❋❊▲■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
❆♦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♦rnˆ✱ ♥♦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~ ·nˆ = 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♥♦H~ext✱ ❡♠ ❖❡rst❡❞s ✭✶ ❖❡≈✼✾✱✻ ❆✴♠✮✱
❛♣❧✐❝❛❞♦ ♥♦ ♣❧❛♥♦ ❞❛ ❛♠♦str❛✱ ❝♦♥❢♦r♠❡ ✐♥❞✐❝❛❞♦✳ ❖ ❝❛♠♣♦ é ❛✉♠❡♥t❛❞♦ s❡❣✉✐❞❛♠❡♥t❡✱ ❝♦♠ ♦s ♠♦♠❡♥t♦s ♠❛❣♥ét✐❝♦s s✉❝❡ss✐✈❛♠❡♥t❡ ♦r✐❡♥t❛♥❞♦✲s❡ ♥❛ ❞✐r❡çã♦ ❞❡ H~ext✳ ▼♦str❛♠♦s t❛♠❜é♠ ❛s ✧❝❛r❣❛s ♠❛❣♥ét✐❝❛s✧σM = M~ ·ˆn ❡
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= 1,5µ♠ ❝♦♠ ✉♠ ❞❡❢❡✐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♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
✹✳✷ ❖ ❝❛♠✐♥❤♦ ❡♠ ✈❡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♦ ❞❡ 2l0 ❡♥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 ♣♦rhext✳ ✳ ✳ ✺✶
❆✳✶ ❊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✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶
❘❡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−x~
0)✱ 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❡✳
❆❜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−x~
0)✱ 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✳
❈❛♣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✉✲
