❛♥t❡ ❞❛ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ q✉❡ ♠❡s♠♦ ❞❡ ❧♦♥❣❡ t♦r❝❡✉ ♣♦r ♠✐♠✱ ❛♣♦✐❛♥❞♦ s❡♠♣r❡

q✉❡ ♣♦❞✐❛✳

❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ♥❛♠♦r❛❞❛ ❘❡♥❛t❛✱ ♣❡❧♦ ❛♠♦r✱ ♣❛❝✐ê♥❝✐❛✱ ❝♦♠♣r❡❡♥sã♦ ❡ ③❡❧♦✱

❜❡♠ ❝♦♠♦ s✉❛ ❢❛♠í❧✐❛✱ q✉❡ ♠❡ ❛❝♦❤❧❡✉ ❝♦♠♦ ✉♠ ✜❧❤♦✱ ❢❛③❡♥❞♦ ❝♦♠ q✉❡ ❡✉ ♠❡ s❡♥t✐ss❡

♠❛✐s ♣❡rt♦ ❞❡ ❝❛s❛✳

❆♦ ♣r♦❢❡ss♦r ❲✐♥❞❡r✱ ♣♦r ❛❝r❡❞✐t❛r q✉❡ ♦ tr❛❜❛❧❤♦ ❡r❛ ❞❡ ❣r❛♥❞❡ r❡❧❡✈â♥❝✐❛ ❡

♣♦r t♦❞♦ ❛♣♦✐♦✱ ♣❛❝✐ê♥❝✐❛ ❡ ❝♦♥✜❛♥ç❛✳

❆♦s ♠❡✉s ❛♠✐❣♦s ❞❛q✉✐ ❞❡ ❱✐ç♦s❛✱ ❋❛❜✐❛♥♦✱ ❋á❜✐♦ ✭❆❣❡✉✮✱ ❙❛✐♠♦♥✱ ❩é✱ ❏ú❧✐♦

✭❇✐①ã♦✮✱ ▼❛✉rí❝✐♦✱ ❚♦❞ã♦✱ ❇r✉♥♦✱ ❩❡③é ✭●✉✐❧❤❡r♠❡✮✳✳✳ s❡♠♣r❡ ❝♦♠ ❝♦♥✈❡rs❛s ❞❡s❝♦♥✲

tr❛í❞❛s ❡ ❛♣r❡♥❞❡♥❞♦ ♠✉✐t❛s ❧✐çõ❡s✳ ❖❜r✐❣❛❞♦ ♣♦r t♦r♥❛r❡♠ ❡ss❡ t❡♠♣♦ ♠❛✐s ❞✐✈❡rt✐❞♦✦

◆ã♦ ♣♦❞❡r✐❛ ❡sq✉❡❝❡r ❞♦ ♣❡ss♦❛❧ ❞❛ ❏✳ ❏✳✿ ❉❛✈✐❞s♦♥✱ ❈♦t♦♥❡t❡✱ ▼♦❝♦tó ❡ ❘❛♣❡✐s✳

➚q✉❡❧❡s ❛♠✐❣♦s q✉❡ ♥ã♦ ♠❛✐s ❡stã♦ ❛q✉✐✱ ❝♦♠♦ ♦ ♠❡str❡ ❙❛♥❞r♦ q✉❡ ❞❡s❞❡ ❝❡❞♦

♠❡ ❛♣♦✐♦✉✱ ❈é③❛r✱ ❏ú❧✐♦ ✭♦ ♣❛r❝❡✐r♦ ❞❛s sér✐❡s ❡ tê♥✐s✮✱ ❋r❛♥❝✐s ✭♠❡str❡ ❞❛ ✐♥❢♦r♠át✐❝❛✮✱

❇❛✐❛♥♦✱ ▲✉❝✐❛♥♦✱ ❏❛♣❛ ✭❝♦♠ s✉❛s ❤✐stór✐❛s✮ ❡ t❛♥t♦s ♦✉tr♦s q✉❡ ♠❛r❝❛r❛♠ é♣♦❝❛✳

❆♦s ❛♠✐❣♦s ❞❡ r❡♣ú❜❧✐❝❛✱ ❋❡r♥❛♥❞♦ 1/2 ❦❣ ❡ ❚♦❢❡✱ s❡♠♣r❡ ❞❡s❝♦♥tr❛✐♥❞♦ ♦ ❛♠✲

❜✐❡♥t❡✳

❆♦s ❝♦❧❡❣❛s q✉❡ t✐✈❡ ♦ ♣r❛③❡r ❞❡ ❢❛③❡r ❛❧❣✉♠❛ ❞✐s❝✐♣❧✐♥❛ ❡ ❛♣r❡♥❞❡r ❛❧❣♦✳

❆♦s ❞❡♠❛✐s ♣r♦❢❡ss♦r❡s ❝♦♠ ♦s q✉❛✐s t✐✈❡ ❝♦♥t❛t♦ ❡ ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❉P❋✳

✐✈

❙✉♠ár✐♦

▲✐st❛ ❞❡ ❋✐❣✉r❛s ✈✐✐

❘❡s✉♠♦ ①✐

❆❜str❛❝t ①✐✐

✶ ■♥tr♦❞✉çã♦ ❡ ▼♦t✐✈❛çã♦ ✶

✷ ❈♦♥s✐❞❡r❛çõ❡s ❣❡r❛✐s s♦❜r❡ ♠❛❣♥❡t✐s♠♦ ❡ ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s ✶✶

✷✳✶ ■♥t❡r❛çã♦ ❞❡ tr♦❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺

✷✳✷ ■♥t❡r❛çã♦ ❞✐♣♦❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

✷✳✸ ❆♥✐s♦tr♦♣✐❛ ♠❛❣♥ét✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✷✳✸✳✶ ❆♥✐s♦tr♦♣✐❛ ♠❛❣♥❡t♦❝r✐st❛❧✐♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✷✳✸✳✷ ❆♥✐s♦tr♦♣✐❛ ❞❡ ❢♦r♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✷✳✹ ❖ ♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ❛♥✐s♦tró♣✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✸ ❊str✉t✉r❛ ❡ ❞✐♥â♠✐❝❛ ❞❡ ✈órt✐❝❡s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s ✷✾

✸✳✶ ❱❡rsã♦ ❝♦♥tí♥✉❛ ❞♦ ♠♦❞❡❧♦ ❳❨ ❡ ✈órt✐❝❡s ♣❧❛♥❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

✸✳✷ ❱órt✐❝❡s ❝♦♠ ❝♦♠♣♦♥❡♥t❡ ❢♦r❛ ❞♦ ♣❧❛♥♦ ✭♦✉t ♦❢ ♣❧❛♥❡✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✸✳✷✳✶ ❉✐♥â♠✐❝❛ ❞❡ ✈órt✐❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✸✳✷✳✷ ❉❡❢❡✐t♦s ❡str✉t✉r❛✐s ❡♠ ♥❛♥♦❞✐s❝♦s ♠❛❣♥ét✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✹ ▼♦❞❡❧♦ ♣❛r❛ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s ✹✺

✹✳✶ ❆♣❧✐❝❛çã♦ ❞♦ ♠♦❞❡❧♦ ❡ r❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽

✺ ❈♦♥❝❧✉sõ❡s ❡ ♣❡rs♣❡❝t✐✈❛s ✺✸

✈

❆ ▼❛❣♥❡t♦r❡s✐stê♥❝✐❛ ❣✐❣❛♥t❡ ✭●▼❘✮ ❡ ❣r❛✈❛çã♦ ♠❛❣♥ét✐❝❛ ✺✺

❆✳✶ ❆ ❞❡s❝♦❜❡rt❛ ❞♦ ❡❢❡✐t♦ ●▼❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

❆✳✷ ●r❛✈❛çã♦ ♠❛❣♥ét✐❝❛ ❡ ●▼❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

❆✳✸ ▼❡♠ór✐❛ ♠❛❣♥ét✐❝❛ ❞❡ ❛❝❡ss♦ ❛❧❡❛tór✐♦ ✭▼❘❆▼✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾

❇ ▲✐♠✐t❡ ❝♦♥tí♥✉♦ ❞❛ ❍❛♠✐❧t♦♥✐❛♥❛ ❞❡ ❍❡✐s❡♥❜❡r❣ ✐s♦tró♣✐❝❛ ✻✵

❈ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❛❧❣✉♠❛s ❡q✉❛çõ❡s ✻✹

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✻✼

✈✐

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✶✳✶ ❇úss♦❧❛ ❛♥t✐❣❛ ✉t✐❧✐③❛❞❛ ♣❡❧♦s ❝❤✐♥❡s❡s ❬✶❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶

✶✳✷ ❖s ♠✐❝r♦❞✐s❝♦s sã♦ ❜✐♦❢✉♥❝✐♦♥❛❧✐③❛❞♦s ❝♦♠ ♦ ❛♥t✐✲❝♦r♣♦ ❛♥t✐✲❤✉♠❛♥✲■▲✶✷α✷❘✱

❡s♣❡❝✐✜❝❛♠❡♥t❡ ♣❛r❛ ❛t✐♥❣✐r❡♠ ❝é❧✉❧❛s ❤✉♠❛♥❛s ❞❡ ❣❧✐♦❜❧❛st♦♠❛ ♠✉❧t✐❢♦r♠❡✳

❆♦ ❛♣❧✐❝❛r✲s❡ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦✱ ♦s ❞✐s❝♦s ♠❛❣♥ét✐❝♦s ♦s❝✐❧❛♠✱ ❝♦♠♣r♦♠✲

❡t❡♥❞♦ ❛ ✐♥t❡❣r✐❞❛❞❡ ❞❛ ♠❡♠❜r❛♥❛✳ ❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢❡rê♥❝✐❛ ❬✻❪✳ ✳ ✳ ✳ ✳ ✸

✶✳✸ ■❧✉str❛çã♦ ❡sq✉❡♠át✐❝❛ ❞❛ ❢❛❜r✐❝❛çã♦ ❞❡ ♥❛♥♦❞✐s❝♦s ♠❛❣♥ét✐❝♦s ✉t✐❧✐③❛♥❞♦✲s❡

❛s três té❝♥✐❝❛s✿ ❊❇▲✱ t❤❡r♠❛❧ ❡✈❛♣♦r❛t✐♦♥ ❡ ❧✐❢t✲♦✛✳ ◆♦ ♣r✐♠✐r♦ q✉❛❞r♦ ❛❝✐♠❛

t❡♠♦s ♦ s✉❜str❛t♦ ❡♠ ❛③✉❧ ❡ ♦ r❡s✐st❡ ❡♠ ❛♠❛r❡❧♦✳ ❊♠ s❡❣✉✐❞❛ é ❡♠✐t✐❞♦ ♦ ❢❡✐①❡

❞❡ ❡❧étr♦♥s ✭❊❇▲✮ ♥❛ ❛♠♦str❛✱ ❛ ✜♠ ✬❧✐♠♣❛r✬ ♦ ❧♦❝❛❧ ♦♥❞❡ s❡rá ❞❡♣♦s✐t❛❞❛ ❛

❛♠♦str❛✳ ❆♣ós ❢❡✐t♦ ✐ss♦✱ ♦ ♠❛t❡r✐❛❧ é ❞❡♣♦s✐t❛❞♦ ✈✐❛ ❛ té❝♥✐❝❛ ❞❡ ❡✈❛♣♦r❛çã♦

tér♠✐❝❛ ✭❚❊✮✳ ❋✐♥❛❧♠❡♥t❡✱ ❤á ✉♠ ❜❛♥❤♦ ❝♦♠ ✉♠ s♦❧✈❡♥t❡ ♣❛r❛ ❛ r❡♠♦çã♦ ❞♦

r❡s✐st❡ ✭❧✐❢t✲♦✛ ✮✱ ❡ ♦ r❡s✉❧t❛❞♦ ✜♥❛❧ é ♦ s❜str❛t♦ ❝♦♠ ♦ ♠❛t❡r✐❛❧ ❞❡♣♦s✐t❛❞♦ ❬✶✼❪✳ ✺

✶✳✹ ✧❊✈♦❧✉çã♦✧ ❞❛ ❣r❛✈❛çã♦ ♠❛❣♥ét✐❝❛✿ ✈❛r✐❛çã♦ ❞❛ ❞❡♥s✐❞❛❞❡ ❞❡ ár❡❛ ✉t✐❧✐③❛❞❛

♥❛ ❣r❛✈❛çã♦ ✭❡♠ ▼❡❣❛❜✐ts✴♣♦❧2✮ ❡♠ ❢✉♥çã♦ ❞♦ ❛♥♦ ❡♠ q✉❡ ♦s ❡q✉✐♣❛♠❡♥t♦s

t♦r♥❛r❛♠✲s❡ ❞✐s♣♦♥í✈❡✐s ♥♦ ♠❡r❝❛❞♦ ✭■❇▼ ✷✵✵✺✮✳ ❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢❡rê♥❝✐❛

❬✷✹❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✷✳✶ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ❝✉r✈❛ ❞❡ ❤✐st❡r❡s❡ ♣❛r❛ ✉♠♠❛t❡r✐❛❧ ❢❡rr♦♠❛❣♥ét✐❝♦ ♠♦str❛♥❞♦

❛ ♠❛❣♥❡t✐③❛çã♦ r❡♠❛♥❡♥t❡✱ Mr✱ ❛ ♠❛❣♥❡t✐③❛çã♦ ❞❡ s❛t✉r❛çã♦✱ Ms✱ ❡ ❛ ❝♦❡r✲

❝✐✈✐❞❛❞❡✱ Hc✳ ❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢❡rê♥❝✐❛ ❬✹✸❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

✷✳✷ ▼❛❣♥❡t✐③❛çã♦ ❞❡ s❛t✉r❛çã♦✱ Ms✱ ❞❡ ✉♠ ❢❡rr♦♠❛❣♥❡t♦ ❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r✲

❛t✉r❛✱ T ✱ ♥♦r♠❛❧✐③❛❞❛ ♣❛r❛ ❛ t❡♠♣❡r❛t✉r❛ ❞❡ ❈✉r✐❡✱ Tc✳ ❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛

r❡❢❡rê♥❝✐❛ ❬✹✸❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✈✐✐

✷✳✸ ■❧✉str❛çã♦ ❡sq✉❡♠át✐❝❛ ❞❡ ♦r❞❛♥❛♠❡♥t♦ ❢❡rr♦♠❛❣♥ét✐❝♦✱ à ❡sq✉❡r❞❛✱ ❡ ❛♥t✐✲

❢❡rr♦♠❛❣♥ét✐❝♦✱ à ❞✐r❡✐t❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✷✳✹ ❉♦♠í♥✐♦s ♠❛❣♥ét✐❝♦s ❝♦♠ ♣❛r❡❞❡s ❞❡ ❞♦♠í♥✐♦s ❢♦r♠❛♥❞♦ ✭❛✮ 180◦ ❡ ✭❜✮ 90♦✳

❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢❡rê♥❝✐❛ ❬✹✽❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✷✳✺ ❘❡♣r❡s❡♥t❛çã♦ ❡sq✉❡♠át✐❝❛ ❞❡ ✉♠ ❢❡rr♦♠❛❣♥❡t♦ ❝♦♠ ❛♥✐s♦tr♦♣✐❛ ❞❡ ❡✐①♦✲❢á❝✐❧✳ ✷✼

✷✳✻ ❘❡♣r❡s❡♥t❛çã♦ ❡sq✉❡♠át✐❝❛ ❞❡ ✉♠ ❢❡rr♦♠❛❣♥❡t♦ ❝♦♠ ❛♥✐s♦tr♦♣✐❛ ❞❡ ♣❧❛♥♦✲❢á❝✐❧✳ ✷✼

✸✳✶ ❊sq✉❡♠❛ ❞❡ s♣✐♥ ❝❧áss✐❝♦ ❝♦♠ |~S| = 1 ❡ ♦r✐❡♥t❛çã♦ ❞❛❞❛ ♣❡❧♦s â♥❣✉❧♦s ❡s❢ér✐❝♦s

θ ❡ φ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✸✳✷ ■❧✉str❛çã♦ ❡sq✉❡♠át✐❝❛ ❞❡ ✉♠ ✈órt✐❝❡ ❝♦♠ Q = +1 ✭à ❡sq✉❡r❞❛✮✱ ✉♠ ❛♥t✐✈ór✲

t✐❝❡ ❝♦♠ Q = −1 ✭❛♦ ❝❡♥tr♦✮ ❡ ✉♠ ♣❛r ✈órt✐❝❡✲❛♥t✐✈órt✐❝❡ ❝♦♠ Q = 0 ✭à

❞✐r❡✐t❛✮✳ ❖s sí♠❜♦❧♦s ⊕ ❡ ⊖ r❡♣r❡s❡♥t❛♠ ♦ ❝❡♥tr♦ ❞❡ ❝❛❞❛ ❡str✉t✉r❛✳ ✳ ✳ ✳ ✳ ✸✷

✸✳✸ ■❧✉str❛çã♦ ❞❡ ✉♠ ❡st❛❞♦ ❞❡ ❞♦♠í♥✐♦ ú♥✐❝♦✱ ♥♦ q✉❛❧ t♦❞♦s ♦s ♠♦♠❡♥t♦s ♠❛❣✲

♥ét✐❝♦s sã♦ ♣❛r❛❧❡❧♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸

✸✳✹ ❊sq✉❡♠❛ ❞❡ ✈órt✐❝❡ ♦✉t ♦❢ ♣❧❛♥❡✱ ♠♦str❛♥❞♦ ❛♦ ❝❡♥tr♦ ♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡✱

♥❡st❡ ❝❛s♦ ❝♦♠ ♣♦❧❛r✐❞❛❞❡ p = +1✳ ❉✐st❛♥t❡ ❞♦ ❝❛r♦ç♦✱ ❛ ♠❛❣♥❡t✐③❛çã♦ ♥❛ s✉✲

♣❡r❢í❝✐❡ ❞❛ ❛♠♦str❛ ❝♦♥t✐♥✉❛♠❡♥t❡ ✬❝✐r❝✉❧❛✬ ❡♠ t♦r♥♦ ❞♦ ❝❡♥tr♦ ❞❛ ❡str✉t✉r❛✳

❊♠ ❞✐r❡çã♦ ❛♦ ❝❡♥tr♦✱ ❛ ♠❛❣♥❡t✐③❛çã♦ ❝♦♠❡ç❛ ❛ r❡✈♦❧✈❡r✲s❡ ♣❡r♣❡♥❞✐❝✉❧❛r✲

♠❡♥t❡ ❛♦ ♣❧❛♥♦ ❞❛ ❛♠♦str❛✳ ❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢❡rê♥❝✐❛ ❬✺✽❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✸✳✺ ■♠❛❣❡♠ ❞❡ ▼❋▼ ❞❡ ✉♠ ❛rr❛② ❞❡ ❞♦ts ❞❡ P❡r♠❛❧❧♦② ♥♦ ❡st❛❞♦ r❡♠❛♥❡♥t❡

✭| ~H| = 0✮✱ ❝♦♠ R = 500 ♥♠ ❡ L = 50 ♥♠✳ ➱ ♠♦str❛❞♦ ♥❛ ✜❣✉r❛ r❡❣✐õ❡s

✭❝❧❛r❛s ❡ ❡s❝✉r❛s✮ q✉❡ ❝♦rr❡s♣♦♥❞❡♠ à ár❡❛ ♦♥❞❡ ❛ ♠❛❣♥❡t✐③❛çã♦ é ❛❧✐♥❤❛❞❛

♣❡r♣❡♥❝✐❞✉❧❛r♠❡♥t❡ ✭✉♣ ♦✉ ❞♦✇♥✮ ❛♦ ♣❧❛♥♦ ❞❛ ❛♠♦str❛✳ ❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛

r❡❢❡rê♥❝✐❛ ❬✻✷❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻

✸✳✻ ❆s s❡t❛s ❛♠❛r❡❧❛s r❡♣r❡s❡♥t❛♠ ♦ ✈❡t♦r n✱ ♥♦r♠❛❧ à s✉♣❡r❢í❝✐❡✱ ❡ ❛s s❡t❛s ❛③✉✐s

❛ ♠❛❣♥❡t✐③❛çã♦ ~M(~r )✳ ➚ ❡sq✉❡r❞❛ r❡♣r❡s❡♥t❛♠♦s ♦ q✉❡ ♦❝♦rr❡ ♥❛ ❢❛❝❡ ❧❛t❡r❛❧

❞♦ ♥❛♥♦❞✐s❝♦✱ ♦♥❞❡ ~M · n = 0✳ ➚ ❞✐r❡✐t❛ ♠♦str❛♠♦s ❛s ❞✉❛s s✐t✉❛çõ❡s q✉❡

❛❝♦♥t❡❝❡♠ ♥❛ ❢❛❝❡ s✉♣❡r✐♦r ❞♦ ♥❛♥♦❞✐s❝♦✿ ❛✮ ♠❛❣♥❡t✐③❛çã♦ ♥♦ ♣❧❛♥♦ xy ❡ ❜✮

♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ ❝♦♠ ♠❛❣♥❡t✐③❛çã♦ t❛♠❜é♠ ♥♦ ❡✐①♦ z✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✈✐✐✐

✸✳✼ ■❧✉str❛çã♦ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ❞❡ ❞♦✐s ♥❛♥♦❞✐s❝♦s ❞❡ P②✱ ❝♦♠ L = 50 ♥♠ ❡ R =

450 ♥♠✱ s✉❜♠❡t✐❞♦ ❛ ✉♠ ❝❛♠♣♦ ❡①t❡r♥♦ ~Hext✱ ❡♠ ❖❡rst❡❞s ✭✶ ❖❡≈ ✼✾✱✻ ❆✴♠✮✱

❛♣❧✐❝❛❞♦ ♥♦ ♣❧❛♥♦ ❞❛ ❛♠♦str❛✱ ❝♦♥❢♦r♠❡ ✐♥❞✐❝❛❞♦✳ ❖ ❝❛♠♣♦ é ❛✉♠❡♥t❛❞♦

s❡❣✉✐❞❛♠❡♥t❡✱ ❝♦♠ ♦s ♠♦♠❡♥t♦s ♠❛❣♥ét✐❝♦s s✉❝❡ss✐✈❛♠❡♥t❡ ♦r✐❡♥t❛♥❞♦✲s❡ ♥❛

❞✐r❡çã♦ ❞❡ ~Hext✳ ▼♦str❛♠♦s t❛♠❜é♠ ❛s ✧❝❛r❣❛s ♠❛❣♥ét✐❝❛s✧σM = ~M · n ❡

ρM = −~∇ · ~M ✳ ❆♦ ❝❡♥tr♦ ❞♦ ✈órt✐❝❡✱ r❡♣r❡s❡♥t❛♠♦s ♦ ❝❛r♦ç♦ ♣♦r ✉♠❛ r❡❣✐ã♦

❝❧❛r❛✱ p = +1✱ ♦✉ ❡s❝✉r❛✱ p = −1✳ ❋✐❣✉r❛ ♠♦❞✐✜❝❛❞❛ r❡t✐r❛❞❛ ❞❛ r❡❢❡rê♥❝✐❛ ❬✶✵❪✳ ✸✽

✸✳✽ ❈✉r✈❛ ❞❡ ❤✐st❡r❡s❡ ♣❛r❛ ✉♠ ❞✐s❝♦ ❞❡ P② ✭R = 300 ♥♠ ❡ L = 60 ♥♠✮ ❢❡✐t❛ ♣♦r

♠❛❣♥❡t♦♠❡tr✐❛ ❍❛❧❧✳ ❋✐❣✉r❛ ♠♦❞✐✜❝❛❞❛ r❡t✐r❛❞❛ ❞❛ r❡❢❡rê♥❝✐❛ ❬✶✵❪✳ ✳ ✳ ✳ ✳ ✳ ✸✾

✸✳✾ ❉♦✐s ♥❛♥♦❞✐s❝♦s ❞❡ P② sã♦ ✐♥✈❡st✐❣❛❞♦s✿ ❡♠ ✭❛✮ ♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ é ♠♦✈✐❞♦

♣♦r ♠❡✐♦ ❞❡ ✉♠ ❝❛♠♣♦ ❡①t❡r♥♦ ♥✉♠ ♥❛♥♦❞✐s❝♦ ❝♦♠ R = 500 ♥♠✳ ➱ ♣♦ssí✈❡❧

✈❡r ♦s ❥✉♠♣s ♥❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦✱ ♦s q✉❛✐s sã♦ ❞❡✈✐❞♦s ❛♦s ♣✐♥♥✐♥❣ s✐t❡s✳

❊♠ ✭❜✮ é ♠♦str❛❞♦ ♦ ❞❡s❧♦❝❛♠❡♥t♦ ❞♦ ❝❛r♦ç♦ ♣♦r ♠❡✐♦ ❞❡ ✉♠ ❝❛♠♣♦✱ ♣❛r❛ ✉♠

♥❛♥♦❞✐s❝♦ ❝♦♠ R = 1, 5 µ♠ ❝♦♠ ✉♠ ❞❡❢❡✐t♦ ❛rt✐✜❝✐❛❧ ❝❡♥tr❛❧✳ ❋✐❣✉r❛ r❡t✐r❛❞❛

❞❛ r❡❢❡rê♥❝✐❛ ❬✶✸❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✹✳✶ ■❧✉str❛çã♦ ❞❡ ❝♦♠♦ ♦❝♦rr❡ ❛ ✐♥t❡r❛çã♦ ❞❡❢❡✐t♦✲❝❛r♦ç♦✿ ✐♥t❡r❛çã♦ é ♣❡q✉❡♥❛ ♥❛

❜♦r❞❛ ❞♦ ❝❛r♦ç♦✱ ♦♥❞❡ ❛ ❡♥❡r❣✐❛ ❞❡ tr♦❝❛ é ❜❛✐①❛❀ ❝♦♥❢♦r♠❡ ♦ ❝❛r♦ç♦ ❞❡s❧♦❝❛✲s❡

❡♠ ❞✐r❡çã♦ ❛♦ ❞❡❢❡✐t♦ ❡♠ ~xi✱ ❛ ✐♥t❡r❛çã♦ ❛✉♠❡♥t❛✱ ❛t✐♥❣✐♥❞♦ s❡✉ ♠á①✐♠♦ ❡♠

~r = ~xi✳ ❖ ❡❢❡t✐✈♦ ✐♥t❡r✈❛❧♦ ❞❡ ❛❧❝❛♥❝❡ é δ ∼ 10 ♥♠ ❡ ❡stá ❡♠ ❜♦♠ ❛❝♦r❞♦

❝♦♠ ❡①♣❡r✐♠❡♥t♦s✳ ❆ ❢♦r♠❛ ✭❣❛✉ss✐❛♥❛✮ ❞❛ ✐♥t❡r❛çã♦ ♣♦❞❡ s❡r ❝♦♥tr♦❧❛❞❛

✈❛r✐❛♥❞♦✲s❡ ♦ ♣❛râ♠❡tr♦ α✱ q✉❡ t♦♠❛♠♦s ❝♦♠♦ α−1 = 2, 5× 104R2✱ ❛ ✜♠ ❞❡

♦❜t❡r ♦ ❛❧❝❛♥❝❡ ❞❡s❡❥❛❞♦ ♣❛r❛ ❛ ✐♥t❡r❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

✐①

✹✳✷ ❖ ❝❛♠✐♥❤♦ ❡♠ ✈❡r♠❡❧❤♦ é ❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡✱ seq✱

❡♠ ❢✉♥çã♦ ❞♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❡①t❡r♥♦✱ hext✳ ❱❡♠♦s ❛í ❝♦♠♦ t❛❧ ❝❛♠✐♥❤♦

é ❛❢❡t❛❞♦ ♣❡❧❛ ✐♥❝❧✉sã♦ ❞❡ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s ✐♥trí♥s❡❝♦s ♥❛ ❛♠♦str❛✳ ❯♠❛

❝♦♠♣❛r❛çã♦ ❝♦♠ ❛ ✜❣✉r❛ ✸✳✾ ♠♦str❛ q✉❡ ♥♦ss♦ r❡s✉❧t❛❞♦ ❡stá ❡♠ ❜♦♠ ❛❝♦r❞♦

q✉❛❧✐t❛t✐✈♦ ❝♦♠ ❛q✉❡❧❡ ❡①♣❡r✐♠❡♥t❛❧ ❬✶✸❪✳ ■♥s❡r✐♠♦s ✷✵ ❞❡❢❡✐t♦s ♥❛ ❛♠♦str❛✱

❝♦♠ ❡s♣❛ç❛♠❡♥t♦ ❞❡ 2 l0 ❡♥tr❡ ❡❧❡s✳ ➱ ♠♦str❛❞♦ ♥♦ ❣rá✜❝♦ ❞❡❢❡✐t♦s ❞♦ t✐♣♦

❛tr❛t✐✈♦ ✭λi < 0✮ ❡ r❡♣✉❧s✐✈♦ ✭λi > 0✮✱ ✐♥❞✐❝❛❞♦s ♣♦r s❡t❛s ❛③✉✐s ❡ ♣r❡t❛s✱

r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖s ✈❛❧♦r❡s ❞❡ λi q✉❡ r❡♣r♦❞✉③❡♠ t❛✐s ✐♥t❡r❛çõ❡s ❡stã♦ ♥✉♠❛

❢❛✐①❛ |λi| ∼ 10−25 − 10−26 ♠2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵

✹✳✸ ❆❝✐♠❛✿ ✐❧✉str❛çã♦ ❡sq✉❡♠át✐❝❛ ❞❛ ✐♥t❡r❛çã♦ ❞♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ ❝♦♠

✉♠ ❞❡❢❡✐t♦ ❡s♣❛❧❤❛❞♦r ✭s❝❛tt❡r✮✱ ♠♦str❛♥❞♦ ♦♥❞❡ t❛❧ ✐♥t❡r❛çã♦ ♣♦❞❡ s❡r

♦❜s❡r✈❛❞❛ ♥♦ ❣rá✜❝♦ ❞❡ s ♣♦r hext✳ ❆❜❛✐①♦✿ ✐❧✉str❛çã♦ ❞❛ ✐♥t❡r❛çã♦

❞♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ ❝♦♠ ✉♠ ❞❡❢❡✐t♦ ❛tr❛t♦r ✭♣✐♥♥✐♥❣✮✱ ✐❧✉str❛♥❞♦ ❛

❝♦rr❡s♣♦♥❞❡♥t❡ ❧♦❝❛❧✐③❛çã♦ ❞❡ t❛❧ ✐♥t❡r❛çã♦ ♥♦ ❣rá✜❝♦ ❞❡ s ♣♦r hext✳ ✳ ✳ ✺✶

❆✳✶ ❊sq✉❡r❞❛✿ ❝❛♠❛❞❛s ❝♦♠ ♠♦♠❡♥t♦s ♦♣♦st♦s ✭❡st❛❞♦ ❛♥t✐❢❡rr♦♠❛❣♥ét✐❝♦✮✱ ♦♥❞❡

❝❛❞❛ ❝❛♥❛❧ ❡♥❝♦♥tr❛ r❡❣✐õ❡s ❝♦♠ ❛❧t❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ✭r❡s✐stê♥✲

❝✐❛ t♦t❛❧ ❛❧t❛✮✳ ❉✐r❡✐t❛✿ ❝❛♠❛❞❛s ❝♦♠ ♠♦♠❡♥t♦s ♣❛r❛❧❡❧♦s ✭❡st❛❞♦ ❢❡rr♦♠❛❣✲

♥ét✐❝♦✮ ❣r❛ç❛s à ♣r❡s❡♥ç❛ ❞❡ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❛♣❧✐❝❛❞♦✳ ❆❣♦r❛ t❡♠♦s q✉❡

✉♠ ❝❛♥❛❧ ❡♥❝♦♥tr❛rá ❛❧t❛ r❡s✐stê♥❝✐❛ ❡ ♦✉tr♦ ❜❛✐①❛ r❡s✐stê♥❝✐❛ ✭r❡s✐stê♥❝✐❛ t♦t❛❧

❜❛✐①❛✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

❆✳✷ ■❧✉str❛çã♦ ❞❡ ✉♠ ♣r♦❝❡ss♦ ❞❡ ❣r❛✈❛çã♦ ❡ ❧❡✐t✉r❛ ♠❛❣♥ét✐❝❛✱ ✉t✐❧✐③❛♥❞♦✱ ❝♦♠✲

♣❛r❛t✐✈❛♠❡♥t❡✱ ✉♠ ❝❛❜❡ç♦t❡ ✐♥❞✉t✐✈♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ✭❡sq✉❡r❞❛✮ ❡ ✉♠ ❝❛❜❡ç♦t❡

♠❛❣♥❡t♦rr❡s✐st✐✈♦ ❝♦♠ t❡❝♥♦❧♦❣✐❛ ♠♦❞❡r♥❛ ✭❞✐r❡✐t❛✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽

❆✳✸ P♦ssí✈❡❧ ♣r♦tót✐♣♦ ❞❡ ✉♠❛ ❝é❧✉❧❛ ❞❛ ❝❤❛♠❛❞❛ ▼❘❆▼✱ ♠❡♠ór✐❛ ♥ã♦ ✈♦❧át✐❧

❡ q✉❡ ✉t✐❧✐③❛ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ●▼❘✳ ❯♠ ♦✉tr♦ ♣r♦tót✐♣♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦

❡♠ ❬✼✻❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾

❇✳✶ ❘❡❞❡ q✉❛❞r❛❞❛ ✐♥❞✐❝❛♥❞♦ ♦ ❡s♣❛ç❛♠❡♥t♦ ❞❡ r❡❞❡ a ❜❡♠ ❝♦♠♦ ♦s q✉❛tr♦ s♣✐♥s

q✉❡ ✐♥t❡r❛❣❡♠ ❝♦♠ ♦ s♣✐♥ ❞♦ sít✐♦ i✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶

①

❘❡s✉♠♦

❆♣♦❧ô♥✐♦✱ ❋❡❧✐♣❡ ❆♥❞r❛❞❡✱ ▼✳ ❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❏✉❧❤♦ ❞❡ 2010✳ ❯♠♠♦❞❡❧♦ ♣❛r❛ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s✳ ❖r✐❡♥t❛❞♦r✿ ❲✐♥❞❡r❆❧❡①❛♥❞❡r ▼♦✉r❛ ▼❡❧♦✳ ❈♦✲❖r✐❡♥t❛❞♦r❡s✿ ❆❢râ♥✐♦ ❘♦❞r✐❣✉❡s P❡r❡✐r❛✱ ❆❧✈❛r♦❱✐❛♥♥❛ ◆♦✈❛❡s ❞❡ ❈❛r✈❛❧❤♦ ❚❡✐①❡✐r❛

▼❛t❡r✐❛✐s ❢❡rr♦♠❛❣♥ét✐❝♦s ♥❛♥♦❡str✉t✉r❛❞♦s✱ ❞❡ ❞✐✈❡rs❛s ❢♦r♠❛s ❡ t❛♠❛♥❤♦s✱

tê♠ s✐❞♦ ❛♠♣❧❛♠❡♥t❡ ✐♥✈❡st✐❣❛❞♦s ❞❡✈✐❞♦ à s✉❛ ✈❛st❛ ❛♣❧✐❝❛çã♦ ❡♠ ❞✐s♣♦s✐t✐✈♦s ♠❛❣✲

♥❡t♦❡❧❡trô♥✐❝♦s✳ ❋❡rr♦♠❛❣♥❡t♦s ♣♦❞❡♠ ❛♣r❡s❡♥t❛r ❞✐✈❡rs❛s ❝♦♥✜❣✉r❛çõ❡s ❞❡ ♠❛❣♥❡t✐✲

③❛çã♦ ❡stá✈❡✐s✳ ❊♠ ♥❛♥♦❞✐s❝♦s ❝♦♠ ❡s♣❡ss✉r❛ ❞❡ L & 15 ♥♠ ❡ r❛✐♦ R & 100 ♥♠✱ ❛

❝♦♥✜❣✉r❛çã♦ ❞♦ t✐♣♦ ✈órt✐❝❡ ❛♣❛r❡❝❡ ❝♦♠♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧✱ ♥❛ ❛✉sê♥❝✐❛ ❞❡ ❝❛♠♣♦

♠❛❣♥ét✐❝♦ ❡①t❡r♥♦✳ P♦r ♠❛✐s ❝✉✐❞❛❞♦ q✉❡ s❡ t❡♥❤❛ ♥❛ ♣r❡♣❛r❛çã♦ ❞❛s ❛♠♦str❛s✱ ✐♠✲

♣✉r❡③❛s ❡ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s ✭♥❛ ❡str✉t✉r❛ ❝r✐st❛❧✐♥❛✱ ♣♦r ❡①❡♠♣❧♦✮ s❡♠♣r❡ ❡st❛rã♦

♣r❡s❡♥t❡s✱ ❡♠ ♠❛✐♦r ♦✉ ♠❡♥♦r q✉❛♥t✐❞❛❞❡✱ s❡♥❞♦ ❣❡r❛❧♠❡♥t❡ ♣✉♥t✐❢♦r♠❡s ✭♦✉ ❜❛st❛♥t❡

❞✐♠✐♥✉t♦s✮ ❡ ❞✐str✐❜✉✐♥❞♦✲s❡ ❛❧❡❛t♦r✐❛♠❡♥t❡ ♣❡❧♦ ♠❛t❡r✐❛❧✳ ➱ s❛❜✐❞♦ ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡

q✉❡ ❡st❡s ❞❡❢❡✐t♦s ♣♦❞❡♠ ♠♦❞✐✜❝❛r ♣r♦❢✉♥❞❛♠❡♥t❡ ❛ ❞✐♥â♠✐❝❛ ❞♦ ✈órt✐❝❡✱ ♥♦t❛❞❛♠❡♥t❡

s✉❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦✱ ❞❡♥tr❡ ♦✉tr♦s ❛s♣❡❝t♦s ❝♦♠♦ s❡✉ ♠♦✈✐♠❡♥t♦ ❣✐r♦tró♣✐❝♦✳ ❊♠

♥♦ss♦ tr❛❜❛❧❤♦✱ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ♣r♦♣♦st❛ ♣❛r❛ ♠♦❞❡❧❛r t❛✐s ❞❡❢❡✐t♦s ❝♦♠♦ s❡♥❞♦

♣✉♥t✐❢♦r♠❡s✱ ✐♥❝❧✉✐♥❞♦ ✉♠ ♣♦t❡♥❝✐❛❧ ❞♦ t✐♣♦ λδ3(~x− ~x0)✱ s❡♥❞♦ λ ✉♠❛ ❝♦♥st❛♥r❡ r❡❛❧✳

❚❛❧ ♠♦❞❡❧♦ ♥♦s ❝♦♥❞✉③✱ ❡❢❡t✐✈❛♠❡♥t❡✱ ❛ ❝♦♥s✐❞❡r❛r s♦♠❡♥t❡ ❛ ✐♥t❡r❛çã♦ ❧♦❝❛❧ ❞♦ ❞❡❢❡✐t♦

❝♦♠ ♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ ✭✈♦rt❡① ❝♦r❡✮✳ ❉❡s❡♥✈♦❧✈❡♠♦s ✉♠ ♠♦❞❡❧♦ ❛♥❛❧ít✐❝♦✱ r❡❧❛t✐✲

✈❛♠❡♥t❡ s✐♠♣❧❡s✱ q✉❡ ♠♦str♦✉✲s❡ ❝❛♣❛③ ❞❡ ❞❡s❝r❡✈❡r ❝❡rt♦s ❛s♣❡❝t♦s ❞❡ t❛❧ ✐♥t❡r❛çã♦✱

❝♦♥❝♦r❞❛♥❞♦✱ q✉❛❧✐t❛t✐✈❛♠❡♥t❡ ❜❡♠✱ ❝♦♠ r❡s✉❧t❛❞♦s ♦❜s❡r✈❛❞♦s ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡✳

①✐

❆❜str❛❝t

❆♣♦❧ô♥✐♦✱ ❋❡❧✐♣❡ ❆♥❞r❛❞❡✱ ▼✳ ❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❏✉❧②✱ 2010✳ ❆♠♦❞❡❧ ❢♦r str✉❝t✉r❛❧ ❞❡❢❡❝ts ✐♥ ♥❛♥♦♠❛❣♥❡ts✳ ❆❞✈✐s❡r✿ ❲✐♥❞❡r ❆❧❡①❛♥❞❡r▼♦✉r❛ ▼❡❧♦✳ ❈♦✲❆❞✈✐s❡rs✿ ❆❢râ♥✐♦ ❘♦❞r✐❣✉❡s P❡r❡✐r❛✱ ❆❧✈❛r♦ ❱✐❛♥♥❛ ◆♦✈❛❡s ❞❡❈❛r✈❛❧❤♦ ❚❡✐①❡✐r❛

✳

◆❛♥♦❡str✉❝t✉r❡❞ ❢❡rr♦♠❛❣♥❡t✐❝ ♠❛t❡r✐❛❧s✱ ✇✐t❤ s❡✈❡r❛❧ s❤❛♣❡s ❛♥❞ s✐③❡s✱ ❤❛✈❡

❜❡❡♥ ✐♥t❡♥s✐✈❡❧② ✐♥✈❡st✐❣❛t❡❞ ❜❡❝❛✉s❡ ♦❢ t❤❡✐r ♣♦t❡♥t✐❛❧ ❛♣♣❧✐❝❛t✐♦♥ ✐♥ ♠❛❣♥❡t♦❡❧❡tr♦♥✐❝

❞❡✈✐❝❡s✳ ❋❡rr♦♠❛❣♥❡ts ♠❛② ❤❛✈❡ ❛ ♥✉♠❜❡r ♦❢ st❛❜❧❡ ♠❛❣♥❡t✐③❛t✐♦♥ ❝♦♥✜❣✉r❛t✐♦♥s✳ ■♥

♥❛♥♦❞✐s❦s ✇✐t❤ t❤✐❝❦♥❡ss L & 15 ♥♠ ❛♥❞ r❛❞✐✉s R & 100 ♥♠✱ ✈♦rt❡① ❝♦♥✜❣✉r❛t✐♦♥

❛♣♣❡❛rs ❛s t❤❡ ❣r♦✉♥❞✲st❛t❡✱ ✐♥ t❤❡ ❛❜s❡♥❝❡ ♦❢ ❡①t❡r♥❛❧ ♠❛❣♥❡t✐❝ ✜❡❧❞✳ ❋♦r ♠♦r❡ ❝❛r❡✲

❢✉❧ ②♦✉ ❛r❡ ✇❤❡♥ ♣r❡♣❛r✐♥❣ s❛♠♣❧❡s✱ ✐♠♣✉r✐t✐❡s ❛♥❞ str✉❝t✉r❛❧ ❞❡❢❡❝ts ✭✐♥ t❤❡ ❝r②st❛❧

str✉❝t✉r❡✱ ❢♦r ❡①❛♠♣❧❡✮ ✇✐❧❧ ❛❧✇❛②s ❜❡ ♣r❡s❡♥t✱ t♦ ❛ ❣r❡❛t❡r ♦r ❧❡ss❡r ❛♠♦✉♥t✱ ❛♥❞ t❤❡②

❛r❡ ❣❡♥❡r❛❧❧② ♣♦✐♥t❧✐❦❡ ✭♦r t✐♥② str✉❝t✉r❡s✮ ❛♥❞ r❛♥❞♦♠❧② ❞✐str✐❜✉t❡❞ t❤r♦✉❣❤♦✉t t❤❡

♠❛t❡r✐❛❧✳ ■t ✐s ❡①♣❡r✐♠❡♥t❛❧❧② ❦♥♦✇♥ t❤❛t t❤❡s❡ ❞❡❢❡❝ts ♠❛② ❞❡❡♣❧② ♠♦❞✐❢② t❤❡ ✈♦rt❡①

❞②♥❛♠✐❝s✱ ♥♦t❛❜❧② t❤❡✐r ❡q✉✐❧✐❜r✐✉♠ ♣♦s✐t✐♦♥✱ ❛♠♦♥❣ ♦t❤❡rs ❛s♣❡❝ts ❧✐❦❡ t❤❡✐r ❣②r♦tr♦♣✐❝

♠♦t✐♦♥✳ ❍❡r❡✱ ✇❡ ♣r❡s❡♥t ❛ ♣r♦♣♦s❛❧ ❢♦r ♠♦❞❡❧✐♥❣ s✉❝❤ ❞❡❢❡❝ts ❛s t❤❡② ✇❡r❡ ♣♦✐♥t❧✐❦❡✱

✉s✐♥❣ ❛ ♣♦t❡♥t✐❛❧ ❧✐❦❡ λδ3(~x− ~x0)✱ λ ❜❡✐♥❣ ❛ r❡❛❧ ❝♦♥st❛♥t✳ ❚❤✐s ♠♦❞❡❧ ②✐❡❧❞s✱ ❛❝t✉❛❧❧②✱

t♦ ❝♦♥s✐❞❡r ♦♥❧② t❤❡ ❧♦❝❛❧ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❞❡❢❡❝t ❛♥❞ t❤❡ ✈♦rt❡① ❝♦r❡✳ ❲❡ ❞❡✲

✈❡❧♦♣❡❞ ❛ r❡❧❛t✐✈❡❧② s✐♠♣❧❡ ❛♥❛❧②t✐❝❛❧ ♠♦❞❡❧✱ ✇❤✐❝❤ ✇❛s ❛❜❧❡ t♦ ❞❡s❝r✐❜❡ s♦♠❡ ❛s♣❡❝ts

♦❢ s✉❝❤ ✐♥t❡r❛❝t✐♦♥✱ ❛❣r❡❡✐♥❣✱ q✉❛❧✐t❛t✐✈❡❧② ✇❡❧❧✱ ✇✐t❤ r❡s✉❧ts ♦❜s❡r✈❡❞ ❡①♣❡r✐♠❡♥t❛❧❧②✳

①✐✐

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦ ❡ ▼♦t✐✈❛çã♦

❖ ❡st✉❞♦ ❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ♠❛❣♥❡t✐s♠♦ ❡ ❞♦s ♠❛t❡r✐❛✐s

♠❛❣♥ét✐❝♦s sã♦✱ s❡♠ ❞ú✈✐❞❛✱ ✐♠♣♦rt❛♥t❡s ❛s♣❡❝t♦s ♥❛ ❋ís✐❝❛✳ ❘❡❧❛t♦s ❞❛ ✉t✐❧✐③❛çã♦

❞❡ss❡s ♠❛t❡r✐❛✐s r❡♠♦♥t❛♠ à ➪s✐❛ ▼❡♥♦r✱ r❡❣✐ã♦ ❞❛ ▼❛❣♥és✐❛ ✭❛t✉❛❧ ●ré❝✐❛✮✱ ♦♥❞❡✱

❡♠ t♦r♥♦ ❞❡ ✻✵✵ ❛✳ ❈✳✱ ❚❛❧❡s ❞❡ ▼✐❧❡t♦ ❞❡s❝♦❜r✐✉ q✉❡ ♣❡q✉❡♥❛s ♣❡❞r❛s ✭♠❛❣♥❡t✐t❛✱

❋❡3❖4✮ ❛tr❛í❛♠ s❡✉ ❝❛❥❛❞♦ ❞❡ ❢❡rr♦✳ ❉✉r❛♥t❡ ♠✉✐t♦ t❡♠♣♦✱ ❛ ú♥✐❝❛ ❛♣❧✐❝❛çã♦ ❞♦s

♠❛t❡r✐❛✐s ♠❛❣♥ét✐❝♦s ❢♦✐ ❛ ❢❛❜r✐❝❛çã♦ ❞❡ ❜úss♦❧❛s✱ ❝♦♠♦ ❛ ♠♦str❛❞❛ ♥❛ ✜❣✉r❛ ✶✳✶✳

❆♣❡s❛r ❞❡ s✉❛ ❤✐stór✐❛ ♠✐❧❡♥❛r✱ ❢♦✐ ♥♦s ú❧t✐♠♦s ❞♦✐s sé❝✉❧♦s q✉❡ ♦ ♠❛❣♥❡t✐s♠♦ ❛❞q✉✐r✐✉

❝❛r❛❝t❡ríst✐❝❛s ❞❡ ✉♠ ✐♠♣♦rt❛♥t❡ ❝❛♠♣♦ ❞❛ ❝✐ê♥❝✐❛✳ ❉❡s❞❡ ❡♥tã♦✱ ♠❛t❡r✐❛✐s ♠❛❣♥ét✐❝♦s

✈ê♠ ❞❡s❡♠♣❡♥❤❛♥❞♦ ✉♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ❡ ❝♦♥tr✐❜✉✐♥❞♦ ❞❡ ❢♦r♠❛ ✈✐t❛❧ ♥❛ ❤✐stór✐❛

❞❛s ❝✐✈✐❧✐③❛çõ❡s ❜❡♠ ❝♦♠♦ ♥♦ s❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ t❡❝♥♦❧ó❣✐❝♦✳

❋✐❣✉r❛ ✶✳✶✿ ❇úss♦❧❛ ❛♥t✐❣❛ ✉t✐❧✐③❛❞❛ ♣❡❧♦s ❝❤✐♥❡s❡s ❬✶❪✳

❆ ♣r✐♠❡✐r❛ ❡①♣❧✐❝❛çã♦ ❝✐❡♥tí✜❝❛ ♣❛r❛ ♦s ❢❡♥ô♠❡♥♦s ❡❧étr✐❝♦s ❡ ♠❛❣♥ét✐❝♦s ✈❡✐♦

♥♦ sé❝✉❧♦ ❳■❳ ❝♦♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❚❡♦r✐❛ ❊❧❡tr♦♠❛❣♥ét✐❝❛✳ ●r❛ç❛s ❛ ❖❡rst❡❞✱

✶

✶✳ ■♥tr♦❞✉çã♦ ❡ ▼♦t✐✈❛çã♦

❆♠♣èr❡✱ ❋❛r❛❞❛②✱ ❞❡♥tr❡ ♦✉tr♦s ❢ís✐❝♦s✱ ❢♦✐ ♣♦ssí✈❡❧ ❛ ❝r✐❛çã♦ ❞❡ ✐♥ú♠❡r♦s ❡q✉✐♣❛♠❡♥t♦s✳

◆♦ ✜♥❛❧ ❞♦ sé❝✉❧♦ ❳■❳✱ ❏✳ ❈✳ ▼❛①✇❡❧❧ s✐♥t❡t✐③♦✉ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞✐s♣♦♥í✈❡❧ à é♣♦❝❛✱

❧❡✈❛♥❞♦ às ❡q✉❛çõ❡s ❞♦ ❊❧❡tr♦♠❛❣♥❡t✐s♠♦✱ ❛♣r❡s❡♥t❛❞❛s ❡♠ s❡✉ ❢❛♠♦s♦ ❧✐✈r♦ ✧❚r❡❛t✐s❡

♦♥ ❊❧❡❝tr✐❝✐t② ❛♥❞ ▼❛❣♥❡t✐s♠✧ ❬✷❪✳

❖ ✜♥❛❧ ❞♦ sé❝✉❧♦ ❳■❳ ❡ ✐♥í❝✐♦ ❞♦ sé❝✉❧♦ ❳❳ ❢♦r❛♠ ♠♦✈✐❞♦s ♣♦r ✉♠❛ ❡♥♦r♠❡

❜✉s❝❛ ♣♦r ♠♦❞❡❧♦s ❝❛♣❛③❡s ❞❡ ❡①♣❧✐❝❛r ♦ ♠❛❣♥❡t✐s♠♦ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❝♦r♣✉s❝✉❧❛r

❞❛ ♠❛tér✐❛✳ P♦❞❡♠♦s ❝✐t❛r ❏❛♠❡s ❊r✇✐♥✱ q✉❡ ♣✉❜❧✐❝❛ ❡♠ 1891 ✉♠ ❧✐✈r♦ ❝♦♠ ❛s ♣r♦✲

♣r✐❡❞❛❞❡s ❝♦♥❤❡❝✐❞❛s ❞♦s ♠❛t❡r✐❛✐s ✉s❛❞♦s ❡♠ r❡❛t♦r❡s✱ ♠♦t♦r❡s ❡ ❣❡r❛❞♦r❡s✳ ◆❛ ♠❡s♠❛

é♣♦❝❛ ❖❜❡r❧✐♥ ❙♠✐t❤ ❡❧❛❜♦r❛ ✉♠❛ ♣r♦♣♦st❛ t❡ór✐❝❛ ❞❡ ❛♣❧✐❝❛çã♦ ❞❡ ♠❛t❡r✐❛✐s ♠❛❣♥ét✐✲

❝♦s ❡♠ ❣r❛✈❛çã♦ ♠❛❣♥ét✐❝❛✳ ❊♠ ✶✾✵✺ P❛✉❧ ▲❛♥❣❡✈✐♥ ❡①♣❧✐❝❛ ♦ ❞✐❛♠❛❣♥❡t✐s♠♦ ❬✷❪✳

❊♠ ✶✾✵✼ P✐❡rr❡ ❲❡✐ss ❢❛③ ♦ ♣r✐♠❡✐r♦ ♠♦❞❡❧♦ ✭❜❛s❡❛❞♦ ❡♠ ♠♦♠❡♥t♦s ♠❛❣♥ét✐❝♦s ♠✐✲

❝r♦s❝ó♣✐❝♦s✮ ♣❛r❛ ✉♠ ♠❛t❡r✐❛❧ ❢❡rr♦♠❛❣♥ét✐❝♦✱ ❧❛♥ç❛♥❞♦ ♠ã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ❝❛♠♣♦

♠♦❧❡❝✉❧❛r ❡ ♣❛r❡❞❡s ❞❡ ❞♦♠í♥✐♦s✳

❆♣❡s❛r ❞❛ ❣❛♠❛ ❞❡ ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ♦s ♠❛t❡r✐❛✐s ♠❛❣♥ét✐❝♦s✱ ❛ ♥❛t✉r❡③❛ ❞♦s

♠♦♠❡♥t♦s ❞❡ ❞✐♣♦❧♦ ♠❛❣♥ét✐❝♦s ❛✐♥❞❛ ❡r❛ ✉♠ ♠✐stér✐♦✳ ❈♦♠ ♦ s✉r❣✐♠❡♥t♦ ❞❛ ▼❡❝â♥✐❝❛

◗✉â♥t✐❝❛✱ ❛✐♥❞❛ ♥♦ ✐♥í❝✐♦ ❞♦ sé❝✉❧♦ ❳❳✱ ❝♦♠❡ç❛ ❛ ✜❝❛r ❝❧❛r♦ q✉❡ ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s

tê♠✱ ❡♠ ú❧t✐♠❛ ✐♥stâ♥❝✐❛✱ ❡①♣❧✐❝❛çã♦ q✉â♥t✐❝❛✳ P♦✉❝♦ ❞❡♣♦✐s ❞❛ ❞❡s❝♦❜❡rt❛ ❞♦ s♣✐♥ ❡♠

1922 ♣♦r ❖tt♦ ❙t❡r♥ ❡ ❲❛❧t❡r ●❡r❧❛❝❤✱ ❍❡✐s❡♥❜❡r❣✱ P❛✉❧✐✱ ❉✐r❛❝ ❡ ✈❛♥ ❱❧❡❝❦ ❡①♣❧✐❝❛♠ ♦

q✉❡ ❛❝♦♥t❡❝✐❛ ♥♦ ✐♥t❡r✐♦r ❞❡ ✉♠ ♠❛t❡r✐❛❧ ♠❛❣♥ét✐❝♦✱ t♦r♥❛♥❞♦ t❛❧ ár❡❛ ✉♠ ❞♦s ❝❛♠♣♦s

♠❛✐s ✐♠♣♦rt❛♥t❡s ❞❡ ❛♣❧✐❝❛çã♦ ❞❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛✳

❉♦s ❛♥♦s 30 ♣❛r❛ ❝á✱ s❡♠ ❞ú✈✐❞❛ ❝r❡s❝❡✉ ♦ ♥ú♠❡r♦ ❞❡ ❛♣❧✐❝❛çõ❡s t❡❝♥♦❧ó❣✐✲

❝❛s ✉s❛♥❞♦✲s❡ ♠❛t❡r✐❛✐s ♠❛❣♥ét✐❝♦s✱ ❝♦♠♦ ❛ ❢❛❜r✐❝❛çã♦ ❞❡ tr❛♥s✐st♦r❡s✱ ❛ t❡❝♥♦❧♦❣✐❛

❞❛ ✐♥❢♦r♠❛çã♦✱ ❣❡r❛❞♦r❡s✱ ♠♦t♦r❡s ♠❛✐s ♣♦t❡♥t❡s✱ ❡t❝✳ ❈♦♥❝r❡t❛♠❡♥t❡✱ ✉♠❛ ❞❛s ♠❛✐s

♣r♦♠✐ss♦r❛s é ❛ ❣r❛✈❛çã♦ ♠❛❣♥ét✐❝❛ ❡♠ ✜❧♠❡s ♠❛❣♥ét✐❝♦s ✭❣❡r❛❧♠❡♥t❡ ❢❡✐t♦s ❞❡ ó①✐❞♦s

❞❡ ❢❡rr♦✱ ♣❧❛t✐♥❛ ❡ ❝r♦♠♦✮✳ ❊ss❛ t❡❝♥♦❧♦❣✐❛ é ❡ss❡♥❝✐❛❧ ♥♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞❡ ❣r❛✈❛❞♦r❡s

❞❡ s♦♠ ❡ ✈í❞❡♦✱ ❛❧é♠ ❞❡ ❞✐✈❡rs♦s ❡q✉✐♣❛♠❡♥t♦s ❛❝✐♦♥❛❞♦s ♣♦r ❝❛rtõ❡s ♠❛❣♥ét✐❝♦s✱

s❡♥❞♦ ✐♠♣r❡s❝✐♥❞í✈❡❧ ♥❛ ❝♦♥❢❡❝çã♦ ❞❡ ❞✐s❝♦s rí❣✐❞♦s ✉t✐❧✐③❛❞♦s ❡♠ ❝♦♠♣✉t❛❞♦r❡s✳ ▼❛✐s

❛❞✐❛♥t❡✱ ❡♠ ♠♦♠❡♥t♦ ♦♣♦rt✉♥♦✱ ✈♦❧t❛r❡♠♦s ❛ ❢❛❧❛r ❞❡ss❡ ❛ss✉♥t♦ ❝♦♠ ♠❛✐s ❝✉✐❞❛❞♦✳

◆ã♦ ♠❡♥♦s ✐♠♣♦rt❛♥t❡✱ sã♦ ♦s ❝✐r❝✉✐t♦s ✐♥t❡❣r❛❞♦s✱ ♦✉ ❝❤✐♣s✱ ♣r❡s❡♥t❡s ❡♠ q✉❛s❡ t♦❞♦s

♦s ❝✐r❝✉✐t♦s ❡❧❡trô♥✐❝♦s✳ ❚❛♥t♦ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❝✐r❝✉✐t♦s ❡❧❡trô♥✐❝♦s ♥♦s ❝❤✐♣s q✉❛♥t♦

✷

✶✳ ■♥tr♦❞✉çã♦ ❡ ▼♦t✐✈❛çã♦

❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❣r❛✈❛çã♦ ♠❛❣♥ét✐❝❛ ♥♦s ❞✐s❝♦s rí❣✐❞♦s ✈ê♠ ❛✉♠❡♥t❛♥❞♦ ♠✉✐t♦ ♥♦s r❡✲

❝❡♥t❡s ❛♥♦s✳ ❆❧é♠ ❞❡ss❛s ❛♣❧✐❝❛çõ❡s✱ ♦ ♠❛❣♥❡t✐s♠♦ ❡stá ♣r❡s❡♥t❡ ❡♠ ♦✉tr❛s ár❡❛s✱ ❝♦♠♦

♥❛ ❣❡♦❧♦❣✐❛ ✭♦♥❞❡ ♦ ❛❧✐♥❤❛♠❡♥t♦ ❞❡ ♠♦♠❡♥t♦s ♠❛❣♥ét✐❝♦s s♦❜ ❛ ✐♥✢✉ê♥❝✐❛ ❞♦ ❝❛♠♣♦

♠❛❣♥ét✐❝♦ ❞❛ ❚❡rr❛ ♣❡r♠✐t❡ ✐♥❢♦r♠❛r ❛ ❡✈♦❧✉çã♦ ❞❛ ❚❡rr❛ ❡ ❛ ❞❛t❛çã♦ ❞❡ r♦❝❤❛s ❬✸❪✮✱

❛té ♦ tr❛♥s♣♦rt❡ ❞❡ ❞r♦❣❛s q✉❡ ♣♦❞❡♠ s❡r ❞✐r❡❝✐♦♥❛❞❛s ❛ ór❣ã♦s ♦✉ t❡❝✐❞♦s ❡s♣❡❝í✜✲

❝♦s ❬✹❪✱ ❛❧é♠ ❞♦ ✉s♦ ❞❡ ♥❛♥♦♣❛rtí❝✉❧❛s ♠❛❣♥ét✐❝❛s ❡♠ ♠ét♦❞♦s t❡r❛♣ê✉t✐❝♦s ❡ ❞✐❛❣♥ós✲

t✐❝♦s ❬✺❪✳ ❊s♣❡❝✐✜❝❛♠❡♥t❡✱ ❝♦♠♦ ♠♦str❛❞♦ r❡❝❡♥t❡♠❡♥t❡ ♥❛ r❡❢❡rê♥❝✐❛ ❬✻❪✱ ♠✐❝r♦❞✐s❝♦s

♠❛❣♥ét✐❝♦s ✭▼❉s✮ ❛♣r❡s❡♥t❛♥❞♦ ❝♦♥✜❣✉r❛çã♦ ♠❛❣♥ét✐❝❛ ❞♦ t✐♣♦✲✈órt✐❝❡ ✭❛ q✉❛❧ s❡rá ❡s✲

t✉❞❛❞❛ ❡♠ ♠♦♠❡♥t♦ ♦♣♦rt✉♥♦✮ ❡ ❜✐♦❢✉♥❝✐♦♥❛❧✐③❛❞♦s ❝♦♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❛♥t✐✲❝♦r♣♦✱

❝❛♣❛③❡s ❞❡ ✧❧✐❣❛r❡♠✲s❡✧ s❡❧❡t✐✈❛♠❡♥t❡ ❛ ❝é❧✉❧❛s ❝❛♥❝❡rí❣❡♥❛s✱ ♦❝❛s✐♦♥❛♠✱ ♣♦r ♠❡✐♦

❞❡ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ♦s❝✐❧❛tór✐♦✱ ❛ ♠♦rt❡ ✐♥ ✈✐tr♦ ❞❡✱ ❛♣r♦①✐♠❛❞❛♠❡♥t❡✱ ✾✵✪ ❞❛s

❝é❧✉❧❛s✳ ❊sq✉❡♠❛t✐❝❛♠❡♥t❡✱ t❛❧ ♣r♦❝❡ss♦ é ❞❡s❝r✐t♦ ♥❛ ✜❣✉r❛ ✶✳✷✳

❋✐❣✉r❛ ✶✳✷✿ ❖s ♠✐❝r♦❞✐s❝♦s sã♦ ❜✐♦❢✉♥❝✐♦♥❛❧✐③❛❞♦s ❝♦♠ ♦ ❛♥t✐✲❝♦r♣♦ ❛♥t✐✲❤✉♠❛♥✲■▲✶✷α✷❘✱

❡s♣❡❝✐✜❝❛♠❡♥t❡ ♣❛r❛ ❛t✐♥❣✐r❡♠ ❝é❧✉❧❛s ❤✉♠❛♥❛s ❞❡ ❣❧✐♦❜❧❛st♦♠❛ ♠✉❧t✐❢♦r♠❡✳ ❆♦ ❛♣❧✐❝❛r✲s❡ ✉♠

❝❛♠♣♦ ♠❛❣♥ét✐❝♦✱ ♦s ❞✐s❝♦s ♠❛❣♥ét✐❝♦s ♦s❝✐❧❛♠✱ ❝♦♠♣r♦♠❡t❡♥❞♦ ❛ ✐♥t❡❣r✐❞❛❞❡ ❞❛ ♠❡♠❜r❛♥❛✳

❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢❡rê♥❝✐❛ ❬✻❪✳

❆♣❧✐❝❛çõ❡s ♥♦ â♠❜✐t♦ ♥❛♥♦♠étr✐❝♦ r❡q✉❡r❡♠ ❛❧t♦ ❣r❛✉ ❞❡ s♦✜st✐❝❛çã♦ ❡♠ té❝✲

♥✐❝❛s ❡①♣❡r✐♠❡♥t❛✐s✳ ❆ ❡✈♦❧✉çã♦ t❡❝♥♦❧ó❣✐❝❛ ❞❡ss❛s ❛♣❧✐❝❛çõ❡s ❢♦✐ ♣♦ssí✈❡❧ ❞❡✈✐❞❛ ❛♦

❛♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❛s té❝♥✐❝❛s ❞❡ ♣r❡♣❛r❛çã♦ ❞❡ ❛♠♦str❛s ❡♠ ❡s❝❛❧❛s ♥❛♥♦♠étr✐❝❛s✱ ❜❡♠

❝♦♠♦ ❞❡ té❝♥✐❝❛s ❞❡ ✐♠❛❣❡♠ ❡♠ ❛❧t❛ r❡s♦❧✉çã♦✳ ◆♦ ❡♥t❛♥t♦✱ ❞✉r❛♥t❡ ❛s ❞✉❛s ú❧t✐♠❛s

✸

✶✳ ■♥tr♦❞✉çã♦ ❡ ▼♦t✐✈❛çã♦

❞é❝❛❞❛s✱ ❛ ♣❡sq✉✐s❛ ❡♠ ♠❛t❡r✐❛✐s ♠❛❣♥ét✐❝♦s s♦❢r❡✉ ✉♠ ❣r❛♥❞❡ ✐♠♣✉❧s♦ ❞❡✈✐❞♦ às

❞❡s❝♦❜❡rt❛s ❢❡✐t❛s ❡♠ ❡str✉t✉r❛s ❛rt✐✜❝✐❛✐s ❞❡ ✜❧♠❡s ✜♥♦s✳ ❈♦♠♦ r❡✈✐s❛❞♦ ♥❛s r❡❢❡✲

rê♥❝✐❛s ❬✼✱ ✺❪✱ ❡①✐st❡♠ três t✐♣♦s ♣r✐♥❝✐♣❛✐s ❞❡ ❢❛❜r✐❝❛çã♦ ❞❡ ♠❛t❡r✐❛✐s ♠❛❣♥ét✐❝♦s✿ ✭✐✮

t♦♣✲❞♦✇♥ ✭✧❝✐♠❛ ♣❛r❛ ❜❛✐①♦✧✮✱ ✭✐✐✮ ❜♦tt♦♠✲✉♣ ✭✧❜❛✐①♦ ♣❛r❛ ❝✐♠❛✧✮✱ ❡ ✭✐✐✐✮ ✈✐rt✉❛❧ ❢❛❜

✭✧❢❛❜r✐❝❛çã♦ ✈✐rt✉❛❧✧✮✳ ❖ ♣r✐♠❡✐r♦ t✐♣♦ é ♦ q✉❡ t❡♠ s✐❞♦ ✉t✐❧✐③❛❞♦ tr❛❞✐❝✐♦♥❛❧♠❡♥t❡

♣❛r❛ ♠♦❞❡❧❛r ❛s ❛♠♦str❛s✱ ✈✐❛ té❝♥✐❝❛s ❞❡ ❧✐t♦❣r❛✜❛✳ ▼❛s ❛ ❧✐t♦❣r❛✜❛ ♥❡❝❡ss✐t❛ ❞❡ ✉♠

❛♣r✐♠♦r❛♠❡♥t♦ ♣❛r❛ ♦❜t❡r ❡s❝❛❧❛s ❞❡ ❝♦♠♣r✐♠❡♥t♦s ♠❡♥♦r❡s ✭. 10 ♥♠✮✳ ◆❡st❡ ♣♦♥t♦

é q✉❡ ❡♥tr❛ ♦ s❡❣✉♥❞♦ t✐♣♦✱ ❜❛s❡❛❞♦ ♥❛ ♦r❣❛♥✐③❛çã♦ ❞❡ ♠✐❝❡❧❛s ❞❡ s✉r❢❛❝t❛♥t❡s ❬✽❪✳ P♦r

✜♠✱ ♦ t❡r❝❡✐r♦ t✐♣♦ é ❜❛s❡❛❞♦ ❡♠ s✐♠✉❧❛çã♦ ❝♦♠♣✉t❛❝✐♦♥❛❧✱ út✐❧ ♥♦ ❡st✉❞♦ ❞❡ ♥♦✈♦s ♠❛✲

t❡r✐❛✐s ❡ ♣r♦♣r✐❡❞❛❞❡s✱ ♦❢❡r❡❝❡♥❞♦ ✉♠ ót✐♠♦ ❝♦♥tr♦❧❡ ❞♦ s✐st❡♠❛✱ ♦ q✉❛❧ é ♣r❡❝✐s❛♠❡♥t❡

❡s♣❡❝✐✜❝❛❞♦ ❞❡ ✐♥í❝✐♦✳

◗✉❛♥❞♦ ♣r♦❞✉③✐❞❛s ❛rt✐✜❝✐❛❧♠❡♥t❡✱ ❛s ♥❛♥♦❡str✉t✉r❛s ♣♦❞❡♠ s❡r ♠♦❞❡❧❛❞❛s

❡♠ ❞✐✈❡rs❛s ❢♦r♠❛s ❣❡♦♠étr✐❝❛s✱ ❝♦♠♦ ✧♥❛♥♦❞♦ts✧ ✿ ❝✐r❝✉❧❛r❡s✱ tr✐❛♥❣✉❧❛r❡s✱ r❡t❛♥❣✉✲

❧❛r❡s✱ ❡❧í♣t✐❝♦s✱ ❡t❝❀ ✧♥❛♥♦✇✐r❡s✧ ✿ r❡t❛♥❣✉❧❛r❡s✱ ❝✐❧í♥❞r✐❝♦s❀ ✧♥❛♥♦r✐♥❣s✧ ✿ ❝✐❧í♥❞r✐❝♦s✱

t♦r♦✐❞❛✐s❀ ✧♥❛♥♦t✉❜♦s✧✱ ❡t❝✳ ◆❛ ❢❛❜r✐❝❛çã♦ ❞❡st❡s ♠❛t❡r✐❛✐s✱ ❛s té❝♥✐❝❛s ♠❛✐s ✉t✐✲

❧✐③❛❞❛s sã♦ ❛ ❧✐t♦❣r❛✜❛ ♣♦r ❢❡✐①❡s ❞❡ ❡❧étr♦♥s ✭❊❇▲✱ ❊❧❡❝tr♦♥ ❇❡❛♠ ▲✐t❤♦❣r❛♣❤②✮ ❬✾✕✶✹❪✱

❧✐t♦❣r❛✜❛ ♣♦r ✐♥t❡r❢❡rê♥❝✐❛ ✭■▲✱ ■♥t❡r❢❡r❡♥❝❡ ▲✐t❤♦❣r❛♣❤②✮ ❬✶✺✱ ✶✻❪✱ q✉❡ é ❛ té❝♥✐❝❛ ❡s✲

❝♦❧❤✐❞❛ q✉❛♥❞♦ ♣r❡t❡♥❞❡✲s❡ ♠♦❞❡❧❛r ❣r❛♥❞❡s ár❡❛s✱ ♣♦r r❡q✉❡r❡r ✉♠ t❡♠♣♦ ♠❡♥♦r q✉❡

❛ ♣r✐♠❡✐r❛✱ ❛❧é♠ ❞❡ ♦✉tr❛s ❝♦♠♦ ♥❛♥♦❧✐t♦❣r❛✜❛ ❞❡ r❛✐♦✲❳ ✭❳✲r❛② ♥❛♥♦❧✐t❤♦❣r❛♣❤②✮ ❡

❧✐t♦❣r❛✜❛ ♣♦r ✐♠♣r❡ssã♦ ✭✐♠♣r✐♥t ❧✐t❤♦❣r❛♣❤②✮ ❬✶✻❪✳ ❈♦♠ ❡st❛s té❝♥✐❝❛s é ♣♦ssí✈❡❧ ♣r♦✲

❞✉③✐r ❛♠♦str❛s ❝♦♠ ❝♦♠♣r✐♠❡♥t♦s ❝❛r❛❝t❡ríst✐❝♦s & 10 ♥♠✳ ❈♦♠♦ s❡rá ❞✐s❝✉t✐❞♦ ❛♦

❧♦♥❣♦ ❞♦ t❡①t♦✱ ♥♦ss♦ tr❛❜❛❧❤♦ é ❜❛s❡❛❞♦ ❡♠ ♠❛t❡r✐❛✐s ❢❡rr♦♠❛❣♥ét✐❝♦s ❝♦♥❤❡❝✐❞♦s

❝♦♠♦ P❡r♠❛❧❧♦② ✭P②✮✱ q✉❡ sã♦ ❧✐❣❛s ❞❡ ♥íq✉❡❧ ❡ ❢❡rr♦ ✭❋❡x◆✐(1−x)✱ x ∼ 18 − 22✪✮ ❡

❛♠♦str❛s ❞❡ ♥íq✉❡❧ ✭◆✐✮ ❡ ❝♦❜❛❧t♦ ✭❈♦✮✳ ❚❛✐s ❧✐❣❛s✱ ❜❡♠ ❝♦♠♦ ❛ ❝❤❛♠❛❞❛ s✉♣❡r♣❡r♠❛❧✲

❧♦② ✭◆✐80 ❋❡14▼♦5✮ ❡ ❛s ❛♠♦str❛s ❞❡ ❈♦ ❡ ◆✐✱ sã♦ ♣r❡♣❛r❛❞❛s ♣❡❧❛ ❝♦♠❜✐♥❛çã♦ ❞❡ três

té❝♥✐❝❛s✿ ❊❇▲✱ ❧✐❢t✲♦✛ ❡ t❤❡r♠❛❧ ❡✈❛♣♦r❛t✐♦♥ ✭❡✈❛♣♦r❛çã♦ tér♠✐❝❛✮ ❬✾✱✶✵❪✳ ❆ ✜❣✉r❛ ✶✳✸

✐❧✉str❛ ❡sq✉❡♠❛t✐❝❛♠❡♥t❡ ❝♦♠♦ s❡ ❞á t❛❧ ♣r♦❝❡ss♦✳

❆ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛s ❛♠♦str❛s é ❢❡✐t❛ ♣♦r ❞✐✈❡rs❛s té❝♥✐❝❛s✱ ❞❡♣❡♥❞❡♥❞♦ ❞❡ s✉❛s

❢✉♥❝✐♦♥❛❧✐❞❛❞❡s✳ ❉❡♥tr❡ ❡❧❛s ♣♦❞❡♠♦s ❝✐t❛r ❛ ♠✐❝r♦s❝♦♣✐❛ ❡❧❡trô♥✐❝❛ ❞❡ tr❛♥s♠✐ssã♦ ❞❡

▲♦r❡♥t③ ✭▲❚❊▼✮ ❬✾✱✶✸❪✱ ♠❛❣♥❡t♦♠❡tr✐❛ ❍❛❧❧ ✭❍❛❧❧ ♠❛❣♥❡t♦♠❡tr②✮ ❬✶✵❪✱ ♠✐❝r♦s❝ó♣✐♦ ❞❡

✹

✶✳ ■♥tr♦❞✉çã♦ ❡ ▼♦t✐✈❛çã♦

❋✐❣✉r❛ ✶✳✸✿ ■❧✉str❛çã♦ ❡sq✉❡♠át✐❝❛ ❞❛ ❢❛❜r✐❝❛çã♦ ❞❡ ♥❛♥♦❞✐s❝♦s ♠❛❣♥ét✐❝♦s ✉t✐❧✐③❛♥❞♦✲s❡ ❛s

três té❝♥✐❝❛s✿ ❊❇▲✱ t❤❡r♠❛❧ ❡✈❛♣♦r❛t✐♦♥ ❡ ❧✐❢t✲♦✛✳ ◆♦ ♣r✐♠✐r♦ q✉❛❞r♦ ❛❝✐♠❛ t❡♠♦s ♦ s✉❜str❛t♦

❡♠ ❛③✉❧ ❡ ♦ r❡s✐st❡ ❡♠ ❛♠❛r❡❧♦✳ ❊♠ s❡❣✉✐❞❛ é ❡♠✐t✐❞♦ ♦ ❢❡✐①❡ ❞❡ ❡❧étr♦♥s ✭❊❇▲✮ ♥❛ ❛♠♦str❛✱ ❛

✜♠ ✬❧✐♠♣❛r✬ ♦ ❧♦❝❛❧ ♦♥❞❡ s❡rá ❞❡♣♦s✐t❛❞❛ ❛ ❛♠♦str❛✳ ❆♣ós ❢❡✐t♦ ✐ss♦✱ ♦ ♠❛t❡r✐❛❧ é ❞❡♣♦s✐t❛❞♦

✈✐❛ ❛ té❝♥✐❝❛ ❞❡ ❡✈❛♣♦r❛çã♦ tér♠✐❝❛ ✭❚❊✮✳ ❋✐♥❛❧♠❡♥t❡✱ ❤á ✉♠ ❜❛♥❤♦ ❝♦♠ ✉♠ s♦❧✈❡♥t❡ ♣❛r❛ ❛

r❡♠♦çã♦ ❞♦ r❡s✐st❡ ✭❧✐❢t✲♦✛ ✮✱ ❡ ♦ r❡s✉❧t❛❞♦ ✜♥❛❧ é ♦ s❜str❛t♦ ❝♦♠ ♦ ♠❛t❡r✐❛❧ ❞❡♣♦s✐t❛❞♦ ❬✶✼❪✳

❢♦rç❛ ♠❛❣♥ét✐❝❛ ✭▼❋▼✱ ♠❛❣♥❡t✐❝ ❢♦r❝❡ ♠✐❝r♦s❝♦♣②✮ ❬✶✵✱✶✽❪✱ q✉❡ sã♦ té❝♥✐❝❛s ✉t✐❧✐③❛❞❛s

♣❛r❛ ♠♦♥✐t♦r❛r✱ ❞❡♥tr❡ ♦✉tr❛s ♣r♦♣r✐❡❞❛❞❡s✱ ❛ r❡✈❡rsã♦ ❞❡ ♠❛❣♥❡t✐③❛çã♦ ❡♠ P❡r♠❛❧✲

❧♦②✱ ❛♠♦str❛s ❞❡ ◆✐ ❡ ❈♦❀ ♦ ♠✐❝r♦s❝ó♣✐♦ ❡❧❡trô♥✐❝♦ ❞❡ tr❛♥s♠✐ssã♦ ✭❚❊▼✱ tr❛♥s♠✐ss✐♦♥

❡❧❡❝tr♦♥ ♠✐❝r♦s❝♦♣②✮ ♣♦❞❡ s❡r ✉s❛❞♦ ♣❛r❛ ♠♦str❛r ❛ ♠✐❝r♦❡str✉t✉r❛ ❞❛ ❛♠♦str❛ ❞❡✲

♣♦s✐t❛❞❛ ❛♦ ♣❛ss♦ q✉❡ ♦ ♠✐❝r♦s❝ó♣✐♦ ❡❧❡trô♥✐❝♦ ❞❡ ✈❛rr❡❞✉r❛ ✭❙❊▼✱ s❝❛♥♥✐♥❣ ❡❧❡❝tr♦♥

♠✐❝r♦s❝♦♣②✮ ♣♦❞❡ s❡r ✉s❛❞♦ ♣❛r❛ ❝❤❡❝❛r ♦ t❛♠❛♥❤♦ ❡ ❢♦r♠❛ ❞♦s ♥❛♥♦♠❛❣♥❡t♦s ❢❛❜r✐❝❛✲

❞♦s ❬✶✷❪❀ ❝✉r✈❛s ❞❡ ❤✐st❡r❡s❡ ❡♠ ❛♠♦str❛s ❞❡ ♣❡q✉❡♥❛ ár❡❛ s✉♣❡r✜❝✐❛❧ ♣♦❞❡♠✱ ♣♦r ❡①✲

❡♠♣❧♦✱ s❡r ♠❡❞✐❞❛s ♣❡❧♦ ❞✐s♣♦s✐t✐✈♦ s✉♣❡r❝♦♥❞✉t♦r ❞❡ ✐♥t❡r❢❡rê♥❝✐❛ q✉â♥t✐❝❛ ✭❙◗❯■❉✱

s✉♣❡r❝♦♥❞✉❝t✐♥❣ q✉❛♥t✉♠ ✐♥t❡r❢❡r❡♥❝❡ ❞❡✈✐❝❡✮ ❬✶✻❪❀ t❡♠♦s t❛♠❜é♠ ♦ ♠✐❝r♦s❝ó♣✐♦ ❞❡

r❡s♦❧✉çã♦ t❡♠♣♦r❛❧ ❑❡rr ✭❚❘❑▼✱ t✐♠❡✲r❡s♦❧✈❡❞ ❑❡rr ♠✐❝r♦s❝♦♣②✮✱ q✉❡ ♣♦❞❡ s❡r ✉s❛❞♦

♣❛r❛ s❡ ❡st✉❞❛r ❛ ❞✐♥â♠✐❝❛ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ❡♠ ♥❛♥♦❞✐s❝♦s ♠❛❣♥ét✐❝♦s ❬✶✾❪❀ ♣♦r ✜♠✱

♣♦❞❡♠♦s ❝✐t❛r ♦ ♠✐❝r♦s❝ó♣✐♦ ♠❛❣♥ét✐❝♦ ❞❡ tr❛♥s♠✐ssã♦ ❞❡ r❛✐♦s✲❳ ✭▼❚❳▼✱ ♠❛❣♥❡t✐❝

tr❛♥s♠✐ss✐♦♥ ❳✲r❛② ♠✐❝r♦s❝♦♣②✮ ❡ ♦ ♠✐❝r♦s❝ó♣✐♦ ❞❡ ❢♦t♦❡♠✐ssã♦ ❞❡ ❡❧étr♦♥s ✭P❊❊▼✱

♣❤♦t♦❡♠✐ss✐♦♥ ❡❧❡❝tr♦♥ ♠✐❝r♦s❝♦♣②✮ ✉t✐❧✐③❛❞♦s✱ ❞❡♥tr❡ ♦✉tr❛s ❝♦✐s❛s✱ ♣❛r❛ ❡st✉❞❛r ♦s ❞❡✲

t❛❧❤❡s ❞❛ ❞✐♥â♠✐❝❛ ❞❛ ❝♦♥✜❣✉r❛çã♦ ♠❛❣♥ét✐❝❛ ❞♦s ♥❛♥♦♠❛❣♥❡t♦s q✉❛♥❞♦ s✉❜♠❡t✐❞♦s

✺

✶✳ ■♥tr♦❞✉çã♦ ❡ ▼♦t✐✈❛çã♦

❛ ♣✉❧s♦s ❞❡ ❝❛♠♣♦s ♠❛❣♥ét✐❝♦s ❡①t❡r♥♦s ❬✷✵❪✳

❆ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❢❛❜r✐❝❛çã♦ ❞❡ ♥❛♥♦❡str✉t✉r❛s ♠❛❣♥ét✐❝❛s ❞❡ ❞✐✈❡rs❛s ❢♦r♠❛s

❡ t❛♠❛♥❤♦s ❧❡✈♦✉ ❛♦ s✉r❣✐♠❡♥t♦ ❞❡ ♥♦✈❛s ár❡❛s ❞❡ ♣❡sq✉✐s❛ ❡♠ ♠❛❣♥❡t✐s♠♦✱ ❡st✐♠✉✲

❧❛❞❛s ♣❡❧❛ ❞❡s❝♦❜❡rt❛ ❞❡ ♥♦✈♦s ❢❡♥ô♠❡♥♦s✳ ❉❡♥tr❡ ❡❧❡s✱ ♣♦❞❡♠♦s ❞❡st❛❝❛r ❛ ❞❡s❝♦❜❡rt❛

❞❛ ♠❛❣♥❡t♦r❡s✐stê♥❝✐❛ ❣✐❣❛♥t❡ ✭●▼❘✱ ❣✐❛♥t ♠❛❣♥❡t♦r❡s✐st❛♥❝❡✮✱ ♥♦ ✜♠ ❞♦s ❛♥♦s ✽✵✳

●▼❘ é ✉♠ ❢❡♥ô♠❡♥♦ ❛ss♦❝✐❛❞♦ ❛ ♠✐❝r♦❡str✉t✉r❛s ♠❛❣♥ét✐❝❛s ❝♦♠♣♦st❛s ❞❡ ♠✉❧t✐❝❛✲

♠❛❞❛s✱ q✉❡ tê♠ s✉❛ r❡s✐stê♥❝✐❛ ❡❧étr✐❝❛ ❛✉♠❡♥t❛❞❛ ❞❡✈✐❞♦ ❛ ♣r❡s❡♥ç❛ ❞❡ ✉♠ ❝❛♠♣♦

♠❛❣♥ét✐❝♦ ❡①t❡r♥♦✳ ❈♦♠♦ ❥á ♠❡♥❝✐♦♥❛♠♦s✱ ❥á ❤á ❛❧❣✉♠ t❡♠♣♦ q✉❡ ❛ ár❡❛ ❞❡ ❛♣❧✐❝❛çõ❡s

❡♠ ♥❛♥♦♠❛❣♥❡t✐s♠♦ ❡stá ❛ss♦❝✐❛❞❛ ❝♦♠ ❛s ♥❡❝❡ss✐❞❛❞❡s ❞❛ ✐♥❞ústr✐❛ ❞❡ ❣r❛✈❛çã♦ ♠❛❣✲

♥ét✐❝❛✶ ✭♣❛r❛ ✉♠❛ ❜r❡✈❡ r❡✈✐sã♦ s♦❜r❡ ❣r❛✈❛çã♦ ♠❛❣♥ét✐❝❛ ❡ ♦ ❢❡♥ô♠❡♥♦ ❞❛ ●▼❘✱ ♦

❧❡✐t♦r é r❡♠❡t✐❞♦ às s❡çõ❡s ❆✳✶ ❡ ❆✳✷ ❞♦ ❆♣ê♥❞✐❝❡ ❆✮✳ P❛r❛ s❡ t❡r ✉♠❛ ✐❞é✐❛✱ ❡♠ ✶✾✾✼✱

♣♦✉❝♦s ❛♥♦s ❞❡♣♦✐s ❞❡ s✉❛ ❞❡s❝♦❜❡rt❛✱ ❛ ■❇▼ ❥á ❧❛♥ç❛✈❛ ♥♦ ♠❡r❝❛❞♦ ♦ ♣r✐♠❡✐r♦ ❞✐s❝♦

rí❣✐❞♦ ❝♦♠ ❝❛❜❡ç❛ ❞❡ ❧❡✐t✉r❛ ❞❡ ❞❛❞♦s ❜❛s❡❛❞❛ ♥♦ ❡❢❡✐t♦ ●▼❘✳ ❊ ❛ ❜✉s❝❛ ♣♦r ♠í❞✐❛s

❞❡ ❛❧tíss✐♠❛s ❞❡♥s✐❞❛❞❡s ❝♦♥t✐♥✉❛ ❡♠ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✿ ♥♦ ✜♠ ❞♦ ❛♥♦ ✷✵✵✵✱ ♣r♦❞✉t♦s

❝♦♠❡r❝✐❛✐s t✐♥❤❛♠ ✉♠❛ ❞❡♥s✐❞❛❞❡ ❞❡ ✐♥❢♦r♠❛çã♦ ❛r♠❛③❡♥❛❞❛ ❞❡ ✷✵ ●❜✐ts✴♣♦❧2 ✭❣✐❣❛✲

❜✐ts ♣♦r ♣♦❧❡❣❛❞❛s2✮ ❬✷✷❪❀ ❡♠ ✷✵✵✻ t❛❧ ❝❛♣❛❝✐❞❛❞❡ ❡r❛ ❞❡ q✉❛s❡ ✶✵✵ ●❜✐ts✴♣♦❧2 ❬✷✸❪✳

❯♠❛ ✐❧✉str❛çã♦ ❞❛ ❡✈♦❧✉çã♦ ❞❛ ❣r❛✈❛çã♦ ♠❛❣♥ét✐❝❛ é ❛♣r❡s❡♥t❛❞❛ ♥❛ ✜❣✉r❛ ✶✳✹✳ ❚❛❧

❢❡♥ô♠❡♥♦ t❡♠ s✐❞♦✱ ❡♥tã♦✱ ❝♦r♦❛❞♦ ❝♦♠♦ ✉♠ ❣r❛♥❞❡ ❛❝❤❛❞♦ ❞❛ ♥❛♥♦t❡❝♥♦❧♦❣✐❛ ❡ r❡♣✲

r❡s❡♥t❛ ✉♠ ♣❛♣❡❧ ❝❡♥tr❛❧ ♥❛ ❞✐r❡çã♦ ❡ ❢♦❝♦ ❞♦ ♠❛❣♥❡t✐s♠♦ ❞❡ s✉♣❡r❢í❝✐❡s✱ q✉❡ ✉♥✐✉ ❛s

❢r♦♥t❡✐r❛s ❡♥tr❡ ♦ ♠❛❣♥❡t✐s♠♦ ♣✉r♦ ❡ ❛♣❧✐❝❛❞♦✳

❉✐s♣♦s✐t✐✈♦s ❡♠ ♥❛♥♦❡s❝❛❧❛ ❡stã♦ s❡ t♦r♥❛♥❞♦ r❛♣✐❞❛♠❡♥t❡ ♦ ❧❛❜♦r❛tór✐♦ ♣r❡❞✐❧❡t♦

❞❡ ♣❡sq✉✐s❛❞♦r❡s✱ ♠♦t✐✈❛♥❞♦ ❛ ♣r♦❝✉r❛ ♣♦r ♥❛♥♦❡str✉t✉r❛s ❝♦♠♦ ✜❧♠❡s ✜♥♦s✱ ✜t❛s ❡

♣♦♥t♦s ✭❞♦ts✮✱ ♣❛r❛ s✐♠✉❧❛r s✉♣❡r❢í❝✐❡s ✷❉✱ ✶❉ ❡ ✵❉✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡ss❛ ♣❡rs♣❡❝✲

t✐✈❛✱ ♦ ♥❛♥♦♠❛❣♥❡t✐s♠♦ s❡❣✉❡ ♦s ❝❛♠✐♥❤♦s ❞❡✐①❛❞♦s ♣❡❧❛ r❡✈♦❧✉çã♦ ♥❛ ♠✐♥✐❛t✉r✐③❛çã♦

❝❛✉s❛❞❛ ♣❡❧♦s s❡♠✐❝♦♥❞✉t♦r❡s✳ ❆ ❡❧❡trô♥✐❝❛ ❞❡ s❡♠✐❝♦♥❞✉t♦r❡s✱ ❡♥tr❡t❛♥t♦✱ ✉t✐❧✐③❛

❛♣❡♥❛s ❛ ❝❛r❣❛ ❞♦ ❡❧étr♦♥✱ ♠❛s ♦ ❡❧étr♦♥ t❛♠❜é♠ ♣♦ss✉✐ s♣✐♥✳ ◆❡st❡ ❝♦♥t❡①t♦✱ ❡ ❧❡✲

✈❛❞❛ ♣❡❧❛ ❞❡s❝♦❜❡rt❛ ❞❛ ●▼❘✱ ✈✐s❧✉♠❜r♦✉✲s❡✱ t❛♠❜é♠✱ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❝♦♥tr♦❧❛r

❡ss❛ ♦✉tr❛ ♣r♦♣r✐❡❞❛❞❡ ❡❧❡trô♥✐❝❛✱ ♦ s♣✐♥✳ ❍á✱ s❡♠ ❞ú✈✐❞❛s✱ ❡s❢♦rç♦s ♥♦ ❞❡s❡♥✈♦❧✈✐✲

✶❖ ♠❡r❝❛❞♦ ♠✉♥❞✐❛❧ ❡♠ ❣r❛✈❛çã♦ ♠❛❣♥ét✐❝❛ é ❡st✐♠❛❞♦ ❡♠ t♦r♥♦ ❞❡ ❯✩✩ ✶✵✵ ❜✐❧❤õ❡s ♣♦r ❛♥♦ ❡

✈❡♠ s❡ ❡①♣❛♥❞✐♥❞♦ ❛ ✉♠❛ t❛①❛ ♣ró①✐♠❛ ❞❡ ✶✼✪ ❛♦ ❛♥♦ ❬✷✶❪✳

✻

✶✳ ■♥tr♦❞✉çã♦ ❡ ▼♦t✐✈❛çã♦

❋✐❣✉r❛ ✶✳✹✿ ✧❊✈♦❧✉çã♦✧ ❞❛ ❣r❛✈❛çã♦ ♠❛❣♥ét✐❝❛✿ ✈❛r✐❛çã♦ ❞❛ ❞❡♥s✐❞❛❞❡ ❞❡ ár❡❛ ✉t✐❧✐③❛❞❛

♥❛ ❣r❛✈❛çã♦ ✭❡♠ ▼❡❣❛❜✐ts✴♣♦❧2✮ ❡♠ ❢✉♥çã♦ ❞♦ ❛♥♦ ❡♠ q✉❡ ♦s ❡q✉✐♣❛♠❡♥t♦s t♦r♥❛r❛♠✲s❡

❞✐s♣♦♥í✈❡✐s ♥♦ ♠❡r❝❛❞♦ ✭■❇▼ ✷✵✵✺✮✳ ❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢❡rê♥❝✐❛ ❬✷✹❪✳

♠❡♥t♦ ❞❛ ❝❤❛♠❛❞❛ ♠❛❣♥❡t♦❡❧❡trô♥✐❝❛✱ ♦✉ s♣✐♥trô♥✐❝❛ ✭s♣✐♥tr♦♥✐❝✮✱ ♦♥❞❡ ✉♠❛ sér✐❡ ❞❡

✐❞❡✐❛s ❡ ♣r♦tót✐♣♦s ✉t✐❧✐③❛♠ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ✐♥t❡r❢❡rê♥❝✐❛s ❡♥tr❡ ❝♦rr❡♥t❡s ❡❧étr✐❝❛s

❝♦♠ ❡❧étr♦♥s ❝♦♠ ✉♠ ú♥✐❝♦ ❡st❛❞♦ ❞❡ s♣✐♥✱ q✉❡ ❛t✉❛❧♠❡♥t❡ ♣♦❞❡♠ s❡r ❜❡♠ ❝♦♥tr♦✲

❧❛❞❛s ❬✷✺❪✳ ❈♦♠♦ ❡①❡♠♣❧♦s✱ ♣♦❞❡♠♦s ❝✐t❛r✿ ♦ tr❛♥s✐st♦r ❞❡ s♣✐♥✱ ♣r♦♣♦st♦ ❡♠ ✶✾✾✵ ❡

❛✐♥❞❛ ❡♠ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✱ q✉❡ ❧❡✈❛ ✈❛♥t❛❣❡♠ s♦❜r❡ ♦ tr❛♥s✐st♦r tr❛❞✐❝✐♦♥❛❧ ❞❡ s❡♠✐✲

❝♦♥❞✉t♦r ♣❡❧♦ ❢❛t♦ ❞❡ ♦s ❡st❛❞♦s ❞❡ s♣✐♥ s❡r❡♠ ❞❡t❡❝t❛❞♦s ❡ ❛❧t❡r❛❞♦s s❡♠ ❛ ❛♣❧✐❝❛çã♦ ❞❡

✉♠❛ ❝♦rr❡♥t❡ ❡❧étr✐❝❛ ❬✷✻❪❀ ❛ ❜✉s❝❛ ♣♦r ♠❛t❡r✐❛✐s q✉❛s❡ ✶✵✵✪ s♣✐♥✲♣♦❧❛r✐③❛❞♦ ♣❛r❛ ❝r✐❛r

❝✐r❝✉✐t♦s ♦♥❞❡ ♦ ✢✉①♦ ❞❡ s♣✐♥s t♦♠❡♠ ❧✉❣❛r ✲ ♦✉✱ t❛❧✈❡③✱ ❝♦❧❛❜♦r❡♠ ✲ ❝♦♠ ♦ ✢✉①♦ ❞❡ ❝❛r✲

❣❛s❀ ❡✱ ♣♦ss✐✈❡❧♠❡♥t❡✱ ❛ ♠❛✐♦r ❛♣❧✐❝❛çã♦ s❡❥❛ ❛ ♠❡♠ór✐❛ ♠❛❣♥ét✐❝❛ ❞❡ ❛❝❡ss♦ ❛❧❡❛tór✐♦

✭▼❘❆▼✱ ♠❛❣♥❡t✐❝ r❛♥❞♦♠ ❛❝❝❡ss ♠❡♠♦r②✮✱ ❛ q✉❛❧ t❡♠ ❛tr❛í❞♦ ❣r❛♥❞❡ ❛t❡♥çã♦ t❛♥t♦

❞❡ ❢❛❜r✐❝❛♥t❡s q✉❛♥t♦ ❞❡ ♣❡sq✉✐s❛❞♦r❡s ♣♦r s❡r ♥ã♦✲✈♦❧át✐❧✷✱ ♦ q✉❡ ❛ ❞✐❢❡r❡♥❝✐❛ ❞❛s ❝♦♥✲

✷❆r♠❛③❡♥❛♠❡♥t♦ ♥ã♦ ✈♦❧át✐❧ é ❛q✉❡❧❡ ♦♥❞❡ ❛ ✐♥❢♦r♠❛çã♦ ♥ã♦ é ♣❡r❞✐❞❛ ♠❡s♠♦ ❛♦ s❡ ❞❡s❧✐❣❛r ❛

❢♦♥t❡ ❞❡ ❡♥❡r❣✐❛✳

✼

✶✳ ■♥tr♦❞✉çã♦ ❡ ▼♦t✐✈❛çã♦

✈❡♥❝✐♦♥❛✐s ♠❡♠ór✐❛s ✈♦❧át❡✐s ❘❆▼ ✭r❛♥❞♦♠ ❛❝❝❡ss ♠❡♠♦r②✮ q✉❡ sã♦ ❢❡✐t❛s à ❜❛s❡ ❞❡

s❡♠✐❝♦♥❞✉t♦r❡s ✭❛♠❜❛s sã♦ ♠❡♠ór✐❛s ❞❡ ❧❡✐t✉r❛ ❡ ❡s❝r✐t❛✮✳ ❖ ❢❛t♦ ❞❡ s❡r ♥ã♦✲✈♦❧át✐❧

t❛❧✈❡③ s❡❥❛ ♦ q✉❡ ♠❛✐s ❡st✐♠✉❧❡ ❛ ♣❡sq✉✐s❛ ❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛s ▼❘❆▼✬s✱ ♣♦✐s

♣♦❞❡♠ ❢✉t✉r❛♠❡♥t❡ t♦r♥❛r r❡❛❧✐❞❛❞❡ ♦ ❛❞✈❡♥t♦ ❞♦ ❝♦♠♣✉t❛❞♦r ❞❡ ✐♥✐❝✐❛❧✐③❛çã♦ ✐♥st❛♥✲

tê♥❡❛ ✭✐♥st❛♥t ❜♦♦t✲✉♣✮✱ ❡♠ ❞❡tr✐♠❡♥t♦ ❞❛ ♠❡♠ór✐❛ ❘❆▼✱ q✉❡ ♣❡r❞❡ ❛ ✐♥❢♦r♠❛çã♦

❛r♠❛③❡♥❛❞❛ q✉❛♥❞♦ ❛ ❝♦rr❡♥t❡ ❡❧étr✐❝❛ ♥❡❝❡ssár✐❛ ❛♦ s❡✉ ♣❡r✐ó❞✐❝♦ ✧r❡❢r❡s❤✧ é ❞❡s❧✐✲

❣❛❞❛ ❡✱ ♣♦rt❛♥t♦✱ t❛♠❜é♠ ❡❝♦♥♦♠✐③❛♥❞♦ ❡♥❡r❣✐❛ ❬✷✷✱✷✸❪✳ ❈♦♠❜✐♥❛♥❞♦✲s❡ ❛s ✈❛♥t❛❣❡♥s

❞❡ ♦♣❡r❛r❡♠ ❡♠ ❛❧t❛ ✈❡❧♦❝✐❞❛❞❡✱ ♣♦ss✉ír❡♠ ❛❧t❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❛r♠❛③❡♥❛♠❡♥t♦ ❞❡ ❞❛❞♦s

❡ s❡r❡♠ ♥ã♦✲✈♦❧át❡✐s✱ t❛✐s ♠❡♠ór✐❛s ❞❡✈❡♠ t♦r♥❛r✲s❡ ❧í❞❡r❡s ♥❛ t❡❝♥♦❧♦❣✐❛ ❞❡ ❛r♠❛③❡♥❛✲

♠❡♥t♦ ❞❡ ❞❛❞♦s ❡ ♣r♦♠❡t❡♠ s✉❜st✐t✉✐r ❛s ❛t✉❛✐s ❬✷✼❪✳ ❊♠❜♦r❛ ❞❡s❞❡ ❛ ❞é❝❛❞❛ ❞❡ ✾✵ ❥á

s❡ ❢❛❧❛✈❛ ❡♠ ▼❘❆▼ ❬✷✽✱✷✾❪✱ ♦s ♣r✐♠❡✐r♦s ❝❤✐♣s ❞✐♣♦♥í✈❡✐s ❛♣❛r❡❝❡r❛♠ ❛♣❡♥❛s ❡♠ ✷✵✵✻✱

♣r♦❞✉③✐❞♦s ♣❡❧❛ ❋r❡❡s❝❛❧❡ ❡✱ ❛♣❡s❛r ❞❡ tã♦ ✐♥t❡r❡ss❛♥t❡s ❡ ✈❛♥t❛❥♦s❛s✱ ❡s❜❛rr❛♠ ♥♦ ❛❧t♦

❝✉st♦✳ ❖ ❧❡✐t♦r ✐♥t❡r❡ss❛❞♦ ♥✉♠ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❡sq✉❡♠át✐❝♦ ❞❛ ▼❘❆▼ é r❡♠❡t✐❞♦ à

s❡çã♦ ❆✳✸ ❞♦ ❆♣ê♥❞✐❝❡ ❆✳ ❆❧é♠ ❞✐ss♦✱ ●✳ ❆✳ Pr✐♥③ r❡❧❛t❛ ♥❛ r❡❢❡rê♥❝✐❛ ❬✸✵❪ ❛ ❢❛❜r✐✲

❝❛çã♦ ❞❡ ❞✐s♣♦s✐t✐✈♦s ❤í❜r✐❞♦s✱ ❡♠♣r❡❣❛♥❞♦ ❡❧❡♠❡♥t♦s ♠❛❣♥ét✐❝♦s ❝♦♠ s❡♠✐❝♦♥❞✉t♦r❡s

❡ s✉♣❡r❝♦♥❞✉t♦r❡s✳

❈♦♠♦ ❥á ❞✐t♦✱ ♦ ✐♥t❡r❡ss❡ ❡ ♣r♦❣r❡ss♦ ❡♠ s✐st❡♠❛s ♠❛❣♥ét✐❝♦s ❝♦♠ r❡❞✉③✐❞❛s

❞✐♠❡♥sõ❡s t❡♠ s✐❞♦ ❡st✐♠✉❧❛❞♦ ♣❡❧❛ rá♣✐❞❛ ❡✈♦❧✉çã♦ ❞❛s ❞✐✈❡rs❛s té❝♥✐❝❛s ❞❡ ♠✐❝r♦✲

❢❛❜r✐❝❛çã♦✱ ♦ q✉❡ ❡stá ❧❡✈❛♥❞♦ ❛ ❞✐✈❡rs♦s ❞✐s♣♦s✐t✐✈♦s✱ ❝♦♠ ✈❛r✐❛❞❛s ❛♣❧✐❝❛çõ❡s✳ ❆❧é♠

❞❛ ▼❘❆▼ ❡ ❞❡ ❞✐s♣♦s✐t✐✈♦s ❞❡ ❣r❛✈❛çã♦ ❞❡ ❛❧t❛ ❞❡♥s✐❞❛❞❡✱ ♣♦❞❡♠♦s ❝✐t❛r s❡♥s♦r❡s ❞❡

❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❞❡ ❛❧t❛ r❡s♦❧✉çã♦ ❬✸✶❪✱ ❤❛✈❡♥❞♦ ❛ss✐♠ ✉♠ ❣r❛♥❞❡ ✐♥t❡r❡ss❡ ♥❛s ♣r♦✲

♣r✐❡❞❛❞❡s ♠✐❝r♦♠❛❣♥ét✐❝❛s ❞♦s ♠❛❣♥❡t♦s ♥❛♥♦❡str✉t✉r❛❞♦s✳ ❙❡♠ ❞ú✈✐❞❛✱ ❛ ❛♣❧✐❝❛çã♦

❞❡ ♣❛rtí❝✉❧❛s ♥❛♥♦♠❛❣♥ét✐❝❛s ✭♥❛♥♦❞♦ts✸✮ ♥❡ss❡s ❞✐s♣♦s✐t✐✈♦s r❡q✉❡r ✉♠ ❝♦♥tr♦❧❡ ❞❛s

❝♦♥✜❣✉r❛çõ❡s ♠❛❣♥ét✐❝❛s✳ ◆❡st❡ ♣♦♥t♦✱ ♣♦❞❡♠♦s ❢❛❧❛r ❞❛ ❝♦♥✜❣✉r❛çã♦ ❞♦ t✐♣♦✲✈órt✐❝❡✱

q✉❡ s❡rá r❡✈✐st❛ ❛❞✐❛♥t❡ ♥♦ tr❛❜❛❧❤♦✳ ❊st❛ é ✉♠❛ ❝♦♥✜❣✉r❛çã♦ ❡♠ q✉❡ ♦s s♣✐♥s ✭♠❛✐s

♣r❡❝✐s❛♠❡♥t❡ ♦s ♠♦♠❡♥t♦s ❞❡ ❞✐♣♦❧♦ ♠❛❣♥ét✐❝♦s✮ ❞✐s♣õ❡♠✲s❡ ❡♠ ✧❧♦♦♣s✧ ❢❡❝❤❛❞♦s ♥♦

♣❧❛♥♦ ❞❛ ❛♠♦str❛ ✭❝✉r❧✐♥❣ s♣✐♥ ❝♦♥✜❣✉r❛t✐♦♥✮ ❡ é ♦❜s❡r✈❛❞❛ ❝♦♠♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥✲

t❛❧ ♥❛ ❛✉sê♥❝✐❛ ❞❡ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ✭❡st❛❞♦ r❡♠❛♥❡♥t❡✮ ❡♠ ❛♠♦str❛s ❢❡rr♦♠❛❣♥ét✐❝❛s

✸❆♣❡s❛r ❞♦ ❡st✉❞♦ ❞❛s ❝♦♥✜❣✉r❛çõ❡s ❞❛s ❞✐✈❡rs❛s ❢♦r♠❛s✱ ♥❡st❡ tr❛❜❛❧❤♦ ♥♦s ❝♦♥❝❡♥tr❛r❡♠♦s ♥♦

❢♦r♠❛t♦ ❝✐❧í♥❞r✐❝♦ ❝✐r❝✉❧❛r✳

✽

✶✳ ■♥tr♦❞✉çã♦ ❡ ▼♦t✐✈❛çã♦

s✉❜♠✐❝r♦♠étr✐❝❛s ❢❡✐t❛s ❞❡ P②✱ ◆✐ ❡ ❈♦✳ ❚❛❧ ❝♦♥✜❣✉r❛çã♦ t❡♠ s✐❞♦ ♦❜s❡r✈❛❞❛ ❡①♣❡r✐✲

♠❡♥t❛❧♠❡♥t❡✱ ❛ss✐♠ ❝♦♠♦ s❡✉ ♣❛♣❡❧ ♥❛ ♠❛❣♥❡t✐③❛çã♦ ❞❡ ♥❛♥♦♣❛rtí❝✉❧❛s ❬✾✱ ✶✶✱ ✶✸✱ ✶✽❪✳

P♦r ❡①❡♠♣❧♦✱ ♥✉♠ ♥❛♥♦❞✐s❝♦ t❛❧ ❝♦♥✜❣✉r❛çã♦ ♦❝♦rr❡ ♣❛r❛ ❛ r❛③ã♦ L/R ≪ 1✱ ♦♥❞❡ L

é ❛ ❡s♣❡ss✉r❛ ❡ R é ♦ r❛✐♦ ❞♦ ♥❛♥♦❞✐s❝♦ ✭❝♦♥❝r❡t❛♠❡♥t❡✱ ✈❛❧♦r❡s tí♣✐❝♦s ♣♦❞❡♠ s❡r✿ ✺

♥♠ ≤ L ≤ ✻✵ ♥♠ ❡ ✶✵✵ ♥♠ ≤ R ≤ ✶ µ♠ ❬✶✵❪✮✳ ▲❡♠❜r❛♠♦s ❛q✉✐ q✉❡ ✈órt✐❝❡s ❡stã♦

♣r❡s❡♥t❡s ❡♠ ❞✐✈❡rs♦s s✐st❡♠❛s ❢ís✐❝♦s ❛❧é♠ ❞♦ ♠❛❣♥❡t✐s♠♦✳ P♦r ❡①❡♠♣❧♦✱ ♣♦❞❡♠ s❡r

♦❜s❡r✈❛❞♦s ♥♦ ❤é❧✐♦ s✉♣❡r✢✉í❞♦✱ s✉♣❡r❝♦♥❞✉t♦r❡s✱ ❡str✉t✉r❛s ❣❛❧át✐❝❛s✱ ❢✉r❛❝õ❡s ❡ ❛té

♠❡s♠♦ ❡♠ s✐t✉❛çõ❡s ♠❛✐s s✐♠♣❧❡s ❝♦♠♦ ♥♦ ❡s❝♦❛♠❡♥t♦ ❞❡ á❣✉❛ ♥✉♠ r❛❧♦✳

❯♠❛ ✈❛♥t❛❣❡♠✱ ❝♦♠♦ ✈❡r❡♠♦s ❛❞✐❛♥t❡✱ é q✉❡ ✉♠❛ ▼❘❆▼✱ q✉❡ ♥❛ ✈❡r❞❛❞❡ é

❝♦♥st✐t✉í❞❛ ♣♦r ✈ár✐❛s ❝é❧✉❧❛s ✭♦✉ ✉♠ ❛rr❛②✮✱ ❡♠ ♣r✐♥❝í♣✐♦ ♣♦❞❡ ❛r♠❛③❡♥❛r ❛té ❞♦✐s

❜✐ts ❞❡ ✐♥❢♦r♠❛çã♦ ❡♠ ❛♣❡♥❛s ✉♠❛ ú♥✐❝❛ ❝é❧✉❧❛ ❝♦♠ ♠❛❣♥❡t✐③❛çã♦ ❞♦ t✐♣♦✲✈órt✐❝❡✱ ❢❛t♦

❡st❡ ❛ss♦❝✐❛❞♦ ❛♦ ♠❡❝❛♥✐s♠♦ ❞❡ s✇✐t❝❤✐♥❣ ✭r❡✈❡rsã♦ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ✲ ♣♦❧❛r✐❞❛❞❡ ✲ ❡

❞❛ ❞✐r❡çã♦ ❞❡ r♦t❛çã♦ ✲ q✉✐r❛❧✐❞❛❞❡ ✲ ❞♦ ✈órt✐❝❡✮ ❡ é ❞❡✈✐❞♦ às q✉❛tr♦ ♣♦ss✐❜✐❧✐❞❛❞❡s

❞✐❢❡r❡♥t❡s ❞❡ ♠❛❣♥❡t✐③❛çã♦ ✭❝♦♠❜✐♥❛♥❞♦✲s❡ ❛ ♣♦❧❛r✐❞❛❞❡ ✲ ✉♣ ♦✉ ❞♦✇♥ ✲ ❝♦♠ ❛ q✉✐r❛❧✐✲

❞❛❞❡ ✲ ❤♦rár✐❛ ♦✉ ❛♥t✐✲❤♦rár✐❛✮ ♥♦ ❡st❛❞♦ r❡♠❛♥❡♥t❡ ❞❡ ✉♠ ✈órt✐❝❡ ♥✉♠ ♥❛♥♦❞✐s❝♦ ❬✶✵❪✳

❋♦✐ ❞❡♠♦♥str❛❞♦ q✉❡ ❛s ▼❘❆▼✬s ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞❛s ♥❛ ♣r♦❞✉çã♦ ❞❡ ❡❧❡♠❡♥t♦s ❧ó❣✐✲

❝♦s ✭❆◆❉✱ ❖❘✱ ◆❆◆❉✱ ◆❖❘✮✱ ♥❡❝❡ssár✐♦s ♥❛ ♣r♦❞✉çã♦ ❞❡ ♣r♦❝❡ss❛❞♦r❡s✳ ▼❛s✱ ♣❛r❛

t❛♥t♦✱ t❛✐s ❡❧❡♠❡♥t♦s r❡q✉❡r❡♠ ✉♠ s✇✐t❝❤✐♥❣ ✭❝♦♠✉t❛çã♦✮ ❡♥tr❡ ❞♦✐s ❡st❛❞♦s ♠❛❣♥ét✐✲

❝♦s ❡stá✈❡✐s✱ ♦ q✉❡ ♣♦❞❡ s❡r ❢❡✐t♦ ♥✉♠ ♥❛♥♦❞✐s❝♦ ❝♦♥t❡♥❞♦ ❞♦✐s ❜✉r❛❝♦s ❧✐t♦❣r❛✜❝❛♠❡♥t❡

✐♥s❡r✐❞♦s ✭♥❛♥♦❞♦ts✮✱ q✉❡ ❛❣❡♠ ❝♦♠♦ ✈❡r❞❛❞❡✐r♦s ❛tr❛t♦r❡s ✭✧♣✐♥♥✐♥❣ s✐t❡s✧✮ ❞♦ ✈ór✲

t✐❝❡ ❬✷✼❪✳

❆♣❡s❛r ❞❛s ♠❛✐s ✈❛r✐❛❞❛s té❝♥✐❝❛s ✉t✐❧✐③❛❞❛s ♥❛ ♣r❡♣❛r❛çã♦ ❡ ❢❛❜r✐❝❛çã♦ ❞❡

❛♠♦str❛s t❡r❡♠ ❡✈♦❧✉✐❞♦ ♠✉✐t♦✱ ♣♦r ♠❛✐s ❝✉✐❞❛❞♦ q✉❡ s❡ t♦♠❡ ♥♦ ♣r♦❝❡ss♦✱ ❛s ❛♠♦str❛s

s❡♠♣r❡ ❛♣r❡s❡♥t❛♠ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s ✐♥trí♥s❡❝♦s ✭✧♣✐♥♥✐♥❣ ♦r s❝❛tt❡r✐♥❣ s✐t❡s✧✮✱ ❛q✉❡✲

❧❡s ❛❞✈✐♥❞♦s ❞♦ ♣r♦❝❡ss♦ ❞❡ ❢♦r♠❛çã♦ ❞♦s ♥❛♥♦♠❛❣♥❡t♦s✳ ❆❝✐♠❛ ❞❡s❝r❡✈❡♠♦s ✉♠ ❡①✲

❡♠♣❧♦ ❞❛ ✐♥s❡rçã♦ ❞❡ ❞♦✐s ❞❡❢❡✐t♦s ✭♥❡st❡ ❝❛s♦ ♣r♦♣♦s✐t❛❧♠❡♥t❡ ✐♥s❡r✐❞♦s✮ ❧❡✈❛♥❞♦ ❛ ✉♠❛

♣♦ssí✈❡❧ ❛♣❧✐❝❛çã♦✳ ❖s ❝❤❛♠❛❞♦s ♣✐♥♥✐♥❣ s✐t❡s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s sã♦✱ ❝♦♥❝r❡t❛♠❡♥t❡✱

✉♠ ❢♦rt❡ ♦❜❥❡t♦ ❞❡ ♣❡sq✉✐s❛✱ ♣♦✐s ❡❧❡s ❡stã♦ ❡♥✈♦❧✈✐❞♦s ❞✐r❡t❛♠❡♥t❡ ♥♦ ♣r♦❝❡ss♦ ❞❛

r❡✈❡rsã♦ ❞❡ ♠❛❣♥❡t✐③❛çã♦✱ q✉❡ ❡stá ✐♥t✐♠❛♠❡♥t❡ ❛ss♦❝✐❛❞♦ ❝♦♠ ❣r❛✈❛çã♦ ♠❛❣♥ét✐❝❛✳

❱ár✐♦s tr❛❜❛❧❤♦s ❡①♣❡r✐♠❡♥t❛✐s tê♠ ❝♦♥s✐❞❡r❛❞♦ ❛ ♣r❡s❡♥ç❛ ❞❡ ❞❡❢❡✐t♦s✱ t❛♥t♦ ♦s ❡str✉✲

✾

✶✳ ■♥tr♦❞✉çã♦ ❡ ▼♦t✐✈❛çã♦

t✉r❛✐s ❬✶✸✱ ✶✹✱ ✸✷❪✱ q✉❛♥t♦ ❛q✉❡❧❡s ✐♥s❡r✐❞♦s ♣♦r ❧✐t♦❣r❛✜❛ ❬✶✶✱ ✸✸✕✸✺❪✱ ❛❧t❡r❛♥❞♦ ♣r♦❢✉♥✲

❞❛♠❡♥t❡ ❛ ❞✐♥â♠✐❝❛ ❞♦ ✈órt✐❝❡✳ ❈❡rt❛♠❡♥t❡ ♠♦t✐✈❛❞♦s ♣♦r t❛❧ ❢❛t♦✱ ♠♦❞❡❧♦s ❛♥❛❧ít✐❝♦s✱

q✉❡ ❝♦♥s❡❣✉❡♠ r❡♣r♦❞✉③✐r t❛✐s ❞❡❢❡✐t♦s ❛rt✐✜❝✐❛✐s✱ ❝♦♥❝♦r❞❛♥❞♦ ♠✉✐t♦ ❜❡♠ ❝♦♠ r❡s✉❧✲

t❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s✱ ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞♦s ❬✸✻✕✹✵❪✳ ◆♦ ❡♥t❛♥t♦✱ ❤á ♥❛ ❧✐t❡r❛t✉r❛ ✉♠❛

❝❛rê♥❝✐❛ ❞❡ ♠♦❞❡❧♦s ❛♥❛❧ít✐❝♦s ❝❛♣❛③❡s ❞❡ ❡①♣❧✐❝❛r ✈❛r✐❛çõ❡s ❡♠ ❝❡rt❛s ♣r♦♣r✐❡❞❛❞❡s

❞♦s ✈órt✐❝❡s ❡♠ ♥❛♥♦❞✐s❝♦s q✉❛♥❞♦ ❡st❡s ✐♥t❡r❛❣❡♠ ❝♦♠ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s✳ ❊♠ ♥♦ss♦

tr❛❜❛❧❤♦ ❞❡s❡♥✈♦❧✈❡♠♦s ✉♠ ♠♦❞❡❧♦ ❛♥❛❧ít✐❝♦ s✐♠♣❧❡s✱ r❡♣r❡s❡♥t❛♥❞♦ ❞❡❢❡✐t♦s ❡str✉t✉✲

r❛✐s ❝♦♠♦ s❡♥❞♦ ♣✉♥t✐❢♦r♠❡s✱ ❡ ❛♥❛❧✐s❛♥❞♦ ❝♦♠♦ ❡❧❡s ✐♥✢✉❡♥❝✐❛♠ ❛ ❞✐♥â♠✐❝❛ ❞♦ ✈órt✐❝❡

❡♠ ✉♠ ♥❛♥♦❞✐s❝♦✳ ▼♦str❛♠♦s q✉❡ ♥♦ ❝❛s♦ ♠❛✐s s✐♠♣❧❡s✱ ❞♦ ♠♦✈✐♠❡♥t♦ ❞♦ ✈órt✐❝❡ s♦❜

❛çã♦ ❞❡ ❝❛♠♣♦ ❡①t❡r♥♦ t❡♠♣♦✲✐♥❞❡♣❡♥❞❡♥t❡✱ ♥♦ss♦ ♠♦❞❡❧♦ r❡♣r♦❞✉③ q✉❛❧✐t❛t✐✈❛♠❡♥t❡

❜❡♠ ♦s r❡s✉❧t❛❞♦s ❡①♣❡r✐♠❡♥t✐❛s ❛tr✐❜✉í❞♦s ❛ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s✳

❆ ❞✐ss❡rt❛çã♦ ❛ s❡❣✉✐r s❡ ❡♥❝♦♥tr❛ ❞✐✈✐❞✐❞❛ ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦✿ ♥♦ ❈❛♣ít✉❧♦ ✷ ✐♥✐✲

❝✐❛♠♦s ❝♦♠ ✉♠❛ ✐♥tr♦❞✉çã♦ q✉❡ ❧❡✈❛ à ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✉♠❛ ❢♦r♠✉❧❛çã♦ q✉â♥t✐❝❛ ♣❛r❛

❡①♣❧✐❝❛r ♦ ❢❡rr♦♠❛❣♥❡t✐s♠♦ ✲ ❛ ✐♥t❡r❛çã♦ ❞❡ tr♦❝❛ r❡♣r❡s❡♥t❛❞❛ ♣❡❧❛ ❝❤❛♠❛❞❛ ❤❛♠✐❧✲

t♦♥✐❛♥❛ ❞❡ ❍❡✐s❡♥❜❡r❣✳ ❈♦♠♣❧❡♠❡♥t❛♥❞♦ ♦ ❡st✉❞♦ ❞❛s ✐♥t❡r❛çõ❡s✱ ❡st✉❞❛♠♦s ♦✉tr❛s

✐♥t❡r❛çõ❡s ♠❛❣♥ét✐❝❛s q✉❡ sã♦ r❡s♣♦♥sá✈❡✐s ♣❡❧❛s ♣r♦♣r✐❡❞❛❞❡s ♠❛❣♥ét✐❝❛s✳ P♦r ✜♠

❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s ✈❡rsõ❡s ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ❛♥✐s♦tró♣✐❝♦✳ ◆♦ ❈❛♣ít✉❧♦

✸✱ ❢❛③❡♠♦s ✉♠ ❛♣❛♥❤❛❞♦ ❣❡r❛❧ s♦❜r❡ ✈órt✐❝❡s ♣❧❛♥❛r❡s ❡✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ s♦❜r❡ ✈órt✐❝❡s

❝♦♠ ❝♦♠♣♦♥❡♥t❡ ❢♦r❛ ❞♦ ♣❧❛♥♦ ✭♦✉t ♦❢ ♣❧❛♥❡✮✱ ❞❡st❛❝❛♥❞♦ ❛ ❞✐♥â♠✐❝❛ ❞❡st❛ ❡str✉t✉r❛✳

❏á ♥♦ ❈❛♣ít✉❧♦ ✹✱ ❛♣r❡s❡♥t❛♠♦s ♥♦ss♦ ♠♦❞❡❧♦ ♣❛r❛ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s ♣✉♥t✐❢♦r♠❡s✱

❜❡♠ ❝♦♠♦ ♦s r❡s✉❧t❛❞♦s ♦r✐✉♥❞♦s ❞❡❧❡✱ ❛❧é♠ ❞❡ ❢❛③❡r♠♦s ✉♠❛ ❝♦♠♣❛r❛çã♦ ❝♦♠ r❡✲

s✉❧t❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s✳ ◆♦ ❈❛♣ít✉❧♦ ✺✱ ❛♣r❡s❡♥t❛♠♦s ❛s ❝♦♥❝❧✉sõ❡s ♦❜t✐❞❛s ❡♠ ♥♦ss♦

tr❛❜❛❧❤♦✱ ❜❡♠ ❝♦♠♦ ♣❡rs♣❡❝t✐✈❛s ❞❡ tr❛❜❛❧❤♦s✳ P♦r ✜♠✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s ❛♣ê♥❞✐❝❡s

❝♦♠ ❛ ✜♥❛❧✐❞❛❞❡ ❞❡ t♦r♥❛r ♦ t❡①t♦ ♠❛✐s r✐❝♦ ❡♠ ❞❡t❛❧❤❡s✱ ♠❛s ♣♦❞❡♠ s❡r ❞❡✐①❛❞♦s ❞❡

❧❛❞♦✱ s❡♠ ❝♦♠♣r♦♠❡t✐♠❡♥t♦ ❞✐r❡t♦ ♥❛ ❝♦♠♣r❡❡♥sã♦ ❞♦ t❡①t♦ ♣r✐♥❝✐♣❛❧✳

✶✵

❈❛♣ít✉❧♦ ✷

❈♦♥s✐❞❡r❛çõ❡s ❣❡r❛✐s s♦❜r❡

♠❛❣♥❡t✐s♠♦ ❡ ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s

➱ ❝♦♥❤❡❝✐❞♦ ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ q✉❡ t♦❞♦ ♠❛t❡r✐❛❧ ♠❛❣♥ét✐❝♦ s✉❜♠❡t✐❞♦ ❛ ✉♠

❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❡①t❡r♥♦✱ ~H✱ ❛❞q✉✐r❡ ✉♠ ♠♦♠❡♥t♦ ♠❛❣♥ét✐❝♦✳ ❖ ♠♦♠❡♥t♦ ❞❡ ❞✐♣♦❧♦

♣♦r ✉♥✐❞❛❞❡ ❞❡ ✈♦❧✉♠❡ é ❞❡✜♥✐❞♦ ❝♦♠♦ s❡♥❞♦ ❛ ♠❛❣♥❡t✐③❛çã♦✱ ❞❡♥♦t❛❞❛ ♣♦r ~M ✳ ◆♦s

♠❛t❡r✐❛✐s ❧✐♥❡❛r❡s ~M é ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ❝❛♠♣♦ ❛♣❧✐❝❛❞♦ ~H✱ s❡♥❞♦ ❛ r❡❧❛çã♦ ❡♥tr❡ ❡❧❡s

❞❛❞❛ ♣♦r✿

~M = χ ~H✱ ✭✷✳✶✮

♦♥❞❡ χ✱ ❣r❛♥❞❡③❛ ❛❞✐♠❡♥s✐♦♥❛❧✱ é ❝❤❛♠❛❞❛ ❞❡ s✉s❝❡♣t✐❜✐❧✐❞❛❞❡ ♠❛❣♥ét✐❝❛ ❞♦ ♠❛t❡r✐❛❧✶✳

❆s ❡q✉❛çõ❡s ❞❡ ▼❛①✇❡❧❧ sã♦ ❣❡r❛❧♠❡♥t❡ ❡s❝r✐t❛s ♣❛r❛ ♦ ✈❡t♦r ✐♥❞✉çã♦ ♠❛❣♥ét✐❝❛ ~B✿

~B = µ0( ~H + ~M)✳ ✭✷✳✷✮

P❛r❛ ♠❛t❡r✐❛✐s q✉❡ ♦❜❡❞❡❝❡♠ à ❡q✉❛çã♦ ✭✷✳✶✮✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ✭✷✳✷✮ ❝♦♠♦✿

~B = µ ~H✱ ✭✷✳✸✮

❡♠ q✉❡ µ = µ0 (1 + χ) é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♣❡r♠❡❛❜✐❧✐❞❛❞❡ ♠❛❣♥ét✐❝❛✳ ❖ ♠❛t❡r✐❛❧ é

❝❧❛ss✐✜❝❛❞♦ ❝♦♠♦ ♣❛r❛♠❛❣♥ét✐❝♦ s❡ χ > 0 ❡ ❝♦♠♦ ❞✐❛♠❛❣♥ét✐❝♦ s❡ χ < 0✳ ❊♥tã♦✱ ♦

♣r✐♠❡✐r♦ t✐♣♦ ❞❡ ♠❛t❡r✐❛❧✱ s♦❜ ✉♠ ❝❛♠♣♦ ~H✱ ✐♥❞✉③ ♠♦♠❡♥t♦s ❞❡ ❞✐♣♦❧♦ ♠❛❣♥ét✐❝♦s

♦r✐❡♥t❛❞♦s ♣❛r❛❧❡❧❛♠❡♥t❡ ❛ ❡❧❡✳ ❏á ♦ s❡❣✉♥❞♦ t✐♣♦ ❛♣r❡s❡♥t❛ ♠❛❣♥❡t✐③❛çã♦ ♦♣♦st❛ ❛♦

❝❛♠♣♦ ❛♣❧✐❝❛❞♦✳ ❍♦❥❡ ❡♠ ❞✐❛✱ s❛❜❡✲s❡ q✉❡ ♦ ♣❛r❛♠❛❣♥❡t✐s♠♦ é ✉♠ ❢❡♥ô♠❡♥♦ q✉❡ ♦❝♦rr❡✶▼❛t❡r✐❛✐s q✉❡ ♦❜❡❞❡❝❡♠ à ❡q✉❛çã♦ ✷✳✶ sã♦ ❝❤❛♠❛❞♦s ❞❡ ♠❛t❡r✐❛✐s ❧✐♥❡❛r❡s✳

✶✶

✷✳ ❈♦♥s✐❞❡r❛çõ❡s ❣❡r❛✐s s♦❜r❡ ♠❛❣♥❡t✐s♠♦ ❡ ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s

♥♦r♠❛❧♠❡♥t❡ ❡♠ ♠❛t❡r✐❛✐s ❝♦♠♣♦st♦s ♣♦r át♦♠♦s ❡✴♦✉ ♠♦❧é❝✉❧❛s ❝♦♠ ✉♠ ♥ú♠❡r♦

í♠♣❛r ❞❡ ❡❧étr♦♥s ❡ q✉❡ ♣♦❞❡ s❡r ❡①♣❧✐❝❛❞♦ ❝♦♠ ❜❛s❡ ♥♦ Pr✐♥❝í♣✐♦ ❞❡ ❊①❝❧✉sã♦ ❞❡ P❛✉❧✐ ✿

❡❧étr♦♥s ❞❡♥tr♦ ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ át♦♠♦ ❡stã♦ ❝♦♥✜♥❛❞♦s ❛♦s ♣❛r❡s✱ ♠❛s ❝♦♠ s♣✐♥s

♦♣♦st♦s✱ ♦ q✉❡ ❢❛③ ❝♦♠ q✉❡✱ ❛♦ ❛♣❧✐❝❛r♠♦s ✉♠ ❝❛♠♣♦ ❡①t❡r♥♦✱ ♦s t♦rq✉❡s ❛ss♦❝✐❛❞♦s

❝♦♠ ♦s ♠♦♠❡♥t♦s ❛♥❣✉❧❛r❡s ❞❡ s♣✐♥ s❡ ❝❛♥❝❡❧❡♠✱ ❝❛❜❡♥❞♦ ❛♦ ❡❧étr♦♥ ❞❡s❡♠♣❛r❡❧❤❛❞♦

s❡r ♦ r❡s♣♦♥sá✈❡❧ ♣❡❧♦ ❢❡♥ô♠❡♥♦ ❬✹✶❪✳ ❖ ❞✐❛♠❛❣♥❡t✐s♠♦✱ ❝❛r❛❝t❡ríst✐❝♦ ❞❡ t♦❞♦s ♦s

át♦♠♦s✱ é ♠❛✐s ❢r❛❝♦ q✉❡ ♦ ♣❛r❛♠❛❣♥❡t✐s♠♦✱ s❡♥❞♦ t❛♠❜é♠ ✉♠ ❢❡♥ô♠❡♥♦ ❡①♣❧✐❝❛❞♦

q✉❛♥t✐❝❛♠❡♥t❡ ❬✹✷❪✳ ◆♦ ❡♥t❛♥t♦✱ ❛té ♦ ✐♥í❝✐♦ ❞♦ sé❝✉❧♦ ❳❳ ♥♦s ❞❡♣❛rá✈❛♠♦s ❝♦♠ t❡♦r✐❛s

s❡♠✐✲❝❧áss✐❝❛s ✲ q✉❡ ✐♠♣✉♥❤❛♠ ❛ ór❜✐t❛ ❞❡ ❇♦❤r ❞♦s ❡❧étr♦♥s ✲ ♣❛r❛ ❡①♣❧✐❝❛r ❛♠❜♦s ♦s

❢❡♥ô♠❡♥♦s✳ ◆ã♦ ✈❛♠♦s ♥♦s ❛t❡r ❛q✉✐ ❛ ❡①♣❧✐❝❛çõ❡s ❞❡t❛❧❤❛❞❛s ❞❡ss❡s ❢❡♥ô♠❡♥♦s✳ P❛r❛

✉♠❛ ❜♦❛ r❡✈✐sã♦ s♦❜r❡ ♠❛t❡r✐❛✐s ♠❛❣♥ét✐❝♦s✱ ♦ ❧❡✐t♦r ♣♦❞❡rá ❝♦♥s✉❧t❛r ❛s r❡❢❡rê♥❝✐❛s

❬✹✶✱✹✸✱✹✹❪ ❡ tr❛❜❛❧❤♦s ❧á ❝✐t❛❞♦s✳

❍á✱ ❝♦♥t✉❞♦✱ ♦s ♠❛t❡r✐❛✐s ❢❡rr♦♠❛❣♥ét✐❝♦s✱ q✉❡ ♣♦ss✉❡♠ ♠❛❣♥❡t✐③❛çã♦ q✉❡ ♥ã♦

s❡ ❝♦♠♣♦rt❛ ❧✐♥❡❛r♠❡♥t❡✷ ❝♦♠ ♦ ❝❛♠♣♦ ❡①t❡r♥♦ ~H✱ ❝♦♠♦ ♦ ❢❛③❡♠ ♦s ❞♦✐s t✐♣♦s ✈✐st♦s

❛❝✐♠❛✳ ❊♠ t❛✐s ♠❛t❡r✐❛✐s✱ ♣♦r ❡①❡♠♣❧♦✱ ♣♦❞❡♠♦s t❡r ♠❛❣♥❡t✐③❛çã♦✱ ~M ✱ ♥ã♦ ♥✉❧❛

♠❡s♠♦ ❡♠ ~H = ~0✱ ♦ q✉❡ r❡✢❡t❡ ♦ ❢❛t♦ ❞❡st❡s ♠❛t❡r✐❛✐s s❡r❡♠ ❝♦♥❤❡❝✐❞♦s ♣♦r t❡r❡♠

♠❛❣♥❡t✐③❛çã♦ ❡s♣♦♥tâ♥❡❛ ✭ ~M 6= ~0✱ ♥❛ ❛✉sê♥❝✐❛ ❞❡ ❝❛♠♣♦ ~H✮✳ ❉❡ ❢❛t♦✱ ~M ❡♠ t❛✐s

♠❛t❡r✐❛✐s ♥ã♦ é ✉♠❛ ❢✉♥çã♦ ✉♥✐❝❛♠❡♥t❡ ✈❛❧♦r❛❞❛ ❞❡ ~H ❡ s❡✉ ✈❛❧♦r ❞❡♣❡♥❞❡ ❞❛ ✧❤✐stór✐❛✧

❞♦ ♠❛t❡r✐❛❧✳ ❯♠ ❝❛s♦ tí♣✐❝♦ é ♠♦str❛❞♦ ♥❛ ❝✉r✈❛ ❞❡ ❤✐st❡r❡s❡ ❞❛ ✜❣✉r❛ ✷✳✶✳

❈♦♥❝❡✐t♦s ❜ás✐❝♦s ♣♦❞❡♠ s❡r t✐r❛❞♦s ❞❡st❛ ✜❣✉r❛✱ ❝♦♠♦ ❛ r❡♠❛♥ê♥❝✐❛ ♦✉ ♠❛❣✲

♥❡t✐③❛çã♦ r❡♠❛♥❡♥t❡✱ Mr✱ q✉❡ é ♦ ✈❛❧♦r ❞❛ ♠❛❣♥❡t✐③❛çã♦ ♥♦ ❧✐♠✐t❡ ❡♠ q✉❡ | ~H| = 0❀

❝♦❡r❝✐✈✐❞❛❞❡✱ Hc✱ é ❛ ♠❛❣♥✐t✉❞❡ ❞❡ ~H ♣❛r❛ ~M = ~0❀ t❡♠♦s t❛♠❜é♠ ❛ ♠❛❣♥❡t✐③❛çã♦

❞❡ s❛t✉r❛çã♦ ♦✉ ❡s♣♦♥tâ♥❡❛✱ Ms✱ q✉❡ é ❞❡✜♥✐❞❛ ❝♦♠♦ ❛ ♠❛❣♥✐t✉❞❡ ❞❡ ~M ♣❛r❛ ❝❛♠♣♦s

♠✉✐t♦ ❣r❛♥❞❡s✳

❆ ♠❛❣♥❡t✐③❛çã♦ ❞❡ s❛t✉r❛çã♦ é ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ✐♥trí♥s❡❝❛ ❞♦ ♠❛t❡r✐❛❧✱ ✐♥❞❡✲

♣❡♥❞❡♥t❡ ❞❛ ❛♠♦str❛✳ ❊❧❛ é ✉♠❛ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛ ❡ ♥❛ ✜❣✉r❛ ✷✳✷ é ❛♣r❡s❡♥t❛❞❛

✉♠❛ ❢♦r♠❛ tí♣✐❝❛ ❞❡st❡ ❝♦♠♣♦rt❛♠❡♥t♦✳ ◆❡st❛ ✜❣✉r❛✱ Ms(T ) é ♥♦r♠❛❧✐③❛❞❛ ❝♦♠ r❡s✲

♣❡✐t♦ ❛♦ s❡✉ ✈❛❧♦r à t❡♠♣❡r❛t✉r❛ ♥✉❧❛✳ ❏á ❛ t❡♠♣❡r❛t✉r❛ é ♥♦r♠❛❧✐③❛❞❛ ❡♠ r❡❧❛çã♦ à

✷❆♣❡s❛r ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ♥ã♦✲❧✐♥❡❛r ❛♣r❡s❡♥t❛❞♦ ♣♦r ♠❛t❡r✐❛✐s ❢❡rr♦♠❛❣♥ét✐❝♦s✱ é ❝♦st✉♠❡ ❡♥✲

❝♦♥tr❛r ❞❡✜♥✐çõ❡s ❞❡ ♣❡r♠❡❛❜✐❧✐❞❛❞❡ ❡❢❡t✐✈❛ ♥✉♠ ✈❛❧♦r ♣❛rt✐❝✉❧❛r ❞❡ H ❝♦♠♦ µef = ∂BH/∂H✱ ♦✉

♠❡s♠♦ s♦❜r❡ ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ ✈❛❧♦r❡s ❞❡ H ❝♦♠♦ µef = ∆BH/∆H✳

✶✷

✷✳ ❈♦♥s✐❞❡r❛çõ❡s ❣❡r❛✐s s♦❜r❡ ♠❛❣♥❡t✐s♠♦ ❡ ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s

❋✐❣✉r❛ ✷✳✶✿ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ❝✉r✈❛ ❞❡ ❤✐st❡r❡s❡ ♣❛r❛ ✉♠ ♠❛t❡r✐❛❧ ❢❡rr♦♠❛❣♥ét✐❝♦ ♠♦str❛♥❞♦

❛ ♠❛❣♥❡t✐③❛çã♦ r❡♠❛♥❡♥t❡✱ Mr✱ ❛ ♠❛❣♥❡t✐③❛çã♦ ❞❡ s❛t✉r❛çã♦✱ Ms✱ ❡ ❛ ❝♦❡r❝✐✈✐❞❛❞❡✱ Hc✳

❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢❡rê♥❝✐❛ ❬✹✸❪✳

❝❤❛♠❛❞❛ t❡♠♣❡r❛t✉r❛ ❞❡ ❈✉r✐❡✱ Tc✱ ❞♦ ♠❛t❡r✐❛❧✱ q✉❡ é ❛ t❡♠♣❡r❛t✉r❛ ❡♠ q✉❡ Ms → 0

♥♦ ❧✐♠✐t❡ H → 0✳ ❚♦❞♦s ♦s ❢❡rr♦♠❛❣♥❡t♦s t♦r♥❛♠✲s❡ ♣❛r❛♠❛❣♥❡t♦s ❡♠ t❡♠♣❡r❛t✉r❛s

❛❝✐♠❛ ❞❡ Tc ✭T > Tc✮ ❡✱ ❛ss✐♠✱ tê♠ ✉♠❛ ♠❛❣♥❡t✐③❛çã♦ ♥ã♦ ♥✉❧❛ ❛♣❡♥❛s ♥❛ ♣r❡s❡♥ç❛ ❞❡

✉♠ ❝❛♠♣♦✳

◗✉❛❧✐t❛t✐✈❛♠❡♥t❡✱ ❛♠❜♦s ♦s ❢❡♥ô♠❡♥♦s ✭♦ ❝♦♠♣♦rt❛♠❡♥t♦ ♠♦str❛❞♦ ♥❛ ❝✉r✈❛

❞❡ ❤✐st❡r❡s❡ ❡ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞❡ Ms ❝♦♠ ❛ t❡♠♣❡r❛t✉r❛✮ ❡r❛♠ ❝♦♠♣r❡❡♥❞✐❞♦s ❛té ❝❡rt♦

♣♦♥t♦ ♣♦r ✉♠❛ ❡①♣❧✐❝❛çã♦ ❞❛❞❛ ♣♦r P✐❡rr❡ ❲❡✐ss ❡♠ ✶✾✵✼✳ ❊❧❡ ❛ss✉♠✐✉ q✉❡ ❤❛✈✐❛

✉♠ ❝❡rt♦ ❝❛♠♣♦ ✐♥t❡r♥♦ ✲ ❝❛♠♣♦ ♠♦❧❡❝✉❧❛r✸ ✲ ❡♠ ♠❛t❡r✐❛✐s ❢❡rr♦♠❛❣♥ét✐❝♦s q✉❡ t❡♥✲

❞✐❛ ❛ ❛❧✐♥❤❛r ♦s ❞✐♣♦❧♦s ♠❛❣♥ét✐❝♦s ❞♦s át♦♠♦s ❡♠ ♦♣♦s✐çã♦ às ✢✉t✉❛çõ❡s tér♠✐❝❛s

q✉❡ ♣r❡❢❡r✐❛♠ ❞❡s♦r❞❡♥á✲❧♦s✳ ❆ ✜♠ ❞❡ ❡①♣❧✐❝❛r ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ❝♦♠

♦ ❝❛♠♣♦✱ ❲❡✐ss ❛ss✉♠✐✉ q✉❡ ♦s ❢❡rr♦♠❛❣♥❡t♦s ❡r❛♠ ❢❡✐t♦s ❞❡ ♠✉✐t♦s ❞♦♠í♥✐♦s ♠❛❣✲

♥ét✐❝♦s ✭r❡❣✐õ❡s ✜♥✐t❛s ❞❡ ✉♠❛ ♣❛rt✐❝✉❧❛r ❞✐r❡çã♦ ❞❡ ♠❛❣♥❡t✐③❛çã♦✮✱ ❝❛❞❛ ✉♠ ❞❡❧❡s

s❡♥❞♦ ♠❛❣♥❡t✐③❛❞♦ ♥♦ ✈❛❧♦r ❞❡ Ms(T )✱ ♠❛s ❛ ❞✐r❡çã♦ ❞♦ ✈❡t♦r ~M ✈❛r✐❛✈❛ ❡♥tr❡ ♦s

❞♦♠í♥✐♦s✱ ♦✉ s❡❥❛✱ ♦r✐❡♥t❛✈❛♠✲s❡ ❛❧❡❛t♦r✐❛♠❡♥t❡✳ ❲❡✐ss ♥ã♦ ❥✉st✐✜❝♦✉ ♥❡♥❤✉♠❛ ❞❡

s✉❛s ❞✉❛s s✉♣♦s✐çõ❡s✱ ♥♦ ❡♥t❛♥t♦ ❡❧❛s ❡r❛♠ ❛té ❝❡rt♦ ♣♦♥t♦ ✈á❧✐❞❛s ❡ ❝♦♥t✐♥❤❛♠ ✉♠❛

✸❆ ❡①♣r❡ssã♦ ❞♦ ❝❛♠♣♦ ❞❛❞❛ ♣♦r ❲❡✐ss ❡r❛ ~Bm = λm~M ✱ s❡♥❞♦ λm ❛ ❝♦♥st❛♥t❡ ❞❡ ❝❛♠♣♦ ♠♦❧❡❝✉❧❛r

❬✹✺❪✳ ❊st❡ ♠♦❞❡❧♦ ❡r❛ s✉✜❝✐❡♥t❡ ♣❛r❛ ❡①♣❧✐❝❛r ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞❡ Ms ❝♦♠ ❛ t❡♠♣❡r❛t✉r❛ ❡ ♦ ♣❛r❛♠❛❣✲

♥❡t✐s♠♦ ❛❝✐♠❛ ❞❡ Tc✱ ❧❡✈❛♥❞♦ ♣♦ré♠ ❛ ✉♠❛ ♠❛❣♥❡t✐③❛çã♦ ❝♦♥st❛♥t❡ ❡♠ t❡♠♣❡r❛t✉r❛s ❛❜❛✐①♦ ❞❡ Tc✳

✶✸

✷✳ ❈♦♥s✐❞❡r❛çõ❡s ❣❡r❛✐s s♦❜r❡ ♠❛❣♥❡t✐s♠♦ ❡ ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s

❋✐❣✉r❛ ✷✳✷✿ ▼❛❣♥❡t✐③❛çã♦ ❞❡ s❛t✉r❛çã♦✱ Ms✱ ❞❡ ✉♠ ❢❡rr♦♠❛❣♥❡t♦ ❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛✱

T ✱ ♥♦r♠❛❧✐③❛❞❛ ♣❛r❛ ❛ t❡♠♣❡r❛t✉r❛ ❞❡ ❈✉r✐❡✱ Tc✳ ❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢❡rê♥❝✐❛ ❬✹✸❪✳

❝♦♠♣r❡❡♥sã♦ ❜ás✐❝❛ ❞♦ ❢❡rr♦♠❛❣♥❡t✐s♠♦✳ ❖ ❝❛♠♣♦ ♠♦❧❡❝✉❧❛r é ❝♦♥❤❡❝✐❞♦ ❤♦❥❡ ❝♦♠♦

✉♠❛ ❝❡rt❛ ❛♣r♦①✐♠❛çã♦ à ❝❤❛♠❛❞❛ ✐♥t❡r❛çã♦ ❞❡ tr♦❝❛ ❡♥tr❡ ♦s s♣✐♥s❀ ❛ ❡①✐stê♥❝✐❛ ❞♦s

❞♦♠í♥✐♦s✱ ♠❛❣♥❡t✐③❛❞♦s ❡♠ ❞✐❢❡r❡♥t❡s ❞✐r❡çõ❡s✱ ♥ã♦ é ♠❛✐s ✉♠❛ s✉♣♦s✐çã♦ ❡ ❝♦♥❤❡❝❡✲

s❡ q✉❡ ❡❧❡s sã♦ ♠❛❣♥❡t✐③❛❞♦s ❡♠ ❝❡rt❛s ❞✐r❡çõ❡s ❡ ♥ã♦ ❛❧❡❛t♦r✐❛♠❡♥t❡ ❝♦♠♦ ♣♦st✉❧♦✉

❲❡✐ss✳ ❈♦♥t✉❞♦✱ ❛♣❡s❛r ❞❡ ♠♦❞❡❧♦s ❝❧áss✐❝♦s ❡①♣❧✐❝❛r❡♠ q✉❛❧✐t❛t✐✈❛♠❡♥t❡ ❜❡♠ ♦ ❢❡r✲

r♦♠❛❣♥❡t✐s♠♦✱ ♣r❡s✐s❛♠♦s ❝♦♥s✐❞❡r❛r ♦ ❢❛♠♦s♦ t❡♦r❡♠❛ ❞❡ ❇♦❤r✲✈❛♥ ▲❡❡✉✇❡♥ q✉❡✱ ❡♠

✶✾✶✾✱ ♠♦str♦✉ q✉❡ ❛ ❋ís✐❝❛ ❈❧áss✐❝❛ ♥ã♦ ♣♦❞✐❛ ❡①♣❧✐❝❛r q✉❛❧q✉❡r ❢❡♥ô♠❡♥♦ ♠❛❣♥ét✐❝♦✱

♣♦✐s ❝❧❛ss✐❝❛♠❡♥t❡ ✉♠ ❡❧étr♦♥ ♥ã♦ ✐♥t❡r❛❣❡ ❝♦♠ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❛♣❧✐❝❛❞♦ ❬✹✸❪✳

❆♦ ❝♦♥trár✐♦ ❞♦ ♣❛r❛♠❛❣♥❡t✐s♠♦✱ ♦ q✉❛❧ ✐♥t❡r❛❣❡ ❛♣❡♥❛s ❝♦♠ ✉♠ ❝❛♠♣♦ ♠❛❣✲

♥ét✐❝♦ ❡①t❡r♥♦✱ ♦s s♣✐♥s ❛tô♠✐❝♦s ♥♦s ❢❡rr♦♠❛❣♥❡t♦s t❛♠❜é♠ ✐♥t❡r❛❣❡♠ ✉♠ ❝♦♠ ♦ ♦✉tr♦✱

❝❛❞❛ ✉♠ ❞❡❧❡s ❛❣❡ t❡♥t❛♥❞♦ ❛❧✐♥❤❛r ♦s ♦✉tr♦s ❡♠ s✉❛ ♣ró♣r✐❛ ❞✐r❡çã♦✳ ❆ ✐♥t❡r❛çã♦ ❡♥tr❡

❡❧❡s ♦r✐❣✐♥❛✲s❡ ❞❛s ♣r♦♣r✐❡❞❛❞❡s q✉â♥t✐❝❛s ❞♦s s♣✐♥s q✉❡✱ s♦♠❛❞❛ à ✐♥t❡r❛çã♦ ❈♦✉❧♦♠✲

❜✐❛♥❛ ✭❡❧❡tr♦♠❛❣♥ét✐❝❛✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡✮ ❡♥tr❡ ♦s ❡❧étr♦♥s✱ ❞ã♦ ♦r✐❣❡♠ à ✐♥t❡r❛çã♦

❞❡ tr♦❝❛✳ ❉❡✈✐❞♦ à s✉❛ ✐♠♣♦rtâ♥❝✐❛ ♥♦ ❡st✉❞♦ ❞❡ s✐st❡♠❛s ♠❛❣♥ét✐❝♦s✱ ❛♣r❡s❡♥t❛r❡♠♦s

✉♠❛ ❞✐s❝✉ssã♦ ❞❡t❛❧❤❛❞❛ ❞❛ ✐♥t❡r❛çã♦ ❞❡ tr♦❝❛✳ P♦st❡r✐♦r♠❡♥t❡✱ ❛♣r❡s❡♥t❛r❡♠♦s ♦✉tr❛s

✐♥t❡r❛çõ❡s t❛♠❜é♠ ✐♠♣♦rt❛♥t❡s ♥♦ ❡st✉❞♦ ❞❡ ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s✳

✶✹

✷✳ ❈♦♥s✐❞❡r❛çõ❡s ❣❡r❛✐s s♦❜r❡ ♠❛❣♥❡t✐s♠♦ ❡ ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s

✷✳✶ ■♥t❡r❛çã♦ ❞❡ tr♦❝❛

❘❡❧❛❝✐♦♥❛❞♦ ❛♦ s♣✐♥ ❡❧❡trô♥✐❝♦ t❡♠♦s ♦ Pr✐♥❝í♣✐♦ ❞❡ ❊①❝❧✉sã♦ ❞❡ P❛✉❧✐✱ ♦ q✉❛❧✱

❝♦♠♦ ✈✐♠♦s✱ ❛✜r♠❛ q✉❡ ♥✉♠ át♦♠♦ ♥✉♥❝❛ ♣♦❞❡ ❤❛✈❡r ♠❛✐s ❞❡ ✉♠ ❡❧étr♦♥ ♦❝✉♣❛♥❞♦ ♦

♠❡s♠♦ ❡st❛❞♦ q✉â♥t✐❝♦✳ ❆ ❝♦♥s❡q✉ê♥❝✐❛ ✐♠❡❞✐❛t❛ ❞❡ t❛❧ ❛✜r♠❛çã♦ é q✉❡ ❛ ❢✉♥çã♦ ❞❡

♦♥❞❛ t♦t❛❧ ✭❡s♣❛❝✐❛❧ ⊕ s♣✐♥✮ ❞♦ ❡❧étr♦♥ ❞❡✈❡ s❡r ❛♥t✐✲s✐♠étr✐❝❛✳ P❛r❛ ❛♥❛❧✐s❛r♠♦s ✐ss♦

♠❛✐s ❛ ❢✉♥❞♦✱ ❝♦♥s✐❞❡r❡ ❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦ ♣❛r❛ ♦ ❝❛s♦

s✐♠♣❧❡s ❞❡ ❞♦✐s ❡❧étr♦♥s ✭♦❜s❡r✈❡ q✉❡ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ❞❡ s♣✐♥ ♥ã♦ ❡♥tr❛ ♥❛ ❡q✉❛çã♦

❞❡ ❙❝❤rö❞✐♥❣❡r ❡ ❞❡✈❡ s❡r ♣♦st❛ à ♣❛rt❡✮✿

− ~2

2m(∇2

1 +∇22)ψ(~r1, ~r2) + V (~r1, ~r2)ψ(~r1, ~r2) = E ψ(~r1, ~r2), ✭✷✳✹✮

❡♠ q✉❡ ~ é ❛ ❝♦♥st❛♥t❡ ❞❡ P❧❛♥❝❦ ❞✐✈✐❞✐❞❛ ♣♦r 2π ❡ m é ❛ ♠❛ss❛ ❞♦ ❡❧étr♦♥✳ ❖s í♥❞✐❝❡s

1 ❡ 2 r❡❢❡r❡♠✲s❡ às ♣♦s✐çõ❡s ❞♦s ❞♦✐s ❡❧étr♦♥s✳ ❊①✐st❡♠ q✉❛tr♦ ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ♦s

s♣✐♥s✱ q✉❡ ✐r❡♠♦s r❡♣r❡s❡♥t❛r ♣♦r✿

| ↑↑〉, | ↑↓〉, | ↓↑〉, | ↓↓〉✱

❡♠ q✉❡ ❛ s❡t❛ ♣❛r❛ ❝✐♠❛ r❡♣r❡s❡♥t❛ ♦ ❡st❛❞♦ ❞❡ s♣✐♥ ✉♣ ✭Sz = +~/2✮ ❡ ❛ s❡t❛ ♣❛r❛

❜❛✐①♦ r❡♣r❡s❡♥t❛ ♦ ❡st❛❞♦ ❞♦✇♥ ✭Sz = −~/2✮✳ ❆ ♣❛rt✐r ❞❡ss❛s ❝♦♠❜✐♥❛çõ❡s✱ ♣♦❞❡♠♦s

❝♦♥str✉✐r ✉♠❛ ❢✉♥çã♦ ❛♥t✐✲s✐♠étr✐❝❛ (χA) ❡ três ❢✉♥çõ❡s s✐♠étr✐❝❛s (χS)✿

χA =1√2(| ↑↓〉 − | ↓↑〉)✱ ✭✷✳✺✮

χS =

| ↑↑〉1√2(| ↑↓〉+ | ↓↑〉)

| ↓↓〉

✭✷✳✻✮

❆ ❡q✉❛çã♦ ✭✷✳✺✮ ❝♦rr❡s♣♦♥❞❡ ❛♦ ❝❛s♦ ❡♠ q✉❡ ♦s s♣✐♥s ❡stã♦ ✧❛♥t✐♣❛r❛❧❡❧♦s✧✱

❝❤❛♠❛❞♦ ❞❡ ❡st❛❞♦ s✐♥❣❧❡t♦✳ ❆s s✐t✉❛çõ❡s ❞❡s❝r✐t❛s ♣❡❧❛ ❡q✉❛çã♦ ✭✷✳✻✮ ❝♦rr❡s♣♦♥❞❡♠

❛♦ ❝❛s♦ ♦♥❞❡ ♦s s♣✐♥s ❡stã♦ ✧♣❛r❛❧❡❧♦s✧✱ ❝❤❛♠❛❞♦ ❞❡ ❡st❛❞♦ tr✐♣❧❡t♦✳

❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s ❝♦♥tr✉✐r ❢✉♥çõ❡s ❡s♣❛❝✐❛✐s s✐♠étr✐❝❛s ❡ ❛♥t✐✲s✐♠étr✐❝❛s✳

❉❡s✐❣♥❛♥❞♦ ♣♦r φ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❡ α ❡ β ❞♦✐s ❡st❛❞♦s ♦r❜✐t❛✐s✱

✶✺

✷✳ ❈♦♥s✐❞❡r❛çõ❡s ❣❡r❛✐s s♦❜r❡ ♠❛❣♥❡t✐s♠♦ ❡ ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s

t❡♠♦s✿ψS(~r1, ~r2) =

1√2[φα(~r1)φβ(~r2) + φα(~r2)φβ(~r1)]✱

ψA(~r1, ~r2) =1√2[φα(~r1)φβ(~r2)− φα(~r2)φβ(~r1)]✳

✭✷✳✼✮

❙❡ ✜③❡r♠♦s✱ ❛❣♦r❛✱ ❛s ♣❛rtí❝✉❧❛s s❡ ❛♣r♦①✐♠❛r❡♠ ✉♠❛ ❞❛ ♦✉tr❛ (~r1 → ~r2)✱ t❡♠♦s

q✉❡ ψS → 2φα φβ✱ ❡♥q✉❛♥t♦ ψA → 0✱ ♠♦str❛♥❞♦ q✉❡ é ✐♠♣r♦✈á✈❡❧ q✉❡ ♣❛rtí❝✉❧❛s ♥♦

❡st❛❞♦ ❛♥t✐✲s✐♠étr✐❝♦ s❡ ❡♥❝♦♥tr❡♠ ♣ró①✐♠❛s✳

❙❛❜❡✲s❡ q✉❡ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ t♦t❛❧ ❞♦ ❡❧étr♦♥ é ❢♦r♠❛❞❛ ♣❡❧♦ ♣r♦❞✉t♦ ❞❛ ♣❛rt❡

❡s♣❛❝✐❛❧ ♣❡❧❛ ♣❛rt❡ ❞❡ s♣✐♥✳ ❈♦♠♦ ❞✐t♦ ❛♥t❡s✱ t❛❧ ❢✉♥çã♦ ❞❡✈❡ s❡r ❛♥t✐✲s✐♠étr✐❝❛✱ ♦

q✉❡ ✐♠♣❧✐❝❛ q✉❡ s❡ ❛ ♣❛rt❡ ❡s♣❛❝✐❛❧ ❢♦r s✐♠étr✐❝❛ (ψS)✱ ❛ ❢✉♥çã♦ ❞❡ s♣✐♥ ❞❡✈❡ s❡r ❛♥t✐✲

s✐♠étr✐❝❛ ✭χA✮✱ ❡ ✈✐❝❡✲✈❡rs❛✳ ❆ss✐♠✱ ❛s ♣♦ssí✈❡✐s ❢✉♥çõ❡s ❞❡ ♦♥❞❛ t♦t❛✐s sã♦✿

Ψtotal = ψS(~r1, ~r2)⊗ χA ♦✉ Ψtotal = ψA(~r1, ~r2)⊗ χS✱ ✭✷✳✽✮

♦♥❞❡ ♦ sí♠❜♦❧♦ ⊗ ❞❡♥♦t❛ ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧✳

❆ ❡①✐❣ê♥❝✐❛ ❞❡ q✉❡ ✉♠❛ ❞❡s❝r✐çã♦ ♣r❡❝✐s❛ ❞♦ s✐st❡♠❛ ❞❡✈❛ ✉t✐❧✐③❛r ✉♠❛ ❢✉♥çã♦ ❞❡

♦♥❞❛ t♦t❛❧ q✉❡ s❡❥❛ ❛♥t✐✲s✐♠étr✐❝❛ ♣❡❧❛ tr♦❝❛ ❞❛s ♣❛rtí❝✉❧❛s ❝♦♥❞✉③ ❛ ✉♠ ❛❝♦♣❧❛♠❡♥t♦

❡♥tr❡ ❛s ✈❛r✐á✈❡✐s ❡s♣❛❝✐❛✐s ❡ ❞❡ s♣✐♥✳ ❆s ♣❛rtí❝✉❧❛s s❡ ❝♦♠♣♦rt❛♠ ❝♦♠♦ s❡ s❡✉ ♠♦✈✐✲

♠❡♥t♦ ❢♦ss❡ ✐♥✢✉❡♥❝✐❛❞♦ ♣♦r ✉♠❛ ❢♦rç❛ ❝✉❥♦ s✐♥❛❧ ❞❡♣❡♥❞❡ss❡ ❞❛ ♦r✐❡♥t❛çã♦ r❡❧❛t✐✈❛

❞❡ s❡✉s s♣✐♥s✳ ❆ t❛❧ ❢❡♥ô♠❡♥♦ ❝❤❛♠❛♠♦s ❞❡ ❢♦rç❛ ❞❡ tr♦❝❛ ✭♦✉ ✐♥t❡r❛çã♦ ❞❡ tr♦❝❛✮✱

❡❢❡✐t♦ ❡ss❡♥❝✐❛❧♠❡♥t❡ q✉â♥t✐❝♦ ❡ q✉❡ ♥ã♦ ♣♦ss✉✐ ❛♥á❧♦❣♦ ❝❧áss✐❝♦✳ ❯t✐❧✐③❛r❡♠♦s ♦ q✉❡

❢♦✐ ❞✐s❝✉t✐❞♦ ❛❝✐♠❛ ♣❛r❛ ❞❡♠♦♥str❛r ❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❞❡ ❍❡✐s❡♥❜❡r❣✳

❍❛♠✐❧t♦♥✐❛♥❛ ❞❡ ❍❡✐s❡♥❜❡r❣

❈♦♥s✐❞❡r❡ ❛ ✐♥t❡r❛çã♦ ❝♦✉❧♦♠❜✐❛♥❛ ❡♥tr❡ ❞♦✐s ❡❧étr♦♥s✿

U(~r1, ~r2) =e2

4πǫ0r12✱ ✭✷✳✾✮

♦♥❞❡ r12 é ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ❡❧étr♦♥s✳ ❆❣♦r❛✱ ❝❛❧❝✉❧❛♠♦s ❛ ❡♥❡r❣✐❛ ♠é❞✐❛✱ ✉s❛♥❞♦ ❛s

❢✉♥çõ❡s ❞❡ ♦♥❞❛ ❡s♣❛❝✐❛✐s ✭✷✳✼✮ ✭❛ ✐♥t❡r❛çã♦ ♥ã♦ ❞❡♣❡♥❞❡ ❞♦ s♣✐♥✮✿

〈U〉 = e2

4πǫ0

∫

ψ∗(~r1, ~r2)1

r12ψ(~r1, ~r2)d

3r1 d3r2✳ ✭✷✳✶✵✮

❙✉❜st✐t✉✐♥❞♦ ψ(~r1, ~r2) = 1√2[φα(~r1)φβ(~r2)± φα(~r2)φβ(~r1)]✱ ♦❜t❡♠♦s✿

〈U〉 = E ± J ✱ ✭✷✳✶✶✮

✶✻

✷✳ ❈♦♥s✐❞❡r❛çõ❡s ❣❡r❛✐s s♦❜r❡ ♠❛❣♥❡t✐s♠♦ ❡ ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s

♦♥❞❡✿

E =e2

4πǫ0

∫

φ∗α(~r1)φ

∗β(~r2)

1

r12φα(~r1)φβ(~r2) d

3r1 d3r2✱ ✭✷✳✶✷✮

J =e2

4πǫ0

∫

φ∗α(~r1)φ

∗β(~r2)

1

r12φα(~r2)φβ(~r1) d

3r1 d3r2✳ ✭✷✳✶✸✮

◆♦t❡♠♦s q✉❡ E r❡♣r❡s❡♥t❛ ❛ ❡♥❡r❣✐❛ ❝♦✉❧♦♠❜✐❛♥❛ ♠é❞✐❛ ❞♦ s✐st❡♠❛✳ ❚❛❧ t❡r♠♦

s❡r✐❛ ♦ ú♥✐❝♦ t❡r♠♦ ♣r❡s❡♥t❡ s❡ ♥ã♦ ❧❡✈áss❡♠♦s ❡♠ ❝♦♥t❛ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ❡①❝❧✉sã♦✳ ❆ss✐♠✱

❛ ✐♠♣♦s✐çã♦ ❞♦ ♣r✐♥❝í♣✐♦ ❞❡ P❛✉❧✐ ♥♦s ❧❡✈♦✉ ❛ ✉♠ t❡r♠♦ ❡①tr❛✱ J ✱ ❝❤❛♠❛❞♦ ❞❡ ✐♥t❡❣r❛❧

♦✉ ❝♦♥st❛♥t❡ ❞❡ tr♦❝❛✹✱ q✉❡ ❝♦rr❡s♣♦♥❞❡ à ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦s ❡st❛❞♦s s✐♥❣❧❡t♦ ❡ tr✐♣❧❡t♦✳

❖ s✐♥❛❧ ± ♥❛ ❡①♣r❡ssã♦ ✭✷✳✶✶✮ ❞❡♣❡♥❞❡ ❞♦ ❡st❛❞♦ ❞❡ s♣✐♥ ✭s❡ s✐♥❣❧❡t♦ ♦✉ tr✐♣❧❡t♦✮✳

❆♣❡s❛r ❞❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ♦s ❡❧étr♦♥s ♥ã♦ ❞❡♣❡♥❞❡r ❞♦ ❡st❛❞♦ ❞❡ s♣✐♥✱ ❛ ❡♥❡r❣✐❛ ♠é❞✐❛

❞❡♣❡♥❞❡rá✱ ♥♦s ❧❡✈❛♥❞♦ ❡♥tã♦ ❛ ♣❡♥s❛r q✉❡ ❛ ❡♥❡r❣✐❛ ❞♦ s✐st❡♠❛ ♣♦ss❛ s❡r ❡s❝r✐t❛ ❡♠

t❡r♠♦s ❞❛s ✈❛r✐á✈❡✐s ❞❡ s♣✐♥✳ P❛r❛ t❛♥t♦✱ ❡s❝r❡✈❡♠♦s 〈U〉 ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

〈U〉 = E ± J ~S1 · ~S2✱ ✭✷✳✶✹✮

❡✱ ❞❡✈✐❞❛ à s✉❛ ❝♦♥✈❡♥✐ê♥❝✐❛✱ ❝♦♥s✐❞❡r❛r❡♠♦s Si ❝♦♠♦ s❡♥❞♦ ❛s ♣r♦❥❡çõ❡s ❞♦s ♠♦♠❡♥t♦s

❛♥❣✉❧❛r❡s ❞❡ s♣✐♥ ✭♥♦r♠❛❧✐③❛❞❛s ♣❡❧♦ ❢❛t♦r ~/2✮✳ ❉❡st❛ ❢♦r♠❛✱ t❡r❡♠♦s q✉❡ ~S1 · ~S2 = +1

❝♦rr❡s♣♦♥❞❡ ❛♦s s♣✐♥s ❡st❛r❡♠ ♣❛r❛❧❡❧♦s ✭❡st❛❞♦ tr✐♣❧❡t♦✮ ❡ ~S1 · ~S2 = −1 ❝♦rr❡s♣♦♥❞❡

❛♦s s♣✐♥s ❡st❛r❡♠ ❛♥t✐♣❛r❛❧❡❧♦s ✭❡st❛❞♦ s✐♥❣❧❡t♦✮✳ ◗✉❡♠ ♣r✐♠❡✐r♦ ❝❤❛♠♦✉ ❛t❡♥çã♦ ❞❛

❡♥❡r❣✐❛ ❞❡ tr♦❝❛ ♣❛r❛ ❡①♣❧✐❝❛r ❛ ♦r❞❡♠ ♠❛❣♥ét✐❝❛ ♥♦s ♠❛t❡r✐❛✐s ❢♦✐ ❲❡r♥❡r ❍❡✐s❡♥❜❡r❣✱

❡♠ 1928✳ ❆ ❤❛♠✐❧t♦♥✐❛♥❛ ❞❡ ❍❡✐s❡♥❜❡r❣ é✱ ❡♥tã♦✱ ❞❛❞❛ ♣♦r✿

H = ± 2 J12 ~S1 · ~S2✱ ✭✷✳✶✺✮

♦♥❞❡ J12✱ ❛ ❝❤❛♠❛❞❛ ❞❡ ❝♦♥t❛♥t❡ ❞❡ tr♦❝❛✱ ❡stá ❛ss♦❝✐❛❞❛ ❛♦s ❡❧étr♦♥s 1 ❡ 2 ❬✹✺❪✳ ❱✐♠♦s

q✉❡ ❡st❛ s✐t✉❛çã♦✱ ❝♦♥s✐❞❡r❛♥❞♦ ❛♣❡♥❛s ❞♦✐s ❡❧étr♦♥s✱ é r❡❧❛t✐✈❛♠❡♥t❡ s✐♠♣❧❡s✳ ❊♥tr❡✲

t❛♥t♦ át♦♠♦s ❡♠ s✐st❡♠❛s ♠❛❣♥ét✐❝♦s tê♠ ♠✉✐t♦s ❡❧étr♦♥s✳ ❆ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r

❞❡st❡ s✐st❡♠❛ ❞❡ ♠✉✐t♦s ❡❧étr♦♥s ♥ã♦ ♣♦❞❡ s❡r r❡s♦❧✈✐❞❛ s❡♠ s✉♣♦s✐çõ❡s✴❛♣r♦①✐♠❛çõ❡s✳

◆♦t❛♠♦s q✉❡ t❛❧ ✐♥t❡r❛çã♦ ❛♣❧✐❝❛✲s❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❡♥tr❡ át♦♠♦s ✈✐③✐♥❤♦s✱ ♣♦✐s ❛ ✐♥✲

t❡r❛çã♦ ❝♦♠♦ ❛♣r❡s❡♥t❛❞❛ ❛q✉✐ ❧❡✈❛ ❡♠ ❝♦♥t❛ ❛ s✉♣❡r♣♦s✐çã♦ ❞❡ ❢✉♥çõ❡s ❞❡ ♦♥❞❛ q✉❡

❛♣❡♥❛s s❡ t♦r♥❛♠ ❛♣r❡❝✐á✈❡✐s ❡♥tr❡ sít✐♦s ✈✐③✐♥❤♦s ❬✹✼❪✳ ❊st❛ ❝♦♥s✐❞❡r❛çã♦ ♥♦s ❧❡✈❛ ❛♦

✹P♦r ❡①❡♠♣❧♦✱ ♣❛r❛ ♦ ❢❡rr♦ ❛ ❝♦♥st❛♥t❡ ❞❡ tr♦❝❛ é J ≈ 3, 45 · 10−21❏ ❬✹✻❪✳

✶✼

✷✳ ❈♦♥s✐❞❡r❛çõ❡s ❣❡r❛✐s s♦❜r❡ ♠❛❣♥❡t✐s♠♦ ❡ ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s

♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ♣❛r❛ s✐st❡♠❛s ❝♦♠ ♠✉✐t♦s át♦♠♦s ❬✹✽❪✿

H = −∑

<i, j>

Jij ~Si · ~Sj✱ ✭✷✳✶✻✮

❝♦♠ Jij s❡♥❞♦ ❛ ❝♦♥st❛♥t❡ ❞❡ tr♦❝❛ ❡♥tr❡ ♦ s♣✐♥ i ❡ s♣✐♥ j ❡ ❛ ♥♦t❛çã♦ < ·, · > ✐♥❞✐❝❛

q✉❡ ❛ s♦♠❛ é ❢❡✐t❛ ❡♥tr❡ ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s✳ ❯♠❛ ❜♦❛ ❛♣r♦①✐♠❛çã♦ é ❢❛③❡r Jij = J ♣❛r❛

✈✐③✐♥❤♦s ♠❛✐s ♣ró①✐♠♦s ❡ Jij = 0 ♣❛r❛ ♦s ❞❡♠❛✐s ✈✐③✐♥❤♦s ✭✐♥t❡r❛çã♦ ❞❡ ❝✉rt♦ ❛❧❝❛♥❝❡✮✳

❆ss✐♠✿

H = − J∑

<i, j>

~Si · ~Sj✳ ✭✷✳✶✼✮

❘❡ss❛❧t❛♠♦s q✉❡ ❡♠ ✭✷✳✶✻✮ ❡ ✭✷✳✶✼✮ ♦ ❢❛t♦r ✷ é ✐♥❝❧✉í❞♦ ♥♦ s♦♠❛tór✐♦✱ ♣♦✐s

❝♦♥t❛♠♦s ❛ ♠❡s♠❛ ✐♥t❡r❛çã♦ ❞✉❛s ✈❡③❡s✳ ❖❜s❡r✈❡ t❛♠❜é♠ q✉❡ ♦ s✐♥❛❧ ±✱ ♣r❡s❡♥t❡

❡♠ ✭✷✳✶✺✮ ♥ã♦ ❛♣❛r❡❝❡ ♠❛✐s✱ ♣♦✐s ❡st❡ ❢♦✐ ✐♥❝❧✉s♦ ❡♠ J ❡ ❛❣♦r❛ ♣♦❞❡♠♦s t❡r J > 0

❡ J < 0✳ ❆♦ ❛♥❛❧✐s❛r♠♦s ✭✷✳✶✼✮✱ ✈❡♠♦s q✉❡ ❛ ❝♦♥✜❣✉r❛çã♦ ❞❡ ♠❡♥♦r ❡♥❡r❣✐❛ ♣❛r❛ ♦

❝❛s♦ ❡♠ q✉❡ J < 0 é t❛❧ q✉❡ ~Si · ~Sj < 0✱ ♦❝♦rr❡♥❞♦ ♦ ❛❧✐♥❤❛♠❡♥t♦ ❛♥t✐♣❛r❛❧❡❧♦ ✭♦r❞❡♠

❛♥t✐✲❢❡rr♦♠❛❣♥ét✐❝❛✮❀ ❥á ❛ ♠❡♥♦r ❡♥❡r❣✐❛ ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ J > 0 é t❛❧ q✉❡ ~Si · ~Sj >

0 ❡ t❡♠♦s ❛❧✐♥❤❛♠❡♥t♦ ♣❛r❛❧❡❧♦ ✭♦r❞❡♠ ❢❡rr♦♠❛❣♥ét✐❝❛✮✳ ❊ss❡s ❝♦♠♣♦rt❛♠❡♥t♦s sã♦

✐❧✉str❛❞♦s ♥❛ ✜❣✉r❛ ✷✳✸✳ ▲❡♠❜r❛♠♦s ❛✐♥❞❛ q✉❡ ♥❡st❡ ♠♦❞❡❧♦ ✉♠❛ ✐♥t❡r❛çã♦ ❝♦♠ ✉♠

❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❡①t❡r♥♦ ~H ♣♦❞❡ s❡r ✐♥tr♦❞✉③✐❞❛ ♣❡❧❛ ❛❞✐çã♦ ❡♠ ✭✷✳✶✼✮ ❞❡ ✉♠ t❡r♠♦

♣r♦♣♦r❝✐♦♥❛❧ ❛∑

i~H · ~Si✳

❋✐❣✉r❛ ✷✳✸✿ ■❧✉str❛çã♦ ❡sq✉❡♠át✐❝❛ ❞❡ ♦r❞❛♥❛♠❡♥t♦ ❢❡rr♦♠❛❣♥ét✐❝♦✱ à ❡sq✉❡r❞❛✱ ❡ ❛♥t✐✲

❢❡rr♦♠❛❣♥ét✐❝♦✱ à ❞✐r❡✐t❛✳

❆ s✐t✉❛çã♦ q✉❡ ❛❝❛❜❛♠♦s ❞❡ ❡st✉❞❛r✱ ❝♦♠ ❡❧étr♦♥s ✐♥t❡r❛❣✐♥❞♦ ❞✐r❡t❛♠❡♥t❡ ❝♦♠

♦s át♦♠♦s ✈✐③✐♥❤♦s é ❝❤❛♠❛❞❛ ❞❡ tr♦❝❛ ❞✐r❡t❛✱ ♣♦✐s ♥ã♦ ❤á ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ át♦♠♦s

✶✽

✷✳ ❈♦♥s✐❞❡r❛çõ❡s ❣❡r❛✐s s♦❜r❡ ♠❛❣♥❡t✐s♠♦ ❡ ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s

✐♥t❡r♠❡❞✐ár✐♦s✳ ❙❡✱ ❡♥tr❡t❛♥t♦✱ ❛ s✉♣❡r♣♦s✐çã♦ ❞❛s ❢✉♥çõ❡s ❞❡ ♦♥❞❛ é ♣❡q✉❡♥❛✱ ❛ tr♦❝❛

❞✐r❡t❛ ♥ã♦ r❡♣r❡s❡♥t❛ ♦ ♠❡❝❛♥✐s♠♦ ❞♦♠✐♥❛♥t❡ ♣❛r❛ ❛s ♣r♦♣r✐❡❞❛❞❡s ♠❛❣♥ét✐❝❛s✳ P❛r❛

❡ss❡ t✐♣♦ ❞❡ s✐st❡♠❛ ❛ tr♦❝❛ ✐♥❞✐r❡t❛ é ❛ r❡s♣♦♥sá✈❡❧ ♣♦r t❛✐s ♣r♦♣r✐❡❞❛❞❡s✳ ❯♠❛ r❡✈✐sã♦

❞❛ tr♦❝❛ ✐♥❞✐r❡t❛ é ❛♣r❡s❡♥t❛❞❛ ♥❛ r❡❢❡rê♥❝✐❛ ❬✹✽❪✳

▼✐❝r♦s❝♦♣✐❝❛♠❡♥t❡✱ ❛❧é♠ ❞♦ ❢❡rr♦♠❛❣♥❡t✐s♠♦ ❡ ❛♥t✐❢❡rr♦♠❛❣♥❡t✐s♠♦✱ ❛ ✐♥t❡r❛✲

çã♦ ❞❡ tr♦❝❛ ♣♦❞❡ ♦r✐❣✐♥❛r ❞✐✈❡rs♦s t✐♣♦s ❞❡ ❛rr❛♥❥♦s ❡s♣❛❝✐❛✐s ❞♦s s♣✐♥s ❛tô♠✐❝♦s✳ ❯♠❛

✈❛r✐❛çã♦ s✐♠♣❧❡s ❞♦ ❛♥t✐✲❢❡rr♦♠❛❣♥❡t✐s♠♦ é ♦ ❢❡rr✐♠❛❣♥❡t✐s♠♦✱ ♦♥❞❡ ♦s s♣✐♥s ❛♣♦♥t❛♠

❡♠ s❡♥t✐❞♦s ♦♣♦st♦s ♠❛s ♣♦ss✉❡♠ ♠❛❣♥✐t✉❞❡s ❞✐❢❡r❡♥t❡s✳ ❍á ❛✐♥❞❛ ❛rr❛♥❥♦s ❝ô♥✐❝♦s✱

♦s ✈✐❞r♦s ❞❡ s♣✐♥ ✭s♣✐♥ ❣❧❛ss❡s✮✱ ❛♥t✐❢❡rr♦♠❛❣♥❡t♦ ✐♥❝❧✐♥❛❞♦✱ ❛rr❛♥❥♦ ❤❡❧✐❝♦✐❞❛❧✳

❉♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❡ ❛♣❧✐❝❛çõ❡s ❞♦ ♠❛❣♥❡t✐s♠♦✱ ♦s ♠❛t❡r✐❛✐s ♣♦❞❡♠ s❡r ❝❧❛s✲

s✐✜❝❛❞♦s ❞❡ ❛❝♦r❞♦ ❝♦♠ s✉❛ ❞✉r❡③❛ ♠❛❣♥ét✐❝❛✳ P♦r ❡①❡♠♣❧♦✱ ♠❛t❡r✐❛✐s ❞✉r♦s ✭❤❛r❞✮

♣♦ss✉❡♠ ❛❧t♦ ✈❛❧♦r ❞❛ ❝♦❡r❝✐✈✐❞❛❞❡ ❡ sã♦ ♥♦r♠❛❧♠❡♥t❡ ✉t✐❧✐③❛❞♦s ❝♦♠♦ ❡❧❡♠❡♥t♦s ❞❡

♠❡♠ór✐❛ ❡♠ ❞✐s❝♦s rí❣✐❞♦s✱ ♦✉ ❝♦♠♦ í♠❛s ♣❡r♠❛♥❡♥t❡s✱ ❞❡♥tr❡ ♦✉tr♦s✳ ❈♦♠♦ ❡①❡♠♣❧♦

t❡♠♦s ♦s ❝♦♠♣♦st♦s ❙♠❈♦5✱ ◆❞2 ❋❡14✱ ❈r❖2✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♠❛t❡r✐❛✐s ♠❛❣♥❡t✐❝❛✲

♠❡♥t❡ ♠♦❧❡s ✭s♦❢t✮✱ ♦✉ s❡❥❛✱ ❞❡ ❜❛✐①❛ ❝♦❡r❝✐✈✐❞❛❞❡✱ ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞♦s ❝♦♠♦ ❡❧❡✲

♠❡♥t♦s ❞❡ ❣r❛✈❛çã♦ ♠❛❣♥ét✐❝❛ ❡♠ ✜t❛s✱ ❝♦♠♦ ♥ú❝❧❡♦s ❞❡ tr❛♥s❢♦r♠❛❞♦r❡s✱ ❡t❝✱ ♦♥❞❡

♣♦❞❡♠♦s ❝✐t❛r ♦ P❡r♠❛❧❧♦②✱ ♦s ♠❛t❡r✐❛✐s ❛♠♦r❢♦s✱ ❧✐❣❛s ❞❡ ❋❡✲❈♦ ❬✹✺❪✳

✷✳✷ ■♥t❡r❛çã♦ ❞✐♣♦❧❛r

❆ ❡♥❡r❣✐❛ ❡♥tr❡ ❞♦✐s ❞✐♣♦❧♦s ♠❛❣♥ét✐❝♦s ~µ1 ❡ ~µ2✱ ♥♦ ✈á❝✉♦ ❡ s❡♣❛r❛❞♦s ♣♦r ✉♠❛

❞✐stâ♥❝✐❛ ~r✱ é ❞❛❞❛ ♣♦r✿

Edip =µ0

4πr3[~µ1 · ~µ2 −

3

r2(~µ1 · ~r)(~µ2 · ~r)]✱ ✭✷✳✶✽✮

♦♥❞❡ µ0 é ❛ ♣❡r♠❡❛❜✐❧✐❞❛❞❡ ♠❛❣♥ét✐❝❛ ❞♦ ✈á❝✉♦✳ ❱❡♠♦s ❛ss✐♠ q✉❡ ❛ ✐♥t❡r❛çã♦ ❡♥tr❡

❞✐♣♦❧♦s✱ Edip✱ ❞❡♣❡♥❞❡ ❞❛ ❞✐stâ♥❝✐❛ ✭✐♥t❡r❛çã♦ ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡✮ ❡ ❞❛ ♦r✐❡♥t❛çã♦ r❡❧❛t✐✈❛

❡♥tr❡ ❡❧❡s✳

P❛r❛ ✉♠❛ ❡st✐♠❛t✐✈❛ ❞❡ Edip✱ ❡s❝♦❧❤❡♠♦s ✈❛❧♦r❡s tí♣✐❝♦s✱ ❝♦♠♦ | ~µ1| = | ~µ2| =1µB

✺ ❡ r = 2➴ ❡✱ ❛❧é♠ ❞✐ss♦✱ ❝♦♥s✐❞❡r❛♠♦s ~µ1 ♣❛r❛❧❡❧♦ ❛ ~µ2 ❡ ❡st❡s ♣❛r❛❧❡❧♦s ❛ ~r✳ ❈♦♠

✺µB = e~2m

✱ ♦♥❞❡ e é ❛ ❝❛r❣❛ ❡ m ❛ ♠❛ss❛ ❞♦ ❡❧étr♦♥✱ é ❝❤❛♠❛❞♦ ❞❡ ♠❛❣♥❡t♦♥ ❞❡ ❇♦❤r✳

✶✾

✷✳ ❈♦♥s✐❞❡r❛çõ❡s ❣❡r❛✐s s♦❜r❡ ♠❛❣♥❡t✐s♠♦ ❡ ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s

❡st❛ s✐t✉❛çã♦✱ ❡♥❝♦♥tr❛♠♦s q✉❡✿

Edip =µ0µ

2B

2πr3= 2, 1 · 10−24❏✳ ✭✷✳✶✾✮

❆ t❡♠♣❡r❛t✉r❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ✭E = kBT ✱ ♦♥❞❡ kB é ❛ ❝♦♥st❛♥t❡ ❞❡ ❇♦❧t③✲

♠❛♥♥✮ ❛ ❡st❡ ✈❛❧♦r ❞❡ ❡♥❡r❣✐❛ é ♣♦✉❝♦ ♠❡♥♦r q✉❡ ✶❑✳ ◆♦ ❡♥t❛♥t♦✱ ❛ t❡♠♣❡r❛t✉r❛ ❞❡

♦r❞❡♥❛♠❡♥t♦ ❡♠ ❢❡rr♦♠❛❣♥❡t♦s ✭t❡♠♣❡r❛t✉r❛ ❞❡ ❈✉r✐❡✮ ❛t✐♥❣❡ ❝❡♥t❡♥❛s ❞❡ ❑❡❧✈✐♥s ✲ t❛❧

t❡♠♣❡r❛t✉r❛ ♣❛r❛ ♦ ❢❡rr♦ é ❡♠ t♦r♥♦ ❞❡ ✶✵✵✵ ❑ ✲ ❡✱ ❞❡st❛ ❢♦r♠❛✱ ✈❡♠♦s q✉❡ ❛ ✐♥t❡r❛çã♦

❞✐♣♦❧❛r é ♠✉✐t♦ ♣❡q✉❡♥❛ ♣❛r❛ ❝❛✉s❛r ♦ ♦r❞❡♥❛♠❡♥t♦ ❢❡rr♦♠❛❣♥ét✐❝♦ ❬✹✽❪✳

✷✳✸ ❆♥✐s♦tr♦♣✐❛ ♠❛❣♥ét✐❝❛

◆❛s s❡çõ❡s ❛♥t❡r✐♦r❡s ❡st✉❞❛♠♦s ♦ s✉r❣✐♠❡♥t♦ ❞❛ ♠❛❣♥❡t✐③❛çã♦ s❡❥❛ ♥❛ ❛♣❧✐❝❛çã♦

❞❡ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦✱ ❝♦♠♦ é ♦ ❝❛s♦ ❞♦ ♣❛r❛♠❛❣♥❡t✐s♠♦✱ s❡❥❛ ❞❡✈✐❞♦ à ✐♥t❡r❛çã♦

❞❡ tr♦❝❛✱ ❝♦♠♦ é ♦ ❝❛s♦ ❞♦ ❢❡rr♦♠❛❣♥❡t✐s♠♦ ❡ ♦✉tr❛s ❝♦♥✜❣✉r❛çõ❡s ♠❛❣♥ét✐❝❛s✳ ◆♦

❡♥t❛♥t♦✱ ❛s ❝♦♥s✐❞❡r❛çõ❡s ❢❡✐t❛s ❛té ❛q✉✐ ❡r❛♠ r❡❧❛❝✐♦♥❛❞❛s ❛♣❡♥❛s ❛ s✐st❡♠❛s ✐s♦tró♣✐✲

❝♦s✱ ✐st♦ é✱ ❛ s✐st❡♠❛s ♦♥❞❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❢ís✐❝❛s sã♦ ✐❞ê♥t✐❝❛s ❡♠ t♦❞❛s ❛s ❞✐r❡çõ❡s✳

◆❡st❛ s❡çã♦✱ ♥ós ✈❛♠♦s ❡st✉❞❛r ♠❛✐s q✉❛❧✐t❛t✐✈❛♠❡♥t❡ ❡❢❡✐t♦s ♠❛❣♥ét✐❝♦s ❞❡♣❡♥❞❡♥t❡s

❞❡ ❞✐r❡çõ❡s ❝r✐st❛❧♦❣rá✜❝❛s✳ P♦r ❡①❡♠♣❧♦✱ ❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❞❡ ❍❡✐s❡♥❜❡r❣ ♥❛ ❢♦r♠❛ ❛✲

♣r❡s❡♥t❛❞❛ ❡♠ ✭✷✳✶✼✮ é ❝♦♠♣❧❡t❛♠❡♥t❡ ✐s♦tró♣✐❝❛✱ ♥ã♦ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥t❛ ✉♠❛ ♣♦ssí✈❡❧

❞✐r❡çã♦ ❡♠ q✉❡ ♦ ♠❛t❡r✐❛❧ ♣♦ss❛ s❡r ❝r✐st❛❧✐③❛❞♦✳

P❛r❛ ✐❧✉str❛r ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❛ ✐♥❝❧✉sã♦ ❞❡ ❛♥✐s♦tr♦♣✐❛s ❡♠ ♠♦❞❡❧♦s q✉❡ ❞❡s✲

❝r❡✈❡♠ ♦ ♠❛❣♥❡t✐s♠♦✱ ❝♦♥s✐❞❡r❡♠♦s ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✉♠ ❢❡rr♦♠❛❣♥❡t♦ ❝♦♠♣❧❡t❛✲

♠❡♥t❡ ✐s♦tró♣✐❝♦✳ ❊♠ t❡♠♣❡r❛t✉r❛s ❜❛✐①❛s✱ ✐st♦ é✱ ❡♠ ❜❛✐①❛s ❡♥❡r❣✐❛s tér♠✐❝❛s✱ ❛ ❢♦rç❛

❞❡ tr♦❝❛ é s✉✜❝✐❡♥t❡ ♣❛r❛ ♠❛♥t❡r ♦s s♣✐♥s ♣❛r❛❧❡❧♦s ❡ ❛ ❞✐r❡çã♦ ❞❡st❡s s♣✐♥s ❞❡✜♥❡ ❛

❞✐r❡çã♦ ♥♦ ❡s♣❛ç♦ ❞♦ ♠♦♠❡♥t♦ ♠❛❣♥ét✐❝♦ ~µ = −gµB

∑ ~S✱ ❡♠ q✉❡ g é ♦ ❢❛t♦r ❞❡ s♣✐♥✳

❈♦♥s✐❞❡r❛♥❞♦ q✉❡ ~µ ❢❛③ ✉♠ â♥❣✉❧♦ θ ❝♦♠ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ✜①♦ ~H✱ ❛ ❡♥❡r❣✐❛ ❞❡

✐♥t❡r❛çã♦ ❡♥tr❡ ♦ ❝❛♠♣♦ ❡ ❛ ♠❛❣♥❡t✐③❛çã♦ é −µH cos θ✳ ◆♦ ❡q✉✐❧í❜r✐♦ tér♠✐❝♦ ❛ ♣r♦✲

❜❛❜✐❧✐❞❛❞❡ ❞❡ t❡r♠♦s ✉♠❛ ❝♦♥✜❣✉r❛çã♦ ❡♠ ✉♠ ♣❛rt✐❝✉❧❛r â♥❣✉❧♦ θ ♥✉♠❛ t❡♠♣❡r❛t✉r❛

T é ♣♦r♣♦r❝✐♦♥❛❧ ❛ e−E/kBT = ex cos θ✱ ♦♥❞❡ x = (µH)/(kBT )✳ ❆ss✐♠✱ ❛ ♠é❞✐❛ ♣❛r❛ ✉♠

❡ss❡♠❜❧❡ ❞❡ ❢❡rr♦♠❛❣♥❡t♦s é✿

✷✵

✷✳ ❈♦♥s✐❞❡r❛çõ❡s ❣❡r❛✐s s♦❜r❡ ♠❛❣♥❡t✐s♠♦ ❡ ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s

〈cos θ〉 =∫ 2π

0

∫ π

0cos θ ex cos θ senθ dθ dφ

∫ 2π

0

∫ π

0ex cos θ senθ dθ dφ

= coth x− 1

x= L(x)✱ ✭✷✳✷✵✮

♦♥❞❡ L(x) é ❝❤❛♠❛❞❛ ❞❡ ❢✉♥çã♦ ❞❡ ▲❛♥❣❡✈✐♥✳ ❖ ❧❛❞♦ ❡sq✉❡r❞♦ ❞❡ ✭✷✳✷✵✮ é✱ ♣❡❧❛ ❞❡✜♥✐çã♦

❞❡ θ✱ ❛ ❝♦♠♣♦♥❡♥t❡ ♣❛r❛❧❡❧❛ ❛ ~H ❞❡ ✉♠ ✈❡t♦r ✉♥✐tár✐♦ ♥❛ ❞✐r❡çã♦ ❞❛ ♠❛❣♥❡t✐③❛çã♦✿

MH

| ~M |= 〈cos θ〉 = L

(µH

kBT

)

✳ ✭✷✳✷✶✮

❆ ❛♥á❧✐s❡ ❞❡ ✭✷✳✷✶✮ ❛ ❝❛♠♣♦s ♥✉❧♦s✱ ❛ s❛❜❡r | ~H| = 0✱ ♠♦str❛ q✉❡ s❡✉ ❧✐♠✐t❡ é

♥✉❧♦ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ q✉❡ t♦❞♦s ❢❡rr♦♠❛❣♥❡t♦s sã♦✱ ❞❡st❡ ♣♦♥t♦ ❞❡ ✈✐st❛✱ ✐❣✉❛✐s

❛♦s ♣❛r❛♠❛❣♥❡t♦s✳ P♦rt❛♥t♦✱ s❡✱ ❞❡ ❢❛t♦✱ ♥ã♦ ❤♦✉✈❡ss❡ ♦✉tr♦ t❡r♠♦ ❞❡ ❡♥❡r❣✐❛ ♣r❡s❡♥t❡

♥❡st❡s ♠❛t❡r✐❛✐s✱ t❡r♠♦ ❡st❡ q✉❡ é ❛❞✐❝✐♦♥❛❞♦ à ❤❛♠✐❧t♦♥✐❛♥❛ ❞❡ ❍❡✐s❡♥❜❡r❣ ✐s♦tró♣✐❝❛✱

t❡r✐❛ s✐❞♦ ✐♠♣♦ssí✈❡❧ ♠❡❞✐r q✉❛❧q✉❡r ♠❛❣♥❡t✐③❛çã♦ ❛ ❝❛♠♣♦ ♥✉❧♦ ❡ ♥ã♦ t❡r✐❛ s❡♥t✐❞♦✱

♣♦r ❡①❡♠♣❧♦✱ ❢❛❧❛r ❡♠ t❡♠♣❡r❛t✉r❛ ❞❡ ❈✉r✐❡✳ ❆ ♠❛❣♥❡t✐③❛çã♦ ❞❛ ❢♦r♠❛ ♠♦str❛❞❛ ❡♠

✭✷✳✷✶✮ ❝♦♥tr❛❞✐③ ♥ã♦ ❛♣❡♥❛s ❡①♣❡r✐♠❡♥t♦s q✉❡ ♣r♦❞✉③❡♠ ❣rá✜❝♦s ❝♦♠♦ ♦ ❞❛ ✜❣✉r❛ ✷✳✶✱

♠❛s t❛♠❜é♠ ❝♦♠ ❛ ❡①♣❡r✐ê♥❝✐❛ ❞✐ár✐❛ ♦♥❞❡✱ ♣♦r ❡①❡♠♣❧♦✱ ✈❡♠♦s q✉❡ ♦s ❞✐s♣♦s✐t✐✈♦s q✉❡

❛r♠❛③❡♥❛♠ ✐♥❢♦r♠❛çã♦ ❛s ♠❛♥tê♠ ❣r❛✈❛❞❛s ♥❡❧❡s ♠❡s♠♦ q✉❛♥❞♦ ♦ ❝❛♠♣♦ r❡s♣♦♥sá✈❡❧

♣❡❧❛ ✬❡s❝r✐t❛✬ é ❞❡s❧✐❣❛❞♦✳ ■st♦ é ❞❡✈✐❞♦ ❛♦ ❢❛t♦ ❞❡ q✉❡ ♠❛t❡r✐❛✐s ♠❛❣♥ét✐❝♦s r❡❛✐s ♥ã♦

sã♦ ✐s♦tró♣✐❝♦s ❡ q✉❡ ♥❡♠ t♦❞♦s ♦s ✈❛❧♦r❡s ❞❡ θ sã♦ ✐❣✉❛❧♠❡♥t❡ ♣r♦✈á✈❡✐s ❬✹✸❪✳ ❆ s❡❣✉✐r

❞✐s❝✉t✐r❡♠♦s ❜r❡✈❡♠❡♥t❡ ❞♦✐s ✐♠♣♦rt❛♥t❡s t✐♣♦s ❞❡ ❛♥✐s♦tr♦♣✐❛✳

✷✳✸✳✶ ❆♥✐s♦tr♦♣✐❛ ♠❛❣♥❡t♦❝r✐st❛❧✐♥❛

❆ ✐♥t❡r❛çã♦ s♣✐♥✲ór❜✐t❛✱ ❛ss♦❝✐❛❞❛ ❛♦ ❛❝♦♣❧❛♠❡♥t♦ ❡♥tr❡ ♦s ♠♦♠❡♥t♦s ❛♥❣✉❧❛r❡s

❞❡ s♣✐♥ ❡ ♦r❜✐t❛❧ ❞♦ ❡❧étr♦♥✱ ❛ss✐♠ ❝♦♠♦ ❛ ✐♥t❡r❛çã♦ ❞✐♣♦❧❛r s♣✐♥✲s♣✐♥ ❞✐s❝✉t✐❞❛ ♥❛ s❡çã♦

✷✳✷✱ é ❜❛st❛♥t❡ ❢r❛❝❛ s❡ ❝♦♠♣❛r❛❞❛ à ✐♥t❡r❛çã♦ ❞❡ tr♦❝❛ ❬✹✾❪✳ ◆♦ ❡♥t❛♥t♦✱ t❛❧ ✐♥t❡r❛çã♦

é r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❝❤❛♠❛❞❛ ❛♥✐s♦tr♦♣✐❛ ♠❛❣♥❡t♦❝r✐st❛❧✐♥❛✳ ❆s ór❜✐t❛s ❞♦ ❡❧étr♦♥ ❡stã♦

r❡❧❛❝✐♦♥❛❞❛s à ❡str✉t✉r❛ ❝r✐st❛❧♦❣rá✜❝❛ ❡✱ ♣♦r s✉❛s ✐♥t❡r❛çõ❡s ❝♦♠ ♦s s♣✐♥s✱ ❡❧❛s ❢❛③❡♠

❝♦♠ q✉❡ ❡st❡s ♣r❡✜r❛♠ ❛❧✐♥❤❛r✲s❡ ❛♦ ❧♦♥❣♦ ❞❡ ❡✐①♦s ❝r✐st❛❧♦❣rá✜❝♦s ❜❡♠ ❞❡✜♥✐❞♦s✳

❊♥tr❡t❛♥t♦✱ ❤á ❞✐r❡çõ❡s ♥♦ ❡s♣❛ç♦ ♥❛s q✉❛✐s ♦ ♠❛t❡r✐❛❧ ♠❛❣♥ét✐❝♦ é ♠❛✐s ❢á❝✐❧ ❞❡ s❡

♠❛❣♥❡t✐③❛r ❞♦ q✉❡ ❡♠ ♦✉tr❛s ✭❡✐①♦s ❢á❝❡✐s✮✱ s❡♥❞♦ q✉❡ t❛❧ ❞✐❢❡r❡♥ç❛ ♣❛r❛ ♦s ❡✐①♦s ♣♦❞❡

s❡r ❡①♣r❡ss❛ ❝♦♠ ✉♠ t❡r♠♦ ❞❡ ❡♥❡r❣✐❛ ❞❡♣❡♥❞❡♥t❡ ❞❛ ❞✐r❡çã♦✳

✷✶

✷✳ ❈♦♥s✐❞❡r❛çõ❡s ❣❡r❛✐s s♦❜r❡ ♠❛❣♥❡t✐s♠♦ ❡ ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s

❆ ❡♥❡r❣✐❛ ♠❛❣♥❡t♦❝r✐st❛❧✐♥❛ é ❝♦rr✐q✉❡✐r❛♠❡♥t❡ ♣❡q✉❡♥❛ q✉❛♥❞♦ ❝♦♠♣❛r❛❞❛ ❝♦♠

❛ ❡♥❡r❣✐❛ ❞❡ tr♦❝❛✱ ❡ ❛ ♠❛❣♥✐t✉❞❡ ❞❛ ♠❛❣♥❡t✐③❛çã♦✱ | ~M |✱ é ❞❡t❡r♠✐♥❛❞❛ q✉❛s❡ q✉❡

❡①❝❧✉s✐✈❛♠❡♥t❡ ♣❡❧❛ ❡♥❡r❣✐❛ ❞❡ tr♦❝❛✱ s❡♥❞♦ ❛ ❝♦♥tr✐❜✉✐çã♦ ❞❛ ❛♥✐s♦tr♦♣✐❛ ♥❡❣❧✐❣❡♥❝✐á✈❡❧

♣❛r❛ q✉❛s❡ t♦❞♦s ♦s ♠❛t❡r✐❛s ❢❡rr♦♠❛❣♥ét✐❝♦s ❝♦♥❤❡❝✐❞♦s✱ ❞❡♥tr❡ ♦s q✉❛✐s ❡stã♦ ♦s

♠❛t❡r✐❛✐s ♠❛❣♥ét✐❝♦s ♠♦❧❡s✱ q✉❡ ❝♦♥tê♠✱ ❝♦♠♦ ✈✐♠♦s✱ ♦ P❡r♠❛❧❧♦② ❬✺✵❪✳ ❊♥tr❡t❛♥t♦✱

✈❛❧❡ r❡ss❛❧t❛r q✉❡ ❛ ❞✐r❡çã♦ ❞❛ ♠❛❣♥❡t✐③❛çã♦ é ❞❡t❡r♠✐♥❛❞❛ ❛♣❡♥❛s ♣♦r t❛❧ ❛♥✐s♦tr♦♣✐❛✱

♣♦✐s ❛ ❡♥❡r❣✐❛ ❞❡ tr♦❝❛ ♥ã♦ ❧❡✈❛ ❡♠ ❝♦♥t❛ q✉❛❧q✉❡r ❞✐r❡çã♦ ♥♦ ❡s♣❛ç♦✳ ❱✐♠♦s ❡♥tã♦ q✉❡

♦ t❡r♠♦ ❞❡ tr♦❝❛ t❡♥t❛ ❛❧✐♥❤❛r t♦❞♦s ♦s s♣✐♥s ♣❛r❛❧❡❧♦s ✉♥s ❛♦s ♦✉tr♦s✱ ❛♦ ♣❛ss♦ q✉❡

t❡r♠♦ ♠❛❣♥❡t♦❝r✐st❛❧✐♥♦ t❡♥t❛ ❛❧✐♥❤á✲❧♦s ❡♠ ❝❡rt❛s ❞✐r❡çõ❡s ❝r✐st❛❧♦❣rá✜❝❛s✳ ❏✉♥t♦s✱

♦s ❞♦✐s t❡r♠♦s t❡♥t❛♠ ❛❧✐♥❤❛r t♦❞♦s s♣✐♥s ♣❛r❛❧❡❧♦s ❡♠ ❞❡t❡r♠✐♥❛❞❛ ❞✐r❡çã♦✱ ♠❛s ♥ã♦

sã♦ r❡s♣♦♥sá✈❡✐s ♣❡❧❛ ❢♦r♠❛çã♦ ❞♦s ❞♦♠í♥✐♦s ♠❛❣♥ét✐❝♦s ❬✹✸❪✳ ❖ t❡r♠♦ q✉❡ ❝❛✉s❛ ♦s

❞♦♠í♥✐♦s s❡rá ✈✐st♦ ❡♠ ♠♦♠❡♥t♦ ♦♣♦rt✉♥♦✳

❆♣❡s❛r ❞❡ ❛ ✐♥t❡r❛çã♦ s♣✐♥✲ór❜✐t❛ ♣♦❞❡r s❡r ❝❛❧❝✉❧❛❞❛ ♣♦r ♣r✐♥❝í♣✐♦s t❡ór✐❝♦s✱

❛s ❡♥❡r❣✐❛s ❞❡ ❛♥✐s♦tr♦♣✐❛ sã♦ ❡s❝r✐t❛s ❝♦♠ ❡①♣r❡ssõ❡s ❢❡♥♦♠❡♥♦❧ó❣✐❝❛s✱ ❛s q✉❛✐s sã♦

❡①♣❛♥sõ❡s ❡♠ sér✐❡s ❞❡ ♣♦tê♥❝✐❛ q✉❡ ❧❡✈❛♠ ❡♠ ❝♦♥t❛ ❛ s✐♠❡tr✐❛ ❞♦ ❝r✐st❛❧✱ s❡♥❞♦ ♦s ❝♦✲

❡✜❝✐❡♥t❡s ♦❜t✐❞♦s ❞❡ ❡①♣❡r✐♠❡♥t♦s✳ ❍á✱ ❡♥tr❡t❛♥t♦✱ ❡①♣r❡ssõ❡s ♣❛r❛ ❞✐❢❡r❡♥t❡s s✐st❡♠❛s

❝r✐st❛❧♦❣rá✜❝♦s✱ ❝♦♠♦ s✐st❡♠❛s ❝ú❜✐❝♦s✱ t❡tr❛❣♦♥❛✐s✱ ❤❡①❛❣♦♥❛✐s ❬✹✽❪✳

✷✳✸✳✷ ❆♥✐s♦tr♦♣✐❛ ❞❡ ❢♦r♠❛

❆♠♦str❛s ♣♦❧✐❝r✐st❛❧✐♥❛s s❡♠ ✉♠❛ ♦r✐❡♥t❛çã♦ ♣r❡❢❡r✐❞❛ ❞❡ s❡✉s ❣rã♦s✱ ♥ã♦ ♣♦s✲

s✉❡♠ q✉❛❧q✉❡r ❛♥✐s♦tr♦♣✐❛ ♠❛❣♥❡t♦❝r✐st❛❧✐♥❛✳ ❯♠ ❝♦♠♣♦rt❛♠❡♥t♦ ✐s♦tró♣✐❝♦✱ ❡♥tr❡✲

t❛♥t♦✱ é s♦♠❡♥t❡ ❞❛❞♦ ♣❛r❛ ❢♦r♠❛s ❡s❢ér✐❝❛s✱ ♣♦✐s s❡ ❛ ❛♠♦str❛ ♥ã♦ é ❡s❢ér✐❝❛✱ ❡♥tã♦ ✉♠❛

♦✉ ♠❛✐s ❞✐r❡çõ❡s ❡s♣❡❝í✜❝❛s ♦❝♦rr❡♠✱ ❛✭s✮ q✉❛❧✭✐s✮ r❡♣r❡s❡♥t❛✭♠✮ ♦s ❝❤❛♠❛❞♦s ❡✐①♦s ❢á✲

❝❡✐s ❞❡ ♠❛❣♥❡t✐③❛çã♦✱ q✉❡ sã♦ ❝❛✉s❛❞♦s ✉♥✐❝❛♠❡♥t❡ ♣❡❧❛ ❢♦r♠❛ ❞❛ ❛♠♦str❛ ❬✹✽❪✳ ❊st❡

t✐♣♦ ❞❡ ❛♥✐s♦tr♦♣✐❛ é t❛♠❜é♠ ❝❤❛♠❛❞♦ ❞❡ ❡♥❡r❣✐❛ ♠❛❣♥❡t♦stát✐❝❛✱ ❡ t❡♠ s✉❛ ♦r✐❣❡♠ ♥❛

✐♥t❡r❛çã♦ ❝❧áss✐❝❛ ❡♥tr❡ ❞✐♣♦❧♦s✳ P❛r❛ ❡♥t❡♥❞❡r♠♦s ♠❡❧❤♦r ❛ ❝❛✉s❛ ❞❡st❛ ❛♥✐s♦tr♦♣✐❛✱

❢❛r❡♠♦s ✉s♦ ❞❡ ❛❧❣✉♠❛s ❞❛s ❡q✉❛çõ❡s ❞❡ ▼❛①✇❡❧❧✳ ❆ ❧❡✐ ❞❡ ❆♠♣èr❡ ❡♠ ♠❛t❡r✐❛✐s ♠❛❣✲

♥❡t✐③❛❞♦s✱ ❡s♣❡❝✐✜❝❛♠❡♥t❡ ❡♠ ❢❡rr♦♠❛❣♥❡t♦s✱ ❡ s❡♠ ❛ ♣r❡s❡♥ç❛ ❞❡ q✉❛❧q✉❡r t✐♣♦ ❞❡

❝♦rr❡♥t❡✱ é t❛❧ q✉❡✿

~∇× ~Hd = ~0✳ ✭✷✳✷✷✮

✷✷

✷✳ ❈♦♥s✐❞❡r❛çõ❡s ❣❡r❛✐s s♦❜r❡ ♠❛❣♥❡t✐s♠♦ ❡ ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s

➱ ❝♦st✉♠❡✱ ♥♦ ❡st✉❞♦ ❞♦ ❢❡rr♦♠❛❣♥❡t✐s♠♦✱ s❡♣❛r❛r ♦s ❝❛♠♣♦s ♠❛❣♥ét✐❝♦s ❡♠

❞✉❛s ❝❛t❡❣♦r✐❛s ❡ tr❛t❛r ♦ ❝❛♠♣♦ ~Hd ❡♠ ✭✷✳✷✷✮ s❡♣❛r❛❞❛♠❡♥t❡ ❞♦ ❝❛♠♣♦ ❛♣❧✐❝❛❞♦

~Hext✱ ♦ q✉❛❧ é ♣r♦❞✉③✐❞♦ ❡①t❡r♥❛♠❡♥t❡ à ❛♠♦str❛✳ ❯♠❛ ❛♠♦str❛ ✜♥✐t❛ ❡①✐❜❡ ♣ó❧♦s

❡♠ s✉❛ s✉♣❡r❢í❝✐❡✱ q✉❡ ❧❡✈❛♠ à ❢♦r♠❛çã♦ ❞❡ ✉♠ ✧❝❛♠♣♦ ♣❡r❞✐❞♦✧ ✭str❛② ✜❡❧❞✮ ❢♦r❛ ❞❛

❛♠♦str❛✳ ❊st❡ ❝❛♠♣♦✱ ❡♥tã♦✱ é r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ♦❝♦rrê♥❝✐❛ ❞❡ ✉♠ ❝❛♠♣♦ ♥♦ ✐♥t❡r✐♦r ❞❛

❛♠♦str❛✱ ❝❤❛♠❛❞♦ ❞❡ ❝❛♠♣♦ ❞❡ ❞❡s♠❛❣♥❡t✐③❛çã♦✱ ~Hd✱ ♣r♦❞✉③✐❞♦ ♣❡❧❛ ❞✐str✐❜✉✐çã♦ ❞❡

♠❛❣♥❡t✐③❛çã♦ ✭❢✉♥çã♦ ❞❡ ~M✮✳ ❆ s♦❧✉çã♦ ❞❡ ✭✷✳✷✷✮ é ❜❡♠ ❝♦♥❤❡❝✐❞❛✱ s❡♥❞♦ ~Hd ❡s❝r✐t♦

❝♦♠♦ ♦ ❣r❛❞✐❡♥t❡ ❞❡ ✉♠ ♣♦t❡♥❝✐❛❧ ❡s❝❛❧❛r✻ Φ✱ ❛♥á❧♦❣♦ ❛♦ ❝❛s♦ ❡❧❡tr♦stát✐❝♦✳ ❯s❛♥❞♦✲s❡

❛ ❝♦♥✈❡♥çã♦ ❞♦ s✐♥❛❧ ♥❡❣❛t✐✈♦ ♥❛ ❞❡✜♥✐çã♦✱ t❡♠♦s✿

~Hd = −~∇Φ✳ ✭✷✳✷✸✮

❯♠❛ ♦✉tr❛ ❡q✉❛çã♦ ❞❡ ▼❛①✇❡❧❧ é✿

~∇ · ~B = 0✱ ✭✷✳✷✹✮

♦♥❞❡ ~B é ❛ ✐♥❞✉çã♦ ♠❛❣♥ét✐❝❛✱ ❞❡✜♥✐❞❛ ❡♠ ✭✷✳✷✮✳ ❆❣♦r❛✱ s✉❜st✐t✉✐♥❞♦ ❛s ❡q✉❛çõ❡s

✭✷✳✷✮ ❡ ✭✷✳✷✸✮ ❡♠ ✭✷✳✷✹✮✱ t❡♠♦s q✉❡✿

∇2Φin = ~∇ · ~M ✱ ✭✷✳✷✺✮

❛ q✉❛❧ ❞❡✈❡ s❡r ✈á❧✐❞❛ ❞❡♥tr♦ ❞♦✭s✮ ♠❛t❡r✐❛❧✭✐s✮ ❢❡rr♦♠❛❣♥ét✐❝♦✭s✮ ❬✹✸❪✳ P♦❞❡♠♦s ✐❞❡♥✲

t✐✜❝❛r✱ ❝♦♠♣❛r❛♥❞♦ ❝♦♠ ❛ ❡❧❡tr♦stát✐❝❛✱

ρM = −~∇ · ~M ✭✷✳✷✻✮

❝♦♠♦ s❡♥❞♦ ❛ ❞❡♥s✐❞❛❞❡ ✈♦❧✉♠étr✐❝❛ ❞❡ ❝❛r❣❛ ♠❛❣♥ét✐❝❛✳ ❆ s♦❧✉çã♦ ♣❛r❛ Φ✱ ♥♦ ❝❛s♦

❡♠ q✉❡ ♥ã♦ ❤á ❝♦♥❞✐çõ❡s ❞♦ ❝♦♥t♦r♥♦ ♥❛ s✉♣❡r❢í❝✐❡ ❞♦ ♠❛t❡r✐❛❧✱ é ❬✺✶❪✿

Φin(~r) = − 1

4π

∫

V

~∇′ · ~M(~r ′)

|~r − ~r ′| d3r′✱ ✭✷✳✷✼✮

♦♥❞❡ V é ♦ ✈♦❧✉♠❡ ❞♦ ❢❡rr♦♠❛❣♥❡t♦✱ ~r é ❛ ♣♦s✐çã♦ ♥♦ ❡s♣❛ç♦ ♦♥❞❡ ♦ ♣♦t❡♥❝✐❛❧ é ❝❛❧❝✉❧❛❞♦

❡ ~r ′ é ❛ ♣♦s✐çã♦ ❞❡ ✉♠❛ ♣♦rçã♦ d3r′ ❞♦ ❢❡rr♦♠❛❣♥❡t♦✳

✻❊♠ ✈❡③ ❞♦ ♣♦t❡♥❝✐❛❧ ❡s❝❛❧❛r✱ ♦ ♣r♦❜❧❡♠❛ ♣♦❞❡ s❡r ❢♦r♠✉❧❛❞♦ ❝♦♠ ❜❛s❡ ♥♦ ✈❡t♦r ♣♦t❡♥❝✐❛❧✱ ~A✱ ❡♠

q✉❡ ~B = ~∇× ~A ❡✱ ❞❛í✱ ❡♥❝♦♥tr❛r ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❡ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ♣❛r❛ ~A✳

✷✸

✷✳ ❈♦♥s✐❞❡r❛çõ❡s ❣❡r❛✐s s♦❜r❡ ♠❛❣♥❡t✐s♠♦ ❡ ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s

❏á ♥❛ r❡❣✐ã♦ ❡①t❡r♥❛ ❛♦ ♠❛t❡r✐❛❧✱ ~M = ~0✱ ❡ ❡♥tã♦ ~B = µ0~Hd✳ ❆ss✐♠ ❛ ❡q✉❛çã♦

❞✐❢❡r❡♥❝✐❛❧ ♥❡ss❛ r❡❣✐ã♦ é✿

∇2Φout = 0✳ ✭✷✳✷✽✮

➱ ❝♦♥❤❡❝✐❞♦ ❛ ♣❛rt✐r ❞❛s ❡q✉❛çõ❡s ❞❡ ▼❛①✇❡❧❧ q✉❡ ❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ ~Hd ♣❛✲

r❛❧❡❧❛s à s✉♣❡r❢í❝✐❡ ✭✐♠❡❞✐❛t❛♠❡♥t❡ ❞❡♥tr♦ ❡ ❢♦r❛ ❞♦ ♠❛t❡r✐❛❧✮✱ ♥ã♦ ❤❛✈❡♥❞♦ ❝♦rr❡♥t❡s

s✉♣❡r✜❝✐❛✐s✱ sã♦ ❝♦♥tí♥✉❛s✱ ❛ss✐♠ ❝♦♠♦ ❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ ~B ♣❡r♣❡♥❞✐❝✉❧❛r❡s à s✉✲

♣❡r❢í❝✐❡ ❬✹✶❪✳ ❊st❡s r❡q✉❡r✐♠❡♥t♦s ❧❡✈❛♠ às ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ♥❛ s✉♣❡r❢í❝✐❡ S ❞♦

❢❡rr♦♠❛❣♥❡t♦✿

Φin

∣∣∣S= Φout

∣∣∣S,

∂Φin

∂n

∣∣∣S− ∂Φout

∂n

∣∣∣S= ~M · n✱ ✭✷✳✷✾✮

♦♥❞❡ n é ✉♠ ✈❡t♦r ✉♥✐tár✐♦ ♥♦r♠❛❧ à s✉♣❡r❢í❝✐❡ ❞♦ ❝♦r♣♦ ❢❡rr♦♠❛❣♥ét✐❝♦✱ s❡♥❞♦ ♣♦s✐t✐✈♦

♥❛ ❞✐r❡çã♦ q✉❡ s❛✐ ❞♦ ❝♦r♣♦✳ ❆♣❡s❛r ❞❡st❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦✱ q✉❡r❡♠♦s ❛✐♥❞❛ q✉❡ ♦

♣♦t❡♥❝✐❛❧ s❡❥❛ r❡❣✉❧❛r ♥♦ ✐♥✜♥✐t♦✱ ♦ q✉❡ s✐❣♥✐✜❝❛ q✉❡ |rΦ| ❡ |r2~∇Φ| ♥ã♦ ❞❡✈❡♠ ❞✐✈❡r❣✐r

❝♦♥❢♦r♠❡ r → ∞✳ ❊st❛ r❡❣✉❧❛r✐❞❛❞❡ s✐❣♥✐✜❝❛ q✉❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ ♣♦t❡♥❝✐❛❧

❛ ❣r❛♥❞❡s ❞✐stâ♥❝✐❛s ❞♦ ❢❡rr♦♠❛❣♥❡t♦ é ♦ ♠❡s♠♦ q✉❡ ♦ ❞♦ ♣♦t❡♥❝✐❛❧ ❞❡ ✉♠❛ ❝❛r❣❛

♣✉♥t✐❢♦r♠❡✱ ♦ q✉❡ ♣♦❞❡ s❡r ❡s♣❡r❛❞♦ s❡ ❛ ♠❛❣♥❡t✐③❛çã♦ ✈❛✐ ❛ ③❡r♦ ❢♦r❛ ❞❡ ✉♠ ❝❡rt♦

✈♦❧✉♠❡ ✜♥✐t♦✳

❆ ❛♣❧✐❝❛çã♦ ❞♦ t❡♦r❡♠❛ ❞❛ ❞✐✈❡r❣ê♥❝✐❛ ✭∫

V~∇ · ~AdV =

∮

S~A · n dS✮ ❡♠ ρM ✱

❡q✉❛çã♦ ✭✷✳✷✻✮✱ ♥✉♠❛ s✉♣❡r❢í❝✐❡ ❣❛✉ss✐❛♥❛ ❞❡ ❡s♣❡ss✉r❛ q✉❡ t❡♥❞❛ ❛ ③❡r♦✱ ❧❡✈❛ ❛ ✉♠❛

❞❡♥s✐❞❛❞❡ s✉♣❡r✜❝✐❛❧ ❞❡ ❝❛r❣❛ ♠❛❣♥ét✐❝❛✿

σM = ~M · n✳ ✭✷✳✸✵✮

❉❡st❛ ❢♦r♠❛✱ ❡♠ ✈❡③ ❞❡ ✉s❛r♠♦s ❛ ❡①♣r❡ssã♦ ✭✷✳✷✼✮ ♣❛r❛ Φ✱ t❡♠♦s q✉❡ ♦ ♣♦t❡♥❝✐❛❧

♥✉♠ ❝❛s♦ ♠❛✐s r❡❛❧íst✐❝♦ ❡♥✈♦❧✈❡♥❞♦ ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥tr♦♥♦ s✉♣❡r✜❝✐❛✐s é ❞❛❞♦ ♣♦r ❬✺✶❪✿

Φ(~r) = − 1

4π

∫

V

~∇′ · ~M(~r ′)

|~r − ~r ′| d3r′ +1

4π

∮

S

n · ~M(~r ′)

|~r − ~r ′| dS✳ ✭✷✳✸✶✮

❆❣♦r❛✱ ❝♦♥❤❡❝❡♥❞♦✲s❡ Φ✱ ~Hd ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❞❡ ✭✷✳✷✸✮ ❡ ❛ ❡♥❡r❣✐❛ ♠❛❣♥❡✲

t♦stát✐❝❛ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞❛✱ r❡s✉❧t❛♥❞♦ ❡♠✿

Emag = −1

2

∫

V

µ0~M · ~Hd d

3r✱ ✭✷✳✸✷✮

✷✹

✷✳ ❈♦♥s✐❞❡r❛çõ❡s ❣❡r❛✐s s♦❜r❡ ♠❛❣♥❡t✐s♠♦ ❡ ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s

♦♥❞❡ ❛ ✐♥t❡❣r❛çã♦ é ❢❡✐t❛ ♥♦ ❢❡rr♦♠❛❣♥❡t♦ ❞❡ ✈♦❧✉♠❡ V ✳ ❆ ♣❛rt✐r ❞❡st❡ r❡s✉❧t❛❞♦✱

♣♦❞❡♠♦s ✐♥❢❡r✐r q✉❡ ❛ ❡♥❡r❣✐❛ ♠❛❣♥❡t♦stát✐❝❛ é ♠✐♥✐♠✐③❛❞❛ q✉❛♥❞♦ ❛ ❛♠♦str❛ ♥ã♦

❡①✐❜❡ ♣ó❧♦s ✭~∇ · ~M = 0 ❡ ~M · n = 0✮✱ q✉❡ sã♦ ❥✉st❛♠❡♥t❡ ♦s ❛❣❡♥t❡s ❢♦r♠❛❞♦r❡s ❞❡

~Hd✳ ❉❛q✉✐ ❡♠ ❞✐❛♥t❡✱ q✉❛♥❞♦ ♥♦s r❡❢❡r✐r♠♦s ❛ ❡st❡ t❡r♠♦ ❞❡ ❡♥❡r❣✐❛✱ ✉s❛r❡♠♦s ♦ ♥♦♠❡

❡♥❡r❣✐❛ ♠❛❣♥❡t♦stát✐❝❛✳

❊st❛ ❡q✉❛çã♦ ♥❛❞❛ ♠❛✐s é ❞♦ q✉❡ ❛ ✐♥t❡r❛çã♦ ❞❡ ❝❛❞❛ ❞✐♣♦❧♦ ❝♦♠ ♦ ❝❛♠♣♦ ~Hd

❝r✐❛❞♦ ♣❡❧♦s ♦✉tr♦s ❞✐♣♦❧♦s✱ s❡♥❞♦ ♦ ❢❛t♦r ✶✴✷ ✐♥tr♦❞✉③✐❞♦ ❝♦♠ ❛ ✜♥❛❧✐❞❛❞❡ ❞❡ ❡✈✐t❛r

❝♦♥t❛r ❞✉❛s ✈❡③❡s ❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ❞♦✐s ❞✐♣♦❧♦s✳ ◆❡st❡ ♣♦♥t♦✱ ❛♣❡♥❛s ❛✜r♠❛r❡♠♦s

s❡r ❛ ❡♥❡r❣✐❛ ♠❛❣♥❡t♦stát✐❝❛ ❛ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❡①✐stê♥❝✐❛ ❞♦s ❞♦♠í♥✐♦s ♠❛❣♥ét✐❝♦s✱

♦✉✱ ♣❡❧♦ ♠❡♥♦s✱ é ❡st❡ t❡r♠♦ ❞❡ ❡♥❡r❣✐❛ q✉❡ ♣r❡❢❡r❡ ❛ s✉❜❞✐✈✐sã♦ ❞❡ ✉♠ ❢❡rr♦♠❛❣♥❡t♦

❡♠ ❞♦♠í♥✐♦s ❬✹✸❪✱ ♦✉ s❡❥❛✱ ♦s ♠❛t❡r✐❛✐s ❢❡rr♦♠❛❣♥ét✐❝♦s✱ ❣❡r❛❧♠❡♥t❡✱ ❢♦r♠❛♠ ❞♦♠í♥✐♦s

♣❛r❛ ♠✐♥✐♠✐③❛r t❛❧ t❡r♠♦ ❞❡ ❡♥❡r❣✐❛✳ ❆ ✜❣✉r❛ ✷✳✹ é ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❡sq✉❡♠át✐❝❛ ❞❡

❞♦♠í♥✐♦s ♠❛❣♥ét✐❝♦s✱ q✉❡ sã♦ s❡♣❛r❛❞♦s ♣❡❧❛s ❝❤❛♠❛❞❛s ♣❛r❡❞❡s ❞❡ ❞♦♠í♥✐♦s✱ ❛s q✉❛✐s

sã♦ r❡❣✐õ❡s ♠✐❝r♦s❝ó♣✐❝❛s q✉❡ tê♠ ✉♠ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ♦r❞❡♠ ❞❡ ✶✵✲✶✵2 ♥♠ ✭❞❛ ♦r❞❡♠

❡ ♣♦✉❝♦ ♠❡♥♦r❡s q✉❡ ♦s ♥❛♥♦❞✐s❝♦s ❛q✉✐ ❝♦♥s✐❞❡r❛❞♦s✮✱ ❞❡♣❡♥❞❡♥❞♦ ❞♦ ♠❛t❡r✐❛❧ ❬✸✵❪✳

❋✐❣✉r❛ ✷✳✹✿ ❉♦♠í♥✐♦s ♠❛❣♥ét✐❝♦s ❝♦♠ ♣❛r❡❞❡s ❞❡ ❞♦♠í♥✐♦s ❢♦r♠❛♥❞♦ ✭❛✮ 180◦ ❡ ✭❜✮ 90♦✳

❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢❡rê♥❝✐❛ ❬✹✽❪✳

❍á ❛✐♥❞❛ ♦✉tr❛s ❢♦♥t❡s ❞❡ ❛♥✐s♦tr♦♣✐❛ ❛❧é♠ ❞❛s ❞✉❛s ❞✐s❝✉t✐❞❛s ❛❝✐♠❛✳ P♦❞❡♠♦s

❝✐t❛r ❛ ♠❛❣♥❡t♦str✐çã♦✱ ♥❛ q✉❛❧✱ q✉❛♥❞♦ ✉♠ ❢❡rr♦♠❛❣♥❡t♦ é ♠❛❣♥❡t✐③❛❞♦ ❡❧❡ s❡ r❡tr❛✐

♦✉ ❡①♣❛♥❞❡ ♥❛ ❞✐r❡çã♦ ❞❛ ♠❛❣♥❡t✐③❛çã♦✱ t♦r♥❛♥❞♦ ✐♥✈á❧✐❞❛ ❛ ❞❡✜♥✐çã♦ ❞❡ ~M ❝♦♠♦ ♦

♠♦♠❡♥t♦ ❞❡ ❞✐♣♦❧♦ ♣♦r ✉♥✐❞❛❞❡ ❞❡ ✈♦❧✉♠❡✱ ♣♦✐s ♦ ✈♦❧✉♠❡ ♠✉❞❛ ❝♦♠ ❛ ♠❛❣♥❡t✐③❛çã♦

q✉❡✱ ♣♦r s✉❛ ✈❡③✱ ♠✉❞❛ ❝♦♠ ♦ ❝❛♠♣♦ ❛♣❧✐❝❛❞♦✳ ➱ ❝♦♥❤❡❝✐❞♦ q✉❡ ♦ P❡r♠❛❧❧♦②✱ ♥❛ ❝♦♠✲

✷✺

✷✳ ❈♦♥s✐❞❡r❛çõ❡s ❣❡r❛✐s s♦❜r❡ ♠❛❣♥❡t✐s♠♦ ❡ ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s

♣♦s✐çã♦ ❝♦♠ ❝❡r❝❛ ❞❡ ✼✽✪ ❞❡ ◆✐✱ ❛❧é♠ ❞❡ ♣♦ss✉✐r ❜❛✐①❛ ❛♥✐s♦tr♦♣✐❛ ♠❛❣♥❡t♦❝rs✐t❛❧✐♥❛✱

t❛♠❜é♠ ♣♦ss✉✐ ❜❛✐①❛ ♠❛❣♥❡t♦str✐çã♦ ❬✺✷❪✳ ❚❡♠♦s t❛♠❜é♠ ❛ ❛♥✐s♦tr♦♣✐❛ ✐♥❞✉③✐❞❛✿ é

♦❜s❡r✈❛❞♦✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❡ q✉❛♥❞♦ ✉♠ ✜❧♠❡ ✜♥♦ é ❞❡♣♦s✐t❛❞♦ ❡♠ ✉♠ â♥❣✉❧♦ ♦❜❧íq✉♦

❛♦ s✉❜str❛t♦✱ ♦✉ ♠❡s♠♦ q✉❛♥❞♦ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ♦✉ ❡❧étr✐❝♦✱ é ❛♣❧✐❝❛❞♦ ❞✉r❛♥t❡

❛ ❞❡♣♦s✐çã♦✱ ✉♠❛ ❛♥✐s♦tr♦♣✐❛ ✉♥✐❛①✐❛❧ ❞❡s❡♥✈♦❧✈❡✲s❡ ✭é ✐♥❞✉③✐❞❛✮ ♥♦ ♣❧❛♥♦ ❞♦ ✜❧♠❡✱

♠❛s t❛❧ ❛♥✐s♦tr♦♣✐❛ é ❝♦♠♣❧❡①❛ ❡ ❞❡ ❞✐❢í❝✐❧ tr❛t❛♠❡♥t♦ q✉❛♥t✐t❛t✐✈♦✳ P♦❞❡♠♦s ❝✐t❛r

❛✐♥❞❛ ❛ ❛♥✐s♦tr♦♣✐❛ ❞❡ s✉♣❡r❢í❝✐❡ ❡ ✐♥t❡r❢❛❝❡✱ q✉❡ ❡stá ❛ss♦❝✐❛❞❛ à ❛ss✐♠❡tr✐❛ ❡①✐❜✐❞❛

♣❡❧❛ s✉♣❡r❢í❝✐❡ ❡♠ ❝♦♠♣❛r❛çã♦ ❝♦♠ ❛ ♣❛rt❡ ✐♥t❡r♥❛ ❞❛ ❛♠♦str❛❀ ♦ s♣✐♥ ♥❛ s✉♣❡r❢í❝✐❡

t❡♠ ✉♠ ✈✐③✐♥❤♦ ♠❛✐s ♣ró①✐♠♦ ❞❡ ✉♠ ❧❛❞♦ ♠❛s ♥ã♦ t❡♠ ✈✐③✐♥❤♦s ♥❛ ♣❛rt❡ ❡①t❡r♥❛ ❞❛

❛♠♦str❛✱ t❛❧ q✉❡ ❛ ❡♥❡r❣✐❛ ❞❡ tr♦❝❛ ♥❡st❡ ❛♠❜✐❡♥t❡ ♥ã♦ ♣♦❞❡ s❡r ✐❣✉❛❧ àq✉❡❧❛ ♥♦ ✐♥t❡r✐♦r

❞❛ ♠❡s♠❛✳ ❯♠ ♠❡t❛❧ ♥ã♦✲♠❛❣♥ét✐❝♦ ✲ ♦✉ ❛té ♠❡s♠♦ ✉♠ ♦✉tr♦ ❝♦r♣♦ ❢❡rr♦♠❛❣♥ét✐❝♦

✲ ❞❡♣♦s✐t❛❞♦ s♦❜r❡ ✉♠ ❢❡rr♦♠❛❣♥❡t♦ ❢♦r♥❡❝❡✱ t❛♠❜é♠✱ ✉♠❛ ✐♥t❡r❢❛❝❡ ❝♦♠ ❛♠❜✐❡♥t❡

❞✐❢❡r❡♥t❡ ♣❛r❛ ♦s s♣✐♥s ♥❛ s✉♣❡r❢í❝✐❡ ❬✹✸❪✳

✷✳✹ ❖ ♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ❛♥✐s♦tró♣✐❝♦

❈♦♠♦ ❞✐t♦✱ ❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❞❡ ❍❡✐s❡♥❜❡r❣ é ✐s♦tró♣✐❝❛ ❡✱ ❛♣ós ❡st✉❞❛r♠♦s ❛❧❣✉♥s

t✐♣♦s ❞❡ ❛♥✐s♦tr♦♣✐❛✱ é ❝♦♥✈❡♥✐❡♥t❡ ❛❝r❡s❝❡♥t❛r t❡r♠♦✭s✮ à ❡①♣r❡ssã♦ ✐s♦tró♣✐❝❛ ✭✷✳✶✼✮

q✉❡ ❧❡✈❡✭♠✮ ❡♠ ❝♦♥t❛ ❛ ♣r❡s❡♥ç❛ ❞❡ t❛❧ ❢❡♥ô♠❡♥♦✱ ♣r♦♣♦r❝✐♦♥❛♥❞♦ ✉♠ ♠♦❞❡❧♦ q✉❡ ❞❡s✲

❝r❡✈❛ ♠❡❧❤♦r ♦s ❞✐✈❡rs♦s ♠❛t❡r✐❛✐s✳ ❉❡st❛ ❢♦r♠❛✱ ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ ❛ ❤❛♠✐❧t♦♥✐❛♥❛✼

❞❡ s♣✐♥ ♥♦ ♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ❛♥✐s♦tró♣✐❝♦ ♣♦❞❡ s❡r ❞❛❞❛ ♣♦r✿

H = −J∑

<i,j>

(Sxi S

xj + Sy

i Syj + λSz

i Szj

)✭✷✳✸✸✮

❡♠ q✉❡ λ é ❝❤❛♠❛❞❛ ❞❡ ❝♦♥st❛♥t❡ ❞❡ ❛♥✐s♦tr♦♣✐❛✳ ❊st❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❝♦♥té♠ ❛❧❣✉♥s

❝❛s♦s ❧✐♠✐t❡ q✉❡ ❞❡♣❡♥❞❡♠ ❞♦ ♣❛râ♠❡tr♦ ❞❡ ❛♥✐s♦tr♦♣✐❛ λ✳ ❆❣♦r❛✱ ❞❡st❛❝❛♠♦s ♦s

s❡❣✉✐♥t❡s ❝❛s♦s✿

• ■s♦tró♣✐❝♦✿ ♣❛r❛ λ = 1 r❡❝✉♣❡r❛♠♦s ♦ ♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ✐s♦tró♣✐❝♦✱ ♦♥❞❡

♦s s♣✐♥s ♥ã♦ tê♠ ❞✐r❡çã♦ ♣r❡❢❡r❡♥❝✐❛❧✳ ❈♦♠♦ ❡①❡♠♣❧♦✱ t❡♠♦s ♦s í♦♥s ▼♥2+ ❡ ❋❡3+ ❬✺✸❪❀

✼❍á ♦✉tr❛s ✈❡rsõ❡s ❛♥✐s♦tró♣✐❝❛s ♠❛✐s ❣❡r❛✐s ❞❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❞❡ ❍❡✐s❡♥❜❡r❣✱ ♦♥❞❡ ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s

❧❡✈❛♠ ❛♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ❡ ❳❨✳ ❱❡❥❛✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ r❡❢❡rê♥❝✐❛ ❬✹✽❪✳

✷✻

✷✳ ❈♦♥s✐❞❡r❛çõ❡s ❣❡r❛✐s s♦❜r❡ ♠❛❣♥❡t✐s♠♦ ❡ ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s

❋✐❣✉r❛ ✷✳✺✿ ❘❡♣r❡s❡♥t❛çã♦ ❡sq✉❡♠át✐❝❛ ❞❡ ✉♠ ❢❡rr♦♠❛❣♥❡t♦ ❝♦♠ ❛♥✐s♦tr♦♣✐❛ ❞❡ ❡✐①♦✲❢á❝✐❧✳

❋✐❣✉r❛ ✷✳✻✿ ❘❡♣r❡s❡♥t❛çã♦ ❡sq✉❡♠át✐❝❛ ❞❡ ✉♠ ❢❡rr♦♠❛❣♥❡t♦ ❝♦♠ ❛♥✐s♦tr♦♣✐❛ ❞❡ ♣❧❛♥♦✲❢á❝✐❧✳

• ❊✐①♦✲❢á❝✐❧✿ ♣❛r❛ λ > 1 ♦s s♣✐♥s tê♠ ❛ ♣r❡❢❡rê♥❝✐❛ ❞❡ ❛♣♦♥t❛r ♥✉♠❛ ❞❛❞❛

❞✐r❡çã♦✱ ❡✐①♦ z ♣♦r ❡①❡♠♣❧♦✳ ■st♦ ❝❛r❛❝t❡r✐③❛ ✉♠❛ s✐♠❡tr✐❛ ❞❡ ❡✐①♦✲❢á❝✐❧ ✭✈❡❥❛ ❛ ✜❣✉r❛

✷✳✺✮✳ ❚❡♠♦s ❝♦♠♦ ❡①❡♠♣❧♦s ❨❋❡❖3✱ ❈r❇r3✱ ▲✐❍♦❋4 ❬✺✹❪❀

• P❧❛♥♦✲❢á❝✐❧✿ ♣❛r❛ 0 < λ < 1 ♦s s♣✐♥s t❡♥❞❡♠ ❛ s❡ ❛❧✐♥❤❛r ❝♦♠ ♦ ♣❧❛♥♦ xy

✭✈❡❥❛ ❛ ✜❣✉r❛ ✷✳✻✮✱ ♦ q✉❡ ❛❝❛rr❡t❛ ♥♦ ♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ❞❡ ♣❧❛♥♦✲❢á❝✐❧✳

❯♠ ❡sq✉❡♠❛ ❞❡ s♣✐♥ ❝❧áss✐❝♦ ❡stá ✐❧✉str❛❞♦ ♥❛ ✜❣✉r❛ ✸✳✶✱ ♥♦ ❝❛♣ít✉❧♦ ✸✳

✷✼

✷✳ ❈♦♥s✐❞❡r❛çõ❡s ❣❡r❛✐s s♦❜r❡ ♠❛❣♥❡t✐s♠♦ ❡ ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s

❖ ♠♦❞❡❧♦ ❳❨ ❡ ♦ ❘♦t♦r P❧❛♥❛r

❆❧é♠ ❞♦s ♠♦❞❡❧♦s ❝✐t❛❞♦s ❛❝✐♠❛✱ t❡♠♦s ❞♦✐s ❞❡ ❣r❛♥❞❡ ✐♥t❡r❡ss❡ ♥♦ ♠❛❣♥❡t✐s♠♦✿

♦ ♠♦❞❡❧♦ ❳❨ ❡ ♦ ❘♦t♦r P❧❛♥❛r✽✳ ❊st❡s ♠♦❞❡❧♦s sã♦ ♦❜t✐❞♦s ❢❛③❡♥❞♦✲s❡ λ = 0 ❡♠ ✭✷✳✸✸✮✿

H = −J∑

<i,j>

(Sxi S

xj + Sy

i Syj

)✳ ✭✷✳✸✹✮

❆♣❡s❛r ❞❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❞♦ ♠♦❞❡❧♦ ❳❨ ❡ ❛ ❞♦ ❘♦t♦r P❧❛♥❛r ♣♦ss✉ír❡♠ ❡①♣r❡ssõ❡s

✐❣✉❛✐s✱ ❛ ❞✐❢❡r❡♥ç❛ ❞♦s ♠♦❞❡❧♦s r❡s✐❞❡ ♥♦ ❢❛t♦ ❞❡ ♦ ♣r✐♠❡✐r♦ ♣♦ss✉✐r ❞✐♥â♠✐❝❛ ❡♠ ❞❡tr✐✲

♠❡♥t♦ ❛♦ s❡❣✉♥❞♦✳ ■st♦ ❞❡❝♦rr❡ q✉❡ ♥♦ ♠♦❞❡❧♦ ❳❨ ♦ ✈í♥❝✉❧♦ é S2 = (Sx)2 + (Sy)2 +

(Sz)2 = 1 ❡ ❡①✐st❡♠ ❛s três ❝♦♠♣♦♥❡♥t❡s ❞❡ s♣✐♥✱ ❛♦ ♣❛ss♦ q✉❡ ♥♦ ♠♦❞❡❧♦ ❞♦ ❘♦t♦r

P❧❛♥❛r t❡♠♦s ♦ ✈í♥❝✉❧♦ S2 = (Sx)2 + (Sy)2 = 1✱ ♥ã♦ ❤❛✈❡♥❞♦ ❛ ❝♦♠♣♦♥❡♥t❡ Sz✳ ❆

❞✐♥â♠✐❝❛ ❞❡ ✉♠ ♠♦❞❡❧♦ ❞❡ s♣✐♥s é ❞❡s❝r✐t❛ ♣❡❧❛ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ ❞❡ ❍❡✐s❡♥❜❡r❣✿

i~d~Sj

dt=

[

~Sj, H]

✱ ✭✷✳✸✺✮

♦♥❞❡ i =√−1 ❡

[

~Sj, H]

= ~Sj H −H ~Sj✳ ❆✐♥❞❛ t❡♠♦s q✉❡ ♦s ♦♣❡r❛❞♦r❡s ❞❡ s♣✐♥ ✭Sx✱

Sy ❡ Sz✮ s❛t✐s❢❛③❡♠ à r❡❣r❛ ❞❡ ❝♦♠✉t❛çã♦ ❞❡ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❞❡ s♣✐♥[

Sαi , S

βj

]

=

i~εαβγδjiSiγ✱ ❡♠ q✉❡ εαβγ é ♦ sí♠❜♦❧♦ ❞❡ ▲❡✈✐✲❈✐✈✐t❛✱ q✉❡ ❛ss✉♠❡ ♦ ✈❛❧♦r +1 s❡ α✱

β ❡ γ ❢♦r❡♠ ❝í❝❧✐❝♦s✱ −1 s❡ α✱ β ❡ γ ❢♦r❡♠ ❛♥t✐❝í❝❧✐❝♦s ❡ 0 ❝❛s♦ ❝♦♥trár✐♦ ❬✺✻❪✳ P♦r

❝♦♥s❡❣✉✐♥t❡✱ t❡♠♦s q✉❡ ♥♦ ❘♦t♦r P❧❛♥❛r [Sx, Sy] = 0✱ ❥á q✉❡ Sz = 0✱ ♦ q✉❡ ✐♠♣❧✐❝❛

❡♠ [Sx, H] = [Sy, H] = 0 ❡✱ ♣♦rt❛♥t♦✱ ♥ã♦ t❡♠♦s ❞✐♥â♠✐❝❛ ✭d~Sj/dt = ~0✮✳ ❏á ♣❛r❛ ♦

♠♦❞❡❧♦ ❳❨✱ Sz 6= 0 ❡ ♣♦❞❡♠♦s ✈❡r q✉❡ d~Sj/dt 6= ~0✳ ❆♣❡s❛r ❞❡ t❡r♠♦s ❛ ❝♦♠♣♦♥❡♥t❡

Sz ♥♦ ♠♦❞❡❧♦ ❳❨✱ ♦ ❢❛t♦ ❞❡st❡ t❡r♠♦ ♥ã♦ ❡st❛r ♣r❡s❡♥t❡ ❡♠ ✭✷✳✸✹✮ ♥♦s ♠♦str❛ ❛ ❣r❛♥❞❡

t❡♥❞ê♥❝✐❛ ❞♦s s♣✐♥s ♣❡r♠❛♥❡❝❡r❡♠ ❛❧✐♥❤❛❞♦s ♥♦ ♣❧❛♥♦✳

❙❡ ❡st✐✈❡r♠♦s ❡st✉❞❛♥❞♦ s✐st❡♠❛s ♠❛❣♥ét✐❝♦s ❞❡ ♠✉✐t❛s ♣❛rtí❝✉❧❛s✱ t❛❧ ❡st✉❞♦

t♦r♥❛✲s❡ ✐♥✈✐á✈❡❧✱ s❡♥ã♦ ✐♠♣♦ssí✈❡❧✱ ❛tr❛✈és ❞❛s ❡①♣r❡ssõ❡s ❞❛❞❛s ♥❡st❛ s❡çã♦✳ P❛r❛

t❛♥t♦✱ ❛❞♦t❛♠♦s ✉♠ ♠♦❞❡❧♦ ❝♦♥tí♥✉♦ ♣❛r❛ ❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❞♦ s✐st❡♠❛✳ ❚❛❧ ♠♦❞❡❧♦✱

♣❛r❛ ♦ ❝❛s♦ ✐s♦tró♣✐❝♦✱ é ❞❡s❡♥✈♦❧✈✐❞♦ ♥♦ ❆♣ê♥❞✐❝❡ ❇✳

✽➚s ✈❡③❡s é ❝♦♠✉♠ ❢❛❧❛r ♥♦ ♠♦❞❡❧♦ ❞❡ P❧❛♥♦✲❢á❝✐❧ ❝♦♠♦ ❡♥❣❧♦❜❛♥❞♦ ♦s ♠♦❞❡❧♦s ❳❨ ❡ ❘♦t♦r P❧❛♥❛r✳

P❛r❛ ✐ss♦✱ ❢❛③❡♠♦s ❝♦♠ q✉❡ ❛ r❡str✐çã♦ ♣❛r❛ λ ♥♦ ♠♦❞❡❧♦ ✐♥❝❧✉❛ λ = 0✱ ♦✉ s❡❥❛✱ 0 ≤ λ < 1✳ ❯♠ t❛❧

tr❛t❛♠❡♥t♦ é ✉s❛❞♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♣❡❧♦s ❛✉t♦r❡s ❞❛ r❡❢❡rê♥❝✐❛ ❬✺✺❪✳

✷✽

❈❛♣ít✉❧♦ ✸

❊str✉t✉r❛ ❡ ❞✐♥â♠✐❝❛ ❞❡ ✈órt✐❝❡s ❡♠

♥❛♥♦♠❛❣♥❡t♦s

❆❧❣✉♥s ❞♦s ♠♦❞❡❧♦s ❞❡s❝r✐t♦s ❛♥t❡r✐♦r♠❡♥t❡✱ ❛♣r❡s❡♥t❛♠ ❝♦♠♦ s♦❧✉çõ❡s ❡①❝✐✲

t❛çõ❡s ♥ã♦✲❧✐♥❡❛r❡s ✭❞❡✈✐❞♦ ❛♦ ✈í♥❝✉❧♦ |~S| = 1✮ ♦s ✈órt✐❝❡s ❞❡ s♣✐♥✳ ❊①✐st❡♠ ❞♦✐s t✐♣♦s

❞❡ ✈órt✐❝❡s✿ ♦ ♣r✐♠❡✐r♦ é ♦ ✈órt✐❝❡ ♣❧❛♥❛r ✭✐♥ ♣❧❛♥❡✮✱ ❛ ❢♦r♠❛ ♠❛✐s s✐♠♣❧❡s ❞❡ ✈órt✐❝❡✱

s✉r❣✐♥❞♦ q✉❛♥❞♦ ❛ ❛♥✐s♦tr♦♣✐❛ é s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ♣❛r❛ ❝♦♥✜♥❛r ♦s s♣✐♥s ❛♦ ♣❧❛♥♦

xy❀ ♦ s❡❣✉♥❞♦ é ♦ ❝❤❛♠❛❞♦ ✈órt✐❝❡ ❢♦r❛ ❞♦ ♣❧❛♥♦ ✭♦✉t ♦❢ ♣❧❛♥❡✮✱ ♦♥❞❡ ♦s s♣✐♥s ♣ró①✐♠♦s

❛♦ ❝❡♥tr♦ ❞♦ ✈órt✐❝❡ ✭✈♦rt❡① ❝♦r❡✮ ♣♦ss✉❡♠ ❝♦♠♣♦♥❡♥t❡s ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❛♦ ♣❧❛♥♦ xy✳

Pr✐♠❡✐r♦✱ ❡❧✉❝✐❞❛r❡♠♦s ✈órt✐❝❡s ♣❧❛♥❛r❡s✳

✸✳✶ ❱❡rsã♦ ❝♦♥tí♥✉❛ ❞♦ ♠♦❞❡❧♦ ❳❨ ❡ ✈órt✐❝❡s ♣❧❛♥❛r❡s

❆♣r♦✈❡✐t❛♥❞♦ ♦ ❡♥s❡❥♦✱ ♣♦❞❡♠♦s✱ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ❛♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❢❡✐t♦

❆♣ê♥❞✐❝❡ ❇✱ ♦❜t❡r✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❛♣r♦①✐♠❛çã♦ ❝♦♥tí♥✉❛ ❞♦ ♠♦❞❡❧♦ ❳❨✶✿

HXY =J

2

∫

S

{2∑

α=1

[(∂Sα

∂x

)2

+

(∂Sα

∂y

)2]

+4

a2(Sz)2

}

d2r✱ ✭✸✳✶✮

♦♥❞❡ ❛ ✐♥t❡❣r❛çã♦ é ❢❡✐t❛ ♥❛ s✉♣❡r❢í❝✐❡ S ❞♦ s✐st❡♠❛ ✲ ❜✐❞✐♠❡♥s✐♦♥❛❧ ✲ ❡ α = x, y✳

◆♦ ❡♥t❛♥t♦✱ ✈✐st♦ q✉❡ ❡①✐st❡ ♦ ✈í♥❝✉❧♦ |~S|2 = 1✱ ♣♦❞❡♠♦s ♣❛r❛♠❡tr✐③❛r ♦s s♣✐♥s

♣♦r ♠❡✐♦ ❞❡ ❞♦✐s ❝❛♠♣♦s ❡s❝❛❧❛r❡s m = cos θ ❡ φ✱ ❞❡ ♠❛♥❡✐r❛ q✉❡ ~S = (√1−m2 cosφ✱

✶❖ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ♣❛ss♦ ❛ ♣❛ss♦ ❞❡st❛ ❡q✉❛çã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♥❛ r❡❢❡rê♥❝✐❛

❬✺✸❪✳

✷✾

✸✳ ❊str✉t✉r❛ ❡ ❞✐♥â♠✐❝❛ ❞❡ ✈órt✐❝❡s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s

❋✐❣✉r❛ ✸✳✶✿ ❊sq✉❡♠❛ ❞❡ s♣✐♥ ❝❧áss✐❝♦ ❝♦♠ |~S| = 1 ❡ ♦r✐❡♥t❛çã♦ ❞❛❞❛ ♣❡❧♦s â♥❣✉❧♦s ❡s❢ér✐❝♦s

θ ❡ φ✳

√1−m2 senφ✱ m)✷✱ ❝♦♥❢♦r♠❡ ✐❧✉str❛❞♦ ♥❛ ✜❣✉r❛ ✸✳✶✳ ❉❡st❛ ❢♦r♠❛✱ ❛ ❤❛♠✐❧t♦♥✐❛♥❛

✭✸✳✶✮ ✜❝❛✿

HXY =J

2

∫

S

[

m2(~∇m)2

1−m2+ (1−m2)(~∇φ)2 + 4

a2m2

]

d2r✱ ✭✸✳✷✮

❡♠ q✉❡ ~∇ = ∂∂xx+ ∂

∂yy✳

❋❛③❡♥❞♦ m = 0 ✭θ = π/2✮ ♥❛ ❡q✉❛çã♦ ✭✸✳✷✮✱ ❣❛♥❤❛♠♦s ❛ ✈❡rsã♦ ❝♦♥tí♥✉❛ ❞♦

♠♦❞❡❧♦ ❞♦ ❘♦t♦r P❧❛♥❛r✿

HRP =J

2

∫

S

(

~∇φ)2

d2r✱ ✭✸✳✸✮

♥♦ q✉❛❧ ♦s s♣✐♥s só ♣♦ss✉❡♠ ❞✉❛s ❝♦♠♣♦♥❡♥t❡s✱ ❛ s❛❜❡r✱ Sx ❡ Sy✳ ❆ ❡q✉❛çã♦ ❞❡

♠♦✈✐♠❡♥t♦ ♣❛r❛ ♦ ❝❛s♦ ❡stát✐❝♦ ✭dφ/dt = 0✮✱ ✈✐❛ ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥ ♣❛r❛ ♦ ❝❛♠♣♦

φ✱ ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❡ HRP ✱ é ❞❛❞❛ ♣♦r✿

δHRP

δφ(~r )= 0 ⇒ ~∇2φ(~r) = 0✱ ✭✸✳✹✮

s❡♥❞♦ ~r = (x, y)✱ q✉❡ é ❛ ❡q✉❛çã♦ ❞❡ ▲❛♣❧❛❝❡✳

❆ ❡q✉❛çã♦ ❞❡ ▲❛♣❧❛❝❡ ♣♦ss✉✐ ❞♦✐s t✐♣♦s ❞❡ s♦❧✉çã♦✿ ♦ ♣r✐♠❡✐r♦ ❝♦rr❡s♣♦♥❞❡

❛♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❡❧✱ ♦✉ s❡❥❛✱ φ(~r ) = constante❀ ♦ s❡❣✉♥❞♦ t✐♣♦ ❝♦♥s✐st❡ ♥✉♠❛

✷❖ ✈❡t♦r ~S✱ q✉❛♥❞♦ ♣❛r❛♠❡tr✐③❛❞♦ ❞❡st❛ ❢♦r♠❛✱ é t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✈❡t♦r ♠❛❣♥❡t✐③❛çã♦

✉♥✐tár✐♦✱ ♥♦r♠❛❧✐③❛❞♦ ♣♦r Ms✱ ~m = ~S = ~M/Ms. ❉❛q✉✐ ❡♠ ❞✐❛♥t❡ ✉s❛r❡♠♦s ♣❛r❛ r❡♣r❡s❡♥t❛r ❛

♠❛❣♥❡t✐③❛çã♦✳

✸✵

✸✳ ❊str✉t✉r❛ ❡ ❞✐♥â♠✐❝❛ ❞❡ ✈órt✐❝❡s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s

❝♦♥✜❣✉r❛çã♦ ❞♦ t✐♣♦ ✈órt✐❝❡✱ ❝✉❥❛ s♦❧✉çã♦ é ❞❛❞❛ ♣♦r✿

φ(x, y) = Q arctan

(y − yvx− xv

)

+ C✱ ✭✸✳✺✮

♦♥❞❡ C é ✉♠❛ ❝♦♥st❛♥t❡✱ ♦ ♣❛r (xv, yv) ✐♥❞✐❝❛ ♦ ❝❡♥tr♦ ❞♦ s✐st❡♠❛✱ ~rv✱ ❡ Q = 0✱ ±1✱

±2✱ ±3✱ . . . é ❛ ❝❛r❣❛ t♦♣♦❧ó❣✐❝❛ ✭♦✉ ✈♦rt✐❝✐❞❛❞❡✮ ❞♦ ✈órt✐❝❡✳

❖✉tr❛ ❢♦r♠❛ ❞❡ ❞❡✜♥✐r ✉♠ ✈órt✐❝❡ ♣❧❛♥❛r é ♣♦r ♠❡✐♦ ❞❛s s❡❣✉✐t❡s ❝♦♥❞✐çõ❡s ❞❡

❝♦♥t♦r♥♦ s♦❜r❡ ❛ ✐♥t❡❣r❛❧ ❞❡ ❝✐r❝✉❧❛çã♦ ❞❡ φ(~r ) ❬✹✾❪✿

i✮ P❛r❛ t♦❞❛s ❛s ❝✉r✈❛s ❢❡❝❤❛❞❛s✱ C✱ q✉❡ ❡♥❣❧♦❜❛♠ ❛ ♣♦s✐çã♦ ~rv = (xv, yv) ❞♦

❝❡♥tr♦ ❞♦ ✈órt✐❝❡✱ t❡♠♦s✿∮

C

~∇φ(~r ) · d~l = 2πQ✱ ✭✸✳✻✮

❞❡✜♥✐♥❞♦✱ ❞❡st❛ ❢♦r♠❛✱ ❛ ❝❛r❣❛ t♦♣♦❧ó❣✐❝❛ Q✳ ❖✉ s❡❥❛✱ ✉♠❛ ❝♦♥✜❣✉r❛çã♦ ❞♦ t✐♣♦ ✈órt✐❝❡

é t❛❧ q✉❡ ❛ s♦♠❛ ❞❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ â♥❣✉❧♦s ❢♦r♠❛❞♦s ♣❡❧♦s s♣✐♥s ✈✐③✐♥❤♦s ❛ ✉♠❛ ♠❡s♠❛

❞✐stâ♥❝✐❛ ❞♦ ❝❡♥tr♦ ❞♦ ✈órt✐❝❡ ♥✉♠❛ r❡❞❡ ❞❡ s♣✐♥s s❡❥❛ ✉♠ ♠ú❧t✐♣❧♦ ✐♥t❡✐r♦ ❞❡ 2π❀

ii✮ P❛r❛ t♦❞♦s ♦s ❝❛♠✐♥❤♦s✱ C✱ q✉❡ ♥ã♦ ❡♥❣❧♦❜❛♠ ~rv✱ t❡♠♦s✿∮

C

~∇φ(~r ) · d~l = 0✳ ✭✸✳✼✮

❈♦♠ Q = 1 ❡♠ ✭✸✳✺✮✱ t❡♠♦s ✉♠ ✈órt✐❝❡✱ ❝♦♠ ♦s ♠♦♠❡♥t♦s ♠❛❣♥ét✐❝♦s ❢♦r♠❛♥❞♦

✧❧♦♦♣s✧ ❢❡❝❤❛❞♦s ❛♦ ❧♦♥❣♦ ❞❡ ~rv✱ ❡ ♦ s❡♥t✐❞♦ ❞❡ r♦t❛çã♦✱ ♦✉ q✉✐r❛❧✐❞❛❞❡ ✭❝❤✐r❛❧✐t②✮ q✱

♣♦❞❡ s❡r ❤♦rár✐❛ ♦✉ ❛♥t✐✲❤♦rár✐❛✱ ❞❡♣❡♥❞❡♥❞♦ ❞❛ ❝♦♥st❛♥t❡ C✱ ❡ ✉♠❛ s♦❧✉çã♦ ❝♦♠

Q = −1 r❡♣r❡s❡♥t❛ ✉♠ ❛♥t✐✈órt✐❝❡✳ ❈♦♠♦ ❛ ❡q✉❛çã♦ ✭✸✳✹✮ é ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧

❧✐♥❡❛r✱ q✉❛❧q✉❡r ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ φ1 = + arctan[(y − yv)/(x − xv)] + C1 ❡ φ2 =

− arctan[(y−yv)/(x−xv)]+C2 é t❛♠❜é♠ s♦❧✉çã♦ ❞❡ ✭✸✳✹✮✱ ♦ q✉❡ r❡s✉❧t❛ ♥❛ ❢♦r♠❛çã♦ ❞❡

✉♠ ♣❛r ✈órt✐❝❡✲❛♥t✐✈órt✐❝❡✳ ❆s três ❝♦♥✜❣✉r❛çõ❡s ❞❡s❝r✐t❛s ❛❝✐♠❛ ❡stã♦ r❡♣r❡s❡♥t❛❞❛s

♥❛ ✜❣✉r❛ ✸✳✷✳

✸✳✷ ❱órt✐❝❡s ❝♦♠ ❝♦♠♣♦♥❡♥t❡ ❢♦r❛ ❞♦ ♣❧❛♥♦ ✭♦✉t ♦❢

♣❧❛♥❡✮

❊♠ ❝♦♥tr❛st❡ ❝♦♠ ♦s ✈órt✐❝❡s ♣❧❛♥❛r❡s✱ q✉❡ ♣♦ss✉❡♠ ❛♥✐s♦tr♦♣✐❛ s✉✜❝✐❡♥t❡♠❡♥t❡

❣r❛♥❞❡ ♣❛r❛ ❝♦♥✜♥❛r ♦s s♣✐♥s ♥♦ ♣❧❛♥♦✱ ❡♠ ❛❧❣✉♥s ♠❛t❡r✐❛s ♠❛❣♥ét✐❝♦s ❞♦❝❡s✱ t❛✐s ❝♦♠♦

✸✶

✸✳ ❊str✉t✉r❛ ❡ ❞✐♥â♠✐❝❛ ❞❡ ✈órt✐❝❡s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s

❋✐❣✉r❛ ✸✳✷✿ ■❧✉str❛çã♦ ❡sq✉❡♠át✐❝❛ ❞❡ ✉♠ ✈órt✐❝❡ ❝♦♠ Q = +1 ✭à ❡sq✉❡r❞❛✮✱ ✉♠ ❛♥t✐✈órt✐❝❡

❝♦♠ Q = −1 ✭❛♦ ❝❡♥tr♦✮ ❡ ✉♠ ♣❛r ✈órt✐❝❡✲❛♥t✐✈órt✐❝❡ ❝♦♠ Q = 0 ✭à ❞✐r❡✐t❛✮✳ ❖s sí♠❜♦❧♦s ⊕❡ ⊖ r❡♣r❡s❡♥t❛♠ ♦ ❝❡♥tr♦ ❞❡ ❝❛❞❛ ❡str✉t✉r❛✳

P❡r♠❛❧❧♦② ❡ ❛♠♦str❛s ❞❡ ◆✐ ❡ ❈♦✱ ❛ ❛♥✐s♦tr♦♣✐❛ ♠❛❣♥❡t♦❝r✐st❛❧✐♥❛✱ ❝♦♠♦ ✈✐♠♦s ♥❛ s❡çã♦

✷✳✸✳✶✱ é ❞❡s♣r❡③í✈❡❧ ❡ ❛ss✐♠✱ ♥❡st❡s ♠❛t❡r✐❛✐s✱ ❛s ♣♦ssí✈❡✐s ❝♦♥✜❣✉r❛çõ❡s s✉r❣❡♠ ❞❡✈✐❞♦

à ❝♦♠♣❡t✐çã♦ ❡♥tr❡ ❛s ❡♥❡r❣✐❛s ❞❡ tr♦❝❛ ❡ ♠❛❣♥❡t♦stát✐❝❛ ❬✺✵❪✳ ❊♥❢❛t✐③❛♥❞♦✱ ❡ss❡s

♠❛t❡r✐❛✐s s❡rã♦ ♦ ♦❜❥❡t♦ ❞❡ ❡st✉❞♦ ♥❛s ♣ró①✐♠❛s ❞✐s❝✉ssõ❡s✳

◆♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❢❡✐t♦ ♥♦ ❆♣ê♥❝✐❝❡ ❇✱ ❡♥❝♦♥tr❛♠♦s q✉❡ ♦ ❧✐♠✐t❡ ❝♦♥tí♥✉♦ ❞❛

❤❛♠✐❧t♦♥✐❛♥❛ ❞❡ ❍❡✐s❡♥❜❡r❣ ✐s♦tró♣✐❝❛ ✸❉ ✭❡q✉❛çã♦ ❇✳✶✹✮ é✿

Etr =A

2

∫

V

3∑

α=1

(~∇mα)2 d3r✱ ✭✸✳✽✮

❡♠ q✉❡ A é ❛ ❝♦♥st❛♥t❡ ❞❡ tr♦❝❛ ❡♠ ❏✴♠ ❡ A ≈ 10−11 ❏✴♠ ♣❛r❛ P❡r♠❛❧❧♦②✳ ❆ ❡♥❡r❣✐❛

❞❡ tr♦❝❛ é ♠✐♥✐♠✐③❛❞❛ q✉❛♥❞♦ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❡stã♦ ❛❧✐♥❤❛❞♦s ♣❛r❛ ❢♦r♠❛r ✉♠

❡st❛❞♦ ❞❡ ❞♦♠í♥✐♦ ú♥✐❝♦ ✭s✐♥❣❧❡ ❞♦♠❛✐♥ st❛t❡✮✱ q✉❡ é ✐❧✉str❛❞♦ ♥❛ ✜❣✉r❛ ✸✳✸✱ ❝♦♠ ❛

♠❛❣♥❡t✐③❛çã♦ ❞❡ s❛t✉r❛çã♦ Ms✳ ◗✉❛❧q✉❡r ♦✉tr❛ ❝♦♥✜❣✉r❛çã♦ q✉❡ ♥ã♦ s❡❥❛ ♦ ❛❧✐♥❤❛✲

♠❡♥t♦ ♣❛r❛❧❡❧♦✱ ❛✉♠❡♥t❛ ❛ ❡♥❡r❣✐❛ ❞❡ tr♦❝❛ ❝♦♠ ✉♠❛ ❞❡♥s✐❞❛❞❡ q✉❡ é ♣r♦♣♦r❝✐♦♥❛❧

t❛♥t♦ à ❝♦♥st❛♥t❡ ❞❡ tr♦❝❛✱ A✱ q✉❛♥t♦ ❝♦♠ ♦ q✉❛❞r❛❞♦ ❞♦ ❣r❛❞✐❡♥t❡ ❞❛ ♠❛❣♥❡t✐③❛çã♦✳

❊♠ s❡ tr❛t❛♥❞♦ ❞❡st❡ t✐♣♦ ❞❡ ✐♥t❡r❛çã♦✱ ✉♠ ✐♠♣♦rt❛♥t❡ ❝♦♠♣r✐♠❡♥t♦ é lex✱ ❝❤❛♠❛❞♦ ❞❡

❝♦♠♣r✐♠❡♥t♦ ❞❡ tr♦❝❛✱ ❡s❝❛❧❛ ❝❛r❛❝t❡ríst✐❝❛ ❛ss♦❝✐❛❞❛ ❛♦ ❛❧❝❛♥❝❡ ❞❛ ✐♥t❡r❛çã♦ ❞❡ tr♦❝❛✱

❞❛❞❛ ♣♦r ❬✺✼❪ lex =√

2A/µ0M2s ✭♦✉ lex =

√

A/M2s ❡♠ ✉♥✐❞❛❞❡s ●❛✉ss✐❛♥❛s✱ µ0 = 1✮✸

❡ lex ≈ 5, 7 ♥♠ ♣❛r❛ ♦ P❡r♠❛❧❧♦② ❬✶✸❪✱ ✈✐st♦ q✉❡ Ms ≈ 8× 105 ❆✴♠ ♣❛r❛ ❡st❡ ♠❛t❡r✐❛❧✳

✸➱ ✐♠♣♦rt❛♥t❡ ❢r✐s❛r q✉❡ ♥ã♦ ❤á ✉♠❛ ❞❡✜♥✐çã♦ ♣❛❞rã♦ ♣❛r❛ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ tr♦❝❛ lex✳ ◆❛

❧✐t❡r❛t✉r❛ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r lex =√

A/4πM2s ✱ lex =

√

4πA/µ0M2s ✳

✸✷

✸✳ ❊str✉t✉r❛ ❡ ❞✐♥â♠✐❝❛ ❞❡ ✈órt✐❝❡s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s

❋✐❣✉r❛ ✸✳✸✿ ■❧✉str❛çã♦ ❞❡ ✉♠ ❡st❛❞♦ ❞❡ ❞♦♠í♥✐♦ ú♥✐❝♦✱ ♥♦ q✉❛❧ t♦❞♦s ♦s ♠♦♠❡♥t♦s ♠❛❣♥ét✐❝♦s

sã♦ ♣❛r❛❧❡❧♦s✳

P❛r❛ ♥❛♥♦♠❛❣♥❡t♦s ♠✉✐t♦ ♣❡q✉❡♥♦s✱ ❝♦♠ ❞✐♠❡♥sõ❡s ❞❡ ❞❡③❡♥❛s ❞❡ ♥❛♥ô♠❡tr♦s✱ ❛

❡♥❡r❣✐❛ ❞❡ tr♦❝❛ ❞♦♠✐♥❛ ❡ ♦ ❡st❛❞♦ ❞❡ ❞♦♠í♥✐♦ ú♥✐❝♦ é ❡♥❡r❣❡t✐❝❛♠❡♥t❡ ❢❛✈♦rá✈❡❧✳

❊♥tr❡t❛♥t♦✱ ❡♠ ♥❛♥♦♠❛❣♥❡t♦s ♠❛✐♦r❡s✱ t❛❧ ❡st❛❞♦ ❞á ❧✉❣❛r ❛ ♦✉tr❛s ❝♦♥✜❣✉r❛çõ❡s✱

❞❡♣❡♥❞❡♥❞♦ ❞❛ ❢♦r♠❛ ❞❛ ❛♠♦str❛✱ ❞❡✈✐❞♦ à ✐♠♣♦rtâ♥❝✐❛ ❞♦ t❡r♠♦ ♠❛❣♥❡t♦stát✐❝♦✳

P♦r ❡①❡♠♣❧♦✱ ♣❛r❛ ♥❛♥♦❞✐s❝♦s ❝✐r❝✉❧❛r❡s✱ ♦ ❞♦♠í♥✐♦ ú♥✐❝♦ é s✉❝❡❞✐❞♦ ♣❡❧♦ ❡st❛❞♦ ❞❡

✈órt✐❝❡✹✳ ❊①♣❡r✐♠❡♥t❛❧♠❡♥t❡ ❢♦✐ ♦❜s❡r✈❛❞♦ q✉❡ ✉♠ ♥❛♥♦❞✐s❝♦ ❝✐❧í♥❞r✐❝♦ ❞❡ P❡r♠❛❧❧♦②

❝♦♠ L = 15 ♥♠ ❝♦♠♣r❡❡♥❞❡ ✉♠ ❡st❛❞♦ ❞❡ ✈órt✐❝❡ ♣❛r❛ R ≥ 50 ♥♠✱ ❡♥q✉❛♥t♦ q✉❡ ♣❛r❛

r❛✐♦s ♠❡♥♦r❡s✱ ✉♠ ❡st❛❞♦ ✐♥t❡r♠❡❞✐ár✐♦ ❡✱ ♣r♦❣r❡ss✐✈❛♠❡♥t❡✱ ♦ ❡st❛❞♦ ❞❡ ❞♦♠í♥✐♦ ú♥✐❝♦

é ♦❜s❡r✈❛❞♦ ❬✶✷❪✱ s❡♥❞♦ ✐♥t❡r❡ss❛♥t❡ ♥♦t❛r q✉❡ ❛ ❝♦♥✜❣✉r❛çã♦ ♠❛❣♥ét✐❝❛ ♥❛ s✉♣❡r❢í❝✐❡

t❛♠❜é♠ é ❛ ❛♣r❡s❡♥t❛❞❛ ❞❡♥tr♦ ❞♦ ❞✐s❝♦✳ ❆ss✐♠✱ ❝♦♥✜❣✉r❛çõ❡s ❞♦ t✐♣♦ ✈órt✐❝❡ sã♦

♦❜t✐❞❛s ❝♦♠ ♣❡q✉❡♥❛ r❛③ã♦ ❡s♣❡ss✉r❛✴r❛✐♦✱ L/R << 1✳ ■♥✈❡rs❛♠❡♥t❡✱ s❡ ❛❣♦r❛ ✜①❛r♠♦s

R ❡ ❛✉♠❡♥t❛r♠♦s L✱ t❛❧ ❡st❛❞♦ ❞❡ ✈órt✐❝❡ ♣r♦❣r❡ss✐✈❡♠❡♥t❡ ❛❧✐♥❤❛ s❡✉s ♠♦♠❡♥t♦s ❞❡

❞✐♣♦❧♦ ♠❛❣♥ét✐❝♦✱ ❞❛♥❞♦ s✉r❣✐♠❡♥t♦ ♥♦✈❛♠❡♥t❡ ❛♦ ❡st❛❞♦ ❞❡ ❞♦♠í♥✐♦ ú♥✐❝♦ ❛❝✐♠❛ ❞❡

✉♠ ❝❡rt♦ ✈❛❧♦r ❞❡ L✱ só q✉❡ ❛❣♦r❛ ♦s ♠♦♠❡♥t♦s ❡stã♦ ❛❧✐♥❤❛❞♦s ♥♦ ❡✐①♦ ❞❡ ❝r❡s❝✐♠❡♥t♦

❞❡ L✳

❚❡♥t❛r❡♠♦s ❛❣♦r❛✱ q✉❛❧✐t❛t✐✈❛♠❡♥t❡✱ ❡①♣❧✐❝❛r ♣♦rq✉ê ✐ss♦ ♦❝♦rr❡✳ ❈♦♥s✐❞❡r❡✱

♣❛r❛ t❛♥t♦✱ ✉♠ ♥❛♥♦❞✐s❝♦ ❝✐r❝✉❧❛r ❝♦♠ r❛③ã♦ L/R ≪ 1✳ ❆ ❡♥❡r❣✐❛ ♠❛❣♥❡t♦stát✐❝❛✱

♦❜t✐❞❛ ♥❛ s❡çã♦ ✷✳✸✳✷✱ é Emag = − (µ0/2)∫

V~M · ~Hd dV ❡✱ ❝♦♠♦ ♣♦❞❡♠♦s ✈❡r✱ ❛✉♠❡♥t❛

✹❈♦♠♦ ♦✉tr♦s ❡①❡♠♣❧♦s✱ ❞✐s❝♦s q✉❛❞r❛❞♦s t❛♠❜é♠ ♣♦❞❡♠ ❛♣r❡s❡♥t❛r ♠❛❣♥❡t✐③❛çã♦ ❞♦ t✐♣♦ ✈órt✐❝❡✱

❢♦r♠❛♥❞♦ q✉❛tr♦ ❞♦♠í♥✐♦s tr✐❛♥❣✉❧❛r❡s q✉❡ s❡ ✐♥t❡r❝❡♣t❛♠ ♥♦ ❝❡♥tr♦ ❞♦ ✈órt✐❝❡✳ ❊st❛ ❡str✉t✉r❛ é

❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❡str✉t✉r❛ ❞❡ ▲❛♥❞❛✉✳ ❯♠ ♣❛r ❞❡ ✈órt✐❝❡s t❛♠❜é♠ ♣♦❞❡ s❡r ❢♦r♠❛❞♦ ♥✉♠ ❞✐s❝♦

❡❧í♣t✐❝♦ ❬✺✵❪✳

✸✸

✸✳ ❊str✉t✉r❛ ❡ ❞✐♥â♠✐❝❛ ❞❡ ✈órt✐❝❡s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s

❝♦♠ ❛s ❞✐♠❡♥sõ❡s ❞♦ ❞✐s❝♦✳ ❆ ❡♥❡r❣✐❛ ❞❡ tr♦❝❛✱ ❞✐♠✐♥✉✐ ❝♦♥❢♦r♠❡ ♥♦s ❛❢❛st❛♠♦s ❞♦

❝❡♥tr♦ ❞♦ ❞✐s❝♦✱ ♦ q✉❡ ♣♦❞❡ s❡r ❝♦♠♣r❡❡♥❞✐❞♦ ✈❡♥❞♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ✜❣✉r❛ ✸✳✷✳ ▲á✱ é

♦❜s❡r✈❛❞♦ q✉❡ ♦s ♠♦♠❡♥t♦s ♠❛❣♥ét✐❝♦s ✈❛r✐❛♠ ❝❛❞❛ ✈❡③ ♠❡♥♦s ❝♦♥❢♦r♠❡ ♥♦s ❛❢❛st❛✲

♠♦s ❞♦ ❝❡♥tr♦ rv ❞♦ ✈órt✐❝❡✳ ❊✱ ❝♦♠♦ ✈✐♠♦s✱ ❛ ❡♥❡r❣✐❛ ❞❡ tr♦❝❛ ♥♦ ❧✐♠✐t❡ ❝♦♥tí♥✉♦ é

♣r♦♣♦r❝✐♦♥❛❧ ❛ (~∇mα)2✳ P♦rt❛♥t♦✱ ❛ ❡♥❡r❣✐❛ ❞❡ tr♦❝❛ t❡♠ ✉♠❛ ❞❡♣❡♥❞ê♥❝✐❛ ♠❛✐s ❢r❛❝❛

❝♦♠ R ❞♦ q✉❡ ❛ ♠❛❣♥❡t♦stát✐❝❛ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ✉♠ ❞✐s❝♦ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡

❛❞q✉✐r❡ ♦ ❡st❛❞♦ ❞❡ ✈órt✐❝❡✳

❆ss✐♠ ❝♦♠♦ ♦s ✈órt✐❝❡s ♣❧❛♥❛r❡s✱ ♦s ✈órt✐❝❡s ❝♦♠ ♠❛❣♥❡t✐③❛çã♦ ♦✉t ♦❢ ♣❧❛♥❡

t❛♠❜é♠ ♣♦ss✉❡♠ ♠♦♠❡♥t♦s ♠❛❣♥ét✐❝♦s q✉❡ ❡stã♦ ❞✐s♣♦st♦s ❡♠ ✧❧♦♦♣s✧ ❢❡❝❤❛❞♦s ❡♠

t♦r♥♦ ❞❡ ~rv✱ ♣♦✐s ❝♦♠♦ ✈❡r❡♠♦s ❛ s❡❣✉✐r✱ ❡ss❛ ❞✐s♣♦s✐çã♦ ❡♠ ✧❧♦♦♣s✧ ♠✐♥✐♠✐③❛ ❛ ❡♥❡r❣✐❛

♠❛❣♥❡t♦stát✐❝❛ ❡✱ ❛ss✐♠✱ ❛ ❡q✉❛çã♦ ✸✳✺ t❛♠❜é♠ é ✈á❧✐❞❛ ❛q✉✐✳ ❆ ❞✐❢❡r❡♥ç❛ r❡s✐❞❡ ♥✉♠❛

♣❡q✉❡♥❛ r❡❣✐ã♦ ❝❡♥tr❛❧ ❞♦ ♥❛♥♦❞✐s❝♦✱ ❞❡♥♦♠✐♥❛❞❛ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ ✭✈♦rt❡① ❝♦r❡✮✱ ♦♥❞❡

❛❣♦r❛ ♥ã♦ ❤á ♠❛✐s ❛ ❞✐s♣♦s✐çã♦ ♣❡r❢❡✐t❛ ♥♦ ♣❧❛♥♦ ❞❛ ❛♠♦str❛ ✭♦ q✉❡ ❡r❛ ❣❛r❛♥t✐❞♦ ♣❡❧❛

❢♦rt❡ ❛♥✐s♦tr♦♣✐❛ ♠❛❣♥❡t♦❝r✐st❛❧✐♥❛ ♥♦ ❝❛s♦ ♣❧❛♥❛r ❬✺✵❪✮✱ ♦ q✉❡ é ♣❡r❝❡❜✐❞♦ ♣❡❧♦ ❢❛t♦ ❞❡

θ 6= π/2 ♥♦ ❝♦r❡✱ ❝♦♠♦ s❡rá ✈✐st♦✳ ■st♦ ♦❝♦rr❡✱ ♠❛✐s ✉♠❛ ✈❡③✱ ❞❡✈✐❞♦ ❛ ❛❧t❛ ❞❡♥s✐❞❛❞❡

❞❡ ❡♥❡r❣✐❛ ❞❡ tr♦❝❛ ♣ró①✐♠♦ à ♦r✐❣❡♠ ✭s✐♥❣✉❧❛r✐❞❛❞❡✮✳ ➱✱ ♣♦rt❛♥t♦✱ ❡♥❡r❣❡t✐❝❛♠❡♥t❡ ❢❛✲

✈♦rá✈❡❧ ❛♦s ♠♦♠❡♥t♦s ♠❛❣♥ét✐❝♦s ♥❛ ♣❡q✉❡♥❛ r❡❣✐ã♦ ❝❡♥tr❛❧ r❡✈♦❧✈❡r❡♠✲s❡ ❡ ❛❧✐♥❤❛r❡♠✲

s❡ ♣❡r♣❡♥❞✐❝✉❧❛r♠❡♥t❡ ❛♦ ♣❧❛♥♦ ❞♦ ❞✐s❝♦✱ ❞✐♠✐♥✉✐♥❞♦ ❛ ❡♥❡r❣✐❛ ❞❡ tr♦❝❛ ❡ ❢♦r♠❛♥❞♦ ✉♠

♣ó❧♦ ♥❡st❛ r❡❣✐ã♦✱ ❝♦♠♦ ♣♦❞❡♠♦s ✈❡r ♥❛ ✜❣✉r❛ ✸✳✹✳ ❆ ♣♦❧❛r✐❞❛❞❡ ❞♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡

♣♦❞❡ s❡r ✉♣ ✭p = +1✮ ♦✉ ❞♦✇♥ ✭p = −1✮✳ ❖ ✈órt✐❝❡ ♦✉t ♦❢ ♣❧❛♥❡ é✱ ❞❡ss❛ ❢♦r♠❛✱ ✉♠❛

❡str✉t✉r❛ ❞❡ s♣✐♥ tr✐❞✐♠❡♥s✐♦♥❛❧✱ q✉❡ ❝♦♠❜✐♥❛ q✉❛tr♦ ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❝♦♥✜❣✉r❛çã♦✱

❥✉♥t❛♥❞♦✲s❡ q✉✐r❛❧✐❞❛❞❡ ❡ ♣♦❧❛r✐❞❛❞❡✱ ✈✐st♦ q✉❡ ❡st❡s sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s ❬✶✽❪✳

❯t✐❧✐③❛♥❞♦✲s❡ ❛ ♣❛r❛♠❡tr✐③❛çã♦ ♣❛r❛ ~m✱ ❝♦♥❢♦r♠❡ ✈✐st♦ ♥❛ s❡çã♦ ✸✳✶✱ ♠♦str❛♠♦s

♥♦ ❆♣ê♥❞✐❝❡ ❈ q✉❡✿

3∑

α=1

(

~∇mα

)2

= sen2θ(

~∇φ)2

+(

~∇θ)2

✳ ✭✸✳✾✮

❆ ✈❛r✐á✈❡❧ θ ❡♠ ✸✳✾ ❞❡s❝r❡✈❡ ♦ ♣❡r✜❧ ❞♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ ✭♠❛❣♥❡t✐③❛çã♦ ❢♦r❛

❞♦ ♣❧❛♥♦✮ ♠❛s ♥♦ ❡♥t❛♥t♦✱ ❛♣❡♥❛s ❝♦♥❤❡❝❡♠♦s s❡✉ ✈❛❧♦r ❛ss✐♥tót✐❝♦✿ cos θ → ±1 ♥♦

❝❡♥tr♦ ❞♦ ✈órt✐❝❡ ✭q✉❡ ❞❛q✉✐ ❡♠ ❞✐❛♥t❡ t♦♠❛r❡♠♦s ❡♠ ~rv = (0, 0)✮✱ ❡♠ q✉❡ ♦ ✈❛❧♦r

+1 r❡♣r❡s❡♥t❛ ✉♠ ✈órt✐❝❡ ❝♦♠ ♣♦❧❛r✐❞❛❞❡ ✉♣ ❡ −1 ❛ ♣♦❧❛r✐❞❛❞❡ ❞♦✇♥✳ ❊st❡ ✈❛❧♦r ✈❛✐

✸✹

✸✳ ❊str✉t✉r❛ ❡ ❞✐♥â♠✐❝❛ ❞❡ ✈órt✐❝❡s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s

❋✐❣✉r❛ ✸✳✹✿ ❊sq✉❡♠❛ ❞❡ ✈órt✐❝❡ ♦✉t ♦❢ ♣❧❛♥❡✱ ♠♦str❛♥❞♦ ❛♦ ❝❡♥tr♦ ♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡✱

♥❡st❡ ❝❛s♦ ❝♦♠ ♣♦❧❛r✐❞❛❞❡ p = +1✳ ❉✐st❛♥t❡ ❞♦ ❝❛r♦ç♦✱ ❛ ♠❛❣♥❡t✐③❛çã♦ ♥❛ s✉♣❡r❢í❝✐❡ ❞❛

❛♠♦str❛ ❝♦♥t✐♥✉❛♠❡♥t❡ ✬❝✐r❝✉❧❛✬ ❡♠ t♦r♥♦ ❞♦ ❝❡♥tr♦ ❞❛ ❡str✉t✉r❛✳ ❊♠ ❞✐r❡çã♦ ❛♦ ❝❡♥tr♦✱ ❛

♠❛❣♥❡t✐③❛çã♦ ❝♦♠❡ç❛ ❛ r❡✈♦❧✈❡r✲s❡ ♣❡r♣❡♥❞✐❝✉❧❛r♠❡♥t❡ ❛♦ ♣❧❛♥♦ ❞❛ ❛♠♦str❛✳ ❋✐❣✉r❛ r❡t✐r❛❞❛

❞❛ r❡❢❡rê♥❝✐❛ ❬✺✽❪✳

❞✐♠✐♥✉✐♥❞♦ ♣r♦❣r❡ss✐✈❛♠❡♥t❡ ✭cos θ → 0✮ ❝♦♥❢♦r♠❡ ♥♦s ❛❢❛st❛♠♦s ❞♦ ❝❡♥tr♦✱ ❡ ❞❡s❛✲

♣❛r❡❝❡ ✭cos θ = 0✮ q✉❛♥❞♦ |~r | > l0✱ ♦♥❞❡ ~r = (x, y) r❡♣r❡s❡♥t❛ ✉♠❛ ❞❛❞❛ ♣♦s✐çã♦

♥♦ ♣❧❛♥♦ ❞♦ ♥❛♥♦❞✐s❝♦ ❡ ❬✺✾❪ l0 ≈ 0, 68lex(L/lex)1/3 ≈ 10 ♥♠ ♣❛r❛ ✉♠ ♥❛♥♦❞✐s❝♦ ❞❡

P❡r♠❛❧❧♦② ❝♦♠ L = 100 ♥♠✱ ✈❛❧♦r ❡st❡ q✉❡ s❡rá ✉s❛❞♦ ❡♠ ♥♦ss♦ tr❛❜❛❧❤♦✳ ❊ss❡ r❡s✉❧✲

t❛❞♦✱ cos θ = 0 ✭θ = π/2✮✱ ❡r❛ ❡s♣❡r❛❞♦✱ ♣♦✐s ♥❡st❛ r❡❣✐ã♦ t❡♠♦s ❛♣❡♥❛s ♠❛❣♥❡t✐③❛çã♦

♥♦ ♣❧❛♥♦ xy✳ ❉❡♥tr❡ ❛s ❞✐✈❡rs❛s ❢✉♥çõ❡s t❡♥t❛t✐✈❛s ✭tr✐❛❧ ❢✉♥❝t✐♦♥s✮ q✉❡ ❛♣r❡s❡♥t❛♠

t❛❧ ❝♦♠♣♦rt❛♠❡♥t♦ ❛ss✐♥tót✐❝♦✱ t♦♠❛r❡♠♦s ❛q✉❡❧❡ ❡♠ q✉❡✿

θ =

π/2 s❡ r > l0

p cos−1(

1− r2

l20

)n

s❡ r ≤ l0✱✭✸✳✶✵✮

♦♥❞❡ p = ±1 é ❛ ♣♦❧❛r✐❞❛❞❡ ❞♦ ✈órt✐❝❡✳ ❖ ♣❛râ♠❡tr♦ n é ❡s❝♦❧❤✐❞♦ ❝♦♠ ❛ ✜♥❛❧✐❞❛❞❡

❞❡ ❛❥✉st❛r ♦ ♣❡r✜❧ ❞♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡✳ ❖ ♠♦❞❡❧♦ ❝♦♠ n = 4 é ❛♣r❡s❡♥t❛❞♦ ♣♦r

❆❤❛r♦♥✐ ❡♠ ❬✻✵❪✱ ❥á ❛q✉❡❧❡ ❡♠ q✉❡ n = 1 é ♣r♦♣♦st♦ ♣♦r ▲❛♥❞❡r♦s ❡t ❛❧✳ ❡♠ ❬✻✶❪✳ P❛r❛

♠❛✐♦r❡s ❞❡t❛❧❤❡s ❞❡st❛ ❡ ❞❡ ♦✉tr❛s ❢✉♥çõ❡s t❡♥t❛t✐✈❛s✱ r❡❝♦♠❡♥❞❛♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ❛

r❡❢❡rê♥❝✐❛ ❬✺✼❪✳

❆♣❡s❛r ❞❡ ❥á ❡①✐st✐r❡♠ ❡st✉❞♦s t❡ór✐❝♦s ❞❡t❛❧❤❛❞♦s ❛❝❡r❝❛ ❞❛ ❧❛r❣✉r❛ ❡ ❡str✉t✉r❛

❞♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡✱ ❛♣❡♥❛s ♥♦ ✐♥í❝✐♦ ❞♦s ❛♥♦s ✷✵✵✵ ❢♦✐ ♦❜s❡r✈❛❞♦ ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡

✸✺

✸✳ ❊str✉t✉r❛ ❡ ❞✐♥â♠✐❝❛ ❞❡ ✈órt✐❝❡s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s

❋✐❣✉r❛ ✸✳✺✿ ■♠❛❣❡♠ ❞❡ ▼❋▼ ❞❡ ✉♠ ❛rr❛② ❞❡ ❞♦ts ❞❡ P❡r♠❛❧❧♦② ♥♦ ❡st❛❞♦ r❡♠❛♥❡♥t❡

✭| ~H| = 0✮✱ ❝♦♠ R = 500 ♥♠ ❡ L = 50 ♥♠✳ ➱ ♠♦str❛❞♦ ♥❛ ✜❣✉r❛ r❡❣✐õ❡s ✭❝❧❛r❛s ❡ ❡s❝✉r❛s✮

q✉❡ ❝♦rr❡s♣♦♥❞❡♠ à ár❡❛ ♦♥❞❡ ❛ ♠❛❣♥❡t✐③❛çã♦ é ❛❧✐♥❤❛❞❛ ♣❡r♣❡♥❝✐❞✉❧❛r♠❡♥t❡ ✭✉♣ ♦✉ ❞♦✇♥✮

❛♦ ♣❧❛♥♦ ❞❛ ❛♠♦str❛✳ ❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢❡rê♥❝✐❛ ❬✻✷❪✳

♦ ❝❛r♦ç♦ ❞❡ ✈órt✐❝❡s ♠❛❣♥ét✐❝♦s ❬✶✽✱ ✻✷❪✳ ❘❡ss❛❧t❛♠♦s t❛♠❜é♠✱ q✉❡ ✉♠ ❞✐s♣♦s✐t✐✈♦ ❞❡

▼❋▼ ♣♦❞❡ ❞❡t❡❝t❛r ❛ ♣♦❧❛r✐❞❛❞❡ ❞♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡✱ ♠❛s ♥ã♦ s✉❛ q✉✐r❛❧✐❞❛❞❡✱ ❛♦

♣❛ss♦ q✉❡ ✉♠ P❊❊▼ ❞❡t❡❝t❛ ❛♣❡♥❛s ❛ q✉✐r❛❧✐❞❛❞❡✳ ◆❛ ✜❣✉r❛ ✸✳✺✱ ♠♦str❛♠♦s ✉♠❛

✐♠❛❣❡♠ ❞❡ ▼❋▼ ❞❡ ✉♠ ❛rr❛② ❞❡ ❞♦ts ❞❡ P❡r♠❛❧❧♦② ♥❛ ❛✉sê♥❝✐❛ ❞❡ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦

❡①t❡r♥♦✳ ❆ ✐♠❛❣❡♠ ♠♦str❛ ❝❧❛r❛♠❡♥t❡ ♦ ❝♦♥tr❛st❡ ♥♦ ❝❡♥tr♦ ❞❡ ❝❛❞❛ ❞♦t✱ s❡♥❞♦ s✉❣❡✲

r✐❞♦ q✉❡ ❡ss❛s r❡❣✐õ❡s ❝❧❛r❛s ✭p = +1✮ ♦✉ ❡s❝✉r❛s ✭p = −1✮ ❛♦ ❝❡♥tr♦ ❝♦rr❡s♣♦♥❞❛♠ à

ár❡❛ ♦♥❞❡ ❛ ♠❛❣♥❡t✐③❛çã♦ é ❛❧✐♥❤❛❞❛ ♣❡r♣❡♥❞✐❝✉❧❛r♠❡♥t❡ ❛♦ ♣❧❛♥♦ ❞❛ ❛♠♦str❛✳ ❊♥tr❡✲

t❛♥t♦✱ ♦ ❝♦♥tr❛st❡ ♠♦str❛❞♦ ♥♦ ❝❡♥tr♦ ❞❡ ❝❛❞❛ ❞♦t ❞á ❛ ✐♠♣r❡ssã♦ ❞❡ ❡st❛r ❞✐str✐❜✉í❞♦

❛❧❡❛t♦r✐❛♠❡♥t❡ ❡♥tr❡ ♦s ❞♦ts q✉❡ ❝♦♠♣õ❡♠ ♦ ❛rr❛②✱ ♦ q✉❡ é ❥✉st✐✜❝❛❞♦ ♣❡❧♦ ❢❛t♦ ❞❡

❛♠❜❛s ❝♦♥✜❣✉r❛çõ❡s s❡r❡♠ ❡q✉✐✈❛❧❡♥t❡s ❡♥❡r❣❡t✐❝❛♠❡♥t❡✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❞❡ s✉❛

♣♦❧❛r✐❞❛❞❡ ❡ q✉✐r❛❧✐❞❛❞❡ ❬✻✷❪✳

▲❡✈❛♥❞♦✲s❡ ❡♠ ❝♦♥t❛ q✉❡ ❡st❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦ ♥❛♥♦❞✐s❝♦s ❝♦♠ ❛ r❛③ã♦ L/R ≪1✱ ♣♦❞❡♠♦s ❛ss✉♠✐r q✉❡ ❛ ♠❛❣♥❡t✐③❛çã♦ ~M = Ms ~m ❛♦ ❧♦♥❣♦ ❞♦ ❡✐①♦ z ♥♦ ✐♥t❡r✐♦r ❞♦

♥❛♥♦❞✐s❝♦ é ✉♥✐❢♦r♠❡ ❡ ~∇ · ~M(~r ) = 0✳ ▼❛s ♥♦t❡ q✉❡ ♦ ♣r♦❞✉t♦ ~M · n ❛♦ ❧♦♥❣♦ ❞❛ ❢❛❝❡

❧❛t❡r❛❧ ❞♦ ♥❛♥♦❞✐s❝♦ ♥ã♦ é ♥✉❧♦✳ ❉❡ ❢❛t♦✱ ♦ ✈❡t♦r ♠❛❣♥❡t✐③❛çã♦ t❡♥❞❡ ❛ ✜❝❛r ♣❛r❛❧❡❧♦ à

s✉♣❡r❢í❝✐❡ ❧❛t❡r❛❧ ❞♦ ❞✐s❝♦ ❛ ✜♠ ❞❡ ♠✐♥✐♠✐③❛r ❛ ❡♥❡r❣✐❛ ♠❛❣♥❡t♦stát✐❝❛ q✉❡✱ ❝♦♠♦ ✈✐♠♦s

♥❛ s❡çã♦ ✸✳✹✱ é ♦❜t✐❞❛ ♣❡❧❛ ❛✉sê♥❝✐❛ ❞❡ ♣ó❧♦s ✭ρM = −~∇ · ~M = 0 ❡ σM = ~M · n = 0✮✳

■st♦ é ❝♦♥s❡❣✉✐❞♦ ❝♦♠ ❛ ❢♦r♠❛çã♦ ❞❡ ✉♠ ✧❧♦♦♣✧ ❢❡❝❤❛❞♦ ♥❛ s✉♣❡r❢í❝✐❡ ❧❛t❡r❛❧✱ ❛rr❛♥❥♦

❡st❡ q✉❡ é s❡❣✉✐❞♦ ♥❛ s✉♣❡r❢í❝✐❡ s✉♣❡r✐♦r✱ à ❡①❝❡çã♦ ❞♦ ❝❛r♦ç♦✱ ♦♥❞❡ ~M · n 6= 0✳ ❱❡♠♦s

✸✻

✸✳ ❊str✉t✉r❛ ❡ ❞✐♥â♠✐❝❛ ❞❡ ✈órt✐❝❡s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s

❋✐❣✉r❛ ✸✳✻✿ ❆s s❡t❛s ❛♠❛r❡❧❛s r❡♣r❡s❡♥t❛♠ ♦ ✈❡t♦r n✱ ♥♦r♠❛❧ à s✉♣❡r❢í❝✐❡✱ ❡ ❛s s❡t❛s ❛③✉✐s

❛ ♠❛❣♥❡t✐③❛çã♦ ~M(~r )✳ ➚ ❡sq✉❡r❞❛ r❡♣r❡s❡♥t❛♠♦s ♦ q✉❡ ♦❝♦rr❡ ♥❛ ❢❛❝❡ ❧❛t❡r❛❧ ❞♦ ♥❛♥♦❞✐s❝♦✱

♦♥❞❡ ~M · n = 0✳ ➚ ❞✐r❡✐t❛ ♠♦str❛♠♦s ❛s ❞✉❛s s✐t✉❛çõ❡s q✉❡ ❛❝♦♥t❡❝❡♠ ♥❛ ❢❛❝❡ s✉♣❡r✐♦r ❞♦

♥❛♥♦❞✐s❝♦✿ ❛✮ ♠❛❣♥❡t✐③❛çã♦ ♥♦ ♣❧❛♥♦ xy ❡ ❜✮ ♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ ❝♦♠ ♠❛❣♥❡t✐③❛çã♦ t❛♠❜é♠

♥♦ ❡✐①♦ z✳

❞❡st❛ ❢♦r♠❛ q✉❡ s♦♠❡♥t❡ σM ❞á ❝♦♥tr✐❜✉✐çõ❡s ♥♦ t❡r♠♦ ♠❛❣♥❡t♦stát✐❝♦✳ ❆ ✜❣✉r❛ ✸✳✻

é ✉♠❛ ✐❧✉str❛çã♦ ❞♦ q✉❡ ❛❝❛❜❛♠♦s ❞❡ ❢❛❧❛r✱ ♠♦str❛♥❞♦ ❝♦♠♦ ❞❡✈❡ s❡r ❛ ♠❛❣♥❡t✐③❛çã♦

♥❛s s✉♣❡r❢í❝✐❡s s✉♣❡r✐♦r ❡ ❧❛t❡r❛❧ ❞♦ ♥❛♥♦❞✐s❝♦✳

❆❣♦r❛ ❡st❛♠♦s ❡♠ ♣♦s✐çã♦ ❞❡ ❞❡t❡r♠✐♥❛r ❛ ❝♦♥st❛♥t❡ C✱ q✉❡ ❛♣❛r❡❝❡ ♥❛ ❡q✉❛çã♦

✭✸✳✺✮ ♣❛r❛ ♦ ❝❛♠♣♦ φ✳ P❛r❛ q✉❡ ❛ ♠❛❣♥❡t✐③❛çã♦ ❛❝✐♠❛ s❡❥❛ ✈❡r✐✜❝❛❞❛✱ C = qπ/2✱ ❡♠

q✉❡ ❛ q✉✐r❛❧✐❞❛❞❡ q = ±1 ✭q = +1 ♣❛r❛ ❝✐r❝✉❧❛çã♦ ❛♥t✐✲❤♦rár✐❛ ❡ q = −1 ♣❛r❛ ❤♦rár✐❛✮✳

✸✳✷✳✶ ❉✐♥â♠✐❝❛ ❞❡ ✈órt✐❝❡s

❊st✉❞❛r❡♠♦s ❛❣♦r❛ ❛ ❞✐♥â♠✐❝❛ ❞♦ ✈órt✐❝❡ q✉❛♥❞♦ s✉❜♠❡t✐❞♦ ❛ ✉♠ ❝❛♠♣♦ ♠❛❣✲

♥ét✐❝♦ ❡①t❡r♥♦ ~Hext ♥♦ ♣❧❛♥♦ xy ❞❛ ❛♠♦str❛✳ ❚❛❧ ❞✐♥â♠✐❝❛ é ❞❛❞❛ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❡❧♦

❞❡s❧♦❝❛♠❡♥t♦ ❞♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡✳ P❛r❛ ❝♦♠♣r❡❡♥❞❡r t❛❧ ❢❡♥ô♠❡♥♦✱ ❢❛r❡♠♦s ✉s♦ ❞❛s

✜❣✉r❛s ✸✳✼ ❡ ✸✳✽✳ ❈♦♠♦ ♣♦❞❡ s❡r ✈✐st♦ ♥❛ ✜❣✉r❛ ✸✳✼✱ ❛ ❝❛♠♣♦ ♥✉❧♦ t❡♠♦s ♦ ❡st❛❞♦

❞❡ ✈órt✐❝❡ ♣❡r❢❡✐t♦ ✭❛✮ ✭❡st❛❞♦ r❡♠❛♥❡♥t❡✮✱ ❝♦♠ ❛ ❝♦♠♣♦♥❡♥t❡ ♣❡r♣❡♥❞✐❝✉❧❛r ♥♦ ❝❡♥tr♦

❞♦ ❞✐s❝♦ ✭r❡❣✐ã♦ ❝❧❛r❛ ❛♦ ❝❡♥tr♦ ❞❛ ✜❣✉r❛✮✳ ◗✉❛♥❞♦ ✉♠ ♣❡q✉❡♥♦ ❝❛♠♣♦ é ❛♣❧✐❝❛❞♦✱

♦ ❝❡♥tr♦ ❞♦ ✈órt✐❝❡ ♠♦✈❡✲s❡ ♥❛ ❞✐r❡çã♦ ♥♦r♠❛❧ ❛♦ ❝❛♠♣♦ ❛♣❧✐❝❛❞♦ ✭❜✮✱ ♣♦✐s ❛ ♣♦rçã♦

❞♦ ✈órt✐❝❡ ❝♦♠ s♣✐♥s ♣❛r❛❧❡❧♦s ❛♦ ❝❛♠♣♦ ❡①♣❛♥❞❡✲s❡✳ ❙❡ ❡❧❡ ♠♦✈❡ ♣❛r❛ ❛ ❡sq✉❡r❞❛ ♦✉

✸✼

✸✳ ❊str✉t✉r❛ ❡ ❞✐♥â♠✐❝❛ ❞❡ ✈órt✐❝❡s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s

❋✐❣✉r❛ ✸✳✼✿ ■❧✉str❛çã♦ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ❞❡ ❞♦✐s ♥❛♥♦❞✐s❝♦s ❞❡ P②✱ ❝♦♠ L = 50 ♥♠ ❡ R = 450

♥♠✱ s✉❜♠❡t✐❞♦ ❛ ✉♠ ❝❛♠♣♦ ❡①t❡r♥♦ ~Hext✱ ❡♠ ❖❡rst❡❞s ✭✶ ❖❡ ≈ ✼✾✱✻ ❆✴♠✮✱ ❛♣❧✐❝❛❞♦ ♥♦ ♣❧❛♥♦

❞❛ ❛♠♦str❛✱ ❝♦♥❢♦r♠❡ ✐♥❞✐❝❛❞♦✳ ❖ ❝❛♠♣♦ é ❛✉♠❡♥t❛❞♦ s❡❣✉✐❞❛♠❡♥t❡✱ ❝♦♠ ♦s ♠♦♠❡♥t♦s

♠❛❣♥ét✐❝♦s s✉❝❡ss✐✈❛♠❡♥t❡ ♦r✐❡♥t❛♥❞♦✲s❡ ♥❛ ❞✐r❡çã♦ ❞❡ ~Hext✳ ▼♦str❛♠♦s t❛♠❜é♠ ❛s ✧❝❛r❣❛s

♠❛❣♥ét✐❝❛s✧σM = ~M · n ❡ ρM = −~∇ · ~M ✳ ❆♦ ❝❡♥tr♦ ❞♦ ✈órt✐❝❡✱ r❡♣r❡s❡♥t❛♠♦s ♦ ❝❛r♦ç♦ ♣♦r

✉♠❛ r❡❣✐ã♦ ❝❧❛r❛✱ p = +1✱ ♦✉ ❡s❝✉r❛✱ p = −1✳ ❋✐❣✉r❛ ♠♦❞✐✜❝❛❞❛ r❡t✐r❛❞❛ ❞❛ r❡❢❡rê♥❝✐❛ ❬✶✵❪✳

❞✐r❡✐t❛✱ ❞❡♣❡♥❞❡ ❞❛ q✉✐r❛❧✐❞❛❞❡ ❞♦ ✈órt✐❝❡✱ s❡ ❤♦rár✐❛ ♦✉ ❛♥t✐✲❤♦rár✐❛✱ ❡✴♦✉ ❞♦ s❡♥t✐❞♦

❞❡ ❛♣❧✐❝❛çã♦ ❞♦ ❝❛♠♣♦✳ ❆✉♠❡♥t❛♥❞♦✲s❡ ♦ ❝❛♠♣♦✱ ❤á ✉♠ ❝♦♥❝♦♠✐t❛♥t❡ ❛✉♠❡♥t♦ ❞❛

♠❛❣♥❡t✐③❛çã♦ ✭❝✮ ❧✐♥❡❛r♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡ ❞♦ ♠ó❞✉❧♦ ❞❡ ~Hext✱ ❢❛t♦ q✉❡ é ✐❧✉str❛❞♦ ♣❡❧❛

❝✉r✈❛ ❞❡ ❤✐st❡r❡s❡ ❞❛ ✜❣✉r❛ ✸✳✽✳ ❖ ♠♦✈✐♠❡♥t♦ ❞♦ ✈órt✐❝❡ ❝♦♥t✐♥✉❛ ❛té q✉❡ ♦ ❝❛r♦ç♦

❛t✐♥❥❛ ❛ ❧❛t❡r❛❧ ❞♦ ❞✐s❝♦ ❡ ❡st❡ ❞❡s❛♣❛r❡❝❡ ♥♦ ❝❛♠♣♦ ❞❡ ❛♥✐q✉✐❧❛çã♦ ✭❛♥♥✐❤✐❧❛t✐♦♥ ✜❡❧❞✮

HA✱ ♥♦ q✉❛❧ M ❝r❡s❝❡ ❛❜r✉♣t❛♠❡♥t❡ ❞❛♥❞♦ ♦r✐❣❡♠ ❛ ✉♠ ❡st❛❞♦ q✉❛s❡ s❛t✉r❛❞♦ ✭❞✮✳

❖ ♠ó❞✉❧♦ HA t❛♠❜é♠ é ♠♦str❛❞♦ ♥❛ ✜❣✉r❛ ✸✳✽✳ ◆✉♠ ❝❛♠♣♦ ❞❡ ♠ó❞✉❧♦ ❧✐❣❡✐r❛♠❡♥t❡

s✉♣❡r✐♦r ❛ HA✱ ♦ ❞✐s❝♦ ❡♥❝♦♥tr❛✲s❡ ♥♦ ❡st❛❞♦ ❞❡ ❞♦♠í♥✐♦ ú♥✐❝♦ ✭♥ã♦ ♠♦str❛❞♦✮✱ ♦♥❞❡

M =Ms✳

❉✐♠✐♥✉✐♥❞♦ ♦ ❝❛♠♣♦ ❡①t❡r♥♦ s✉❝❡ss✐✈❛♠❡♥t❡✱ ♦ ❝❛r♦ç♦ ♥ã♦ ✧r❡❛♣❛r❡❝❡✧ ❛té q✉❡

Hext ❛t✐♥❥❛ ♦ ❝❛♠♣♦ ❞❡ ♥✉❝❧❡❛çã♦ ✭♥✉❝❧❡❛t✐♦♥ ✜❡❧❞✮ HN ✱ ♣♦♥t♦ ♦♥❞❡ ❛ ♠❛❣♥❡t✐③❛çã♦ ❝❛✐

❞r❛st✐❝❛♠❡♥t❡ ✭❡✮✱ ❝♦♠♦ ✐♥❞✐❝❛❞♦ ♥❛ ✜❣✉r❛ ✸✳✽✳ ❙❡ ❝♦♥t✐♥✉❛r♠♦s ❞✐♠✐♥✉✐♥❞♦ ♦ ❝❛♠♣♦

✭❢✮✱ r❡t♦♠❛r❡♠♦s ♦ ❡st❛❞♦ ❞❡ ✈órt✐❝❡ ♣❡r❢❡✐t♦ ✭❣✮ ❬✶✵❪✳ ❙❡ ❝♦♥t✐♥✉❛r♠♦s ❛ ❞✐♠✐♥✉✐r ♦

❝❛♠♣♦✱ ❛❣♦r❛ ❝♦♠ ✈❛❧♦r❡s ♥❡❣❛t✐✈♦s✱ ✉♠ ❝♦♠♣♦rt❛♠❡♥t♦ ❛♥á❧♦❣♦ é ✈❡r✐✜❝❛❞♦✱ ❝♦♠ ♦

❝❛r♦ç♦ ❞❡s❧♦❝❛♥❞♦✲s❡ ♣❡r♣❡♥❞✐❝✉❧❛r♠❡♥t❡ ❛♦ ❝❛♠♣♦ ♠❛s ❞❡st❛ ✈❡③ ♥♦ s❡♥t✐❞♦ ♦♣♦st♦✳

❊♥tr❡t❛♥t♦✱ é ✐♠♣♦rt❛♥t❡ ♦❜s❡r✈❛r q✉❡ ♦s ✈❛❧♦r❡s ❞❡ HN ❡ HA ❞❡♣❡♥❞❡♠ ❢♦rt❡♠❡♥t❡ ❞♦

✸✽

✸✳ ❊str✉t✉r❛ ❡ ❞✐♥â♠✐❝❛ ❞❡ ✈órt✐❝❡s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s

❋✐❣✉r❛ ✸✳✽✿ ❈✉r✈❛ ❞❡ ❤✐st❡r❡s❡ ♣❛r❛ ✉♠ ❞✐s❝♦ ❞❡ P② ✭R = 300 ♥♠ ❡ L = 60 ♥♠✮ ❢❡✐t❛ ♣♦r

♠❛❣♥❡t♦♠❡tr✐❛ ❍❛❧❧✳ ❋✐❣✉r❛ ♠♦❞✐✜❝❛❞❛ r❡t✐r❛❞❛ ❞❛ r❡❢❡rê♥❝✐❛ ❬✶✵❪✳

r❛✐♦✱ ❡s♣❡ss✉r❛✺✱ ♠❛t❡r✐❛❧ ❡ ❞❡♥s✐❞❛❞❡ ❞❡ ❞❡❢❡✐t♦s ❞♦ ❞✐s❝♦✳ ❯♠ r❡♣r❡s❡♥t❛t✐✈♦ ✐♥t❡r✈❛❧♦

♣❛r❛ HA ❡ HN é ❡♠ t♦r♥♦ ❞❡ 10−2 − 10−1 ❚ ❬✺✵❪✳

❈♦♠♣❛r❛♥❞♦ ♦ ❡st❛❞♦ ✐♥✐❝✐❛❧ ✭❛✮ ❞♦s ❞♦✐s ❞✐s❝♦s ❝♦♠ ♦ ❡st❛❞♦ ✜♥❛❧ ✭❣✮✱ ✈❡♠♦s

q✉❡ ♦ ❞✐s❝♦ ❞❛ ❡sq✉❡r❞❛ ♠✉❞♦✉ ❛ q✉✐r❛❧✐❞❛❞❡ ❞♦ ✈órt✐❝❡ ❛♣❡♥❛s✱ ❥á ♦ ❞❛ ❞✐r❡✐t❛ ✐♥✈❡rt❡✉

❛♣❡♥❛s s✉❛ ♣♦❧❛r✐❞❛❞❡✱ ❢❡♥ô♠❡♥♦ ❡st❡ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ s✇✐t❝❤✐♥❣✳ ❇✉ss♠❛♥♥ ❡ ❝♦❧❛❜♦✲

r❛❞♦r❡s ♠♦str❛r❛♠ ❬✻✸❪ q✉❡ é ♣♦ssí✈❡❧ ✐♥✈❡rt❡r t❛♠❜é♠ ❛ q✉✐r❛❧✐❞❛❞❡ ❞♦ ✈órt✐❝❡✱ ♦ q✉❡

é ❢❡✐t♦ ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ ❝♦rr❡♥t❡ ❡❧étr✐❝❛ ♥❛ ✈✐③✐♥❤❛♥ç❛ ❞♦ ♥❛♥♦❞✐s❝♦ ❡ ❛♣❧✐❝❛❞❛ ♣❡r✲

♣❡♥❞✐❝✉❧❛r♠❡♥t❡ ❛tr❛✈és ❞♦ ♥❛♥♦❞✐s❝♦✱ q✉❡ sã♦ ❛s ❝é❧✉❧❛s q✉❡ ❝♦♠♣õ❡♠ ❛s ▼❘❆▼✬s✳

❆ss✐♠✱ t❛♥t♦ ♦ ❢❡♥ô♠❡♥♦ ❞❡ s✇✐t❝❤✐♥❣ ❞❛ ♣♦❧❛r✐❞❛❞❡ q✉❛♥t♦ ♦ ❞❛ q✉✐r❛❧✐❞❛❞❡ ❛♣❛r❡❝❡♠

❝♦♠♦ ✉♠ ❞♦s ✐♠♣♦rt❛♥t❡s ❛s♣❡❝t♦s ❛ss♦❝✐❛❞♦s à ♠❛❣♥❡t✐③❛çã♦ ❞♦ t✐♣♦ ✈órt✐❝❡✱ ✉♠❛ ✈❡③

q✉❡ ❛ r❡✈❡rsã♦ ❝♦♥tr♦❧❛❞❛ ❢♦r♥❡❝❡ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❛r♠❛③❡♥❛♠❡♥t♦ ❞❡ ❞♦✐s ❜✐ts ❞❡

✐♥❢♦r♠❛çã♦ q✉❛♥❞♦ t❛❧ ❝♦♥✜❣✉r❛çã♦ é ✉s❛❞❛ ♥✉♠❛ ❝é❧✉❧❛ ❞❡ ▼❘❆▼ ❬✶✵❪✳

❉❡t❡r♠✐♥❛çã♦ ❞❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞♦ ✈órt✐❝❡

❊✈✐❞❡♥t❡♠❡♥t❡✱ t❡♠♦s t❛♠❜é♠ ✉♠ t❡r♠♦ ❞❡ ❡♥r❣✐❛ ❛ss♦❝✐❛❞♦ à ✐♥t❡r❛çã♦ ❞♦

❝❛♠♣♦ ❡①t❡r♥♦ ❛♣❧✐❝❛❞♦✱ ~Hext✱ ❝♦♠ ♦s ♠♦♠❡♥t♦s ❞❡ ❞✐♣♦❧♦✳ ❚❛❧ t❡r♠♦ é ❝❤❛♠❛❞♦ ❞❡

✺❆ r❡❢❡rê♥❝✐❛ ❬✶✵❪✱ ♣♦r ❡①❡♠♣❧♦✱ ❛♣r❡s❡♥t❛ ✉♠ ❣rá✜❝♦ ♠♦str❛♥❞♦ ❛ ✈❛r✐❛çã♦ ❞♦ ❝❛♠♣♦ HA ❡♠

❢✉♥çã♦ ❞❛ r❛③ã♦ 2R/L✳ ❖ ❝❛♠♣♦ ❞❡ ♥✉❝❧❡❛çã♦ ♣♦❞❡ ❛té ♠❡s♠♦ s❡r ♥❡❣❛t✐✈♦ ♣❛r❛ ❞✐s❝♦s ❞❡ ❞✐♠❡♥sõ❡s

♠❡♥♦r❡s✱ ❝♦♠♦ ♦ é✱ ♣♦r ❡①❡♠♣❧♦✱ ♣❛r❛ ✉♠ ❞✐s❝♦ ❝♦♠ R = 50 ♥♠ ❬✺✵❪✳

✸✾

✸✳ ❊str✉t✉r❛ ❡ ❞✐♥â♠✐❝❛ ❞❡ ✈órt✐❝❡s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s

❡♥❡r❣✐❛ ❩❡❡♠❛♥✱ ♣♦r s❡r ❡st❛ ✐♥t❡r❛çã♦ ❛ r❡s♣♦♥sá✈❡❧ ♣❡❧♦ ❡❢❡✐t♦ ❩❡❡♠❛♥ ❡♠ át♦♠♦s

q✉❛♥❞♦ s✉❜♠❡t✐❞♦s ❛ ❝❛♠♣♦s ♠❛❣♥ét✐❝♦s✳ ❚❛❧ ❡♥❡r❣✐❛ é ❞❛❞❛ ♣♦r✿

EH = −∫

V

µ0~M(~r) · ~Hext d

3r✳ ✭✸✳✶✶✮

❊♥tr❡t❛♥t♦✱ s❡ ♥♦s r❡str✐♥❣✐r♠♦s s♦♠❡♥t❡ ❛ ♣❡q✉❡♥♦s ❞❡s❧♦❝❛♠❡♥t♦s ❞♦ ❝❛r♦ç♦ ❞♦

✈órt✐❝❡ ✭s = |~s| = |~r|/R ≪ 1✮ ❡ s❡ ❝♦♥s✐❞❡r❛r♠♦s ❛✐♥❞❛ q✉❡ ♦ ❝❛r♦ç♦ ♠♦✈❡✲s❡ ♠❛♥t❡♥❞♦

s✉❛ ❝♦♥✜❣✉r❛çã♦ ❞❡ s♣✐♥s✱ ❡st❛r❡♠♦s ♥♦ ❝❤❛♠❛❞♦ ♠♦❞❡❧♦ ❞❡ ✈órt✐❝❡ rí❣✐❞♦ ❬✻✹❪✳ ◆❡st❡

♠♦❞❡❧♦✱ t❡♠♦s q✉❡ ❛ ❡♥❡r❣✐❛ ❩❡❡♠❛♥ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞❛✱ r❡s✉❧t❛♥❞♦ ❡♠✿

wH =EH

µ0πR2LM2s

= −hext s+O(s3)✱ ✭✸✳✶✷✮

❡♠ q✉❡ ~hext = ~Hext/Ms✳

❈♦♠♦ ❡st❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦ ❛ ♠❛❣♥❡t✐③❛çã♦ s❡♥❞♦ ❞✐str✐❜✉í❞❛ ✉♥✐❢♦r♠❡♠❡♥t❡

❞❡♥tr♦ ❞♦ ♥❛♥♦❞✐s❝♦✱ ♣♦❞❡♠♦s ❢❛③❡r ❛ ✐♥t❡❣r❛çã♦ ❡♠ z ❡♠ ✭✸✳✽✮✱ ♦ q✉❡ ♥♦s ❢♦r♥❡❝❡✿

Etr =1

2AL

∫

S

3∑

α=1

(

~∇mα

)2

d2r✳ ✭✸✳✶✸✮

❉❡♥tr♦ ❞❡ss❡ r❡❣✐♠❡ ❞❡ ✈órt✐❝❡ rí❣✐❞♦✱ t❡♠♦s q✉❡ ❛ ❡♥❡r❣✐❛ ❞❡ tr♦❝❛ ♥♦r♠❛❧✐③❛❞❛

é✿

wtr =Etr

µ0πR2LM2s

= wtr(0) +1

2

(lexR

)2

ln(1− s2) +O(s4)✱ ✭✸✳✶✹✮

♦♥❞❡ wtr(0) é ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ❞❡ tr♦❝❛ ♥❛ ♦r✐❣❡♠ ❡ t❡♠♦s ✉s❛❞♦ ❛ ❞❡✜♥✐çã♦

❞❡ lex ❞❛❞❛ ❛♥t❡r✐♦r♠❡♥t❡ ♥♦ t❡①t♦✳ ❆ ❡♥❡r❣✐❛ ♠❛❣♥❡t♦stát✐❝❛ ❡stá ❛ss♦❝✐❛❞❛ s♦♠❡♥t❡

às ❝❛r❣❛s ♠❛❣♥ét✐❝❛s s✉♣❡r✜❝✐❛✐s✱ σ = ~M · n✱ ❝♦♠♦ ❥á ❞✐s❝✉t✐♠♦s ❡✱ ♥❡st❡ ♠♦❞❡❧♦✱ ❛

❡q✉❛çã♦ ✭✷✳✸✷✮ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞❛✱ r❡s✉❧t❛♥❞♦ ♥❛ ❡♥❡r❣✐❛ ✭♥♦r♠❛❧✐③❛❞❛✮✿

wmag =Emag

µ0πR2LM2s

= wm(0) + 2πF1(β)s2 +O(s4)✱ ✭✸✳✶✺✮

s❡♥❞♦ wm(0) ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ♠❛❣♥❡t♦stát✐❝❛ ♥❛ ♦r✐❣❡♠✱ β = L/R ❡✿

Fµ(x) =

∫ ∞

0

f(xt)J2µ(t)

tdt✱ f(x) = 1− 1− e−x

x✱ ✭✸✳✶✻✮

♦♥❞❡ Jµ(x) sã♦ ❛s ❢✉♥çõ❡s ❞❡ ❇❡ss❡❧✳

P♦❞❡♠♦s ♦❜t❡r ❛❣♦r❛ ❛ ❡♥❡r❣✐❛ t♦t❛❧ ❞♦ ♥❛♥♦❞✐s❝♦ ❡♠ ❢✉♥çã♦ ❞❛s ❣r❛♥❞❡③❛s L✱ R

❡ H ❡✱ ♠✐♥✐♠✐③❛♥❞♦ ❛s ❡♥❡r❣✐❛s ✭✸✳✶✷✮✱ ✭✸✳✶✹✮ ❡ ✭✸✳✶✺✮✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ❛ ♣♦s✐çã♦ ❞❡

✹✵

✸✳ ❊str✉t✉r❛ ❡ ❞✐♥â♠✐❝❛ ❞❡ ✈órt✐❝❡s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s

❡q✉✐❧í❜r✐♦ ❞♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡✱ s✳ ❯s❛♥❞♦ ❛ ❡①♣❛♥sã♦ ❡♠ sér✐❡ ❞❡ ❚❛②❧♦r ❡♠ ✭✸✳✶✹✮ ❡♠

t♦r♥♦ ❞❡ s = 0✱ ❡♥❝♦♥tr❛♠♦s q✉❡ ln(1− s2) ≈ −s2 ❡ ❛❣♦r❛ ❞❡s❝♦♥s✐❞❡r❛♥❞♦ ♦s t❡r♠♦s

O(s3) ✭♣♦✐s s ≪ 1✮✱ ❝❤❡❣❛♠♦s à ❡♥❡r❣✐❛ t♦t❛❧ W = µ0πR2LM2

s w(s)✱ ❝✉❥❛ ❞❡♥s✐❞❛❞❡

é ❬✻✹❪✿

w(s) = wtr(s) + wm(s) + wH(s)

= w(0) + 2πF1(β)s2 − 1

2

(lexR

)

s2 − hext s+O(s3)✳ ✭✸✳✶✼✮

❋❛③❡♥❞♦ dw(s)/ds = 0 ❡ ♠♦str❛♥❞♦ q✉❡ é ♠í♥✐♠♦ ❞❡ ❡♥❡r❣✐❛✱ ❡♥❝♦♥tr❛♠♦s ❛

♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ sh ❡♠ ❢✉♥çã♦ ❞❡ hext✿

sh =xhR

=−q hext

4πF1(β)− (lex/R)2. ✭✸✳✶✽✮

♦♥❞❡ ❡st❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦ ✉♠ ❝❛♠♣♦ ♣❛r❛❧❡❧♦ ❛♦ ♣❧❛♥♦ ❞♦ ♥❛♥♦❞✐s❝♦ ♥❛ ❞✐r❡çã♦ y✱

~hext = (0, hext)✱ ❡ ❛ q✉✐r❛❧✐❞❛❞❡ q é q✉❡♠ ❞❡✜♥❡ ♦ s❡♥t✐❞♦ ❞♦ ❞❡s❧♦❝❛♠❡♥t♦ ❞♦ ❝❛r♦ç♦

♥♦ ❡✐①♦ x✱ ~sh = (xh/R, 0)✳ ❆ss✐♠✱ ✈❡♠♦s ❝❧❛r❛♠❡♥t❡ q✉❡ ❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ sh

❞♦ ❝❛r♦ç♦ ❝♦♠♣♦rt❛✲s❡✴❞❡s❧♦❝❛✲s❡ ❧✐♥❡❛r♠❡♥t❡ ❡♠ ❢✉♥çã♦ ❞❡ hext ✭♣❡❧♦ ♠❡♥♦s ♣❛r❛ s

♣❡q✉❡♥♦✮✳

▼♦✈✐♠❡♥t♦ ●✐r♦tró♣✐❝♦

❙❡ ❛❣♦r❛ ❛♣❧✐❝❛r♠♦s ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ♦s❝✐❧❛tór✐♦ ❞❡ ♣❡q✉❡♥❛s ❛♠♣❧✐t✉❞❡s✱

♦ ✈órt✐❝❡ ♥ã♦ ❛♣❡♥❛s ♦s❝✐❧❛✱ ♠❛s t❛♠❜é♠ ❡①❡❝✉t❛ ✉♠ ♠♦✈✐♠❡♥t♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦

♠♦✈✐♠❡♥t♦ ❣✐r♦tró♣✐❝♦✳ ❆s ♣r♦♣r✐❡❞❛❞❡s ❞❡ss❛ ❞✐♥â♠✐❝❛ sã♦ ❞❡s❝r✐t❛s ♣❡❧❛ ❡q✉❛çã♦

❞❡ ▲❛♥❞❛✉✲▲✐❢s❤✐t③✲●✐❧❜❡rt ❬✺✺❪✿

~G× ~v − k~r = mv ~a✱ ✭✸✳✶✾✮

♦♥❞❡ ~r✱ ~v ❡ ~a sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛ ♣♦s✐çã♦✱ ✈❡❧♦❝✐❞❛❞❡ ❡ ❛❝❡❧❡r❛çã♦ ❞♦ ❝❡♥tr♦ ❞♦ ✈ór✲

t✐❝❡ ✭❛ ú♥✐❝❛ r❡❣✐ã♦ q✉❡ ❝♦♥❢❡r❡ ❞✐♥â♠✐❝❛✮✱ ❛♦ ♣❛ss♦ q✉❡mv é ❛ ♠❛ss❛ ❞♦ ✈órt✐❝❡✱ q✉❡ ❡♠

❣❡r❛❧ é ✉♠ t❡♥s♦r✱ ~G é ❝❤❛♠❛❞♦ ❞❡ ❣✐r♦✈❡t♦r ❞♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡✳ ❆ ❝♦s♥t❛♥t❡ k é ✉♠

♣❛râ♠❡tr♦ ❞♦ t✐♣♦ ❝♦♥t❛♥t❡ ❡❧ást✐❝❛✱ ❛ss♦❝✐❛❞♦ à ❢♦rç❛ r❡st❛✉r❛❞♦r❛ ♠❛❣♥❡t♦stát✐❝❛✱ q✉❡

t❡♥❞❡ ❛ ♣✉①❛r ♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ ♣❛r❛ ♦ ❝❡♥tr♦ ❞♦ ❞✐s❝♦✱ ❛♥á❧♦❣♦ à ❢♦rç❛ r❡st❛✉r❛❞♦r❛

❞❛ ❧❡✐ ❞❡ ❍♦♦❦❡✱ ❡ ❡stá ❛ss♦❝✐❛❞♦ ❛♦ ♣♦t❡♥❝✐❛❧ ❡①♣❡r✐♠❡♥t❛❞♦ ♣❡❧♦ ✈órt✐❝❡ ♥❛ ❛✉sê♥❝✐❛

✹✶

✸✳ ❊str✉t✉r❛ ❡ ❞✐♥â♠✐❝❛ ❞❡ ✈órt✐❝❡s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s

❞❡ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ✭❡q✉❛çã♦ ✭✸✳✶✼✮ ❝♦♠ ❤❂✵✮✱ q✉❡ ♣❛r❛ ♣❡q✉❡♥♦s ❞❡s❧♦❝❛♠❡♥t♦s ♣♦❞❡

s❡r ♦❜t✐❞♦ ❞❡W∣∣h=0

= µ0πR2LM2

s [wtr(s)+wms]✱ ❢❛③❡♥❞♦✲s❡ ~Frest = −k~r = −~∇W∣∣h=0

✱

❞♦♥❞❡ r❡s✉❧t❛ q✉❡✿

k = µ0πR2LM2

s

[

4πF1(β)−l2exR2

]

. ✭✸✳✷✵✮

❆ ❢♦rç❛ ❣✐r♦tró♣✐❝❛ s❡♥t✐❞❛ ♣❡❧♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ é ❞❛❞❛ ♣♦r ~Fgir = ~G × ~v✱

♣❡r♣❡♥❞✐❝✉❧❛r à ✈❡❧♦❝✐❞❛❞❡ ❞♦ ❝❡♥tr♦ ❞♦ ✈órt✐❝❡ ❡ ❛♥á❧♦❣❛ à ❢♦rç❛ ♠❛❣♥ét✐❝❛ s❡♥t✐❞❛

♣♦r ✉♠❛ ❝❛r❣❛ ❡❧étr✐❝❛ ❡♠ ♠♦✈✐♠❡♥t♦✳ ◆✉♠ ♥❛♥♦❞✐s❝♦ ❝♦♠ ♠❛❣♥❡t✐③❛çã♦ ✉♥✐❢♦r♠❡

❛♦ ❧♦♥❣♦ ❞♦ ❡✐①♦ z✱ t❡♠♦s q✉❡ ❬✺✵❪ ~G = −2πpMsL/γz✱ ❡♠ q✉❡ γ é ❛ t❛①❛ ❣✐r♦♠❛❣✲

♥ét✐❝❛ ❡ ✈❛❧❡ ❝❡r❝❛ ❞❡ 104 ❍③ ♠✴❆ ♣❛r❛ ♥❛♥♦❞✐s❝♦s ❞❡ P②✳ ❙❡♥❞♦ ❛ ♠❛ss❛ ❡❢❡t✐✈❛ ❞♦

❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ mv ♣❡q✉❡♥❛ ❬✻✺❪✱ s✉❛ tr❛❥❡tór✐❛ é ✉♥✐❝❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦ ❜❛✲

❧❛♥ç♦ ❞❛s ❢♦rç❛s ❣✐r♦tró♣✐❝❛ ❡ r❡st❛✉r❛❞♦r❛✳ ❆ ❢r❡q✉ê♥❝✐❛ ❞❡ r❡ss♦♥â♥❝✐❛ ❞♦ ♠♦✈✐♠❡♥t♦

❣✐r♦tró♣✐❝♦ ❞❡♣❡♥❞❡ ❞❛ ♠❛❣♥✐t✉❞❡ ❞❛s ❞✉❛s ❢♦rç❛s✱ ❡ é ❞❛❞❛ ♣♦r ❬✺✺❪✿

ωG =k

| ~G|=γMs

2

[

4πF1(β)−l2exR2

]

✱ ✭✸✳✷✶✮

s❡♥❞♦ ❡st❡ r❡s✉❧t❛❞♦ ❡str✐t❛♠❡♥t❡ ✈á❧✐❞♦ ❞❡♥tr♦ ❞♦ r❡❣✐♠❡ ❞❡ ✈órt✐❝❡ rí❣✐❞♦✳ ❈♦♠♦

✈✐♠♦s✱ ❛ ❢r❡q✉ê♥❝✐❛ ❞❡♣❡♥❞❡ ❞❛ t❛①❛ β = L/R ❡ t❛❧ ❞❡♣❡♥❞ê♥❝✐❛ é q✉❛s❡ ❧✐♥❡❛r ♣❛r❛

✉♠ ✈❛st♦ ✐♥t❡r✈❛❧♦ ❞❡ β ❬✺✺❪✳ ❊s♣❡❝✐✜❝❛♠❡♥t❡✱ ωG ≈ 100 ▼❍③ ♣❛r❛ ✉♠ ♥❛♥♦❞✐s❝♦ ❞❡

P② ❝♦♠ L = 20 ♥♠ ❡ R = 1 µ♠✳

❆ r❡✈❡rsã♦ ❞❛ ♣♦❧❛r✐❞❛❞❡ é ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❡ ❛♣❧✐✲

❝❛çõ❡s t❡❝♥♦❧ó❣✐❝❛s✱ ♣♦✐s s❡✉ ❝♦♥tr♦❧❡ t♦r♥❛✲s❡ ✐♥❞✐s♣❡♥sá✈❡❧ q✉❛♥❞♦ ❝é❧✉❧❛s ❝♦♠ ❝♦♥✲

✜❣✉r❛çã♦ ❞♦ t✐♣♦✲✈órt✐❝❡ sã♦ ✉s❛❞❛s ♣❛r❛ ❛r♠❛③❡♥❛♠❡♥t♦ ❞❡ ❞❛❞♦s✳ ❆t✉❛❧♠❡♥t❡ ❤á

❞✉❛s ❢♦r♠❛s ❞❡ r❡❛❧✐③❛r ♦ ❝♦♥tr♦❧❡ ❞❛ r❡✈❡rsã♦ ❞❛ ♣♦❧❛r✐❞❛❞❡✿ ♣♦r ♠❡✐♦ ❞❡ ❝♦rr❡♥t❡s

❞❡ s♣✐♥✲♣♦❧❛r✐③❛❞♦✱ q✉❡ t❡♠ ♦ ♠❡s♠♦ ❡❢❡✐t♦ ❞❡ s❡ ❛♣❧✐❝❛r ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ♥♦

♥❛♥♦❞✐s❝♦ ❬✻✻✱ ✻✼❪✱ ♦✉ ❛♣❧✐❝❛♥❞♦ ♣✉❧s♦s ❞❡ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦s ♦s❝✐❧❛tór✐♦s ✲ ❝❛♣❛③❡s ❞❡

❡①❝✐t❛r ♦ ♠♦✈✐♠❡♥t♦ ❣✐r♦tró♣✐❝♦ ✲ ❞❡ ❛♣❡♥❛s 1, 5 ♠❚ ❬✺✵✱ ✻✽❪✱ q✉❡ é ♠✉✐t♦ ♠❡♥♦r q✉❡

❛q✉❡❧❡s ✈❛❧♦r❡s ❞❡ ❝❛♠♣♦ ♥❡❝❡ssár✐♦s à ✐♥✈❡rsã♦ ❞❡ ♣♦❧❛r✐❞❛❞❡ ❞✐s❝✉t✐❞❛ ❛♥t❡r✐♦r♠❡♥t❡ ❡

❞❛ ♦r❞❡♠ ❞❡ 10−1 ❚✳ ❊♥tr❡t❛♥t♦✱ t❡♠ s✐❞♦ ♣r♦♣♦st♦ ❡ ✈❡r✐✜❝❛❞♦ ❡♠ s✐♠✉❧❛çõ❡s q✉❡ ❛ r❡✲

✈❡rsã♦ ❞❛ ♣♦❧❛r✐❞❛❞❡ ♣♦❞❡ s❡r ✐♥❞✉③✐❞❛ ♣❡❧❛ ✐♥t❡r❛çã♦ ✈órt✐❝❡✲❜✉r❛❝♦ ❬✻✾❪ ✭♥❡st❡ s❡♥t✐❞♦✱

❜✉r❛❝♦s sã♦ ❞❡❢❡✐t♦s ❛rt✐✜❝✐❛❧♠❡♥t❡ ✲ ❧✐t♦❣r❛✜❝❛♠❡♥t❡ ✲ ✐♥s❡r✐❞♦s ♥❛ ❛♠♦str❛✮ ♦✉ ♠❡s♠♦

♣♦r ♠❡✐♦ ❞❡ ✉♠❛ ✐♥t❡r❢❛❝❡ q✉❡ ❞✐✈✐❞❡ ❛ ❛♠♦str❛ ❡♠ ❞♦✐s ♠❡✐♦s ❬✼✵❪ ✭❝❛❞❛ ♠❡✐♦ ❝♦♠ ✉♠❛

✹✷

✸✳ ❊str✉t✉r❛ ❡ ❞✐♥â♠✐❝❛ ❞❡ ✈órt✐❝❡s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s

❋✐❣✉r❛ ✸✳✾✿ ❉♦✐s ♥❛♥♦❞✐s❝♦s ❞❡ P② sã♦ ✐♥✈❡st✐❣❛❞♦s✿ ❡♠ ✭❛✮ ♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ é ♠♦✈✐❞♦

♣♦r ♠❡✐♦ ❞❡ ✉♠ ❝❛♠♣♦ ❡①t❡r♥♦ ♥✉♠ ♥❛♥♦❞✐s❝♦ ❝♦♠ R = 500 ♥♠✳ ➱ ♣♦ssí✈❡❧ ✈❡r ♦s ❥✉♠♣s ♥❛

♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦✱ ♦s q✉❛✐s sã♦ ❞❡✈✐❞♦s ❛♦s ♣✐♥♥✐♥❣ s✐t❡s✳ ❊♠ ✭❜✮ é ♠♦str❛❞♦ ♦ ❞❡s❧♦❝❛♠❡♥t♦

❞♦ ❝❛r♦ç♦ ♣♦r ♠❡✐♦ ❞❡ ✉♠ ❝❛♠♣♦✱ ♣❛r❛ ✉♠ ♥❛♥♦❞✐s❝♦ ❝♦♠ R = 1, 5 µ♠ ❝♦♠ ✉♠ ❞❡❢❡✐t♦

❛rt✐✜❝✐❛❧ ❝❡♥tr❛❧✳ ❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢❡rê♥❝✐❛ ❬✶✸❪✳

❝♦♥st❛♥t❡ ❞❡ tr♦❝❛ J ❞✐❢❡r❡♥t❡✮✳ ❈♦♥❝r❡t❛♠❡♥t❡✱ t❡♠ s✐❞♦ ✈❡r✐✜❝❛❞♦ r❡❝❡♥t❡♠❡♥t❡ ❬✼✶❪

q✉❡ ♦ ♣r♦❝❡ss♦ ❞❡ r❡✈❡rsã♦ ❞♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡✱ ❛♦ ♠❡♥♦s ❡♠ ❛♠♦str❛s ♣✉r❛s✱ t♦♠❛

❧✉❣❛r s❡♠♣r❡ q✉❡ ♦ ✈órt✐❝❡ ❛t✐♥❣❡ ✉♠❛ ❝❡rt❛ ✈❡❧♦❝✐❞❛❞❡ ❝rít✐❝❛ vcr = γ√

2A/µ0✱ q✉❡

♣❛r❛ P② é ❡♥❝♦♥tr❛❞♦ s❡r ∼ 340 ♠✴s✱ s❡♥❞♦ ✉♠❛ ❣r❛♥❞❡③❛ ❝❛r❛❝t❡ríst✐❝❛ ❞♦ ♠❛t❡r✐❛❧✳

❊♥tr❡t❛♥t♦✱ s❡ ❡ ❝♦♠♦ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s ❛❢❡t❛♠ t❛❧ ❞✐♥â♠✐❝❛ ❞❡ s✇✐t❝❤✐♥❣ ❛✐♥❞❛ é

❞❡s❝♦♥❤❡❝✐❞♦✳

✸✳✷✳✷ ❉❡❢❡✐t♦s ❡str✉t✉r❛✐s ❡♠ ♥❛♥♦❞✐s❝♦s ♠❛❣♥ét✐❝♦s

❏á ♠❡♥❝✐♦♥❛♠♦s ♥❡st❡ tr❛❜❛❧❤♦ q✉❡ t❛♥t♦ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s q✉❛♥t♦ ❛q✉❡❧❡s

✐♥s❡r✐❞♦s ❛rt✐✜❝✐❛❧♠❡♥t❡ ♥❛s ❛♠♦str❛s ❡stã♦ r❡q✉❡r❡♥❞♦ ❝❛❞❛ ✈❡③ ♠❛✐s ❛t❡♥çã♦ ❡♥tr❡ ♦s

♣❡sq✉✐s❛❞♦r❡s✳ ❆♣❡s❛r ❞♦ ✉s♦ ❞❛s té❝♥✐❝❛s ♠❛✐s r❡✜♥❛❞❛s ♥❛ ♣r❡♣❛r❛çã♦ ❞❡ ❛♠♦str❛s✱ é

✐♠♣♦rt❛♥t❡ ♥♦t❛r q✉❡ ♠❡s♠♦ ♥❛q✉❡❧❛s ♠❛✐s ♣✉r❛s✱ ❞❡❢❡✐t♦s ❡stã♦ ♣r❡s❡♥t❡s✱ ❣❡r❛❧♠❡♥t❡

❞✐str✐❜✉í❞♦s ❛❧❡❛t♦r✐❛♠❡♥t❡ ♥♦ ♠❛t❡r✐❛❧✳ ❖s ❛✉t♦r❡s ❞❛ r❡❢❡rê♥❝✐❛ ❬✶✹❪ ❡st✐♠❛♠ ✉♠❛

❞❡♥s✐❞❛❞❡ ❞❡ ∼ 1011 ❞❡❢❡✐t♦s/❝♠2 ❡♠ ❞✐s❝♦s ❞❡ P②✱ ❝♦♥❝❧✉✐♥❞♦ ♥❡st❡ ♠❡s♠♦ tr❛❜❛❧❤♦

q✉❡ t❛✐s ❞❡❢❡✐t♦s ♣♦❞❡♠ ♠♦❞✐✜❝❛r s❡r✐❛♠❡♥t❡ ❛ ❞✐♥â♠✐❝❛ ❞♦ ✈órt✐❝❡✳

❋♦✐ r❡❛❧✐③❛❞♦ ♥❛ r❡❢❡rê♥❝✐❛ ❬✶✸❪ ✉♠ ❡①♣❡r✐♠❡♥t♦ ♣❛r❛ ✈❡r✐✜❝❛r ❝♦♠♦ ❛ tr❛❥❡tór✐❛

✹✸

✸✳ ❊str✉t✉r❛ ❡ ❞✐♥â♠✐❝❛ ❞❡ ✈órt✐❝❡s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s

❞♦ ✈órt✐❝❡ é ❛❢❡t❛❞❛✱ ♣♦ss✐✈❡❧♠❡♥t❡✱ ♣♦r ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s✿ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦

é ❛❞❡q✉❛❞❛♠❡♥t❡ ❛♣❧✐❝❛❞♦✱ ♣r♦♣♦r❝✐♦♥❛♥❞♦ ♦ ❞❡s❧♦❝❛♠❡♥t♦ ❞♦ ✈órt✐❝❡❀ s❡ ❛ ❛♠♦str❛

❢♦ss❡ ♣✉r❛ ✭s❡♠ ❞❡❢❡✐t♦s✮ ❡s♣❡r❛rí❛♠♦s ✉♠ ❣rá✜❝♦ ❧✐♥❡❛r ❞❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❝♦♠

♦ ❝❛♠♣♦✱ ❝♦♥❢♦r♠❡ ♠♦str❛❞♦ ♣❡❧❛ ❡q✉❛çã♦ ✸✳✶✽✳ ❊♥tr❡t❛♥t♦✱ ❝♦♠♦ ♣♦❞❡ s❡r ✈✐st♦

♥❛ ✜❣✉r❛ ✸✳✾✱ ❡♠ ✭❛✮✱ ❢♦✐ ❡♥❝♦♥tr❛❞♦ ✉♠❛ tr❛❥❡tór✐❛ ❝♦♠♣❧✐❝❛❞❛✱ ❝♦♠ r✉❣♦s✐❞❛❞❡s✱

❥✉♥t❛♠❡♥t❡ ❝♦♠ s✉❝❡ss✐✈♦s ✧♣✉❧♦s✧ ✭❥✉♠♣s✮ ♥❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦✳ ❆s ♣♦s✐çõ❡s

♦♥❞❡ ❛❝♦♥t❡❝❡♠ t❛✐s ❢❡♥ô♠❡♥♦s sã♦ ❝❤❛♠❛❞❛s ❞❡ ♣✐♥♥✐♥❣ s✐t❡s✱ ♣♦✐s ❛í ♦ ❝❛r♦ç♦ ❞♦

✈órt✐❝❡ ❡♥❝♦♥tr❛✲s❡ ❧✐❣❡✐r❛♠❡♥t❡ ✧♣r❡s♦✧✳ ❆té ❡♥tã♦ ♥ã♦ t✐♥❤❛ s✐❞♦ ♣♦ssí✈❡❧ ❛ss♦❝✐❛r

❛s ♣♦s✐çõ❡s ❞♦s ♣✐♥♥✐♥❣ s✐t❡s ❛ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s✳ ◆❛ ♣❛rt❡ ✭❜✮ ❞❛ ✜❣✉r❛ ✸✳✾ é

♠♦str❛❞♦ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ ❝❛r♦ç♦✱ t❛♠❜é♠ ♥✉♠ ♥❛♥♦❞✐s❝♦ ❞❡ P②✱ só q✉❡ ❝♦♠

✉♠ ❞❡❢❡✐t♦ ❛rt✐✜❝✐❛❧ ❝❡♥tr❛❧ ❝♦♥s✐❞❡rá✈❡❧ ❞❡ ✸✵✵ ♥♠ ❞❡ ❞✐â♠❡tr♦✳ ➱ ♦❜s❡r✈❛❞♦ q✉❡

♦ ❝❛r♦ç♦ ❡♥❝♦♥tr❛✲s❡ ✧♣r❡s♦✧ ❛♦ ❛t✐♥❣✐r ✉♠❛ ♠❛r❣❡♠ ❞♦ ❞❡❢❡✐t♦ ❡ ♣❡r♠❛♥❡❝❡ ❛ss✐♠

❞✉r❛♥t❡ ✉♠ ❝❡rt♦ ✐♥t❡r✈❛❧♦ ❞♦ ❝❛♠♣♦ ❡①t❡r♥♦✱ ❛té q✉❡ ❡st❡ ❛t✐♥❥❛ ❛ ♠❛r❣❡♠ ♦♣♦st❛ à

q✉❡ ❢♦✐ ❝❛♣t✉r❛❞♦ ♣❡❧♦ ❞❡❢❡✐t♦✱ ♦♥❞❡ ♥♦✈❛♠❡♥t❡ ♣♦❞❡ s❡r ✈❡r✐✜❝❛❞♦ ✭❞❡♣✐♥♥❡❞✮✳

❆♣r❡s❡♥t❛r❡♠♦s ♥♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦ ✉♠❛ ♣r♦♣♦st❛ q✉❡ ♠♦❞❡❧❛ t❛✐s ❞❡❢❡✐t♦s

❡str✉t✉r❛✐s ❝♦♠♦ s❡♥❞♦ ♣✉♥t✐❢♦r♠❡s✳ ❆♣❡s❛r ❞❡ t❡♦r✐❝❛♠❡♥t❡ s✐♠♣❧❡s✱ ♥♦ss♦ ♠♦❞❡❧♦

❧❡✈❛ ❛ ✉♠❛ tr❛❥❡tór✐❛ ❞♦ ❝❛r♦ç♦ q✉❡ ❡stá ❡♠ ❛❝♦r❞♦ q✉❛❧✐t❛t✐✈♦ ❝♦♠ ❛q✉❡❧❛ ❡①♣❡r✐♠❡♥t❛❧

❞❡s❝r✐t❛ ❛❝✐♠❛✳ ❚❛♠❜é♠ ❞✐s❝✉t✐♠♦s ❞♦✐s t✐♣♦s ♣♦ssí✈❡✐s ❞❡ ❞❡❢❡✐t♦s✱ q✉❡ ❛❣❡♠ ❝♦♠♦ ✉♠

i✮ ❞❡❢❡✐t♦ ❛tr❛t✐✈♦ ✭♣✐♥♥✐♥❣ s✐t❡✮ ♦✉ ii✮ r❡♣✉❧s✐✈♦ ✭s❝❛tt❡r✮✱ ❡ ❛✐♥❞❛ ♦ ✐♠♣♦rt❛♥t❡ ♣❛♣❡❧

q✉❡ ❡st❡s ♣♦❞❡♠ t❡r ❝♦♠ ❛s ♦s❝✐❧❛çõ❡s ❣✐r♦tró♣✐❝❛s ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♥❛ ❞✐♥â♠✐❝❛

❞❡ r❡✈❡rsã♦ ❞❛ ♣♦❧❛r✐❞❛❞❡ ✈✐❛ ✐♥t❡r❛çã♦ ❝♦♠ ♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡✳

✹✹

❈❛♣ít✉❧♦ ✹

▼♦❞❡❧♦ ♣❛r❛ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s ❡♠

♥❛♥♦♠❛❣♥❡t♦s

❈♦♠♦ q✉❡r❡♠♦s ❡st✉❞❛r ♦ ❡❢❡✐t♦ ❞❡ ♣❡q✉❡♥♦s ❞❡❢❡✐t♦s ✐♥tr✐♥s❡❝❛♠❡♥t❡ ❢♦r♠❛❞♦s

❡♠ ♥❛♥♦❞✐s❝♦s ❝♦♠ ❛ ❝♦♥✜❣✉r❛çã♦ ❞❡ ✈órt✐❝❡✱ ♣r❡❝✐s❛♠♦s r❡♣r❡s❡♥t❛r t❛✐s ❡str✉t✉r❛s ❡♠

♥♦ss♦ ♠♦❞❡❧♦✳ ❋❛r❡♠♦s ✐ss♦ ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ ❛♣r♦①✐♠❛çã♦✱ s✉♣♦♥❞♦ q✉❡ t❛✐s ❡str✉t✉r❛s

s❡❥❛♠ ♣✉♥t✐❢♦r♠❡s ❡ r❡♣r❡s❡♥t❛❞❛s ♣♦r ✉♠ ♣♦t❡♥❝✐❛❧ ❞♦ t✐♣♦ λiδ3(~x−~xi)✱ ❡♠ q✉❡ δ3(~x)

é ❛ ✧❢✉♥çã♦✲❞❡❧t❛✧ ❞❡ ❉✐r❛❝✶✱ λi é ✉♠ ♣❛râ♠❡tr♦ r❡❛❧ ❝♦♠ ❞✐♠❡♥sã♦ ❞❡ ♠3✱ t❛❧ q✉❡

λi > 0 ✐♠♣❧✐❝❛ ♥✉♠ ❞❡❢❡✐t♦ r❡♣✉❧s✐✈♦ ❡ λi < 0 ♥✉♠ ❞❡❢❡✐t♦ ❛tr❛t✐✈♦✱ ❝♦♠♦ ✈❡r❡♠♦s ❛

s❡❣✉✐r✳ ❱❛r✐❛♥❞♦ λi ♣♦❞❡♠♦s ❝♦♥tr♦❧❛r ❛ ✐♥t❡♥s✐❞❛❞❡ ❞❛ ✐♥t❡r❛çã♦ ✲ ❡ss❡♥❝✐❛❧♠❡♥t❡ ❞♦

❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ ✲ ❝♦♠ ♦ ❞❡❢❡✐t♦ ❧♦❝❛❧✐③❛❞♦ ♥❛ ♣♦s✐çã♦ ~xi✳

P♦r t❡r♠♦s ❛ss✉♠✐❞♦ ✉♠ ❝❛rát❡r ♣✉♥t✐❢♦r♠❡ ♣❛r❛ ♦s ❞❡❢❡✐t♦s✱ ❛✜r♠❛♠♦s q✉❡

❡st❡s ♥ã♦ ✈ã♦ ❛❧t❡r❛r ❛♣r❡❝✐❛✈❡❧♠❡♥t❡ ❛ ❡♥❡r❣✐❛ ♠❛❣♥❡t♦stát✐❝❛✱ ♣♦✐s t❛✐s ❞❡❢❡✐t♦s ♣♦s✲

s✉❡♠ ár❡❛ ❡ ✈♦❧✉♠❡ ✲ ♣r❛t✐❝❛♠❡♥t❡ ✲ ♥✉❧♦s ❡✱ ♣♦r ❝♦♥s❡❣✉✐♥t❡✱ ♥ã♦ ✐♥❞✉③✐rã♦ ❝❛r❣❛s

♠❛❣♥ét✐❝❛s s✉♣❡r✜❝✐❛✐s ♦✉ ✈♦❧✉♠étr✐❝❛s ✭q✉❡ ❝♦♠♦ ✈✐♠♦s ❞❛ ❞✐s❝✉ssã♦ ❢❡✐t❛ ♥❛ s❡çã♦

✷✳✸✳✷ sã♦ ❛s r❡s♣♦♥sá✈❡✐s ♣♦r t❛❧ ✐♥t❡r❛çã♦✮✳ ❚❡♠♦s✱ t❛♠❜é♠✱ q✉❡ t❛❧ ❞❡❢❡✐t♦ ♥ã♦ ♠✉❞❛

s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡ ❛ ❝♦♥✜❣✉r❛çã♦ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ♥❛ r❡❣✐ã♦ à s✉❛ ✈♦❧t❛✳ ❊♥tr❡t❛♥t♦✱

❛ ❡♥❡r❣✐❛ ❞❡ tr♦❝❛ é ♠♦❞✐✜❝❛❞❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡✱ ♦♥❞❡ ❛ ❞❡♥✲

✶❆ ❢✉♥çã♦ δ✲❉✐r❛❝ ✸❉ é ❞❡✜♥✐❞❛ ❞❡ ❢♦r♠❛ q✉❡✿

∫

V

f(~x) δ3(~x− ~x0) d3x =

{

f(~x0)✱ s❡ ~x0 ∈ V

0✱ ❝❛s♦ ❝♦♥trár✐♦✳

✹✺

✹✳ ▼♦❞❡❧♦ ♣❛r❛ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s

s✐❞❛❞❡ ❞❡ t❛❧ ✐♥t❡r❛çã♦ é ♠❛✐s ❛♣r❡❝✐á✈❡❧✳ ◆❛ r❡❣✐ã♦ ❡①t❡r♥❛ ❛♦ ❝❛r♦ç♦✱ ❛ ✐♥t❡r❛çã♦

❞❡❢❡✐t♦✲✈órt✐❝❡ é ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ❝♦♥st❛♥t❡ ❡ ❝❡rt❛♠❡♥t❡ é ♠❡♥♦s ♣r♦♥✉♥❝✐❛❞❛ ❞♦

q✉❡ ❛ ❞❡❢❡✐t♦✲❝❛r♦ç♦✳ ❆ss✐♠✱ ❝♦♥❝❧✉í♠♦s q✉❡ ♦ t❡r♠♦ ✐♠♣♦rt❛♥t❡ ❛ s❡ ❝❛❧❝✉❧❛r ♥❛ ✈❛r✐✲

❛çã♦ ❞❛ ❡♥❡r❣✐❛ t♦t❛❧ s❡♠♣r❡ q✉❛♥❞♦ ❤á ✐♥t❡r❛çã♦ ❝♦♠ ✉♠ ♦✉ ♠❛✐s ❞❡❢❡✐t♦s✱ é ❛q✉❡❧❡

r❡❧❛❝✐♦♥❛❞♦ à ✐♥t❡r❛çã♦ ❞❡❢❡✐t♦✲❝❛r♦ç♦✳ ❯♠❛ ✈❡③ q✉❡ ♦ ✈órt✐❝❡ ✈❛✐ ✐♥t❡r❛❣✐r ❛♣❡♥❛s ❝♦♠

❛q✉❡❧❡s ❞❡❢❡✐t♦s ❧♦❝❛❧✐③❛❞♦s ♥❛ ❢❛❝❡ s✉♣❡r✐♦r ❞♦ ♥❛♥♦❞✐s❝♦✱ t❡♠♦s ❡ss❡♥❝✐❛❧♠❡♥t❡ ✉♠

♣r♦❜❧❡♠❛ ❜✐❞✐♠❡♥s✐♦♥❛❧✱ t❛❧ q✉❡ λi é ♠❡❞✐❞♦ ❡♠ ✉♥✐❞❛❞❡s ❞❡ ár❡❛✳ ❱❛❧♦r❡s tí♣✐❝♦s ❞♦

♣❛râ♠❡tr♦ λi ❞❡✈❡♠ s❡r t♦♠❛❞♦s ❞❡ ❡①♣❡r✐♠❡♥t♦s ❡✱ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ t❡ór✐❝♦✱ sã♦

❜❛st❛♥t❡ ♣❡q✉❡♥♦s q✉❛♥❞♦ ❝♦♠♣❛r❛❞♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ❝♦♠ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ tr♦❝❛✱ ❞❡

❢♦r♠❛ t❛❧ q✉❡ λi ≪ l2ex✱ ❝♦♠♦ ✈❡r❡♠♦s ❛❞✐❛♥t❡ ❡♠ ♥♦ss❛ ❞✐s❝✉ssã♦✳ ❊♥tã♦✱ ❡♥q✉❛♥t♦

♦❝♦rr❡ ❛ ✐♥t❡r❛çã♦ ❞❡❢❡✐t♦✲❝❛r♦ç♦✱ ♦ ♣❛râ♠❡tr♦ λi ✐♥❞✐❝❛ ❛ ✐♥t❡♥s✐❞❛❞❡ ✲ ♣♦r ♠❡✐♦ ❞♦

s❡✉ ♠ó❞✉❧♦ ✲ ❡ t❛♠❜é♠ s❡ ♦ ❞❡❢❡✐t♦ é ❛tr❛t✐✈♦ ♦✉ r❡♣✉❧s✐✈♦ ✲ ♣♦r ♠❡✐♦ ❞❡ s❡✉ s✐♥❛❧✳

P♦rt❛♥t♦✱ ✉♠ ❞❡❢❡✐t♦ ❧♦❝❛❧✐③❛❞♦ ❡♠ ~xi ❛❧t❡r❛ ❛ ❡♥❡r❣✐❛ ❞❡ tr♦❝❛✱ ❡q✉❛çã♦ ✭✸✳✶✸✮✱ ♣♦r

✉♠❛ q✉❛♥t✐❞❛❞❡✿

∆Etr = Etr − Etr

∣∣∣def

= −Etr(~xi)ξi✱ ✭✹✳✶✮

♣♦✐s ❡st❛♠♦s s✉♣♦♥❞♦ q✉❡ ❛ ❡♥❡r❣✐❛ ❞❡ ✐♥t❡r❛çã♦ ❛ss♦❝✐❛❞❛ ❛♦ ❞❡❢❡✐t♦ ❡♠ xi✱ ♣♦r r❛③õ❡s

❞✐s❝✉t✐❞❛s ❛❝✐♠❛✱ é ♣r♦♣♦r❝✐♦♥❛❧ à ❡♥❡r❣✐❛ ❞❡ tr♦❝❛ ♥♦ ❧♦❝❛❧ ❡♠ q✉❡ ♦ ❞❡❢❡✐t♦ s❡ ❡♥❝♦♥tr❛✱

s❡♥❞♦ t❛❧ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ❞❛❞❛ ♣♦r✿

ξi = Gi(~r − ~xi)

∫

λi δ2(~x ′ − ~xi)

3∑

α=1

[

~∇mα(~x′)]2

d2x′✳ ✭✹✳✷✮

❖❜s❡r✈❡ q✉❡ ❛ ❢✉♥çã♦ δ3(~r−~xi) ❞❡♥tr♦ ❞❛ ✐♥t❡❣r❛❧ ♠✉❧t✐♣❧✐❝❛ ♦ t❡r♠♦ [~∇mα(~x′)]2✱

t❡r♠♦ ❡st❡ ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ à ❡♥❡r❣✐❛ ❞❡ tr♦❝❛ ❡♠ ~x ′✳ ◆❛ ❡①♣r❡ssã♦ ❛❝✐♠❛✱ Gi

é ✉♠❛ ❢✉♥çã♦ q✉❡ ❧❡✈❛ ❡♠ ❝♦♥t❛ ❝♦♠♦ ♦ ❞❡❢❡✐t♦ ❧♦❝❛❧✐③❛❞♦ ❡♠ ~xi ✐♥t❡r❛❣❡ ❝♦♠ ♦ ❝❛r♦ç♦

❞♦ ✈órt✐❝❡✳ ❆♣❡s❛r ❞❡ ♦ ♣❡r✜❧ ❞♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ s❡r ❞✐❢í❝✐❧ ❞❡ s❡r ❞❡t❡r♠✐♥❛❞♦ ♥❛

♣rát✐❝❛ ✭♥ã♦ ❤á ❡①♣r❡ssõ❡s ❛♥❛❧ít✐❝❛s ❡①❛t❛s✮✱ ❢❛r❡♠♦s✱ ♣❛r❛ t❛❧✱ ✉♠❛ ❡s❝♦❧❤❛ r❛③♦á✈❡❧

❡ ❝♦♥s✐❞❡r❛r❡♠♦s ✉♠❛ ❢✉♥çã♦ ❣❛✉ss✐❛♥❛✿

Gi(~r − ~xi) = e−α(~r−~xi)2

✳ ✭✹✳✸✮

❆♣❡s❛r ❞❡ss❛ ❡s❝♦❧❤❛✱ ♦✉tr❛s ❢✉♥çõ❡s ♣♦❞❡r✐❛♠ s❡r ❡s❝♦❧❤✐❞❛s ♣❛r❛ s✐♠✉❧❛r ❡st❛

✐♥t❡r❛çã♦✱ ❝♦♠♦ ❛ ❢✉♥çã♦ θ ❞❛❞❛ ♣❡❧❛ ❡q✉❛çã♦ ✭✸✳✶✵✮✳ ❖❜s❡r✈❡ t❛♠❜é♠ q✉❡ ❤á ✉♠

✹✻

✹✳ ▼♦❞❡❧♦ ♣❛r❛ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s

❋✐❣✉r❛ ✹✳✶✿ ■❧✉str❛çã♦ ❞❡ ❝♦♠♦ ♦❝♦rr❡ ❛ ✐♥t❡r❛çã♦ ❞❡❢❡✐t♦✲❝❛r♦ç♦✿ ✐♥t❡r❛çã♦ é ♣❡q✉❡♥❛ ♥❛

❜♦r❞❛ ❞♦ ❝❛r♦ç♦✱ ♦♥❞❡ ❛ ❡♥❡r❣✐❛ ❞❡ tr♦❝❛ é ❜❛✐①❛❀ ❝♦♥❢♦r♠❡ ♦ ❝❛r♦ç♦ ❞❡s❧♦❝❛✲s❡ ❡♠ ❞✐r❡çã♦

❛♦ ❞❡❢❡✐t♦ ❡♠ ~xi✱ ❛ ✐♥t❡r❛çã♦ ❛✉♠❡♥t❛✱ ❛t✐♥❣✐♥❞♦ s❡✉ ♠á①✐♠♦ ❡♠ ~r = ~xi✳ ❖ ❡❢❡t✐✈♦ ✐♥t❡r✈❛❧♦

❞❡ ❛❧❝❛♥❝❡ é δ ∼ 10 ♥♠ ❡ ❡stá ❡♠ ❜♦♠ ❛❝♦r❞♦ ❝♦♠ ❡①♣❡r✐♠❡♥t♦s✳ ❆ ❢♦r♠❛ ✭❣❛✉ss✐❛♥❛✮

❞❛ ✐♥t❡r❛çã♦ ♣♦❞❡ s❡r ❝♦♥tr♦❧❛❞❛ ✈❛r✐❛♥❞♦✲s❡ ♦ ♣❛râ♠❡tr♦ α✱ q✉❡ t♦♠❛♠♦s ❝♦♠♦ α−1 =

2, 5× 104R2✱ ❛ ✜♠ ❞❡ ♦❜t❡r ♦ ❛❧❝❛♥❝❡ ❞❡s❡❥❛❞♦ ♣❛r❛ ❛ ✐♥t❡r❛çã♦✳

♣❛râ♠❡tr♦ ❧✐✈r❡ α✱ ❝♦♠ ❞✐♠❡♥sã♦ ❞❡ ♠2✳ ❆ ♣r❡s❡♥ç❛ ❞❡st❡ t❡r♠♦ ❛ss❡❣✉r❛ q✉❡ ❛

✐♥t❡r❛çã♦ ❞❡❢❡✐t♦✲❝❛r♦ç♦ é✱ ❞❡ ❝❡rt❛ ❢♦r♠❛✱ s✉❛✈❡✳ ❖❜s❡r✈❛♥❞♦ ❛ ✜❣✉r❛ ✹✳✶✱ ✈❡♠♦s q✉❡

t❛❧ ✐♥t❡r❛çã♦ é ✐♥✐❝✐❛❞❛✱ ❡❢❡t✐✈❛♠❡♥t❡✱ q✉❛♥❞♦ ❛ ♣❡r✐❢❡r✐❛ ❞♦ ❝❛r♦ç♦ ❛❧❝❛♥ç❛ ❛ ♣♦s✐çã♦

~xi ❞❡ ❞❡t❡r♠✐♥❛❞♦ ❞❡❢❡✐t♦✱ ❡ ✈❛✐ ❛✉♠❡♥t❛♥❞♦ ♣r♦❣r❡ss✐✈❛♠❡♥t❡ ❛té ✉♠ ♠á①✐♠♦✱ ♦♥❞❡

♦ ❝❡♥tr♦ ❞♦ ❝❛r♦ç♦ ❛t✐♥❣❡ ♦ ❞❡❢❡✐t♦ ❡ s✉❛s ♣♦s✐çõ❡s ❝♦✐♥❝✐❞❡♠✱ ~r = ~xi✳ ❆♣ós ✐ss♦✱ ❛

✐♥t❡r❛çã♦ ✈♦❧t❛ ❛ ❞✐♠✐♥✉✐r ❛té ❡①t✐♥❣✉✐r✲s❡ q✉❛♥❞♦ ❡♥❝♦♥tr❛ ❛ ❜♦r❞❛ ♣♦st❡r✐♦r ❞♦ ❝❛r♦ç♦✳

P♦❞❡♠♦s ❡①tr❛✐r t❛♠❜é♠ ❞❡ Gi ✉♠ ✐♥t❡r✈❛❧♦ r❛③♦á✈❡❧ ♣❛r❛ ♦ ❛❧❝❛♥❝❡ ❞❡ t❛❧ ✐♥t❡r❛çã♦

q✉❡✱ ♣❛r❛ ♥♦ss♦ ♠♦❞❡❧♦✱ é ❞❛❞♦ ♣♦r δ = 2l0/e ≈ 10 ♥♠✱ ♦ q✉❡ ❡stá ❡♠ ❜♦♠ ❛❝♦r❞♦

❝♦♠ r❡s✉❧t❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❬✶✸✱✶✹❪✱ s❡ ✉s❛r♠♦s ♦s ✈❛❧♦r❡s ❞❡ l0 ∼ 8− 15 ♥♠✱ ✈❛❧♦r❡s

tí♣✐❝♦s ♣❛r❛ ♥❛♥♦❞✐s❝♦s ❞❡ P②✳ ❉❡st❛ ❢♦r♠❛✱ ✈❡♠♦s q✉❡ ♦ t❡r♠♦ ❞❛ ✐♥t❡❣r❛❧ ♥❛ ❡q✉❛çã♦

✭✹✳✷✮✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ♣r❡s❡♥ç❛ ❞❡ Gi✱ ♥♦s ❞✐③ q✉❡ ξi ❧❡✈❛ ❡♠ ❝♦♥t❛ q✉❛♥t✐t❛t✐✈❡♠❡♥t❡

❜❡♠ ♦ ❡❢❡✐t♦ ❞❛ ♣r❡s❡♥ç❛ ❞❡ ✉♠ ❞❡❢❡✐t♦ ❡♠ ~xi✳

❚❡♠♦s✱ ♣r✐♠❡✐r❛♠❡♥t❡✱ q✉❡ ❝❛❧❝✉❧❛r ❛ q✉❛♥t✐❞❛❞❡∑

α(~∇mα)

2 q✉❡ ❛♣❛r❡❝❡ ♥❛

❞❡✜♥✐çã♦ ❞❡ ξi ♥❛ ❡q✉❛çã♦ ✭✹✳✷✮✳ ❈♦♠ ❛ ❡q✉❛çã♦ ✭✸✳✾✮ ❡♠ ♠ã♦s✱ ♣♦❞❡♠♦s✱ ❞❡ ❢❛t♦✱

✹✼

✹✳ ▼♦❞❡❧♦ ♣❛r❛ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s

♦❜t❡r s✉❛ ❡①♣r❡ssã♦ ♣❛r❛ ✉♠ ♥❛♥♦❞✐s❝♦ ❝♦♠ ❝♦♥✜❣✉r❛çã♦ ❞♦ t✐♣♦ ✈órt✐❝❡✱ ✈✐❛ ✉s♦ ❞❛s

❡q✉❛çõ❡s ♣❛r❛ φ ❡ θ✱ ❝♦♥❢♦r♠❡ ✈✐♠♦s ❡✱ ❛ ♣❛rt✐r ❞❡ t❛✐s✱ ❡♥❝♦♥tr❛r ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛

ξi ♥♦ ❝❛s♦ ❞❡ ✉♠ ✈órt✐❝❡ ❝♦♠ ✈♦rt✐❝✐❞❛❞❡ Q = 1 ❡ ♣♦❧❛r✐❞❛❞❡ p = 1✳ ▼❛✐s ✉♠❛ ✈❡③✱

❝♦♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❢❡✐t♦ ♥♦ ❆♣ê♥❞✐❝❡ ❈✱ ❡♥❝♦♥tr❛♠♦s q✉❡✿

ξi =(2 l20 − x2i )

2 + 4 l40l40 (2 l

20 − x2i )

λi Gi(~r − ~xi)✳ ✭✹✳✹✮

❆❣♦r❛ q✉❡ s❛❜❡♠♦s ❝♦♠♦ ♦ ❞❡❢❡✐t♦ ❛❢❡t❛ ❛ ❡♥❡r❣✐❛ ✭❡ss❡♥❝✐❛❧♠❡♥t❡ ❛ ❞❡ tr♦❝❛✮ ❞❡

✉♠ ♥❛♥♦❞✐s❝♦ ❝♦♠ ❝♦♥✜❣✉r❛çã♦ ❞♦ t✐♣♦ ✈órt✐❝❡✱ ❡s❝r❡✈❡♠♦s ❛ ❡♥❡r❣✐❛ t♦t❛❧ ♥♦r♠❛❧✐③❛❞❛

✭wdef (s) = Wdef (s)/µ0πR2LM2

s ✮ ❞❡st❛ ❝♦♥✜❣✉r❛çã♦ ❝♦♠♦✿

wdef (s) = 2πF1(β)s2 +

1

2

(lexR

)2

ln(1− s2) [1− ξi]− hext s+O(s3)✳ ✭✹✳✺✮

❖❜s❡r✈❡ q✉❡ ❛ ❡♥❡r❣✐❛ ♥♦r♠❛❧✐③❛❞❛ ❛❝✐♠❛ é r❡❧❛t✐✈❛ à ♦r✐❣❡♠✱ ❡♠ ~s = ~r/R = ~0✱

♣♦✐s ❡❧❛ ♥ã♦ q✉❛♥t✐✜❝❛ ❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ✉♠ ❞❡❢❡✐t♦ ❡ ♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ ♥❡st❡ ❧♦❝❛❧ ✲

✈❡❥❛ q✉❡ s = 0 ❡♠ ✭✹✳✺✮ r❡s✉❧t❛ ♥✉♠❛ ✐♥t❡r❛çã♦ ❞❡❢❡✐t♦✲❝❛r♦ç♦ ♥✉❧❛ ✲✱ ♠❛s t❛❧ ✐♥t❡r❛çã♦

♣♦❞❡✱ ❞❡ ❢❛t♦✱ s❡r ❝❛❧❝✉❧❛❞❛ ❝♦♠♦ ❞❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ♣❛r❛ ✉♠ ❞❡❢❡✐t♦ ❧♦❝❛❧✐③❛❞♦ ❡♠

~xi 6= ~0✳

◆♦✈❛♠❡♥t❡✱ t♦♠❛♥❞♦ ❛ ❡①♣❛♥sã♦ ❡♠ ❚❛②❧♦r ❡♠ t♦r♥♦ ❞❡ s = 0 ♣❛r❛ ♦ t❡r♠♦ ❞❡

tr♦❝❛ ❡ ✜❝❛♥❞♦ ❝♦♠ t❡r♠♦s ❛té s2✱ ♣♦❞❡♠♦s✱ ❞❛ ♠❡s♠❛ ♠❛♥❡✐r❛ ❢❡✐t❛ ♥❛ s❡çã♦ ✸✳✷✳✶✱

❡♥❝♦♥tr❛r ❛ ♥♦✈❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ ♥❛ ♣r❡s❡♥ç❛ ❞❡ ❞❡❢❡✐t♦s✿

seq =xeqR

=−q hext

4πF1(β)− (1− ξi) (lex/R)2✱ ✭✹✳✻✮

♦♥❞❡ ♥♦✈❛♠❡♥t❡ t♦♠❛♠♦s ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❡①t❡r♥♦ ♣❛r❛❧❡❧♦ ❛♦ ♣❧❛♥♦ ❞♦ ♥❛♥♦❞✐s❝♦✱

♥❛ ❞✐r❡çã♦ y✱ hext = (0, hext)✳ ❖❜✈✐❛♠❡♥t❡✱ ❛s ❡q✉❛çõ❡s ✭✹✳✺✮ ❡ ✭✹✳✻✮ r❡❝♦❜r❡♠ ❛q✉❡❧❛s

s❡♠ ❛ ♣r❡s❡♥ç❛ ❞❡ ❞❡❢❡✐t♦s ✭ξi = 0✮✱ ✭✸✳✶✼✮ ❡ ✭✸✳✶✽✮ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

✹✳✶ ❆♣❧✐❝❛çã♦ ❞♦ ♠♦❞❡❧♦ ❡ r❡s✉❧t❛❞♦s

❈♦♥s✐❞❡r❛♠♦s ❛❣♦r❛ ✉♠ ♥ú♠❡r♦ ❞❡ ❞❡❢❡✐t♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠❛

❧✐♥❤❛ r❡t❛✳ ❆♦ ❛♣❧✐❝❛r♠♦s ❡♠ s❡❣✉✐❞❛ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ♥❛ ❞✐r❡çã♦ y ♣♦s✐t✐✈❛✱ ♦

❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ ♠♦✈❡rá ♥♦ s❡♥t✐❞♦ ♥❡❣❛t✐✈♦ ✭q = +1✮ ♦✉ ♣♦s✐t✐✈♦ ✭q = −1✮✱ ❡♥❝♦♥✲

tr❛♥❞♦ ❡✱ ♣♦ss✐✈❡❧♠❡♥t❡✱ ✐♥t❡r❛❣✐♥❞♦ ❝♦♠ ♦s ❞❡❢❡✐t♦s ❛♦ ❧♦♥❣♦ ❞♦ ❝❛♠✐♥❤♦✳ ❉❡st❛ ❢♦r♠❛✱

✹✽

✹✳ ▼♦❞❡❧♦ ♣❛r❛ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s

❡st❛r❡♠♦s ❛♣t♦s ❛ ♠♦str❛r ❝♦♠♦ ♥♦ss♦ ♠♦❞❡❧♦ ❞❡s❝r❡✈❡ ❛ ✐♥t❡r❛çã♦ ❞❡❢❡✐t♦✲❝❛r♦ç♦ ♠❛✐s

❝❧❛r❛♠❡♥t❡✱ t♦r♥❛♥❞♦ ♣♦ssí✈❡❧ ✉♠❛ ❝♦♠♣❛r❛çã♦ ❝♦♠ ♦s r❡s✉❧t❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ♠❡♥✲

❝✐♦♥❛❞♦s ♥❛ s❡çã♦ ✸✳✷✳✷✳ P❛r❛ t❛♥t♦✱ ❡ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ ✐♥s❡r✐♠♦s ✷✵ ❞❡❢❡✐t♦s

♣✉♥t✐❢♦r♠❡s ❛♦ ❧♦♥❣♦ ❞♦ ❡✐①♦ x✱ ❝♦♠ ❞✐❢❡r❡♥t❡s ♣❛râ♠❡tr♦s ❞❡ ✐♥t❡r❛çã♦ λi ❡✱ ❝♦♠♦

❞✐ss❡♠♦s✱ ❛♥❛❧✐s❛♠♦s s✉❛ ❞✐♥â♠✐❝❛ s♦❜ ❛ ♣r❡s❡♥ç❛ ❞❡ ~Hext✳ ❉❡✈✐❞♦ à r❡❧❛t✐✈❛ ❝♦♠♣❧❡✲

①✐❞❛❞❡ ♥❛ ❞❡t❡r♠✐♥❛çã♦ ❡①❛t❛ ❞❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡✱ ✉t✐❧✐③❛♠♦s

✉♠ ♣r♦❣r❛♠❛ ❡s❝r✐t♦ ♥❛ ❧✐♥❣✉❛❣❡♠ ❋❖❘❚❆◆ ✾✵✳ ❇❛s✐❝❛♠❡♥t❡✱ t❛❧ ♣r♦❣r❛♠❛ t❡♠ ❝♦♠♦

❡♥tr❛❞❛s ❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❛t✉❛❧ ❞♦ ❝❛r♦ç♦✱ ❜❡♠ ❝♦♠♦ ♦ ✈❛❧♦r ❞❡ | ~Hext| ❡ ♦s ❛s✲

♣❡❝t♦s r❡❧❛t✐✈♦s ❛♦s ❞❡❢❡✐t♦s ✭~xi✱ λi ❡ α✮❀ ❛ s❡❣✉✐r✱ é ❝❛❧❝✉❧❛❞♦ ❛ ✐♥t❡r❛çã♦ ❞♦ ❞❡❢❡✐t♦

❝♦♠ ♦ ❝❛r♦ç♦ ✲ ♠❡❞✐❛♥t❡ ♦ ❝á❧❝✉❧♦ ❞❡ G(~r − ~xi) ❡ ❞❡ ξi ✲ ❡✱ ❛ ♣❛rt✐r ❞❛í✱ ❡♥❝♦♥tr❛✲s❡

❛ ♥♦✈❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦❀ ♦ ♣r♦❝❡ss♦ s❡ r❡♣❡t❡ ❝♦♠ t❛❧ ♣♦s✐çã♦ ❝♦♠♦ ♥♦✈❛ ❡♥tr❛❞❛✳

❚❛❧ ♣❛ss♦ é ❢❡✐t♦ ✶✵✵ ✈❡③❡s ❡♠ ❝❛❞❛ ❞❡❢❡✐t♦ ❡✱ ❡♠ ❝❛❞❛ ✉♠ ❞❡❧❡s✱ ❣r❛✈❛♠♦s ❛ ♣♦s✐çã♦

❞❡ ❡q✉✐❧í❜r✐♦ ❞♦ ✈órt✐❝❡✱ seq✳ ❯♠ ❣rá✜❝♦ ❞❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ✈❡rs✉s ♦ ❝❛♠♣♦ ♠❛❣✲

♥ét✐❝♦ ❡①t❡r♥♦ hext ♣❛r❛ ✉♠ ✈órt✐❝❡ ❝♦♠ q✉✐r❛❧✐❞❛❞❡ q = +1 é ❛♣r❡s❡♥t❛❞♦ ♥❛ ✜❣✉r❛

✹✳✷✳

❖❜s❡r✈❡ q✉❡ ❡♠ ✈✐rt✉❞❡ ❞❛ ✐♥t❡r❛çã♦ ❞♦s ❞❡❢❡✐t♦s ❝♦♠ ♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡✱ s❡✉

❝❛♠✐♥❤♦ ♥ã♦ é ♠❛✐s ✉♠❛ ❧✐♥❤❛ r❡t❛✱ ❝♦♠♦ ❡♥❝♦♥tr❛❞♦ ♣❛r❛ ✉♠❛ ❛♠♦str❛ ♣✉r❛✱ ❝♦♠♦

❞❡s❝r✐t♦ ♣❡❧❛ ❡q✉❛çã♦ ✭✸✳✶✽✮✳ P♦❞❡♠♦s ♣❡r❝❡❜❡r ❛♦ ♦❜s❡r✈❛r♠♦s ❛ ✜❣✉r❛ ✹✳✷✱ q✉❡ ❛

tr❛❥❡tór✐❛ ❛♣r❡s❡♥t❛ ❝♦♥s✐❞❡rá✈❡✐s r✉❣♦s✐❞❛❞❡s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❡♠ t♦r♥♦ ❞❛ ♣♦s✐çã♦ ~xi

❞❡ ❝❛❞❛ ❞❡❢❡✐t♦✱ q✉❡ t♦♠❛♠♦s ❝♦♠♦ seq = |~xi|/R = −0, 19, −0, 17, . . . , 0, 17, 0, 19✳

P♦❞❡♠♦s ♥♦t❛r t❛♠❜é♠ ❛tr❛✈és ❞❛ ✜❣✉r❛ ✹✳✷✱ q✉❡ ❤á ❞♦✐s t✐♣♦s ❞❡ ❞❡❢❡✐t♦s✿ i✮ r❡♣❡♥t✐✲

♥❛s ♠✉❞❛♥ç❛s ❡♠ seq q✉❡ ♦❝♦rr❡♠✱ ♣r❛t✐❝❛♠❡♥t❡ ♣❛r❛ ♦ ♠❡s♠♦ ❝❛♠♣♦ hext✱ ♦ q✉❡ ♥♦s

♠♦str❛ q✉❡ ♦ ❝❛r♦ç♦ ❡stá ✐♥t❡r❛❣✐♥❞♦ ❝♦♠ ✉♠ ❞❡❢❡✐t♦ r❡♣✉❧s✐✈♦ ✭λi > 0✮✳ ■st♦ é✱ ❛tr✐❜✉í✲

♠♦s t❛✐s ♠✉❞❛♥ç❛s ❛❜r✉♣t❛s ❡ss❡♥❝✐❛❧♠❡♥t❡ à ✐♥t❡r❛çã♦ ❝❛r♦ç♦✲❞❡❢❡✐t♦✱ ♣♦✐s ❤á ✉♠❛

♣❡q✉❡♥❛ ✈❛r✐❛çã♦ ❡♠ hext q✉❡ ♥ã♦ é ❝❛♣❛③ ❞❡ ❛❢❡t❛r ❝♦♥s✐❞❡r❛✈❡❧♠❡♥t❡ ♦ ♠♦✈✐♠❡♥t♦

✭t❛✐s ❞❡❢❡✐t♦s sã♦ r❡♣r❡s❡♥t❛❞♦s ♣♦r s❡t❛s ♣r❡t❛s✮❀ ii✮ ♦❜s❡r✈❛♠♦s ❛✐♥❞❛ ♥❛ ✜❣✉r❛ ✹✳✷

q✉❡ ❡♠ ♦✉tr♦s ❧✉❣❛r❡s seq ♣r❛t✐❝❛♠❡♥t❡ ♥ã♦ ✈❛r✐❛✱ ❛♦ ♣❛ss♦ q✉❡ hext ♠✉❞❛ ❝♦♥s✐❞❡r❛✈❡❧✲

♠❡♥t❡✱ ❡✈✐❞❡♥❝✐❛♥❞♦ ✉♠❛ ✐♥t❡r❛çã♦ ❞♦ ❝❛r♦ç♦ ❝♦♠ ✉♠ ❞❡❢❡✐t♦ ❞♦ t✐♣♦ ❛tr❛t✐✈♦ ✭λi < 0✮✳

❈♦♥s✐❞❡r❛♠♦s t❛❧ ❞❡❢❡✐t♦ ❝♦♠♦ ❛tr❛t✐✈♦ ♣♦✐s ❡❧❡ ♠❛♥té♠ ✧♣r❡s♦✧ ✭seq ≈ ❝♦♥st❛♥t❡✮ ♦

❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ ❞✉r❛♥t❡ ✉♠ ✐♥t❡r✈❛❧♦ ❝♦♥s✐❞❡rá✈❡❧ ❞♦ ❝❛♠♣♦ ❡①t❡r♥♦ hext✳ ❆s s✐t✉✲

✹✾

✹✳ ▼♦❞❡❧♦ ♣❛r❛ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s

❋✐❣✉r❛ ✹✳✷✿ ❖ ❝❛♠✐♥❤♦ ❡♠ ✈❡r♠❡❧❤♦ é ❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡✱ seq✱ ❡♠

❢✉♥çã♦ ❞♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❡①t❡r♥♦✱ hext✳ ❱❡♠♦s ❛í ❝♦♠♦ t❛❧ ❝❛♠✐♥❤♦ é ❛❢❡t❛❞♦ ♣❡❧❛ ✐♥❝❧✉sã♦

❞❡ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s ✐♥trí♥s❡❝♦s ♥❛ ❛♠♦str❛✳ ❯♠❛ ❝♦♠♣❛r❛çã♦ ❝♦♠ ❛ ✜❣✉r❛ ✸✳✾ ♠♦str❛ q✉❡

♥♦ss♦ r❡s✉❧t❛❞♦ ❡stá ❡♠ ❜♦♠ ❛❝♦r❞♦ q✉❛❧✐t❛t✐✈♦ ❝♦♠ ❛q✉❡❧❡ ❡①♣❡r✐♠❡♥t❛❧ ❬✶✸❪✳ ■♥s❡r✐♠♦s ✷✵

❞❡❢❡✐t♦s ♥❛ ❛♠♦str❛✱ ❝♦♠ ❡s♣❛ç❛♠❡♥t♦ ❞❡ 2 l0 ❡♥tr❡ ❡❧❡s✳ ➱ ♠♦str❛❞♦ ♥♦ ❣rá✜❝♦ ❞❡❢❡✐t♦s ❞♦

t✐♣♦ ❛tr❛t✐✈♦ ✭λi < 0✮ ❡ r❡♣✉❧s✐✈♦ ✭λi > 0✮✱ ✐♥❞✐❝❛❞♦s ♣♦r s❡t❛s ❛③✉✐s ❡ ♣r❡t❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❖s ✈❛❧♦r❡s ❞❡ λi q✉❡ r❡♣r♦❞✉③❡♠ t❛✐s ✐♥t❡r❛çõ❡s ❡stã♦ ♥✉♠❛ ❢❛✐①❛ |λi| ∼ 10−25 − 10−26 ♠2✳

❛çõ❡s i✮ ❡ ii✮ q✉❡ ❛❝❛❜❛♠♦s ❞❡ ❞❡s❝r❡✈❡r ❡stã♦ ✐❧✉str❛❞❛s ♥❛ ✜❣✉r❛ ✹✳✸✳ ❆♦ ❝♦♠♣❛r❛r♠♦s

t❛❧ r❡s✉❧t❛❞♦ ❝♦♠ ❛q✉❡❧❡ ❡①♣❡r✐♠❡♥t❛❧ ❬✶✸❪ ❛♣r❡s❡♥t❛❞♦ ♥❛ ✜❣✉r❛ ✸✳✾✱ ✈❡♠♦s ✉♠ ❜♦♠

❛❝♦r❞♦ q✉❛❧✐t❛t✐✈♦ ❡♥tr❡ ❛♠❜♦s r❡s✉❧t❛❞♦s✳

❆❧é♠ ❞❛ ✐♥t❡r❛çã♦ ❞❡❢❡✐t♦✲❝❛r♦ç♦ ❡ ❛ ❝♦♥s❡q✉❡♥t❡ ❛❧t❡r❛çã♦ ♥❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐✲

❧í❜r✐♦ ❞♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡✱ ♣♦❞❡♠♦s ✐♥❞❛❣❛r s❡ ❡ ❝♦♠♦ t❛✐s ❞❡❢❡✐t♦s ❛❢❡t❛♠ ♦ ♠♦✈✐✲

♠❡♥t♦ ❣✐r♦tró♣✐❝♦ ✭❡ ❛ss✐♠ s✉❛ ❢r❡q✉ê♥❝✐❛ ωG✮✳ ◗✉❛❧✐t❛t✐✈❡♠❡♥t❡✱ é ❡s♣❡r❛❞♦ q✉❡ t❛❧

♠♦✈✐♠❡♥t♦ ✭tr❛❥❡tór✐❛ ❞♦ ♠♦✈✐♠❡♥t♦ ❣✐r♦tró♣✐❝♦ ❞♦ ❝❛r♦ç♦ ♥♦ ♥❛♥♦❞✐s❝♦✮ s❡❥❛ ♠❛✐s

❝♦♠♣❧✐❝❛❞♦ ❞♦ q✉❡ ❛ tr❛❥❡tór✐❛ s✉❛✈❡ ♦❜s❡r✈❛❞❛ ❡♠ ❛♠♦str❛s ♣✉r❛s✳ ❆ r✉❣♦s✐❞❛❞❡ ❛♦

❧♦♥❣♦ ❞♦ ❝❛♠✐♥❤♦ ❛ss❡♠❡❧❤❛✲s❡ ❛ ❛❧❣♦ ❝♦♠♦ ✧♠♦✈✐♠❡♥t♦ ❇r♦✇♥✐❛♥♦✧✱ ❛✉♠❡♥t❛♥t♦ ❛

❞✐stâ♥❝✐❛ ♣❡r❝♦rr✐❞❛ ♣❡❧♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ ❛♦ ❝♦♠♣❧❡t❛r ✉♠❛ r❡✈♦❧✉çã♦✳ ❊♥tr❡t❛♥t♦✱

s❡ ❤á ✉♠❛ ❝♦♥❝❡♥tr❛çã♦ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ s✐♠✐❧❛r ❞❡ ❞❡❢❡✐t♦s ❛tr❛t✐✈♦s ❡ r❡♣✉❧s✐✈♦s✱

é ❡s♣❡r❛❞♦ ✉♠❛ ❝♦♠♣❡♥s❛çã♦ ♥❛ tr❛❥❡tór✐❛✱ t❛❧ q✉❡ ❛ ❢r❡q✉ê♥❝✐❛ ❣✐r♦tró♣✐❝❛ ωG ♣❡r✲

♠❛♥❡ç❛ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✐♥❛❧t❡r❛❞❛ ♣❡❧❛ ♣r❡s❡♥ç❛ ❞❡ t❛✐s ❞❡❢❡✐t♦s✱ ✈✐st♦ q✉❡ ♦ rá♣✐❞♦

❞❡s❧♦❝❛♠❡♥t♦ ♣r♦♠♦✈✐❞♦ ♣♦r ❞❡❢❡✐t♦s ❞♦ t✐♣♦ s❝❛tt❡r ✭λi > 0✮ é ❡q✉✐❧✐❜r❛❞♦ ♣❡❧♦ ❧❡♥t♦

✺✵

✹✳ ▼♦❞❡❧♦ ♣❛r❛ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s

❋✐❣✉r❛ ✹✳✸✿ ❆❝✐♠❛✿ ✐❧✉str❛çã♦ ❡sq✉❡♠át✐❝❛ ❞❛ ✐♥t❡r❛çã♦ ❞♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ ❝♦♠

✉♠ ❞❡❢❡✐t♦ ❡s♣❛❧❤❛❞♦r ✭s❝❛tt❡r✮✱ ♠♦str❛♥❞♦ ♦♥❞❡ t❛❧ ✐♥t❡r❛çã♦ ♣♦❞❡ s❡r ♦❜s❡r✈❛❞❛ ♥♦

❣rá✜❝♦ ❞❡ s ♣♦r hext✳ ❆❜❛✐①♦✿ ✐❧✉str❛çã♦ ❞❛ ✐♥t❡r❛çã♦ ❞♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ ❝♦♠ ✉♠

❞❡❢❡✐t♦ ❛tr❛t♦r ✭♣✐♥♥✐♥❣✮✱ ✐❧✉str❛♥❞♦ ❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❧♦❝❛❧✐③❛çã♦ ❞❡ t❛❧ ✐♥t❡r❛çã♦ ♥♦

❣rá✜❝♦ ❞❡ s ♣♦r hext✳

❞❡s❧♦❝❛♠❡♥t♦ ♣r♦♣♦r❝✐♦♥❛❞♦ ♣♦r ❞❡❢❡✐t♦s ❞♦ t✐♣♦ ♣✐♥♥✐♥❣ ✭λi < 0✮✳ ◆♦ ❡♥t❛♥t♦✱ s❡ ❤á

✉♠ ❞❡s❜❛❧❛♥❝❡❛♠❡♥t♦ ♥❛s ❝♦♥❝❡♥tr❛çõ❡s ❞❡ ❛♠❜♦s✱ é ❡s♣❡r❛❞♦ ✉♠ r❡t❛r❞❛♠❡♥t♦ ♥♦

♠♦✈✐♠❡♥t♦ ❞♦ ✈órt✐❝❡ ✲ s❡ ❤á ♠❛✐s ❞❡❢❡✐t♦s ❛tr❛t✐✈♦s q✉❡ r❡♣✉❧s✐✈♦s ✲ ♦✉ ✉♠ ❛✈❛♥ç♦ ✲

❝❛s♦ t❡♥❤❛♠♦s ♠❛✐s ❞❡❢❡✐t♦s r❡♣✉❧s✐✈♦s ✲ ♦ q✉❡ é ❡✈✐❞❡♥❝✐❛❞♦ ♣♦r ✉♠❛ ❞✐♠✐♥✉✐çã♦ ♦✉

❛✉♠❡♥t♦ ♥❛ ❢r❡q✉ê♥❝✐❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❙❡ ♣r♦❝❡❞❡r♠♦s ❞❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ♥❛ ♦❜t❡♥çã♦ ❞❛ ❡q✉❛çã♦ ✭✸✳✷✵✮✱ ♦✉ s❡❥❛

~Frest = −k~r = −~∇Wdef

∣∣h=0

✱ ❡♥❝♦♥tr❛r❡♠♦s q✉❡✿

k = µ0πLM2s

[

4πF1(β)− (1− ξi)lexR2

]

✳ ✭✹✳✼✮

❆ss✐♠✱ ❝♦♠♦ k é ❛❢❡t❛❞♦ ♣❡❧♦ ❢❛t♦r ξi✱ ✈❡♠♦s ❝❧❛r❛♠❡♥t❡ q✉❡ ❞❡❢❡✐t♦s ❡str✉t✉✲

r❛✐s ❛❧t❡r❛♠ ♦ ♠♦✈♦♠❡♥t♦ ❣✐r♦tró♣✐❝♦ ❞♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡✳ ❊♥tr❡t❛♥t♦✱ ♥ã♦ ♣♦❞❡♠♦s

♣r♦❝❡❞❡r ❛❧é♠ ❝♦♠ ♥♦ss❛ ❛♥á❧✐s❡✱ ♣♦✐s ♥ã♦ ♣♦❞❡♠♦s ❛ss✉♠✐r q✉❡ ♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ r❡✲

❛❧✐③❛ ♦s❝✐❧❛çõ❡s ❤❛r♠ô♥✐❝❛s ❡♠ t♦r♥♦ ❞❡ s✉❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦✱ ✉♠❛ ✈❡③ q✉❡ ❞❡s✈✐♦s

❞❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ sã♦ ❛♣r❡❝✐á✈❡✐s s❡♠♣r❡ q✉❡ ❞❡❢❡✐t♦s ❡stã♦ ♣r❡s❡♥t❡s✱ t♦r♥❛♥❞♦

✐♥❡✜❝✐❡♥t❡ ♥♦ss❛ ❛♣r♦①✐♠❛çã♦ ❛♥❛❧ít✐❝❛✳ ◆❡st❡ ❝❡♥ár✐♦✱ s✐♠✉❧❛çõ❡s ♥✉♠ér✐❝❛s ❛♣❛r❡✲

❝❡♠ ❝♦♠♦ ♣♦ssí✈❡✐s ❛❧t❡r♥❛t✐✈❛s ♣❛r❛ ♦ ❡st✉❞♦ ❞❛ ❛❧t❡r❛çã♦ ❞♦ ♠♦✈✐♠❡♥t♦ ❣✐r♦tró♣✐❝♦

✺✶

✹✳ ▼♦❞❡❧♦ ♣❛r❛ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s ❡♠ ♥❛♥♦♠❛❣♥❡t♦s

❡♠ ✈✐rt✉❞❡ ❞❛ ♣r❡s❡♥ç❛ ❞❡ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s ❬✼✷❪✳ ❆❧é♠ ❞✐ss♦✱ ♣♦❞❡♠♦s ❡s♣❡r❛r q✉❡

t❛✐s ❞❡❢❡✐t♦s✱ q✉❛♥❞♦ ✐♥t❡r❛❣✐♥❞♦ ❝♦♠ ♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ ♥♦ ♠♦✈✐♠❡♥t♦ ❣✐r♦tró♣✐❝♦✱

t❛♠❜é♠ ❛❧t❡r❡♠ ❛ s✐t✉❛çã♦ ♦♥❞❡ ❛ r❡✈❡rsã♦ ❞❛ ♣♦❧❛r✐③❛çã♦ ❛❝♦♥t❡❝❡✱ ♣♦✐s ♠❡s♠♦ t❛✐s

❞❡❢❡✐t♦s ♥ã♦ ❛❧t❡r❛♥❞♦ ❛ ✈❡❧♦❝✐❞❛❞❡ ❝rít✐❝❛ vcr ✭❝❛r❛❝t❡ríst✐❝❛ ❞♦ ♠❛t❡r✐❛❧✮✱ ❡❧❡s ♣♦❞❡♠

s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡ ❛❢❡t❛r ♦ ♠❡❝❛♥✐s♠♦ ❞❡ s✇✐t❝❤✐♥❣✳ P♦r ❡①❡♠♣❧♦✱ s❡ ♦ ❝❛r♦ç♦ ❞♦ ✈ór✲

t✐❝❡ ❡stá s❡ ♠♦✈❡♥❞♦ ♥✉♠❛ r❡❣✐ã♦ ♦♥❞❡ ❛ ❝♦♥❝❡♥tr❛çã♦ ❞❡ ❞❡❢❡✐t♦s ❛tr❛t✐✈♦s é ♠❛✐♦r✱

t❡r❡♠♦s ♠❛✐s r❡❣✐õ❡s ♦♥❞❡ ♦ ❝❛r♦ç♦ ✜❝❛rá ✧♣r❡s♦✧ ❞♦ q✉❡ r❡❣✐õ❡s q✉❡ ♦ ❡s♣❛❧❤❡♠ ❡✱

❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ t❛❧ ❡str✉t✉r❛ ❞❡s❛❝❡❧❡r❛rá✱ ❞✐ss✐♣❛♥❞♦ ❡♥❡r❣✐❛ ❝✐♥ét✐❝❛ ♥❛ ❢♦r♠❛ ❞❡

♦♥❞❛s ❞❡ s♣✐♥s ❡✴♦✉ ❝❛❧♦r ❡✱ ♥❡st❡ ❝❛s♦✱ r❡t❛r❞❛♥❞♦ ♦ ❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ ❛ ❡♥❝♦♥tr❛r s✉❛

✈❡❧♦❝✐❞❛❞❡ ❝rít✐❝❛✳ ❚❛❧ q✉❛❧ ♥♦ ❝❛s♦ ❞♦ ♠♦✈✐♠❡♥t♦ ❣✐r♦tró♣✐❝♦✱ ♦ ♣r♦❝❡ss♦ ❞❡ ✐♥✈❡rsã♦

❞❛ ♣♦❧❛r✐❞❛❞❡ ❝❛r❡❝❡ ❞❡ r❡s♣♦st❛s ❡ ♠❛✐s ✐♥✈❡st✐❣❛çõ❡s ✈✐❛ s✐♠✉❧❛çõ❡s ♥✉♠ér✐❝❛s✳ ❖s

r❡s✉❧t❛❞♦s ❛q✉✐ ♦❜t✐❞♦s ❢♦r❛♠ ♣✉❜❧✐❝❛❞♦s ♥♦ ❏♦✉r♥❛❧ ♦❢ ❆♣♣❧✐❡❞ P❤②s✐❝s ❬✼✸❪✳

✺✷

❈❛♣ít✉❧♦ ✺

❈♦♥❝❧✉sõ❡s ❡ ♣❡rs♣❡❝t✐✈❛s

❆♣r❡s❡♥t❛♠♦s ❛q✉✐ ✉♠ ♠♦❞❡❧♦ s✐♠♣❧❡s ♣❛r❛ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s ♣✉♥t✐❢♦r♠❡s

❡♠ s✐st❡♠❛s ❢❡rr♦♠❛❣♥ét✐❝♦s ❡♠ ♥❛♥♦❡s❝❛❧❛✳ ◆♦ ❡♥t❛♥t♦✱ ❛♣❡s❛r ❞❡ s✐♠♣❧❡s ❡♠ s✉❛

❡str✉t✉r❛ ♠❛t❡♠át✐❝❛✱ t❛❧ ♠♦❞❡❧♦ ♥♦s ❧❡✈♦✉ ❛ r❡s✉❧t❛❞♦s ✐♥t❡r❡ss❛♥t❡s✳ ❈♦♠♦ s✉♣♦♠♦✲

♦s ♣✉♥t✐❢♦r♠❡s✱ ❡❧❡s ♠♦❞✐✜❝❛♠ ♣r✐♥❝✐♣❛❧♠❡♥t❡ s✉❛ ❡♥❡r❣✐❛ ❞❡ tr♦❝❛✱ ♥ã♦ ❛❢❡t❛♥❞♦ ♦

t❡r♠♦ ♠❛❣♥❡t♦stát✐❝♦ ♣♦✐s✱ ♣♦r ♥ã♦ ♣♦ss✉ír❡♠ ❝♦♥s✐❞❡rá✈❡✐s ár❡❛ ❡ ✈♦❧✉♠❡✱ ♥ã♦ sã♦

❝❛♣❛③❡s ❞❡ ✐♥❞✉③✐r ❝❛r❣❛s ♠❛❣♥ét✐❝❛s s✉♣❡r✜❝✐❛✐s ❡✴♦✉ ✈♦❧✉♠étr✐❝❛s✳ ❯s❛♠♦s ♥♦ss♦

♠♦❞❡❧♦ ♣❛r❛ ✉♠ ♥❛♥♦❞✐s❝♦ ❢❡rr♦♠❛❣♥ét✐❝♦ ✭❝♦♠ r❛③ã♦ L/R ≪ 1✮ ❝♦♠ ❝♦♥✜❣✉r❛çã♦

❞♦ t✐♣♦ ✈órt✐❝❡✳ ❖❜t✐✈❡♠♦s ✉♠❛ ❡①♣r❡ssã♦ ❛♥❛❧ít✐❝❛ ♣❛r❛ ❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞♦

❝❛r♦ç♦ ❞♦ ✈órt✐❝❡ ♥✉♠❛ ❛♠♦str❛ ❝♦♠ ❞❡❢❡✐t♦s✱ q✉❛♥❞♦ ❡st❛ é s✉❜♠❡t✐❞❛ ❛ ✉♠ ❝❛♠♣♦

♠❛❣♥ét✐❝♦ ❡①t❡r♥♦ ❡ ✈❡♠♦s ❛✐♥❞❛ q✉❡ t❛❧ ❡q✉❛çã♦✱ ♥✉♠ ♠♦❞❡❧♦ s❡♠ ❛ ♣r❡s❡♥ç❛ ❞❡

❞❡❢❡✐t♦s✱ é ✐♠❡❞✐❛t❛♠❡♥t❡ r❡❝✉♣❡r❛❞❛ s❡ ❛ ✐♥✢✉ê♥❝✐❛ ❞♦s ❞❡❢❡✐t♦s é r❡t✐r❛❞❛✳ ❆✐♥❞❛ ♥❡st❡

❝❡♥ár✐♦✱ ♦❜s❡r✈❛♠♦s ♥✉♠ ❣rá✜❝♦ ❞❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ✈❡rs✉s ❝❛♠♣♦ ❡①t❡r♥♦✱ ✉♠

❜♦♠ ❛❝♦r❞♦ q✉❛❧✐t❛t✐✈♦ ❡♥tr❡ r❡s✉❧t❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❬✶✸❪ ❡ ❛q✉❡❧❡s t❡ór✐❝♦s ♦❜t✐❞♦s

♣❡❧♦ ♥♦ss♦ ♠♦❞❡❧♦✱ ♠♦str❛♥❞♦ ❝♦♠♦ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s ♠♦❞✐✜❝❛♠ ♦ ♠♦✈✐♠❡♥t♦ ❞♦

❝❛r♦ç♦ ❞♦ ✈órt✐❝❡✳

P♦❞❡♠♦s ❝✐t❛r ❞❡♥tr❡ ❛s ♣❡rs♣❡❝t✐✈❛s ❛ s❡r❡♠ ✐♥✈❡st✐❣❛❞❛s✱ ❛❧❣✉♥s ❡❧❡♠❡♥t♦s ❛s✲

s♦❝✐❛❞♦s ❛♦ t✐♣♦ ❞❡ ❞❡❢❡✐t♦ ✭♣♦r ❡①❡♠♣❧♦✱ ❛❣♦r❛ ❤❛✈❡♥❞♦ ✉♠❛ ✐♥t❡r❛çã♦ ❞♦ ❞❡❢❡✐t♦ ❝♦♠

♦ t❡r♠♦ ♠❛❣♥❡t♦stát✐❝♦ ❞❛ ❡♥❡r❣✐❛✮✱ ✐♥t❡♥s✐❞❛❞❡ ❞❛ ✐♥t❡r❛çã♦✱ ❞❡♣❡♥❞ê♥❝✐❛ ❞❛ ❝♦♥✲

✜❣✉r❛çã♦ ❞❡ ♠❛❣♥❡t✐③❛çã♦ ❡✴♦✉ ❢♦r♠❛ ❞❛ ❛♠♦str❛ ♥❛ ✐♥t❡r❛çã♦ ❝♦♠ ❞❡❢❡✐t♦s✱ ❡t❝✳ ❆♦

❛♣❧✐❝❛r♠♦s ♥♦ss♦ ♠♦❞❡❧♦ ❡ ❝♦♥s✐❞❡r❛r♠♦s ❝❡rt❛ ❛♥✐s♦tr♦♣✐❛✱ ❡s♣❡r❛♠♦s q✉❡ ♦ ♣❛râ♠❡tr♦

✺✸

✺✳ ❈♦♥❝❧✉sõ❡s ❡ ♣❡rs♣❡❝t✐✈❛s

λi ❛ss♦❝✐❛❞♦ ❛♦ ❞❡❢❡✐t♦✱ ❞ê ❧✉❣❛r ❛ ✉♠❛ ❢✉♥çã♦ ✈❡t♦r✐❛❧ ~Λi✱ ❛ q✉❛❧ ❞❡✈❡ q✉❛♥t✐✜❝❛r ❛

♣r❡❢❡rê♥❝✐❛ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ❡♠ ❛♣♦♥t❛r ❡♠ ❞❡t❡r♠✐♥❛❞❛ ❞✐r❡çã♦ ❡♠ ❝❛❞❛ r❡❣✐ã♦ ❞♦

♠❛❣♥❡t♦✳ ❊s♣❡r❛♠♦s q✉❡ t❛❧ ❛♣❡r❢❡✐ç♦❛♠❡♥t♦ ❝♦rr♦❜♦r❡ ♥❛ ❝♦♠♣r❡❡♥sã♦ ❞❛ ❢♦rt❡ ❛s✲

s✐♠❡tr✐❛ ♦❜s❡r✈❛❞❛ ♥❛ ❢r❡q✉ê♥❝✐❛ ❣✐r♦tró♣✐❝❛ ❡♥tr❡ ♣♦❧❛r✐❞❛❞❡s ✉♣ ❡ ❞♦✇♥✱ ❝♦♠♦ ♦❜s❡r✲

✈❛❞♦ ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ ❡♠ ❬✼✹❪ ❡ r❡❝❡♥t❡♠❡♥t❡ ❡st✉❞❛❞♦ ❡ ❡①♣❧❛♥❛❞♦ ❡♠ ❬✼✺❪✳ P♦❞❡✲

♠♦s t❛♠❜é♠ ✐♥✈❡st✐❣❛r✱ ✈✐❛ s✐♠✉❧❛çõ❡s ♥✉♠ér✐❝❛s ❬✼✷❪✱ ♦ ❡❢❡✐t♦ ❞❡ ❞❡❢❡✐t♦s ❡str✉t✉r❛✐s

♥❛ ❢r❡q✉ê♥❝✐❛ ❣✐r♦tró♣✐❝❛✳

❈♦♠♦ ♠❡♥❝✐♦♥❛❞♦ ❛♥t❡s✱ ❡st❡ tr❛❜❛❧❤♦ ❢♦✐ ♣✉❜❧✐❝❛❞♦ r❡❝❡♥t❡♠❡♥t❡ ♥♦ ❏♦✉r♥❛❧

♦❢ ❆♣♣❧✐❡❞ P❤②s✐❝s ❬✼✸❪✳

✺✹

❆♣ê♥❞✐❝❡ ❆

▼❛❣♥❡t♦r❡s✐stê♥❝✐❛ ❣✐❣❛♥t❡ ✭●▼❘✮ ❡

❣r❛✈❛çã♦ ♠❛❣♥ét✐❝❛

❆✳✶ ❆ ❞❡s❝♦❜❡rt❛ ❞♦ ❡❢❡✐t♦ ●▼❘

❖s ❛♥♦s ✽✵ ❢♦r❛♠ ♠❛r❝❛❞♦s ♣❡❧♦ ❡♥♦r♠❡ ❡s❢♦rç♦ ❞❡✈♦t❛❞♦ à ❛r❡❛ ❞❡ ♠❛t❡r✐❛✐s

♠❛❣♥ét✐❝♦s ♥❛♥♦❡str✉t✉r❛❞♦s✱ ❥á s❡♥❞♦ ♣♦ssí✈❡❧✱ ♣♦r ❡①❡♠♣❧♦✱ ❝r❡s❝❡r ❝♦♥tr♦❧❛❞❛♠❡♥t❡

✜❧♠❡s ✉❧tr❛✲✜♥♦s✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❞♦✐s ❣r✉♣♦s ❞❡ ♣❡sq✉✐s❛ s❡ ❞❡st❛❝❛r❛♠✿ ❆❧❜❡rt ❋❡rt

❡ ❝♦❧❛❜♦r❛❞♦r❡s✶ ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ P❛r✐s✲❙✉❧ ✭❖rs❛②✱ ❋r❛♥ç❛✮ ❡ P❡t❡r ●rü♥❜❡r❣ ❡

s❡✉ ❣r✉♣♦ ♥♦ ❈❡♥tr♦ ❞❡ P❡sq✉✐s❛s ◆✉❝❧❡❛r❡s ✭❏ü❧✐❝❤✱ ❆❧❡♠❛♥❤❛✮✳ ❊ss❡s ❞♦✐s ❣r✉♣♦s

❡st✉❞❛✈❛♠✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡✱ ✜❧♠❡s ✜♥♦s✱ ✉❧tr❛✲✜♥♦s ❡ ❞❡ ♠✉❧t✐❝❛♠❛❞❛s✳ ❋❡rt ✐♥✲

✈❡st✐❣❛✈❛ ♦ ❡❢❡✐t♦ ❞❛ ❛♣❧✐❝❛çã♦ ❞❡ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ♥❛ ❝♦rr❡♥t❡ ❡❧étr✐❝❛ ❛tr❛✈és ❞❡

♠✉❧t✐❝❛♠❛❞❛s ❞❡ ❢❡rr♦ ❡ ❝r♦♠♦ ✭✈ár✐❛s ❝❛♠❛❞❛s ❛❧t❡r♥❛❞❛s ❞❡ ❋❡ ❡ ❈r✱ ❝❛❞❛ ✉♠❛ ❝♦♠

❡s♣❡ss✉r❛ ❞❡ ♣♦✉❝♦s át♦♠♦s✮❀ ●rü♥❜❡r❣ ❡st✉❞❛✈❛ ❡❢❡✐t♦s s❡♠❡❧❤❛♥t❡s ❡♠ ❝❛♠❛❞❛s ❞❡

❋❡ ❡ ❈r✱ só q✉❡ ✉t✐❧✐③❛♥❞♦ ❛♠♦str❛s ❝♦♠ ❛♣❡♥❛s ❞✉❛s ❝❛♠❛❞❛s ❞❡ ❢❡rr♦ ✐♥t❡r❝❛❧❛❞❛s ♣♦r

✉♠❛ ✜♥❛ ❝❛♠❛❞❛ ❞❡ ❝r♦♠♦✳ ❆♠❜♦s ♦❜t✐✈❡r❛♠ r❡s✉❧t❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s q✉❡ ❡✈✐❞❡♥✲

❝✐❛✈❛♠ ✉♠❛ ♠✉❞❛♥ç❛ ❡♥♦r♠❡ ♥❛ r❡s✐stê♥❝✐❛ ❡❧étr✐❝❛ ❡♠ ❢✉♥çã♦ ❞♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦

❛♣❧✐❝❛❞♦✳ ❖s ✈❛❧♦r❡s ♦❜s❡r✈❛❞♦s ❝❤❡❣❛✈❛♠ ❛ ✈❛r✐❛çõ❡s ❞❡ ❛té ✺✵✪✱ q✉❡ ❡r❛♠✱ à é♣♦❝❛✱

♠✉✐t♦ s✉♣❡r✐♦r❡s ❛♦s ✈❛❧♦r❡s ❝♦♥❤❡❝✐❞♦s ❞❛ ♠❛❣♥❡t♦r❡s✐stê♥❝✐❛ tr❛❞✐❝✐♦♥❛❧ ❛♣r❡s❡♥t❛✲

✶▲❡♠❜r❛♠♦s q✉❡ ♦ tr❛❜❛❧❤♦ r❡❛❧✐③❛❞♦ ♥♦ ❧❛❜♦r❛tór✐♦ ❞❡ ❆✳ ❋❡rt ❝♦♥t♦✉ ❝♦♠ ❛ ❝♦❧❛❜♦r❛çã♦ ❞❡ ▼ár✐♦

◆✳ ❇❛✐❜✐❝❤✱ ♣❡sq✉✐s❛❞♦r ❜r❛s✐❧❡✐r♦ ❡ ❛t✉❛❧♠❡♥t❡ ❞♦❝❡♥t❡ ♥♦ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧

❞♦ ❘✐♦ ●r❛♥❞❡ ❞♦ ❙✉❧ ✭❯❋❘●❙✮✳

✺✺

❆♣ê♥❞✐❝❡ ❆

❞❛ ♣♦r ♠❡t❛✐s✱ q✉❡ é ❞❡ ♥♦ ♠á①✐♠♦ ✷✪✳ ❉❛í ❡♠ ❞✐❛♥t❡ ❡st❡ ❢❡♥ô♠❡♥♦ ♣❛ss♦✉ ❛ s❡r

❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♠❛❣♥❡t♦r❡s✐stê♥❝✐❛ ❣✐❣❛♥t❡ ✭●▼❘✮✳

❆ ❡①♣❧✐❝❛çã♦ ♣❛r❛ ♦ ❢❡♥ô♠❡♥♦ é r❛③♦❛✈❡❧♠❡♥t❡ ❝♦♠♣❧❡①❛ ❡ ❡stá r❡❧❛❝✐♦♥❛❞❛ ❝♦♠

❛ ♠♦❜✐❧✐❞❛❞❡ ❡❧❡trô♥✐❝❛ ❡♠ ♠❛t❡r✐❛✐s ♠❛❣♥ét✐❝♦s✳ ❱❛r✐❛♥❞♦ ❛ ❡s♣❡ss✉r❛ ❞❛ ❝❛♠❛❞❛ ❞❡

❈r✱ ❢♦✐ ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ❞✐❢❡r❡♥t❡s ❡s♣❡ss✉r❛s ♣❛r❛ ❛s q✉❛✐s ❛s ❝❛♠❛❞❛s ❡①t❡r♥❛s ❞❡

❋❡ ♣♦ss✉í❛♠ ♠♦♠❡♥t♦s ♠❛❣♥ét✐❝♦s ❛♣♦♥t❛♥❞♦ ❡♠ s❡♥t✐❞♦s ❝♦♥trár✐♦s✳ ❆í✱ ❛ ❛♣❧✐❝❛çã♦

❞♦ ❝❛♠♣♦ ❧❡✈❛ ❛ ✉♠❛ ❝♦♥✜❣✉r❛çã♦ ♦♥❞❡ ♦s ♠♦♠❡♥t♦s ♠❛❣♥ét✐❝♦s ❞❡ ❛♠❜❛s ❝❛♠❛❞❛s

s❡ ❛❧✐♥❤❛♠✳ ❈♦♠♦ ♦s ❡❧étr♦♥s ♣♦ss✉❡♠ ❞♦✐s ❡st❛❞♦s ❞❡ s♣✐♥✱ ✉♣ ❡ ❞♦✇♥✱ ♣♦❞❡♠♦s

❝♦♥s✐❞❡r❛r q✉❡ ❛ ❝♦rr❡♥t❡ ❡❧étr✐❝❛ é ♦ r❡s✉❧t❛❞♦ ❞❡ ❞✉❛s ❝♦rr❡♥t❡s ♣❛r❛❧❡❧❛s✱ ✉♠❛ ❞❡✈✐❞❛

❛ ❡❧étr♦♥s ♥♦ ❡st❛❞♦ ✉♣ ❡ ♦✉tr❛ ❛ ❡❧étr♦♥s ♥♦ ❡st❛❞♦ ❞♦✇♥✳ ❖ q✉❡ ❧❡✈❛ à ♠❛❣♥❡t♦r❡✲

s✐stê♥❝✐❛ é ♦ ❢❛t♦ ❞❛ r❡s✐st✐✈✐❞❛❞❡ ❡❧étr✐❝❛ ❞❡♣❡♥❞❡r ❞❛ ♦r✐❡♥t❛çã♦ r❡❧❛t✐✈❛ ❡♥tr❡ ♦ s♣✐♥

❡❧❡trô♥✐❝♦ ❡ ❛ ♠❛❣♥❡t✐③❛çã♦ ❞♦ ♠❛t❡r✐❛❧✳ ❆ ✜❣✉r❛ ❆✳✶ ✐❧✉str❛ ♦ ❡❢❡✐t♦✳ ◆♦ ❡sq✉❡♠❛ à

❡sq✉❡r❞❛ t❡♠♦s ❝❛♠❛❞❛s ♠❛❣♥ét✐❝❛s ♥♦ ❡st❛❞♦ ❛♥t✐❢❡rr♦♠❛❣♥ét✐❝♦✳ ◆❡st❡ ❝❛s♦✱ ❝❛❞❛

❝❛♥❛❧ ❞❡ ❡❧étr♦♥s ❡♥❝♦♥tr❛ r❡❣✐õ❡s ❝♦♠ ❛❧t❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❡✱ ❛ss✐♠✱ ♦s

❝❛♥❛✐s tê♠ ✉♠❛ r❡s✐stê♥❝✐❛ s✐♠✐❧❛r✱ ✐♠♣❧✐❝❛♥❞♦ ♥✉♠❛ r❡s✐stê♥❝✐❛ t♦t❛❧ ❛❧t❛✳ ❏á ♥♦ ❡s✲

q✉❡♠❛ ❞❛ ❞✐r❡✐t❛ t❡♠♦s ❛ s✐t✉❛çã♦ ♦♣♦st❛ ✭❡st❛❞♦ ❢❡rr♦♠❛❣♥ét✐❝♦✮✱ ♦♥❞❡ ♦s ♠♦♠❡♥t♦s

♠❛❣♥ét✐❝♦s ❞❛s ❝❛♠❛❞❛s ❛♣♦♥t❛♠ ♥♦ ♠❡s♠♦ s❡♥t✐❞♦ ❞❡✈✐❞♦ à ❛♣❧✐❝❛çã♦ ❞❡ ✉♠ ❝❛♠♣♦

♠❛❣♥ét✐❝♦✳ ❆❣♦r❛✱ ❛♣❡s❛r ❞❡ ✉♠ ❞♦s ❝❛♥❛✐s ❞❡ ❡❧étr♦♥s t❡r ✉♠❛ r❡s✐stê♥❝✐❛ ❛❧t❛✱ ♦ ♦✉tr♦

❝❛♥❛❧ ❝❡rt❛♠❡♥t❡ s♦❢r❡rá ♣♦✉❝♦ ❡s♣❛❧❤❛♠❡♥t♦✱ ✐♠♣❧✐❝❛♥❞♦ ♥✉♠❛ r❡s✐stê♥❝✐❛ ❜❛✐①❛ ♣❛r❛

❡ss❡ ❝❛♥❛❧ ❡✱ t❛❧ q✉❛❧ ♥✉♠ s✐st❡♠❛ ❞❡ r❡s✐st♦r❡s ❡♠ ♣❛r❛❧❡❧♦✱ ❛ r❡s✐stê♥❝✐❛ ❞♦ s✐st❡♠❛ é

❜❡♠ ♠❡♥♦r ❞♦ q✉❡ ♥♦ ♣r✐♠❡✐r♦ ❝❛s♦ ❬✷✺❪✳

❖ ✐♠❡♥s♦ ♣♦t❡♥❝✐❛❧ ❞❛ ●▼❘ ♣❛r❛ ❛♣❧✐❝❛çõ❡s t❡❝♥♦❧ó❣✐❝❛s ❢♦✐ ✐♠❡❞✐❛t❛♠❡♥t❡

♣❡r❝❡❜✐❞♦ t❛♥t♦ ♣❡❧♦s ❞❡s❝♦❜r✐❞♦r❡s ❡ ♣❡sq✉✐s❛❞♦r❡s ❞❛ ár❡❛ q✉❛♥t♦ ♣❡❧❛ ✐♥❞ústr✐❛ ❞❡

❣r❛✈❛çã♦ ♠❛❣♥ét✐❝❛✳ ❆ ♣❛rt✐r ❞❡ss❛ ❞❡s❝♦❜❡rt❛ ♦ t❛♠❛♥❤♦ ❢ís✐❝♦ ❞♦s ❞✐s♣♦s✐t✐✈♦s ❞❡

❛r♠❛③❡♥❛♠❡♥t♦ ♥ã♦ ♣❛r♦✉ ❞❡ ❞✐♠✐♥✉✐r✳ ❆ s❡❣✉✐r ❢❛r❡♠♦s ✉♠❛ ❜r❡✈❡ ❞✐s❝✉ssã♦ ❞❡ ❝♦♠♦

❛ ●▼❘ ❛t✉♦✉ ♥❡ss❛ ár❡❛✳

✺✻

❆♣ê♥❞✐❝❡ ❆

❋✐❣✉r❛ ❆✳✶✿ ❊sq✉❡r❞❛✿ ❝❛♠❛❞❛s ❝♦♠ ♠♦♠❡♥t♦s ♦♣♦st♦s ✭❡st❛❞♦ ❛♥t✐❢❡rr♦♠❛❣♥ét✐❝♦✮✱ ♦♥❞❡

❝❛❞❛ ❝❛♥❛❧ ❡♥❝♦♥tr❛ r❡❣✐õ❡s ❝♦♠ ❛❧t❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ✭r❡s✐stê♥❝✐❛ t♦t❛❧ ❛❧t❛✮✳

❉✐r❡✐t❛✿ ❝❛♠❛❞❛s ❝♦♠ ♠♦♠❡♥t♦s ♣❛r❛❧❡❧♦s ✭❡st❛❞♦ ❢❡rr♦♠❛❣♥ét✐❝♦✮ ❣r❛ç❛s à ♣r❡s❡♥ç❛ ❞❡ ✉♠

❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❛♣❧✐❝❛❞♦✳ ❆❣♦r❛ t❡♠♦s q✉❡ ✉♠ ❝❛♥❛❧ ❡♥❝♦♥tr❛rá ❛❧t❛ r❡s✐stê♥❝✐❛ ❡ ♦✉tr♦

❜❛✐①❛ r❡s✐stê♥❝✐❛ ✭r❡s✐stê♥❝✐❛ t♦t❛❧ ❜❛✐①❛✮✳

❆✳✷ ●r❛✈❛çã♦ ♠❛❣♥ét✐❝❛ ❡ ●▼❘

❖ ♣r✐♥❝í♣✐♦ ❞❛ ❣r❛✈❛çã♦ ❡ ❧❡✐t✉r❛ ♠❛❣♥ét✐❝❛ é r❡❧❛t✐✈❛♠❡♥t❡ s✐♠♣❧❡s✳ ◆❛ ❣r❛✈❛çã♦

♠❛❣♥ét✐❝❛ ❝♦♥✈❡♥❝✐♦♥❛❧✱ ✉♠ ❝❛❜❡ç♦t❡ ♠❛❣♥ét✐❝♦ ✐♥❞✉t✐✈♦ é ✉s❛❞♦ ♣❛r❛ ✧❡s❝r❡✈❡r✧ ❛

✐♥❢♦r♠❛çã♦ ❡♠ ✉♠ ♠❡✐♦ ❞❡ ❣r❛✈❛çã♦ ♠❛❣♥ét✐❝❛ ✭✜t❛ ♦✉ ❞✐s❝♦✮✳ ❊ss❡ ♠❡✐♦ s❡ ♠♦✈❡

❝♦♠ r❡❧❛çã♦ ❛♦ ❝❛❜❡ç♦t❡✱ ❡ ❛ss✐♠ ♦s ❜✐ts ✭♣❡q✉❡♥❛s r❡❣✐õ❡s ♠❛❣♥❡t✐③❛❞❛s ✲ ❞♦♠í♥✐♦s

♠❛❣♥ét✐❝♦s ✲ ❡♠ s❡♥t✐❞♦s ♦♣♦st♦s✱ ✐♥❞✐❝❛♥❞♦ ❛ ❧✐♥❣✉❛❣❡♠ ❜✐♥ár✐❛ ❵✵✬ ❡ ❵✶✬✮ sã♦ ❣r❛✈❛✲

❞♦s ❛♦ ❛♣❧✐❝❛r ♣✉❧s♦s ❞❡ ❝♦rr❡♥t❡s ♣♦s✐t✐✈❛s ♦✉ ♥❡❣❛t✐✈❛s à ❜♦❜✐♥❛ q✉❡ ❢❛③ ♣❛rt❡ ❞♦

❝❛❜❡ç♦t❡✳ ❖ ♠❡s♠♦ ❝❛❜❡ç♦t❡ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞♦ ♣❛r❛ ❧❡r ❛ ✐♥❢♦r♠❛çã♦✱ ♣♦✐s ♦ ♠♦✈✐✲

♠❡♥t♦ ❞❛s r❡❣✐õ❡s ♠❛❣♥❡t✐③❛❞❛s ❞❛ ♠í❞✐❛ s♦❜r❡ ❡❧❡ ✐♥❞✉③ ♣❡q✉❡♥íss✐♠❛s ❝♦rr❡♥t❡s ♥❛

❜♦❜✐♥❛ s❡♥s♦r❛✳ ❊st❛s ❝♦rr❡♥t❡s sã♦ ❞❡t❡❝t❛❞❛s ❛♣ós ✉♠❛ ❝✉✐❞❛❞♦s❛ ❛♠♣❧✐✜❝❛çã♦ ❡

♣r♦❝❡ss❛♠❡♥t♦✳ ❖ s✐♥❛❧ ♦❜t✐❞♦ ❡stá ❞✐r❡t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ❛ ✈❡❧♦❝✐❞❛❞❡ r❡❧❛t✐✈❛

❞♦ ❝❛❜❡ç♦t❡ ❡ ❝♦♠ ♦ t❛♠❛♥❤♦ ❞♦ ❜✐t✳

❆♠❜❛s ❛s t❡❝♥♦❧♦❣✐❛s✱ ❝♦♥✈❡♥❝✐♦♥❛❧ ❡ ♠❛❣♥❡t♦rr❡s✐st✐✈❛✱ ✉t✐❧✐③❛♠ ♦ ❝❛❜❡ç♦t❡

✐♥❞✉t✐✈♦ ♣❛r❛ ❛❧t❡r❛r ❛ ♠❛❣♥❡t✐③❛çã♦ ❞❛ ♠í❞✐❛ ❡ ❣r❛✈❛r ❛ ✐♥❢♦r♠❛çã♦✳ ◆♦ ❡♥t❛♥t♦✱ ❞✐✲

❢❡r❡♥t❡♠❡♥t❡ ❞❛ ❧❡✐t✉r❛ ❝♦♥✈❡♥❝✐♦♥❛❧ q✉❡ ✉s❛ ♦ ♠❡s♠♦ ❝❛❜❡ç♦t❡✱ ♥♦ ♣r♦❝❡ss♦ ❞❡ ❧❡✐t✉r❛

♠❛❣♥❡t♦rr❡s✐st✐✈♦ ❡①✐st❡ ✉♠ s❡❣✉♥❞♦ ❝❛❜❡ç♦t❡ ❜❛s❡❛❞♦ ♥♦ ♣r✐♥❝í♣✐♦ ❞❛ ●▼❘✱ q✉❡ é

♠✉✐t♦ ♠❛✐s s❡♥sí✈❡❧ ♣❛r❛ ❞❡t❡❝t❛r ❛ r❡❣✐ã♦ ♠❛❣♥❡t✐③❛❞❛ ✭✐♥❢♦r♠❛çã♦ ❣r❛✈❛❞❛✮✱ ♣♦✐s ❡st❛

r❡❣✐ã♦ ❝❛✉s❛ ♥♦ ❝❛❜❡ç♦t❡ ✉♠❛ ❛❧t❡r❛çã♦ ♥❛ r❡s✐stê♥❝✐❛ ❡❧étr✐❝❛✱ t❛❧ ❝♦♠♦ ✈✐♠♦s ❛❝✐♠❛✱

✺✼

❆♣ê♥❞✐❝❡ ❆

♦♥❞❡ ♦ ❡st❛❞♦ ❞❡ ❜❛✐①❛ r❡s✐stê♥❝✐❛ r❡♣r❡s❡♥t❛ ♦ ❵✵✬ ❜✐♥ár✐♦ ❡ ♦ ❞❡ ❛❧t❛ r❡s✐stê♥❝✐❛ ♦ ❵✶✬

❜✐♥ár✐♦ ❬✹✽❪✳ P❛r❛ ✉♠ ❡st✉❞♦ ❞❡t❛❧❤❛❞♦ ❞❡ ❝♦♠♦ ♦❝♦rr❡ ❛ ❛❧t❡r❛çã♦ ❞❛ r❡s✐stê♥❝✐❛ ♥✉♠

❝❛❜❡ç♦t❡ q✉❡ ✉t✐❧✐③❛ ❛ t❡❝♥♦❧♦❣✐❛ ♠❛❣♥❡t♦rr❡s✐st✐✈❛✱ r❡❝♦♠❡♥❞❛♠♦s ❧❡r ❛ r❡❢❡rê♥❝✐❛ ❬✸✵❪✳

❖ q✉❡ ❞✐s❝✉t✐♠♦s ❛q✉✐ ♣♦❞❡ s❡r ✈✐st♦ ♥❛ ✜❣✉r❛ ❆✳✷✳

❋✐❣✉r❛ ❆✳✷✿ ■❧✉str❛çã♦ ❞❡ ✉♠ ♣r♦❝❡ss♦ ❞❡ ❣r❛✈❛çã♦ ❡ ❧❡✐t✉r❛ ♠❛❣♥ét✐❝❛✱ ✉t✐❧✐③❛♥❞♦✱ ❝♦♠♣❛✲

r❛t✐✈❛♠❡♥t❡✱ ✉♠ ❝❛❜❡ç♦t❡ ✐♥❞✉t✐✈♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ✭❡sq✉❡r❞❛✮ ❡ ✉♠ ❝❛❜❡ç♦t❡ ♠❛❣♥❡t♦rr❡s✐st✐✈♦

❝♦♠ t❡❝♥♦❧♦❣✐❛ ♠♦❞❡r♥❛ ✭❞✐r❡✐t❛✮✳

❖ ♦❜❥❡t✐✈♦ ❞❡ss❡ ❝❛❜❡ç♦t❡ ❞❡ ❧❡✐t✉r❛ é ♦❜t❡r ❛ ♠❛✐♦r t❛①❛ ❞❡ ✈❛r✐❛çã♦ ❞❡ s✉❛

r❡s✐stê♥❝✐❛ ♣❡❧❛ ✈❛r✐❛çã♦ ❞♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ♠❡❞✐❞♦ ♥♦s ❞✐s❝♦s ❞❡ ❣r❛✈❛çã♦✳ ❚❡r

✉♠ ❡❢❡✐t♦ ♠❛✐♦r ♥❛ r❡s✐stê♥❝✐❛ s✐❣♥✐✜❝❛ ♣♦❞❡r ❧❡r ❝♦✐s❛s ♠❡♥♦r❡s ❡ ❝♦♠ ♠❛✐s ♣r❡❝✐sã♦✳

❆ ✉t✐❧✐③❛çã♦ ❞❛ ●▼❘ ❡♠ t❛✐s ❞✐s♣♦s✐t✐✈♦s ♣❡r♠✐t✐✉ q✉❡ s❡ ❝♦♥✈❡rt❡ss❡♠ ❛❧t❡r❛çõ❡s

♠í♥✐♠❛s ❞❡ ❝❛♠♣♦s ♠❛❣♥ét✐❝♦s ❡♠ ❞✐❢❡r❡♥ç❛s s✐❣♥✐✜❝❛t✐✈❛s ♥❛ r❡s✐stê♥❝✐❛ ❡❧étr✐❝❛ ❡✱

♣♦r s✉❛ ✈❡③✱ ❡♠ ❞✐❢❡r❡♥ç❛s ❞❡ s✐♥❛❧ ❡❧étr✐❝♦ ❢❛❝✐❧♠❡♥t❡ ♦❜s❡r✈á✈❡✐s ♣❡❧♦ ❝❛❜❡ç♦t❡ ❞❡

❧❡✐t✉r❛✳ ❆❧é♠ ❞✐ss♦✱ t❛✐s ❝❛❜❡ç♦t❡s ♥ã♦ ♣r❡❝✐s❛♠ t❡r ✉♠❛ ❣❡♦♠❡tr✐❛ ❝♦♠♣❧✐❝❛❞❛ ❡✱

❥✉♥t❛♥❞♦✲s❡ t♦❞♦s ❡ss❡s ❢❛t♦r❡s✱ ❛❥✉❞❛r❛♠ ❡ ❛✐♥❞❛ ❛❥✉❞❛♠ ❛ ❛✉♠❡♥t❛r ❛ ❞❡♥s✐❞❛❞❡

❞❡ ✐♥❢♦r♠❛çã♦ ❝♦♥t✐❞❛ ♥♦s ❞✐s❝♦s ♠❛❣♥ét✐❝♦s✱ ♣♦✐s sã♦ ❝❛♣❛③❡s ❞❡ ❧❡r ❛s ✐♥❢♦r♠❛çõ❡s

♠❡s♠♦ ❡♠ ♠❛✐♦r ❞❡♥s✐❞❛❞❡ ❬✷✶❪✳

✺✽

❆♣ê♥❞✐❝❡ ❆

❆✳✸ ▼❡♠ór✐❛ ♠❛❣♥ét✐❝❛ ❞❡ ❛❝❡ss♦ ❛❧❡❛tór✐♦ ✭▼❘❆▼✮

P❛r❛ ♠❛✐s ✉♠❛ ✐♠♣♦rt❛♥t❡ ❛♣❧✐❝❛çã♦ ❞♦ ❡❢❡✐t♦ ♠❛❣♥❡t♦r❡s✐st✐✈♦✱ ✈❛♠♦s ❡①♣❧✐❝❛r✱

❡sq✉❡♠❛t✐❝❛♠❡♥t❡✱ ❝♦♠♦ s❡ ❞á ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞❡ ✉♠❛ ▼❘❆▼✱ q✉❡ t❡♠ ❛ ✐♠♣♦rt❛♥t❡

♣r♦♣r✐❡❞❛❞❡ ❞❡ s❡r ♥ã♦ ✈♦❧át✐❧✱ ❡ q✉❡ ♣♦❞❡ s✉❜st✐t✉✐r ❡♠ ❜r❡✈❡ ❛s tr❛❞✐❝✐♦♥❛✐s ♠❡♠ór✐❛s

❘❆▼✳ ❆❜❛✐①♦✱ ♥❛ ✜❣✉r❛ ❆✳✸✱ ❛♣r❡s❡♥t❛♠♦s ✉♠ ♣♦ssí✈❡❧ ❡sq✉❡♠❛ ❞❡ ▼❘❆▼✳ ❆s ❧✐♥❤❛s

❞❡ ❝♦rr❡♥t❡ ❡❧étr✐❝❛ ❢♦r♠❛♠ ✉♠❛ ❡s♣é❝✐❡ ❞❡ t❡❝✐❞♦✱ ♦♥❞❡ ❞✉❛s ♠❛tr✐③❡s ♣❡r♣❡♥❞✐❝✉❧❛r❡s✱

❝♦♥❡❝t❛❞❛s ♣♦r ❥✉♥çõ❡s✱ ❛♣r❡s❡♥t❛♠ ●▼❘✳ ➱ ♣♦ssí✈❡❧ ❝♦♥tr♦❧❛r ❛ ❧❡✐t✉r❛ ❞❡ ❝❛❞❛ ❧✐♥❤❛

❞❡ss❛ ❝♦♠♣❧✐❝❛❞❛ ♠❛tr✐③ ❡✱ ❛ss✐♠✱ ❞❡s❝♦❜r✐r s❡ ❛ r❡s✐stê♥❝✐❛ ❞❛ ❥✉♥çã♦ é ❛❧t❛ ♦✉ ❜❛✐①❛✱

✐♥❞✐❝❛♥❞♦ ❛ss✐♠ ♦ ❡st❛❞♦ ♠❛❣♥ét✐❝♦ ❞❡ s✉❛s ❝❛♠❛❞❛s✱ q✉❡ ♣♦r s✉❛ ✈❡③ r❡♣r❡s❡♥t❛♠ ♦s

❵③❡r♦s✬ ❡ ❵✉♥s✬ ♦♥❞❡ ❛ ✐♥❢♦r♠❛çã♦ é ❛r♠❛③❡♥❛❞❛✳

❋✐❣✉r❛ ❆✳✸✿ P♦ssí✈❡❧ ♣r♦tót✐♣♦ ❞❡ ✉♠❛ ❝é❧✉❧❛ ❞❛ ❝❤❛♠❛❞❛ ▼❘❆▼✱ ♠❡♠ór✐❛ ♥ã♦ ✈♦❧át✐❧ ❡

q✉❡ ✉t✐❧✐③❛ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ●▼❘✳ ❯♠ ♦✉tr♦ ♣r♦tót✐♣♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❬✼✻❪✳

✺✾

❆♣ê♥❞✐❝❡ ❇

▲✐♠✐t❡ ❝♦♥tí♥✉♦ ❞❛ ❍❛♠✐❧t♦♥✐❛♥❛ ❞❡

❍❡✐s❡♥❜❡r❣ ✐s♦tró♣✐❝❛

❊♠❜♦r❛ ❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❞❡ ❍❡✐s❡♥❜❡r❣ ♣♦ss❛ ❞❡s❝r❡✈❡r ❜❡♠ s✐st❡♠❛s ♠❛❣♥ét✐❝♦s✱

♦ ❡st✉❞♦ ❞❡ s✐st❡♠❛s ♠❛❝r♦s❝ó♣✐❝♦s ❝♦♠ 1023 ♣❛rtí❝✉❧❛s s❡ t♦r♥❛ ❞✐❢í❝✐❧ ✈✐❛ ✉s♦ ❞❛s

❡①♣r❡ssõ❡s ✐s♦tró♣✐❝❛ ✭✷✳✶✼✮ ❡ ♠❡s♠♦ ❞❛ ❛♥✐s♦tró♣✐❝❛ ✭✷✳✸✸✮✳ ❊♥tr❡t❛♥❞♦✱ r❡❝♦rr❡✲s❡

♠✉✐t❛s ✈❡③❡s à té❝♥✐❝❛s ❞❡ ❛♣r♦①✐♠❛çã♦✱ ❝♦♠♦ ♦ ❝❛♠♣♦ ♠♦❧❡❝✉❧❛r ✭♦♥❞❡ s✉❜st✐t✉í♠♦s ♦

♦♣❡r❛❞♦r q✉â♥t✐❝♦ ❞❡ s♣✐♥ ♣❡❧♦ s❡✉ ✈❛❧♦r ♠é❞✐♦ tér♠✐❝♦ ❬✹✺❪✮ ❡ ♠ét♦❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s

❝♦♠♦ ♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ❬✸✻❪✳ ❆♥❛❧✐t✐❝❛♠❡♥t❡✱ ♣♦❞❡♠♦s tr❛❜❛❧❤❛r ❝♦♠ t❡♦r✐❛s ❞❡ ❝❛♠♣♦s

❬✺✻❪✱ ❛s q✉❛✐s ❞❡s❝r❡✈❡♠ s✐st❡♠❛s ❝♦♠ ✐♥✜♥✐t♦s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❡✱ ❡♠ ✉♠❛ ♣r✐♠❡✐r❛

❛♣r♦①✐♠❛çã♦✱ ✉♠ s✐st❡♠❛ ❝♦♠ 1023 ♣❛rtí❝✉❧❛s ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ t❡♥❞♦ ✐♥✜♥✐t♦s

❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡✳ P❛r❛ ♦❜t❡r♠♦s ✉♠❛ t❡♦r✐❛ ❞❡ ❝❛♠♣♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ♠♦❞❡❧♦ ❞❡

❍❡✐s❡♥❜❡r❣ ❜✐❞✐♠❡♥s✐♦♥❛❧ ✐s♦tró♣✐❝♦✶ ✭✷✳✶✼✮✱ ❝♦♥s✐❞❡r❛r❡♠♦s ♦ ❧✐♠✐t❡ ❝♦♥tí♥✉♦✳ ◆❡st❡

❧✐♠✐t❡✱ ❛ss✉♠✐♠♦s q✉❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s s♣✐♥s ✈✐③✐♥❤♦s ♠❛✐s ♣ró①✐♠♦s ✭❡s♣❛ç❛♠❡♥t♦

❞❡ r❡❞❡ a❀ ✈❡r ✜❣✉r❛ ❇✳✶✮ é ♠✉✐t♦ ♣❡q✉❡♥❛ ❝♦♠♣❛r❛❞❛ ❛ ♦✉tr❛s ❡s❝❛❧❛s ❡s♣❛❝✐❛✐s✱ ❛❧é♠

❞❡ q✉❡ ❛ ❞✐r❡çã♦ ❞♦s s♣✐♥s ✈✐③✐♥❤♦s ✈❛r✐❛ ❧❡♥t❛♠❡♥t❡✳

P❛rt✐♥❞♦ ❞❛ ❤❛♠✐❧t♦♥✐❛♥❛ ✭✷✳✶✼✮✱ ❝♦♥s✐❞❡r❛♠♦s q✉❡ ♦ s♣✐♥ ♥♦ sít✐♦ i ✐♥t❡r❛❣❡

❝♦♠ ♦s s♣✐♥s ✈✐③✐♥❤♦s ❧♦❝❛❧✐③❛❞♦s ♥♦s sít✐♦s i + 1 ✭à ❞✐r❡✐t❛✮✱ i − 1 ✭à ❡sq✉❡r❞❛✮✱ i + 2

✭❛❝✐♠❛✮ ❡ i− 2 ✭❛❜❛✐①♦✮✱ ❝♦♥❢♦r♠❡ ♠♦str❛❞♦ ♥❛ r❡❞❡ ❞❛ ✜❣✉r❛ ❇✳✶✳ ❉❡✜♥✐♠♦s✿

T α = Sαi

(Sαi+1 + Sα

i−1

)+ Sα

i

(Sαi+2 + Sα

i−2

)✱ ✭❇✳✶✮

✶❆♣❡s❛r ❞❡ ❡①✐st✐r❡♠ ✈❡rsõ❡s ❝♦♥tí♥✉❛s ♣❛r❛ ♠♦❞❡❧♦s ❛♥✐s♦tró♣✐❝♦s✱ ❛q✉✐ s♦♠❡♥t❡ ❞❡♠♦♥str❛r❡♠♦s

❛ ✈❡rsã♦ ❝♦♥tí♥✉❛ ❞♦ ♠♦❞❡❧♦ ✐s♦tró♣✐❝♦✳

✻✵

❆♣ê♥❞✐❝❡ ❇

❋✐❣✉r❛ ❇✳✶✿ ❘❡❞❡ q✉❛❞r❛❞❛ ✐♥❞✐❝❛♥❞♦ ♦ ❡s♣❛ç❛♠❡♥t♦ ❞❡ r❡❞❡ a ❜❡♠ ❝♦♠♦ ♦s q✉❛tr♦ s♣✐♥s

q✉❡ ✐♥t❡r❛❣❡♠ ❝♦♠ ♦ s♣✐♥ ❞♦ sít✐♦ i✳

♦♥❞❡ α = x✱ y ❡ z✳ ❊①♣❛♥❞✐♥❞♦ ❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ s♣✐♥ ❡♠ sér✐❡s ❞❡ ❚❛②❧♦r✱ ❝♦♠ ✉♠❛

❜♦❛ ❛♣r♦①✐♠❛çã♦ ❛té s❡❣✉♥❞❛ ♦r❞❡♠✱ t❡♠♦s✿

Sαi+1 = Sα

i + a∂Sα

i

∂x+a2

2

∂2Sαi

∂x2+ . . . ✱ ✭❇✳✷✮

Sαi−1 = Sα

i − a∂Sα

i

∂x+a2

2

∂2Sαi

∂x2− . . . ✱ ✭❇✳✸✮

Sαi+2 = Sα

i + a∂Sα

i

∂y+a2

2

∂2Sαi

∂y2+ . . . ✱ ✭❇✳✹✮

Sαi−2 = Sα

i − a∂Sα

i

∂y+a2

2

∂2Sαi

∂y2− . . . ✭❇✳✺✮

❉❛í✱ t✐r❛♠♦s q✉❡✿

T α = 2Sαi S

αi + a2

∂2Sαi

∂x2Sαi + 2Sα

i Sαi + a2

∂2Sαi

∂y2Sαi + . . . ✭❇✳✻✮

⇒ T α

2= 2 (Sα

i )2 +

a2

2

[∂2Sα

i

∂x2+∂2Sα

i

∂y2

]

Sαi ✳ ✭❇✳✼✮

❆❣♦r❛✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦s três t❡r♠♦s ❞❡ ✐♥t❡r❛çã♦✱ T x✱ T y ❡ T z ❡ ❛♣r♦①✐♠❛♥❞♦ ♦s

s♦♠❛tór✐♦s ❡♠ i ❡ j q✉❡ ❛♣❛r❡❝❡♠ ♥❛ ❤❛♠✐❧t♦♥✐❛♥❛ ✭✷✳✶✼✮ ♣♦r ✐♥t❡❣r❛✐s ❡♠ dx ❡ dy✿∑

i,j

7−→∫ ∫

dx dy

a2=

∫

S

d2r

a2✱ ✭❇✳✽✮

✻✶

❆♣ê♥❞✐❝❡ ❇

♦❜t❡♠♦s q✉❡✿

H = 2J

∫

S

[(Sx)

2 + (Sy)2 + (Sz)

2] d2r

a2− J

2

∫ 3∑

α=1

[∂2Sα

i

∂x2+∂2Sα

i

∂y2

]

Sαd2r

= −2J

∫

|~S|2d2r

a2− J

2

∫ 3∑

α=1

[∂2Sα

i

∂x2+∂2Sα

i

∂y2

]

Sαd2r, ✭❇✳✾✮

s❡♥❞♦ ❛ ✐♥t❡❣r❛çã♦ ❢❡✐t❛ ♥❛ s✉♣❡r❢í❝✐❡ S ❞♦ ♠❛❣♥❡t♦ ❡ q✉❡ S1 = Sx✱ S2 = Sy ❡ S3 = Sz✳

❖ í♥❞✐❝❡ i ❢♦✐ ♦♠✐t✐❞♦ ♣♦✐s t♦❞♦s ♦s sít✐♦s ❢♦r❛♠ ❝♦♥s✐❞❡r❛❞♦s ❡ ❛ss✐♠✱ ♣❛r❛ ❡✈✐t❛r♠♦s ❛

❝♦♥t❛❣❡♠ ❞✉♣❧❛ ❞❡ ❝❛❞❛ ♣❛r ❞❡ sít✐♦s ~Si · ~Sj✱ ✉♠❛ ❞✐✈✐sã♦ ♣♦r ✷ ❢♦✐ ❢❡✐t❛ ♥❛ ♦❜t❡♥çã♦ ❞❛

❡q✉❛çã♦ ❛❝✐♠❛✳ ❋❛③❡♥❞♦ ✉♠❛ ✐♥t❡❣r❛çã♦ ♣♦r ♣❛rt❡s ♥♦ ú❧t✐♠♦ t❡r♠♦ ❞❡ ✭❇✳✾✮✱ t❡♠♦s✿

∫

dx∂2Sα

∂x2Sα = −

∫ (∂Sα

∂x

)2

dx ❡∫

dy∂2Sα

∂y2Sα = −

∫ (∂Sα

∂y

)2

dy✳ ✭❇✳✶✵✮

❖ t❡r♠♦ −2J∫|~S|2 d2r

a2♣r❡s❡♥t❡ ❡♠ ✭❇✳✾✮ é ❛ ❡♥❡r❣✐❛ E0 ❞♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧

❞♦ s✐st❡♠❛✱ ❡ é ✐♥✜♥✐t♦ ♣❛r❛ ✉♠❛ r❡❞❡ ✐♥✜♥✐t❛✱ ♣♦✐s ❡st❛♠♦s s✉♣♦♥❞♦ ♦ ✈í♥❝✉❧♦ ♥ã♦✲

❧✐♥❡❛r |~S|2 = 1 ❡✱ ♣♦rt❛♥t♦✱ t❛❧ t❡r♠♦ ❞❡✈❡ s❡r s✉❜tr❛í❞♦ ❞♦ r❡s✉❧t❛❞♦ ✜♥❛❧ ✭❇✳✾✮❀

❛ ❡♥❡r❣✐❛ ❞♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ é✱ ❡♥tã♦✱ r❡♥♦r♠❛❧✐③❛❞❛ ♣❛r❛ ③❡r♦✳ ❉❡st❛ ❢♦r♠❛✱ ❛

❛♣r♦①✐♠❛çã♦ ❝♦♥tí♥✉❛ ❞❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❞❡ ❍❡✐s❡♥❜❡r❣ ❜✐❞✐♠❡♥s✐♦♥❛❧ ✐s♦tró♣✐❝❛ é ❞❛❞❛

♣♦r✿

H =J

2

∫

S

3∑

α=1

[(∂Sα

∂x

)2

+

(∂Sα

∂y

)2]

d2r =J

2

∫

S

3∑

α=1

(

~∇mα

)2

d2r✱ ✭❇✳✶✶✮

❡♠ q✉❡ ~∇ = ∂∂xx + ∂

∂yy ❡ ♦♥❞❡ t❡♠♦s ✉s❛❞♦ ❝♦♠♦ ♣❛r❛♠❡tr✐③❛çã♦ ~m = ~S✳ ❊st❛

❤❛♠✐❧t♦♥✐❛♥❛ ♣♦❞❡ s❡r r❡❡s❝r✐t❛ ❝♦♠♦ ❬✼✼❪✿

H =J

2

∫

S

(∂µ ~m) · (∂µ ~m) d2r✱ µ = 1✱ 2✱ ✭❇✳✶✷✮

♦♥❞❡ ∂1 = ∂∂x

❡ ∂2 = ∂∂y✳ ❖ ♠♦❞❡❧♦ ❞❡s❝r✐t♦ ♣♦r ✭❇✳✶✷✮ é ✉♠ ♠♦❞❡❧♦ ♠✉✐t♦ ❝♦♥❤❡❝✐❞♦

❡♠ t❡♦r✐❛ ❞❡ ❝❛♠♣♦s✱ ❝❤❛♠❛❞♦ ❞❡ ♠♦❞❡❧♦ σ ♥ã♦✲❧✐♥❡❛r ✭❞❡✈✐❞♦ ❛♦ ✈í♥❝✉❧♦ |~S|2 = 1✮✳

❚❡♥❞♦ ❢❡✐t♦ ✐st♦✱ ♥ã♦ é ❞❡ ❞✐❢í❝✐❧ ✈✐s✉❛❧✐③❛çã♦ q✉❡ s❡ ✉sáss❡♠♦s ✉♠❛ r❡❞❡ q✉❛❞r❛❞❛

tr✐❞✐♠❡♥s✐♦♥❛❧ ❡♠ ✈❡③ ❞❛ ❜✐❞✐♠❡♥s✐♦♥❛❧✱ ❛ ❡q✉❛çã♦ ✭❇✳✽✮ ❞❛r✐❛ ❧✉❣❛r ❛✿

∑

i,j,k

7−→∫ ∫ ∫

dx dy dz

a3=

∫

V

d3r

a3✱ ✭❇✳✶✸✮

✻✷

❆♣ê♥❞✐❝❡ ❇

♦♥❞❡ ❛❣♦r❛ ❛ ✐♥t❡❣r❛çã♦ s❡r✐❛ ❢❡✐t❛ ❡♠ t♦❞♦ ♦ ✈♦❧✉♠❡ ❞♦ ♠❛❣♥❡t♦✱ ❡ ❝❤❡❣❛rí❛♠♦s à

✈❡rsã♦ ❝♦♥tí♥✉❛ ❞❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❞❡ ❍❡✐s❡♥❜❡r❣ ✸❉ ✐s♦tró♣✐❝❛✱ ❞❛❞❛ ♣♦r✿

H =A

2

∫

V

3∑

α=1

(

~∇mα

)2

d3r✱ ✭❇✳✶✹✮

♦♥❞❡ ❛❣♦r❛ ~∇ = ∂∂xx+ ∂

∂yy + ∂

∂zz✱ A é ❛ ❝♦♥st❛♥t❡ ❞❡ tr♦❝❛ ❡①♣r❡ss❛ ❡♠ ❥♦✉❧❡s✴♠❡tr♦✳

❊st❛ ❡q✉❛çã♦ ♣♦❞❡ ❛✐♥❞❛ s❡r r❡❡s❝r✐t❛ ❝♦♠♦✿

H =A

2

∫

V

(∂µ ~m) · (∂µ ~m) d3r µ = 1✱ 2✱ 3✳ ✭❇✳✶✺✮

✻✸

❆♣ê♥❞✐❝❡ ❈

❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❛❧❣✉♠❛s ❡q✉❛çõ❡s

◆❡st❡ ❆♣ê♥❞✐❝❡✱ ❞❡❞✉③✐r❡♠♦s ❛s ❡q✉❛çõ❡s ✭✸✳✾✮ ❡ ✭✹✳✹✮✳

❉❡❞✉çã♦ ❞❛ ❡q✉❛çã♦ ✭✸✳✾✮

Pr✐♠❡✐r♦✱ t♦♠❛♥❞♦✲s❡ ❛ ♣❛r❛♠❡tr✐③❛çã♦ ❞❡ ~m = (senθ cosφ, senθ senφ, cos θ)✱

t❡♠♦s q✉❡✿3∑

α=1

(

~∇mα

)2

=[

~∇ (senθ cosφ)]2

+[

~∇ (senθ senφ)]2

+[

~∇ cos θ]2

=[

senθ ~∇ cosφ+ cosφ ~∇senθ]2

+[

senθ ~∇senφ+ senφ ~∇senθ]2

+[

~∇ cos θ]2

✳ ✭❈✳✶✮

P♦❞❡♠♦s ❞❡s❡♥✈♦❧✈❡r ❛ ❡q✉❛çã♦ ✭❈✳✶✮ ❡ ❝❤❡❣❛r ❡♠✿

3∑

α=1

(

~∇mα

)2

= sen2θ

[(

~∇ cosφ)2

+(

~∇senφ)2]

+(

~∇ cos θ)2

+(

~∇sen θ)2

+

+2 sen θ ~∇sen θ ·(

cosφ+ senφ ~∇senφ)

✳ ✭❈✳✷✮

❙❡ ♦❜s❡r✈❛r♠♦s ❛s ❞❡✜♥✐çõ❡s ❞❡ φ✱ ♥❛ ❡q✉❛çã♦ ✭✸✳✺✮✱ ❡ ❞❡ θ✱ ♥❛ ❡q✉❛çã♦ ✭✸✳✶✵✮✱

✈❡♠♦s q✉❡ ❛♠❜❛s sã♦ ❢✉♥çõ❡s ❞❡ ✭x✱ y✮✱ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ♣❧❛♥♦ ❞♦ ♥❛♥♦❞✐s❝♦✳ ❆ss✐♠✿

~∇ cos θ(x, y) = −sen θ(x, y) ~∇θ(x, y)

⇒[

~∇ cos θ(x, y)]2

= sen2θ[

~∇θ(x, y)]2

✱ ✭❈✳✸✮

~∇sen θ(x, y) = cos θ(x, y) ~∇θ(x, y)

⇒[

~∇sen θ(x, y)]2

= cos2 θ[

~∇θ(x, y)]2

✱ ✭❈✳✹✮

✻✹

❆♣ê♥❞✐❝❡ ❈

~∇ cosφ(x, y) = −senφ(x, y) ~∇φ(x, y)

⇒[

~∇ cosφ(x, y)]2

= sen2φ[

~∇φ(x, y)]2

✱ ✭❈✳✺✮

~∇senφ(x, y) = cosφ(x, y) ~∇φ(x, y)

⇒[

~∇senφ(x, y)]2

= cos2 φ[

~∇φ(x, y)]2

✳ ✭❈✳✻✮

❉❡ ♣♦ss❡ ❞❛s ❡q✉❛çõ❡s ✭❈✳✸✮ à ✭❈✳✻✮✱ s✉❜st✐t✉✐♥❞♦✲❛s ❡♠ ✭❈✳✷✮ ❡ r❡❛rr❛♥❥❛♥❞♦

♦s t❡r♠♦s✱ t❡♠♦s q✉❡✿3∑

α=1

(

~∇mα

)2

= sen2θ(sen2φ+ cos2 φ

) (

~∇φ)2

+(sen2θ + cos2 θ

) (

~∇φ)2

+

+2 sen θ cos θ ~∇θ ·(

cosφ~∇ cosφ+ senφ~∇senφ)

︸ ︷︷ ︸

= ~I

✱ ✭❈✳✼✮

♦♥❞❡ ~∇ = ∂∂xx+ ∂

∂yy✳

❆❣♦r❛✱ ♣r♦❝❡❞❡r❡♠♦s ♣❛r❛ ❝❛❧❝✉❧❛r ♦ t❡r♠♦ ❞❡st❛❝❛❞♦ ❡♠ ✭❈✳✼✮✿

~I = cosφ(

−senφ ~∇φ)

+ senφ(

cosφ ~∇φ)

≡ ~0✳ ✭❈✳✽✮

▲❡✈❛♥❞♦ ♦ r❡s✉❧t❛❞♦ ✭❈✳✽✮ ♥❛ ❡q✉❛çã♦ ✭❈✳✼✮✱ ❡♥❝♦♥tr❛♠♦s q✉❡✿

3∑

α=1

(

~∇mα

)2

= sen2θ(

~∇φ)2

+(

~∇θ)2

✱ ✭❈✳✾✮

q✉❡ é ❛ ❡q✉❛çã♦ ✭✸✳✾✮✱ ♣r✐♠❡✐r♦ r❡s✉❧t❛❞♦ ♦ q✉❛❧ ♥♦s ♣r♦♣♦♠♦s ♣r♦✈❛r✳

❉❡❞✉çã♦ ❞❛ ❡q✉❛çã♦ ✭✹✳✹✮

❚♦♠❛♥❞♦ ❛s ❞❡✜♥✐çõ❡s ❞❡ φ ❡ θ ♥❛s ❡q✉❛çõ❡s ✭✸✳✺✮ ❡ ✭✸✳✶✵✮✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱

❧❡♠❜r❛♠♦s q✉❡ ❡st❛♠♦s t♦♠❛♥❞♦ Q = 1✱ ~rv = (0, 0) ❡♠ ✭✸✳✺✮ ❡ p = 1✱ n = 1 ❡♠ ✭✸✳✶✵✮✳

❆ s❡❣✉✐r✱ ♣r♦❝❡❞❡♠♦s ♣❛r❛ ❝❛❧❝✉❧❛r ♦s t❡r♠♦s ❞❡ ✭❈✳✾✮✳ Pr✐♠❡✐r♦✿

∂φ

∂x=

−yx2 + y2

❡∂φ

∂y=

x

x2 + y2✳ ✭❈✳✶✵✮

❆❣♦r❛✱ ❞❡ ✭❈✳✶✵✮✱ ✜❝❛♠♦s ❝♦♠✿

~∇φ =−y

x2 + y2x+

x

x2 + y2y

⇒(

~∇φ)2

=

(x

x2 + y2

)2

=x2 + y2

(x2 + y2)2=

1

r2✳ ✭❈✳✶✶✮

✻✺

❆♣ê♥❞✐❝❡ ❈

❈♦♠ θ = θ(r)✱ t❡♠♦s q✉❡✿

dθ

dr=

2 r√

2 r2 l20 − r4⇒

(

~∇θ)2

=2 r

2 l20 − r2✳ ✭❈✳✶✷✮

❚❡♠♦s t❛♠❜é♠ q✉❡✿

sen2θ = 1−(l20 − r2

l20

)2

=2 r2 l20 − r4

l40✳ ✭❈✳✶✸✮

❙✉❜st✐t✉✐♥❞♦ ❛s ❡q✉❛çõ❡s ✭❈✳✶✶✮✱ ✭❈✳✶✷✮ ❡ ✭❈✳✶✸✮ ♥❛ ❡q✉❛çã♦ ✭❈✳✾✮✱ ❡♥❝♦♥tr❛♠♦s✿

3∑

α=1

(

~∇mα

)2

=

(2 r2 l20 − r4

l40

)1

r2+

4

2 l20 − r2=

(2 l20 − r2)2 + 4 l40l40 (2 l

20 − r2)

✳ ✭❈✳✶✹✮

❆❣♦r❛✱ s✉❜st✐t✉✐♥❞♦ ♦ r❡s✉❧t❛❞♦ ♦❜t✐❞♦ ❡♠ ✭❈✳✶✹✮ ♥❛ ❞❡✜♥✐çã♦ ❞❡ ξi✱ ❞❛❞❛ ❡♠

✭✹✳✷✮✱ ❡ ❧❡♠❜r❛♥❞♦ ❞❛ ❞❡✜♥✐çã♦ ❞❛ ❢✉♥çã♦ δ✲❉✐r❛❝✱ ♦❜t❡♠♦s✿

ξi =(2 l20 − x2i )

2 + 4 l40l40 (2 l

20 − x2i )

λi Gi(~r − ~xi)✱ ✭❈✳✶✺✮

q✉❡ é ❥✉st❛♠❡♥t❡ ❛ ❡q✉❛çã♦ ✭✹✳✹✮✱ s❡❣✉♥❞♦ r❡s✉❧t❛❞♦ q✉❡ ♥♦s ♣r♦♣♦♠♦s ♣r♦✈❛r✳

✻✻

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

❬✶❪ ❈✳✲❲✳ ❈❤❡♥✱ ▼❛❣♥❡t✐s♠ ❛♥❞ ▼❡t❛❧✉r❣② ♦❢ ❙♦❢t ▼❛❣♥❡t✐❝ ▼❛t❡r✐❛❧s✱ ❉♦✈❡r P✉❜❧✐✲

❝❛t✐♦♥s✱ ◆❡✇ ❏❡rs❡② ✭✶✾✽✻✮✳

❬✷❪ ❙✳ ▼✳ ❘❡③❡♥❞❡✱ ❘❡✈✳ ❇r❛s✳ ❊♥s✳ ❋ís✳ ✷✷✱ ✷✾✸ ✭✷✵✵✵✮✳

❬✸❪ ▼✳ ❊✳ ❊✈❛♥s ❛♥❞ ❋✳ ❍❡❧❧❡r✱ ❊♥✈✐r♦♠❡♥t❛❧ ▼❛❣♥❡t✐s♠✿ Pr✐♥❝✐♣❧❡s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s

♦❢ ❊♥✈✐r♦♠❛❣♥❡ts✱ ❆❝❛❞❡♠✐❝ Pr❡ss✱ ❙❛♥ ❉✐❡❣♦ ✭✷✵✵✸✮✳

❬✹❪ ❆✳ ❑✳ ❙❛❧❡♠✱ P✳ ❈✳ ❙❡❛rs♦♥✱ ❑✳ ❲✳ ▲❡♦♥❣✱ ◆❛t✳ ▼❛t❡r✳ ✷✱ ✻✻✽ ✭✷✵✵✸✮✳

❬✺❪ ❑✳ ▼✳ ❑r✐s❤♥❛♥✱ ❆✳ ❇✳ P❛❦❤♦♠♦✈✱ ❨✳ ❇❛♦✱ P✳ ❇❧♦♠q✈✐st✱ ❨✳ ❈❤✉♥✱ ▼✳ ●♦♥③❛❧❡③✱

❑✳ ●r✐✣♥✱ ❳✳ ❏✐✱ ❇✳ ❑✳ ❘♦❜❡rts✱ ❏✳ ▼❛t❡r✳ ❙❝✐✳ ✹✶✱ ✼✾✸ ✭✷✵✵✻✮✳

❬✻❪ ❉✳✲❍✳ ❑✐♠✱ ❡t ❛❧✳✱ ◆❛t✉r❡ ▼❛t✳ ✾✱ ✶✻✺ ✭✷✵✶✵✮✳

❬✼❪ ❏✳ ■✳ ▼❛rtí♥✱ ❡t ❛❧✳✱ ❏✳ ▼❛❣♥✳ ▼❛❣♥✳ ▼❛t❡r✳ ✷✺✻✱ ✹✹✾ ✭✷✵✵✸✮✳

❬✽❪ ❙✳ ❉✳ ❇❛❞❡r✱ ❑✳ ❙✳ ❇✉❝❤❛♥❛♥✱ ❙✳✲❍✳ ❈❤✉♥❣✱ ❑✳ ❨✳ ●✉s❧✐❡♥❦♦✱ ❆✳ ❍♦✛♠❛♥♥✱ ❨✐ ❏✐

❛♥❞ ❱✳ ◆♦✈♦s❛❞✱ ❙✉♣❡r❧❛tt✐❝❡s ❛♥❞ ▼✐❝r♦str✉❝t✉r❡s ✹✶✱ ✼✷ ✭✷✵✵✼✮✳

❬✾❪ ▼✳ ❙❝❤♥❡✐❞❡r✱ ❍✳ ❍♦✛♠❛♥✱ ❛♥❞ ❏✳ ❩✇❡❝❦✱ ❆♣♣❧✳ P❤②s✳ ▲❡tt✳ ✼✼✱ ✷✾✵✾ ✭✷✵✵✵✮✳

❬✶✵❪ ❘✳ P✉❧✇❡②✱ ▼✳ ❘❛❤♠✱ ❏✳ ❇✐❜❡r❣❡r✱ ❛♥❞ ❉✳ ❲❡✐ss✱ ■❊❊❊ ❚r❛♥s✳ ▼❛❣♥✳ ✸✼✱ ✷✵✼✻

✭✷✵✵✶✮✳

❬✶✶❪ ▼✳ ❘❛❤♠✱ ❘✳ ❍ö❧❧✐♥❣❡r✱ ❱✳ ❯♠❛♥s❦②✱ ❛♥❞ ❉✳ ❲❡✐ss✱ ❏✳ ❆♣♣❧✳ P❤②s✳ ✾✺✱ ✻✼✵✽

✭✷✵✵✹✮✳

❬✶✷❪ ❘✳ P✳ ❈♦✇❜✉r♥✱ ❉✳ ❑✳ ❑♦❧st♦✈✱ ❆✳ ❖✳ ❆❞❡②❡②❡✱ ▼✳ ❊✳ ❲❡❧❧❛♥❞✱ ❛♥❞ ❉✳ ▼✳ ❚r✐❝❦❡r✱

P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✽✸✱ ✶✵✹✷ ✭✶✾✾✾✮✳

✻✼

✻✳ ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

❬✶✸❪ ❚✳ ❯❤❧✐❣ ❡t ❛❧✳✱ P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✾✺✱ ✷✸✼✷✵✺ ✭✷✵✵✺✮✳

❬✶✹❪ ❘✳ ▲✳ ❈♦♠♣t♦♥ ❛♥❞ P✳ ❆✳ ❈r♦✇❡❧❧✱ P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✾✼✱ ✶✸✼✷✵✷ ✭✷✵✵✻✮✳

❬✶✺❪ ❈✳ ❆✳ ❘♦ss ❡t ❛❧✳✱ P❤②s✳ ❘❡✈✳ ❇ ✻✺✱ ✶✹✹✹✶✼ ✭✷✵✵✷✮✳

❬✶✻❪ ❈✳ ❆✳ ❘♦ss ❡t ❛❧✳✱ ❏✳ ❆♣♣❧✳ P❤②s✳ ✽✾✱ ✶✸✶✵ ✭✷✵✵✶✮✳

❬✶✼❪ ❘✳ ▲✳ ❙✐❧✈❛✱ ✧❉✐♥â♠✐❝❛ ❞❡ s♣✐♥s ♥❛ ♣r❡s❡♥ç❛ ❞❡ ❞❡❢❡✐t♦s ❡♠ ❡str✉t✉r❛s ♠❛❣♥ét✐❝❛s

❝♦♥✜♥❛❞❛s✿ ❛♣❧✐❝❛çõ❡s ❛ ♥❛♥♦❞✐s❝♦s ♠❛❣♥ét✐❝♦s✧✱ ❚❡s❡ ❞❡ ❉♦✉t♦r❛❞♦✱ ❯❋❱ ✭✷✵✵✾✮✳

❬✶✽❪ ❏✳ ❘❛❛❜❡✱ ❘✳ P✉❧✇❡②✱ ❆✳ ❙❛tt❧❡r✱ ❚✳ ❙❝❤✇❡✐♥❜ö❝❦✱ ❏✳ ❩✇❡❝❦ ❛♥❞ ❉✳ ❲❡✐ss✱ ❏✳ ❆♣♣❧✳

P❤②s✳ ✽✽✱ ✹✹✸✼ ✭✷✵✵✵✮✳

❬✶✾❪ ❲✳ ❑✳ ❍✐❡❜❡rt✱ ❆✳ ❙t❛♥❦✐❡✇✐❝③✱ ❛♥❞ ▼✳ ❘✳ ❋r❡❡♠❛♥✱ P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✼✾✱ ✶✶✸✹

✭✶✾✾✼✮✳

❬✷✵❪ ❑✳ ❲✳ ❈❤♦✉ ❡t ❛❧✳✱ ❆♣♣❧✳ P❤②s✳ ▲❡tt✳ ✾✵✱ ✷✵✷✺✵✺ ✭✷✵✵✼✮✳

❬✷✶❪ ▼✳ ❑♥♦❜❡❧✱ ❈✐ê♥❝✐❛ ❍♦❥❡ ✸✻✱ ✶✽ ✭✷✵✵✺✮✳

❬✷✷❪ ❙✳ ❉✳ ❇❛❞❡r✱ ❙✉r❢✳ ❙❝✐✳ ✺✵✵✱ ✶✼✷ ✭✷✵✵✶✮✳

❬✷✸❪ ❙✳ ❉✳ ❇❛❞❡r✱ ❘❡✈✳ ▼♦❞✳ P❤②s✳ ✼✽✱ ✶ ✭✷✵✵✻✮✳

❬✷✹❪ ❆✳ P✳ ●✉✐♠❛rã❡s✱ ✧■♥tr♦❞✉çã♦ ❛♦ ♥❛♥♦♠❛❣♥❡t✐s♠♦✧✱ ❈❇P❋ ✭✷✵✵✻✮✳

❬✷✺❪ ▼✳ ❑♥♦❜❡❧✱ ❲✳ ❆✳ ❆✳ ▼❛❝❡❞♦✱ ❋ís✐❝❛ ♥❛ ❊s❝♦❧❛ ✽✱ ✸✸ ✭✷✵✵✼✮✳

❬✷✻❪ ❙✳ ❆✳ ❲♦❧❢✱ ❉✳ ❉✳ ❆✇s❝❤❛❧♦♠✱ ❘✳ ❆✳ ❇❤✉r♠❛♥✱ ❏✳ ▼✳ ❉❛✉❣❤t♦♥✱ ❙✳ ✈♦♥ ▼♦❧♥ár✱

▼✳ ▲✳ ❘♦✉❦❡s✱ ❆✳ ❨✳ ❈❤t❝❤❡❧❦❛♥♦✈❛✱ ❛♥❞ ❉✳ ▼✳ ❚r❡❣❡✱ ❙❝✐❡♥❝❡ ✷✾✹✱ ✶✹✽✽ ✭✷✵✵✶✮✳

❬✷✼❪ ▼✳ ❘❛❤♠✱ ❏✳ ❙t❛❤❧✱ ❛♥❞ ❉✳ ❲❡✐ss✱ ❆♣♣❧✳ P❤②s✳ ▲❡tt✳ ✽✼✱ ✶✽✷✶✵✼ ✭✷✵✵✺✮✳

❬✷✽❪ ▼✳ ❚❡❤r❛♥✐ ❊✳ ❈❤❡♥✱ ▼✳ ❉✉r❧❛♠✱ ▼✳ ❉❡❍❡rr❡r❛✱ ❏✳ ▼✳ ❙❧❛✉❣❤t❡r✱ ❏✳ ❙❤✐✱ ❛♥❞ ●✳

❑❡r③②❦♦✇s❦✐✱ ❏✳ ❆♣♣❧✳ P❤②s✳ ✽✺✱ ✺✽✷✷ ✭✶✾✾✾✮✳

❬✷✾❪ ❙✳ ❙✳ P✳ P❛r❦✐♥ ❡t ❛❧✳✱ ❏✳ ❆♣♣❧✳ P❤②s✳ ✽✺✱ ✺✽✷✽ ✭✶✾✾✾✮✳

❬✸✵❪ ●✳ ❆✳ Pr✐♥③✱ ❏✳ ▼❛❣♥✳ ▼❛❣♥✳ ▼❛t✳ ✷✵✵✱ ✺✼ ✭✶✾✾✾✮✳

✻✽

✻✳ ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

❬✸✶❪ ❏✳ ◆✳ ❈❤❛♣♠❛♥ ❡t ❛❧✳✱ ❏✳ ❆♣♣❧✳ P❤②s✳ ✽✺✱ ✺✽✷✽ ✭✶✾✾✾✮✳

❬✸✷❪ ▲✳ ❈❤❡♥✱ ❋✳ ▲✐✉✱ ❑✳ ❙t♦❡✈✱ ❙✳ ▲✐✱ ▼✳ ❍♦✱ ❛♥❞ ❙✳ ▼❛♦✱ ❏✳ ❆♣♣❧✳ P❤②s✳ ✶✵✺✱ ✵✼❇✵✼✸✵

✭✷✵✵✾✮✳

❬✸✸❪ ▼✳ ❘❛❤♠✱ ❏✳ ❙t❛❤❧✱ ❲✳ ❲❡❣s❝❤❡✐❞❡r✱ ❛♥❞ ❉✳ ❲❡✐ss✱ ❆♣♣❧✳ P❤②s✳ ▲❡tt✳ ✽✺✱ ✶✺✺✸

✭✷✵✵✹✮✳

❬✸✹❪ ▼✳ ❘❛❤♠✱ ❏✳ ❇✐❜❡r❣❡r✱ ❱✳ ❯♠❛♥s❦②✱ ❛♥❞ ❉✳ ❲❡✐ss✱ ❏✳ ❆♣♣❧✳ P❤②s✳ ✾✸✱ ✼✹✷✾ ✭✷✵✵✸✮✳

❬✸✺❪ ▼✳❘❛❤♠✱ ❏✳ ❙t❛❤❧✱ ❲✳ ❲❡❣s❝❤❡✐❞❡r✱ ❛♥❞ ❉✳ ❲❡✐ss✱ ❆♣♣❧✳ P❤②s✳ ▲❡tt✳ ✽✺✱ ✶✺✺✸

✭✷✵✵✹✮✳

❬✸✻❪ ❆✳ ❘✳ P❡r❡✐r❛✱ ▲✳ ❆✳ ❙✳ ▼ó❧✱ ❙✳ ❆✳ ▲❡♦♥❡❧✱ P✳ ❩✳ ❈♦✉r❛✱ ❛♥❞ ❇✳ ❱✳ ❈♦st❛✱ P❤②s✳

❘❡✈✳ ❇ ✻✽✱ ✶✸✷✹✵✾ ✭✷✵✵✸✮✳

❬✸✼❪ ❆✳ ❘✳ P❡r❡✐r❛✱ P❤②s✳ ❘❡✈✳ ❇ ✼✶✱ ✷✷✹✹✵✹ ✭✷✵✵✺✮✳

❬✸✽❪ ❆✳ ❘✳ P❡r❡✐r❛✱ ❏✳ ❆♣♣❧✳ P❤②s✳ ✾✼✱ ✵✾✹✸✵✸ ✭✷✵✵✺✮✳

❬✸✾❪ ❆✳ ❘✳ P❡r❡✐r❛✱ ❆✳ ❘✳ ▼♦✉r❛✱ ❲✳ ❆✳ ▼♦✉r❛✲▼❡❧♦✱ ❉✳ ❋✳ ❈❛r♥❡✐r♦✱ ❙✳ ❆✳ ▲❡♦♥❡❧✱ ❛♥❞

P✳ ❩✳ ❈♦✉r❛✱ ❏✳ ❆♣♣❧✳ P❤②s✳ ✶✵✶✱ ✵✸✹✸✶✵ ✭✷✵✵✼✮✳

❬✹✵❪ ❋✳ ▼✳ P❛✉❧❛✱ ❆✳ ❘✳ P❡r❡✐r❛✱ ❛♥❞ ▲✳ ❆✳ ❙✳ ▼ó❧✱ P❤②s✳ ▲❡tt✳ ❆ ✸✷✾✱ ✶✺✺ ✭✷✵✵✹✮✳

❬✹✶❪ ❉✳ ❏✳ ●r✐✣t❤s✱ ✧■♥tr♦❞✉❝t✐♦♥ t♦ ❊❧❡❝tr♦❞②♥❛♠✐❝s✧✱ Pr❡♥t✐❝❡ ❍❛❧❧✱ ◆❡✇ ❏❡rs❡②✱

❚❤✐r❞ ❊❞✐t✐♦♥ ✭✶✾✾✾✮✳

❬✹✷❪ ❙✳ ▲✳ ❖✬❉❡❧❧ ❛♥❞ ❘✳ ❑✳ P✳ ❩✐❛✱ ❆♠✳ ❏✳ P❤②s✳ ✺✹✱ ✸✷ ✭✶✾✽✻✮✳

❬✹✸❪ ❆✳ ❆❤❛r♦♥✐✱ ✧■♥tr♦❞✉❝t✐♦♥ t♦ t❤❡ ❚❤❡♦r② ♦❢ ❋❡rr♦♠❛❣♥❡t✐s♠✧✱ ❖①❢♦r❞ ❙❝✐❡♥❝❡ P✉❜✲

❧✐❝❛t✐♦♥s✱ ❙❡❝♦♥❞ ❊❞✐t✐♦♥ ✭✷✵✵✵✮✳

❬✹✹❪ ❱✳ ▲✳ ❈✳ ❙❛♥t♦s✱ ✧❙♦❧✉çõ❡s t♦♣♦❧ó❣✐❝❛s ❞❡ s♣✐♥s ♥♦ t♦r♦✧✱ ❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦✱

❯❋❱ ✭✷✵✵✽✮✳

❬✹✺❪ ■✳ ❙✳ ❖❧✐✈❡✐r❛✱ ❱✳ ▲✳ ❇✳ ❞❡ ❏❡s✉s✱ ✧■♥tr♦❞✉çã♦ à ❋ís✐❝❛ ❞♦ ❊st❛❞♦ ❙ó❧✐❞♦✧✱ ❊❞✐t♦r❛

▲✐✈r❛r✐❛ ❞❛ ❋ís✐❝❛✱ ❙ã♦ P❛✉❧♦✱ ✶❛ ❊❞✐çã♦ ✭✷✵✵✺✮✳

✻✾

✻✳ ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

❬✹✻❪ ▼✳ ❆✳ ◆♦✈❛❦✱ ❁❤tt♣✿✴✴✇✇✇✳❝❜♣❢✳❜r✴ ❧❛❜♠❛❣✴♠✐❣✉❡❧✳♣❞❢❃✳

❬✹✼❪ ❙✳ ❘✳ ❆✳ ❙❛❧✐♥❛s✱ ✧■♥tr♦❞✉çã♦ à ❋ís✐❝❛ ❊st❛tíst✐❝❛✧✱ ❊❞✐t♦r❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡

❙ã♦ P❛✉❧♦✱ ❙ã♦ P❛✉❧♦ ✭✶✾✾✾✮✳

❬✹✽❪ ▼✳ ●❡t③❧❛✛✱ ✧❋✉♥❞❛♠❡♥t❛❧s ♦❢ ▼❛❣♥❡t✐s♠✧✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥ ✭✷✵✵✽✮✳

❬✹✾❪ ❋✳ ▼✳ P❛✉❧❛✱ ✧■♥✢✉ê♥❝✐❛ ❞❡ ✈❛❝â♥❝✐❛s ❞❡ s♣✐♥ ♥❛ ❞✐♥â♠✐❝❛ ❞❡ ✈órt✐❝❡s ❡♠ s✐st❡♠❛s

♠❛❣♥ét✐❝♦s ❜✐❞✐♠❡♥s✐♦♥❛✐s✧✱ ❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦✱ ❯❋❱ ✭✷✵✵✻✮✳

❬✺✵❪ ❈✳ ▲✳ ❈❤✐❡♥✱ ❋✳ ◗✳ ❩❤✉✱ ❛♥❞ ❏✳✲●✳ ❩❤✉✱ P❤②s✳ ❚♦❞❛② ✵✻✱ ✹✵ ✭✷✵✵✼✮✳

❬✺✶❪ ❏✳ ❉✳ ❏❛❝❦s♦♥✱ ✧❈❧❛ss✐❝❛❧ ❊❧❡❝tr♦❞②♥❛♠✐❝s✧✱ ❲✐❧❡②✱ ◆❡✇ ❨♦r❦✱ ❚❤✐r❞ ❊❞✐t✐♦♥

✭✶✾✾✾✮✳

❬✺✷❪ ❏✳ P✳ ❙✐♥♥❡❝❦❡r✱ ❘❡✈✳ ❇r❛s✳ ❊♥s✳ ❋ís✳✱ ✷✷✱ ✸✾✻ ✭✷✵✵✵✮✳

❬✺✸❪ ❉✳ ❋✳ ❈❛r♥❡✐r♦✱ ✧❈♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✈órt✐❝❡s ❡♠ ♥❛♥♦❞✐s❝♦s ♠❛❣♥ét✐❝♦s ♥❛ ♣r❡✲

s❡♥ç❛ ❞❡ ❞❡❢❡✐t♦s ♥ã♦✲♠❛❣♥ét✐❝♦s✧✱ ❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦✱ ❯❋❏❋ ✭✷✵✵✻✮✳

❬✺✹❪ ❊✳ ▼✐r❛♥❞❛✱ ❁❤tt♣✿✴✴✇✇✇✳✐✜✳✉♥✐❝❛♠♣✳❜r✴ ❡♠✐r❛♥❞❛✴♥♦t❡s✴r❣✳♣❞❢❃✳

❬✺✺❪ ❆✳ ❘✳ P❡r❡✐r❛ ❛♥❞ ❲✳ ❆✳ ▼♦✉r❛✲▼❡❧♦✱ ✧❱♦rt❡① ❜❛❤❛✈✐♦r ✐♥ ❢❡rr♦♠❛❣♥❡t✐❝ s②s✲

t❡♠s ✇✐t❤ s♠❛❧❧ ❞❡❢❡❝ts✿ ❋r♦♠ ♠❛❝r♦ t♦ ♥❛♥♦str✉❝t✉r❡❞ ♠❛❣♥❡ts✧✱ ❡♠ ✧❊❧❡tr♦✲

♠❛❣♥❡t✐❝✱ ▼❛♥❡t♦st❛t✐❝✱ ❛♥❞ ❊①❝❤❛♥❣❡✲■♥t❡r❛❝t✐♦♥ ❱♦rt✐❝❡s ✐♥ ❈♦♥✜♥❡❞ ▼❆❣♥❡t✐❝

❙tr✉❝t✉r❡s✧✱ ❡❞✐t❛❞♦ ♣♦r ❊✳ ❖✳ ❑❛♠❡♥❡ts❦✐✐✱ ❚r❛♥s✇♦r❧❞ ❘s❡❛r❝❤ ◆❡t✇♦r❦✱ ❑❡r❛❧❛✱

❮♥❞✐❛ ✭✷✵✵✾✮✳

❬✺✻❪ ▲✳ ❆✳ ❙✳ ▼ó❧✱ ✧❊st✉❞♦s ❞❡ ♠❛t❡r✐❛✐s ♠❛❣♥ét✐❝♦s ❞❡ ❜❛✐①❛ ❞✐♠❡♥s✐♦♥❛❧✐❞❛❞❡ ❞♦♣❛❞♦s

❝♦♠ ✐♠♣✉r❡③❛s ♥ã♦ ♠❛❣♥ét✐❝❛s✧✱ ❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦✱ ❯❋❱ ✭✷✵✵✹✮✳

❬✺✼❪ ❉✳ ❆❧t❜✐r✱ ❏✳ ❊s❝r✐❣✱ P✳ ▲❛♥❞❡r♦s✱ ❋✳ ❙✳ ❆♠❛r❛❧✱ ❛♥❞ ▼✳ ❇❛❤✐❛♥❛✱ ◆❛♥♦t❡❝❤♥♦❧♦❣②

✶✽✱ ✹✽✺✼✵✼ ✭✷✵✵✼✮✳

❬✺✽❪ ❆✳ ❲❛❝❤♦✇✐❛❦✱ ❏✳ ❲✐❡❜❡✱ ▼✳ ❇♦❞❡✱ ❖✳ P✐❡t③s❝❤✱ ▼✳ ▼♦r❣❡♥st❡r♥✱ ❘✳ ❲✐❡s❡♥❞❛♥✲

❣❡r✱ ❙❝✐❡♥❝❡ ✷✾✽✱ ✺✼✼ ✭✷✵✵✷✮✳

❬✺✾❪ ◆✳ ❆✳ ❯s♦✈ ❛♥❞ ▲✳ ●✳ ❑✉r❦✐♥❛✱ ❏✳ ▼❛❣♥✳ ▼❣❛♥✳ ▼❛t❡r✳ ✷✹✷✲✷✹✺✱ ✶✵✵✺ ✭✷✵✵✷✮✳

✼✵

✻✳ ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

❬✻✵❪ ❆✳ ❆❤❛r♦♥✐✱ ❏✳ ❆♣♣❧✳ P❤②s✳ ✻✽✱ ✷✽✾✷ ✭✶✾✾✵✮✳

❬✻✶❪ P✳ ▲❛♥❞❡r♦s ❡t ❛❧✳✱ P❤②s✳ ❘❡✈✳ ❇ ✼✶✱ ✵✾✹✹✸✺ ✭✷✵✵✺✮✳

❬✻✷❪ ❚✳ ❙❤✐♥❥♦ ❡t ❛❧✳✱ ❙❝✐❡♥❝❡ ✷✽✾✱ ✾✸✵ ✭✷✵✵✵✮✳

❬✻✸❪ ❑✳ ❇✉ss♠❛♥♥✱ ●✳ ❆✳ Pr✐♥③✱ ❙✳✲❋✳ ❈❤❡♥❣✱ ❛♥❞ ❉✳ ❲❛♥❣✱ ❆♣♣❧✳ P❤②s✳ ▲❡tt✳ ✼✺✱ ✷✹✼✻

✭✶✾✾✾✮✳

❬✻✹❪ ❑✳ ❨✳ ●✉s❧✐❡♥❦♦✱ ❱✳ ◆♦✈♦s❛❞✱ ❨✳ ❖t❛♥✐✱ ❍✳ ❙❤✐♠❛✱ ❛♥❞ ❑✳ ❋✉❦❛♠✐❝❤✐✱ ❆♣♣❧✳ P❤②s✳

▲❡tt✳ ✼✽✱ ✸✽✹✽ ✭✷✵✵✶✮✳

❬✻✺❪ ❑✳ ❨✳ ●✉s❧✐❡♥❦♦✱ ❳✳ ❋✳ ❍❛♥✱ ❉✳ ❏✳ ❑❡❛✈♥❡②✱ ❘✳ ❉✐✈❛♥✱ ❛♥❞ ❙✳ ❉✳ ❇❛❞❡r✱ P❤②s ❘❡✈✳

▲❡tt✳ ✾✻✱ ✵✻✼✷✵✺ ✭✷✵✵✻✮✳

❬✻✻❪ ❏✳✲●✳ ❈❛♣✉t♦✱ ❨✳ ●❛✐❞✐❞❡✐✱ ❋✳ ●✳ ▼❡rt❡♥s✱ ❛♥❞ ❉✳ ❉✳ ❙❤❡❦❛✱ P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✾✽✱

✵✺✻✻✵✹ ✭✷✵✵✼✮✳

❬✻✼❪ ❑✳ ❨❛♠❛❞❛✱ ❙✳ ❑❛s❛✐✱ ❨✳ ◆❛❦❛t❛♥✐✱ ❑✳ ❑♦❜❛②❛s❤✐✱ ❍✳ ❑♦❤♥♦✱ ❆✳ ❚❤✐❛✈✐❧❧❡✱ ❛♥❞ ❚✳

❖♥♦✱ ◆❛t✉r❡ ▼❛t❡r✳ ✻✱ ✷✻✾ ✭✷✵✵✼✮✳

❬✻✽❪ ❇✳ ❱❛♥ ❲❛❡②❡♥❜❡r❣❡ ❡t ❛❧✳✱ ◆❛t✉r❡ ✹✹✹✱ ✹✻✶ ✭✷✵✵✻✮✳

❬✻✾❪ ❘✳ ▲✳ ❙✐❧✈❛ ❡t ❛❧✳✱ P❤②s✳ ❘❡✈✳ ❇ ✼✽✱ ✵✺✹✹✷✸ ✭✷✵✵✽✮✳

❬✼✵❪ ❘✳ ▲✳ ❙✐❧✈❛✱ ❆✳ ❘✳ P❡r❡✐r❛✱ ❛♥❞ ❲✳ ❆✳ ▼♦✉r❛✲▼❡❧♦✱ ❏✳ ❆♣♣❧✳ P❤②s✳ ✶✵✺✱ ✵✶✹✸✶✹

✭✷✵✵✾✮✳

❬✼✶❪ ❑✳ ❨✳ ●✉s❧✐❡♥❦♦✱ ❑✳✲❙✳ ▲❡❡✱ ❛♥❞ ❙✳✲❑✳ ❑✐♠✱ P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✶✵✵✱ ✵✷✼✷✵✸ ✭✷✵✵✽✮✳

❬✼✷❪ ❘✳ ●♦❜❜✐✱ ❋✳ ❆✳ ❆♣♦❧♦♥✐♦✱ ❘✳ ▲✳ ❙✐❧✈❛✱ ❲✳ ❆✳ ▼♦✉r❛✲▼❡❧♦ ❡ ❆✳ ❘✳ P❡r❡✐r❛✱ tr❛❜❛❧❤♦

❡♠ ♣r❡♣❛r❛çã♦✳

❬✼✸❪ ❋✳ ❆✳ ❆♣♦❧♦♥✐♦✱ ❲✳ ❆✳ ▼♦✉r❛✲▼❡❧♦✱ ❋✳ P✳ ❈r✐s❛❢✉❧✐✱ ❆✳ ❘✳ P❡r❡✐r❛✱ ❛♥❞ ❘✳ ▲✳ ❙✐❧✈❛✱

❏✳ ❆♣♣❧✳ P❤②s✳ ✶✵✻✱ ✵✽✹✸✷✵ ✭✷✵✵✾✮✳

❬✼✹❪ ❋✳ ❍♦✛♠❛♥♥ ❡t ❛❧✳✱ P❤②s✳ ❘❡✈✳ ❇ ✼✻✱ ✵✶✹✹✶✻ ✭✷✵✵✼✮✳

✼✶

✻✳ ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

❬✼✺❪ ❆✳ ❱❛♥st❡❡♥❦✐st❡✱ ▼✳ ❲❡✐❣❛♥❞✱ ▼✳ ❈✉r❝✐❝✱ ❍✳ ❙t♦❧❧✱ ●✳ ❙❝❤üt③✱ ❛♥❞ ❇✳ ❱❛♥❲❛❡②❡♥✲

❜❡r❣❡✱ ◆✳ ❏✳ P②s✳ ✶✶✱ ✵✻✸✵✵✻ ✭✷✵✵✾✮✳

❬✼✻❪ ❏✳ ❇✳ ❑♦rtr✐❣❤t✱ ❉✳ ❉✳ ❆✇s❝❤❛❧♦♠ ❡t ❛❧✳✱ ❏✳ ▼❛❣♥✳ ▼❛❣♥✳ ▼❛t❡r✳ ✷✵✼✱ ✼ ✭✶✾✾✾✮✳

❬✼✼❪ ❆✳ ❘✳ P❡r❡✐r❛ ❡ ❆✳ ❙✳ ❚✳ P✐r❡s✱ ✧❚❡♦r✐❛ ◗✉â♥t✐❝❛ ❞❡ ❈❛♠✲

♣♦s ❡♠ ❙✐st❡♠❛s ▼❛❣♥ét✐❝♦s ❞❡ ❇❛✐①❛s ❉✐♠❡♥sõ❡s✧✱ ❞✐s♣♦♥í✈❡❧ ❡♠

❁❤tt♣✿✴✴✇✇✇✳✉❢✈✳❜r✴❞♣❢✴tq❝s♠❜❞✳♣❞❢❃✱ ✭✷✵✵✽✮✳

✼✷