Estudo de nanoestruturas de nitreto de boro hexagonal por microscopia de varredura por sonda

❊st✉❞♦ ❞❡ ♥❛♥♦❡str✉t✉r❛s ❞❡ ◆✐tr❡t♦ ❞❡ ❇♦r♦

❍❡①❛❣♦♥❛❧ ♣♦r ▼✐❝r♦s❝♦♣✐❛ ❞❡ ❱❛rr❡❞✉r❛ ♣♦r ❙♦♥❞❛

❊st✉❞♦ ❞❡ ♥❛♥♦❡str✉t✉r❛s ❞❡

◆✐tr❡t♦ ❞❡ ❇♦r♦ ❍❡①❛❣♦♥❛❧ ♣♦r

▼✐❝r♦s❝♦♣✐❛ ❞❡ ❱❛rr❡❞✉r❛ ♣♦r

❙♦♥❞❛

❈❛♠✐❧❧❛ ❑❛r❧❛ ❇r✐t❡s ◗✉❡✐r♦③ ▼❛rt✐♥s ❞❡ ❖❧✐✈❡✐r❛

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❇❡r♥❛r❞♦ ❘✳ ❆✳ ◆❡✈❡s

❚❡s❡ ❛♣r❡s❡♥t❛❞❛ à ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ❉♦✉t♦r ❡♠ ❈✐ê♥❝✐❛s ✕ ❋ís✐❝❛

✐✈

❙✉♠ár✐♦

❆❣r❛❞❡❝✐♠❡♥t♦s ✈✐

❘❡s✉♠♦ ✈✐✐✐

❆❜str❛❝t ✐①

✶ ■♥tr♦❞✉çã♦ ✶

✷ ▼❛t❡r✐❛❧ ❊st✉❞❛❞♦ ✺

✷✳✶ ◆✐tr❡t♦ ❞❡ ❇♦r♦ ❍❡①❛❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✶✳✶ Pr♦♣r✐❡❞❛❞❡s ❊str✉t✉r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✶✳✷ Pr♦♣r✐❡❞❛❞❡s ❊❧❡trô♥✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✸ ▼❡t♦❞♦❧♦❣✐❛s ❊①♣❡r✐♠❡♥t❛✐s ❡ ❚❡ór✐❝❛s ✶✶

✸✳✶ ❚é❝♥✐❝❛s ❊①♣❡r✐♠❡♥t❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✸✳✶✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✸✳✶✳✷ ▼✐❝r♦s❝♦♣✐❛ ❞❡ ❋♦rç❛ ❆tô♠✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✸✳✶✳✸ ▼✐❝r♦s❝♦♣✐❛ ❞❡ ❋♦rç❛ ❊❧étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✶✳✹ ▼✐❝r♦s❝♦♣✐❛ ❞❡ ❱❛rr❡❞✉r❛ ❞❡ ❈❛♣❛❝✐tâ♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✷ ❈á❧❝✉❧♦s ❞❡ Pr✐♠❡✐r♦s Pr✐♥❝í♣✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✷✳✶ Pr♦❜❧❡♠❛ ❞❡ ♠✉✐t♦s ❝♦r♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✷✳✷ ❚❡♦r✐❛ ❞♦ ❋✉♥❝✐♦♥❛❧ ❞❛ ❉❡♥s✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✷✳✸ ❉✐♥â♠✐❝❛ ▼♦❧❡❝✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸

✹ ❈❛r❛❝t❡r✐③❛çã♦ ❊❧étr✐❝❛ ❞❡ ♥✐tr❡t♦ ❞❡ ❜♦r♦ ❤❡①❛❣♦♥❛❧ ♣♦r ▼✐✲

❝r♦s❝♦♣✐❛ ❞❡ ❋♦rç❛ ❊❧étr✐❝❛ ✸✺

❙❯▼➪❘■❖ ✈

✹✳✷ ❉❡t❛❧❤❡s ❊①♣❡r✐♠❡♥t❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✷✳✶ Pr❡♣❛r❛çã♦ ❞❡ ❆♠♦str❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✷✳✷ ❘❡s✉❧t❛❞♦s ❡ ❉✐s❝✉ssõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✹✳✸ ❘❡s✉❧t❛❞♦s ❚❡ór✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✹ ❈♦♥❝❧✉sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷

✺ ◆❛♥♦❡str✉t✉r❛s ❝r✐st❛❧♦❣r❛✜❝❛♠❡♥t❡ ♦r✐❡♥t❛❞❛s s♦❜r❡ ♥✐tr❡t♦

❞❡ ❜♦r♦ ❤❡①❛❣♦♥❛❧ ✺✹

✺✳✶ ▼♦t✐✈❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✺✳✷ ❉❡t❛❧❤❡s ❊①♣❡r✐♠❡♥t❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✺✳✷✳✶ Pr❡♣❛r❛çã♦ ❞❡ ❆♠♦str❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✺✳✷✳✷ ❘❡s✉❧t❛❞♦s ❡ ❉✐s❝✉ssõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✺✳✸ ❘❡s✉❧t❛❞♦s ❚❡ór✐❝♦s Pr❡❧✐♠✐♥❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✺✳✹ ❈♦♥❝❧✉sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻

❆ ▲✐st❛ ❞❡ P✉❜❧✐❝❛çõ❡s ✻✼

❇ P❡rs♣❡❝t✐✈❛s ✻✽

✈✐

❆❣r❛❞❡❝✐♠❡♥t♦s

• ➚ ❉❡✉s✳

• ➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ♣❛♣✐t♦ ❡ ♠❛♠ã❡✳✳✳ ▲é♦✱ ▼✐❝❤❡❧❧❡ ❡ ▲é♦✳ ❖❜r✐❣❛❞❛ ♣❡❧❛ ❢♦rç❛ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✳ ❖❜r✐❣❛❞❛ ❛ t♦❞❛ ❢❛♠í❧✐❛ ❇r✐t❡s ◗✉❡✐r♦③ ❡ t❛♠❜é♠ ❛ ❢❛♠í❧✐❛ ❖❧✐✈❡✐r❛✳

• ❆♦ ♣r♦❢❡ss♦r ❇❡r♥❛r❞♦ ◆❡✈❡s ♣♦r t❡r ❛❝❡✐t❛❞♦ ❛ t❛r❡❢❛ ❞❡ ♠❡ ♦r✐❡♥t❛r✱ ♠❡s♠♦ ❞❡♣♦✐s ❞❡ ✉♠ ❛♥♦ ❡ ♠❡✐♦ ❞❡ ❞♦✉t♦r❛❞♦✳ ❖❜r✐❣❛❞❛ ♣❡❧❛ ♦r✐✲ ❡♥t❛çã♦ ❡♥❝♦r❛❥❛❞♦r❛✱ ♠❡s♠♦ ❝♦♠ t♦❞♦s ♦s ✧❞❡s✈✐♦s ❝♦♠ ❡♠♦çã♦ ♥❛ ❢❡rr♦✈✐❛✧✳

• ❆♦ ♣r♦❢❡ss♦r ❖s❝❛r ▼❡sq✉✐t❛ ♣❡❧❛ ♦r✐❡♥t❛çã♦ ❡ ❝♦♥s❡❧❤♦s✳

• ❆♦s ♣r♦❢❡ss♦r❡s ▼ár✐♦ ▼❛③③♦♥✐ ❡ ❍é❧✐♦ ❈❤❛❝❤❛♠ ♣❡❧❛ ❝♦❧❛❜♦r❛çã♦ q✉❡ ❢♦✐ ❡ss❡♥❝✐❛❧ ♣❛r❛ ❡ss❛ t❡s❡✳ ❆♦ Pr♦❢❡ss♦r ❘♦♥❛❧❞♦ ❇❛t✐st❛ ✭❯❋❖P✮ ♣❡❧♦s ❝á❧❝✉❧♦s ♥♦ ♣r♦❥❡t♦ ❞❛s ♥❛♥♦❡str✉t✉r❛s ♦r✐❡♥t❛❞❛s✳ ❆♦ ❛❧✉♥♦ ▼❛t❤❡✉s ▼❛t♦s ♣♦r t♦❞♦s ♦s ❝á❧❝✉❧♦s r❡❛❧✐③❛❞♦s ♥♦s ♣r♦❥❡t♦s✳

• ❆♦s ♣r♦❢❡ss♦r❡s✿ ▲❡❛♥❞r♦ ▼❛❧❛r❞ ✭❘❛♠❛♥✴▲❛❜♦r❛tór✐♦ ❞❡ ✷ ❋ót♦♥s✮✱ ▲✉✐③ ●✉st❛✈♦ ❈❛♥ç❛❞♦ ✭❘❛♠❛♥✮✱ ▼❛r❝♦s P✐♠❡♥t❛ ✭❘❛♠❛♥✮✱ ❘♦❞r✐❣♦ ▲❛❝❡r❞❛ ✭◆❛♥♦♠❛t❡r✐❛✐s✮✱ ❯❜✐r❛❥❛r❛ ❆❣❡r♦ ✭▲❛❜♦r❛tór✐♦ ❞❡ ❋ís✐❝❛ ❞❡ ❙✐st❡♠❛s ❇✐♦❧ó❣✐❝♦s✮✱ ▼❛r❝❡❧♦ ❋r❛♥ç❛ ✭■♥❢♦r♠❛çã♦ ◗✉â♥t✐❝❛✮✱ ❏♦sé ❙❛♠✲ ♣❛✐♦ ✭❙❡♠✐❝♦♥❞✉t♦r❡s✮ q✉❡ ❞✐s♣♦♥✐❜✐❧✐③❛r❛♠ ♦s ❧❛❜♦r❛tór✐♦s✱ ❛❧é♠ ❞❡ t❡♠♣♦✱ ❡♥s✐♥❛♠❡♥t♦s ❡ ❝♦♥s❡❧❤♦s✳

❙❯▼➪❘■❖ ✈✐✐

• Pós✲❞♦❝s✿ ▲❡♦♥❛r❞♦ ❈❛♠♣♦s ✭▼■❚✮✱ ❊✈❛♥❞r♦ ▼♦r❛✐s ✭◆❛♥♦♠❛t❡r✐❛✐s✮✱ ▲✉❝✐❛♥♦ ▼♦✉r❛ ✭❘❛♠❛♥✮✳

• ❚é❝♥✐❝♦s✿ ❆♥tô♥✐♦ ❞❡ Pá❞✉❛ ✭❙❛❧❛ ▲✐♠♣❛✮ ❡ ❙ér❣✐♦ ✭◆❛♥♦♠❛t❡r✐❛✐s✮✳ ❆ ❛❥✉❞❛ ❞❡ ✈♦❝ês ❢♦✐ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞♦s ♣r♦❥❡t♦s✳

• ❆♦s ❛♠✐❣♦s ❞♦ ▲❛❜ ◆❛♥♦✿ ❊❧✐s✱ ●✐❛♥✐♥♥✐✱ ❇ár❜❛r❛✳ ▼❛s ♣r✐♥❝✐♣❛❧✲ ♠❡♥t❡✱ ♦❜r✐❣❛❞❛ ❛s ❞✉❛s ❇❡r♥❛❞❡t❡s ❆♥❛ ❡ ▼❛r✐✱ s❡♠ ❛s q✉❛✐s ❡ss❛ t❡s❡ ♥ã♦ t❡r✐❛ s✐❞♦ ♣♦ssí✈❡❧✳ ❖❜r✐❣❛❞❛ ♣❡❧♦s ❡♥s✐♥❛♠❡♥t♦s✱ ❜r✐❣❛❞❡✐r♦s✱ ♠✐❧❦s❤❛❦❡s✱ ✈í❞❡♦s ❡ ♣✐❛❞❛s r✉✐♥s✳✳✳ ❊ ♦ ♦❜r✐❣❛❞❛ ♣♦r t♦❞❛ ❛ s✉❛ ❛❥✉❞❛✱ P❛rá❜♦❧❛s✦✦✦

• ➚ t✉r♠❛ q✉❡ ✧♥ã♦ ❡①✐st❡✧✳ ❱✐✈❛ ❛ t✉r♠❛ ❞❡ ✷✵✵✷✦

• ➚ t♦❞♦s ♦s ♦✉tr♦s ❛♠✐❣♦s q✉❡ ✜③ ♥❡ss❡s ✶✵ ❛♥♦s ❞❡ ❢ís✐❝❛✳✳✳ ♣r❛ ♥ã♦ ❡sq✉❡❝❡r ♥✐♥❣✉é♠✱ ♦ ❛❣r❛❞❡❝✐♠❡♥t♦ ✈❛✐ s❡r ❣❡r❛❧✦✦✦✦

• ❆♦ s❡❧❡t♦ ❣r✉♣♦ ❞❡♥♦♠✐♥❛❞♦ ●✉❛r❞✐õ❡s ❈❡❣♦s✿ ✈♦❝ês ❝♦♥s❡❣✉❡♠ ❞✐✈❡r✲ t✐r ❛té q✉❡♠ t❡♠ ✉♠ s❡♥s♦ ❞❡ ❤✉♠♦r ❝♦♠♦ ♦ ♠❡✉✦ ❊ ❛♦ ♠❡✉ ♣❛✐ ❘❛❢❛✱ q✉❡ ♣❛rt✐❝✐♣❛ ❞❡ t❛❜❡❧❛ ✭❝♦✐t❛❞♦✦✮ ❞❡ss❡ ❣r✉♣♦ ❞❡ ♠❛❧✉❝♦s✳

• ➚s ♣❡ss♦✐♥❤❛s q✉❡r✐❞❛s ❞♦ ♠❡✉ ❝♦r❛çã♦✿ ❉✉✱ ▲✐❝♦ ❡ ❘✉❞á✱ ❇r✉♥✐♥❤❛ ✭❙✐s✮✱ ❚❛❞❡✉③✐t❝❤♦✱ ❚❤✐✱ ●❧✐❦ ✭✈❛❧❡✉ ♣♦r t❡r ♣❛❣♦ ♠❡✉ s❛❧ár✐♦ t♦❞♦s ❡ss❡s ❛♥♦s✦✮✱ ▼❛rq✉✐♠ ❡ ▲✉✱ ❳✉①✉✱ ▲❛✉r✐♥❤❛ ✭✇❤❛ts❛♣♣♣✦✮✱ ▲✉✐③✳✳✳ ♦❜r✐❣❛❞❛ ♣❡❧♦s ❛❜r❛❝✐♥❤♦s✱ ♣✐q✉❡♥✐q✉❡s✱ ❛❧♠♦ç♦s✱ ❝✐♥❡♠✐♥❤❛s✱ ♣❡❧❛ ❢♦rç❛ ♥♦s ♠♦✲ ♠❡♥t♦s ❞❛ t❡s❡ ❡ ❡♠ t♦❞♦s ♦s ♦✉tr♦s✳✳✳

• ➚ Pós✲●r❛❞✉❛çã♦ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛ ❞❛ ❯❋▼●✳

• ❆♦s ♣r♦❢❡ss♦r❡s ❡ ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛ ❞❛ ❯❋▼●✳

• ❆♦ ❈◆Pq ♣❡❧❛ ❜♦❧s❛ ❞❡ ❡st✉❞♦s✳

✈✐✐✐

❘❡s✉♠♦

◆❡st❛ t❡s❡ ❛♣r❡s❡♥t❛r❡♠♦s ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♥♦ ▲❛❜♦r❛tór✐♦ ❞❡ ◆❛♥♦s✲ ❝♦♣✐❛ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s ✭❉❋ ✴ ❯❋▼●✮ ❛ r❡s♣❡✐t♦ ❞♦ ❡st✉❞♦ ❞❡ ♥❛♥♦❡str✉t✉r❛s ❞❡ ♥✐tr❡t♦ ❞❡ ❜♦r♦ ❤❡①❛❣♦♥❛❧✳ ❉✉r❛♥t❡ ❛ r❡❛❧✐③❛çã♦ ❞❡ss❡s ♣r♦❥❡t♦s✱ ❝♦♥t❛♠♦s ❝♦♠ ✉♠❛ ✐♥t❡♥s❛ ❝♦❧❛❜♦r❛çã♦ ❝♦♠ ♦ ●r✉♣♦ ❞❡ ❊str✉t✉r❛ ❊❧❡trô♥✐❝❛ ❞♦ ❉❋✴❯❋▼●✳ ❆ r❡s♣♦st❛ ❞❡ ❝❛♠❛❞❛s ❞❡ ♥✐tr❡t♦ ❞❡ ❜♦r♦ ❤❡①❛❣♦♥❛❧ ❛♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ❛♣❧✐❝❛❞♦ ♣♦r ✉♠❛ s♦♥❞❛ ❞❡ ❆❋▼ ❢♦✐ ❡st✉❞❛❞❛ ✉s❛♥❞♦ ❛ té❝♥✐❝❛ ❞❡ ▼✐❝r♦s❝♦♣✐❛ ❞❡ ❋♦rç❛ ❊❧étr✐❝❛ ✭❊❋▼✮✳ ◆♦ss♦s r❡s✉❧t❛❞♦s ♠♦str❛♠ ✉♠ ❝♦♠♣♦rt❛♠❡♥t♦ ❛♥ô♠❛❧♦ ♥❛ r❡s♣♦st❛ ❞✐❡❧étr✐❝❛ ♣❛r❛ ❛s ❝❛♠❛❞❛s ❞❡ ❤✲❇◆ ♣❛r❛ ❛s ❞✐❢❡r❡♥t❡s ♦r✐❡♥t❛çõ❡s ❞❛ t❡♥sã♦ ❛♣❧✐❝❛❞❛✳ ❆tr❛✈és ❞❡ ❝á❧❝✉❧♦s ❞❡ ♣r✐♠❡✐r♦s ♣r✐♥❝í♣✐♦s✱ ❝♦♥s❡❣✉✐♠♦s ♠♦str❛r q✉❡ ❡ss❛ r❡s♣♦st❛ ❛♥ô♠❛❧❛ ♣♦❞❡ s❡r ✐♥t❡r♣r❡t❛❞❛❞❛ ❝♦♠♦ ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ♠❛❝r♦s❝ó♣✐❝❛ ❞♦ ❝♦♥✜♥❛♠❡♥t♦ ❞❡ ✉♠❛ ✜♥❛ ❝❛♠❛❞❛ ❞❡ á❣✉❛ ❡♥tr❡ ❛ ❝❛♠❛❞❛ ❞❡ ❤✲❇◆ ❡ ♦ s✉❜str❛t♦ ❞❡ ❙✐❖x✳ ❊ss❡s r❡s✉❧t❛❞♦s ❢♦r❛♠ ❝♦♠♣r♦✈❛❞♦s ❛tr❛✲

✐①

❆❜str❛❝t

✶

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❖ ♥❛♥♦♠❡tr♦ é ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠ ♠✐✲ ❧✐♦♥és✐♠♦ ❞❡ ♠✐❧í♠❡tr♦✱ ♦✉ s❡❥❛✱ s✐❣♥✐✜❝❛ ❞✐✈✐❞✐r♠♦s ✉♠ ♠✐❧í♠❡tr♦ ❞❡ ✉♠❛ ré❣✉❛ ❡s❝♦❧❛r ❡♠ ✶✵✵✵✵✵✵ ❞❡ ♣❡❞❛ç♦s✳ ❊ss❛ é ❛ ❡s❝❛❧❛ ❞❡ t❛♠❛♥❤♦ ❝♦♠✉♠❡♥t❡ ✉s❛❞❛ ♣❛r❛ ♠❡❞✐r ♦ t❛♠❛♥❤♦ ❞❡ ✉♠❛ ✜t❛ ❞✉♣❧❛ ❞❡ ❉◆❆ ❡ t❛♠❜é♠ ❞❡ ❛❧❣✉♥s ✈ír✉s ❡ ❜❛❝tér✐❛s✳ ◆♦✈❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ❢❡♥ô♠❡♥♦s ❢♦r❛♠ ❞❡s❝♦❜❡rt♦s ♥❡ss❛ ❡s❝❛❧❛ ♥❛♥♦♠étr✐❝❛✱ ❡ ❛ ❜✉s❝❛ ❞♦ s❡✉ ❡♥t❡♥❞✐♠❡♥t♦ t❡♠ ♠♦t✐✈❛❞♦✱ ♥♦s ú❧t✐✲ ♠♦s ❛♥♦s✱ ❣r❛♥❞❡ ♣❛rt❡ ❞❛s ♣❡sq✉✐s❛s ❝✐❡♥tí✜❝❛s ❡♠ t♦❞♦ ♦ ♠✉♥❞♦✳ ❆ss✐♠✱ ❛ ♥❛♥♦t❡❝♥♦❧♦❣✐❛ ❜✉s❝❛ s❡ ❛♣r♦✈❡✐t❛r ❞❡ss❛s ♥♦✈❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ s✉r❣❡♠ ♥♦s ♠❛t❡r✐❛✐s ♣❛r❛ ♣r♦❞✉③✐r ♥♦✈♦s ❞✐s♣♦s✐t✐✈♦s t❡❝♥♦❧ó❣✐❝♦s✳

❆❧é♠ ❞❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛s ❡s❝❛❧❛s ❞❡ t❛♠❛♥❤♦✱ ❛ ❞✐♠❡♥s✐♦♥❛❧✐❞❛❞❡ t❛♠✲ ❜é♠ é ✐♠♣♦rt❛♥t❡ ♣❛r❛ ❞❡✜♥✐r ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ✉♠ ♠❛t❡r✐❛❧✱ ✈✐st♦ q✉❡ ✉♠ ♠❡s♠♦ ❝♦♠♣♦st♦ q✉í♠✐❝♦ ♣♦❞❡ ♣♦ss✉✐r ❞✐❢❡r❡♥t❡s ❝❛r❛❝t❡ríst✐❝❛s s✐♠✲ ♣❧❡s♠❡♥t❡ ♣♦r ❛♣r❡s❡♥t❛r✲s❡ ❡♠ ✉♠❛ ❡st✉t✉r❛ ✵❉ ✶❉✱ ✷❉✱ ♦✉ ✸❉ ❬✶❪✳ ❯♠ ❡①❡♠♣❧♦ ♥♦tá✈❡❧ é ❛ ❣❛♠❛ ❞❡ ❞✐❢❡r❡♥t❡s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♠❛t❡r✐❛✐s ❢❡✐t♦s s♦✲ ♠❡♥t❡ ❞❡ ❝❛r❜♦♥♦✿ ❢✉❧❡r❡♥♦s ✭✵❉✮✱ ♥❛♥♦t✉❜♦s ❞❡ ❝❛r❜♦♥♦ ✭✶❉✮✱ ❣r❛❢❡♥♦ ✭✷❉✮ ❡ ♦ ❣r❛✜t❡ ❡ ♦ ❞✐❛♠❛♥t❡ ✭✸❉✮✳

❈❆P❮❚❯▲❖ ✶ ✷

t❡s✱ ❡♥tr❡ ❛s ❝❛♠❛❞❛s ❡ss❛s ❧✐❣❛çõ❡s sã♦ ❛s ❢r❛❝❛s ❧✐❣❛çõ❡s ❞❡ ✈❛♥ ❞❡r ❲❛❛❧s✳ ❊ss❛s ❝❛♠❛❞❛s ❢r❛❝❛♠❡♥t❡ ❧✐❣❛❞❛s ❢♦r❛♠ s❡♣❛r❛❞❛s ❛tr❛✈és ❞❡ ✉♠ ♠ét♦❞♦ ❞❡ ❡s❢♦❧✐❛çã♦ ♠❡❝â♥✐❝❛ ❝♦♠ ♦ ✉s♦ ❞❡ ✉♠❛ ✜t❛ ❛❞❡s✐✈❛ ❬✷❪✳

❉❛❞❛ ❛ s✐♠♣❧✐❝✐❞❛❞❡ ❞♦ ♠ét♦❞♦ ❞❡ ❡s❢♦❧✐❛çã♦✱ ♦✉tr♦s ♠❛t❡r✐❛✐s q✉❡ ♣♦s✲ s✉❡♠ ❡ss❛ ♠❡s♠❛ ❡str✉t✉r❛ ❡♠ ❝❛♠❛❞❛s ♣❛ss❛r❛♠ ❛ s❡r ❡st✉❞❛❞♦s✱ ❡ ❡♥tr❡ ❡❧❡s✱ ❡stá ♦ ♥✐tr❡t♦ ❞❡ ❜♦r♦✳ ❖ ♥✐tr❡t♦ ❞❡ ❜♦r♦ é ✉♠ ❝♦♠♣♦st♦ ❞❛ ❢❛♠í❧✐❛ ■■■✲❱✱ q✉❡ ❛♣r❡s❡♥t❛ q✉❛tr♦ ❞✐❢❡r❡♥t❡s ❢❛s❡s ❬✸❪✿ ❝ú❜✐❝♦ ✭❝✲❇◆✮✱ ✇✉rt③ít✐❝♦ ✭✇✲❇◆✮✱ ❤❡①❛❣♦♥❛❧ ✭❤✲❇◆✮ ❡ r♦♠❜♦é❞r✐❝♦ ✭r✲❇◆✮✳ ❆s ❢❛s❡s ❝✲❇◆ ❡ ✇✲❇◆ ♣♦ss✉❡♠ ❧✐❣❛çõ❡s ❝♦♠ ❤✐❜r✐❞✐③❛çã♦ s♣3_{✱ ❢♦r♠❛♥❞♦ ❡str✉t✉r❛s s✐♠✐❧❛r❡s ❛♦ ❞✐✲}

❛♠❛♥t❡✳ ❏á ❛s ❢❛s❡s ❤✲❇◆ ❡ r✲❇◆✱ ❝♦♥s✐st❡♠ ❞❡ ❝❛♠❛❞❛s s✐♠✐❧❛r❡s ❛♦ ❣r❛✜t❡✱ ❝♦♠ ♦s át♦♠♦s ❞❡ ❇ ❡ ◆ ❢❛③❡♥❞♦ ❧✐❣❛çõ❡s ❝♦♠ ❤✐❜r✐❞✐③❛çã♦ s♣2 _{♥♦ ♣❧❛♥♦✳ ❊♠}

❛♠❜♦s ♦s ❝❛s♦s✱ é ♦ ❡♠♣✐❧❤❛♠❡♥t♦ ❞♦s át♦♠♦s ❞❡ ❇ ❡ ◆ q✉❡ ❞❡t❡r♠✐♥❛♠ ❛s ❞✐❢❡r❡♥ç❛s ❡♥tr❡ ❝❛❞❛ ✉♠ ❞♦s ❝♦♠♣♦st♦s✳

◆❡ss❛ t❡s❡✱ tr❛❜❛❧❤❛♠♦s s♦♠❡♥t❡ ❝♦♠ ♦ ♥✐tr❡t♦ ❞❡ ❜♦r♦ ❤❡①❛❣♦♥❛❧ ✭❤✲ ❇◆✮✳ ❖ ❤✲❇◆ ❝♦♥s✐st❡ ❞❡ ❝❛♠❛❞❛s ❜✐❞✐♠❡♥s✐♦♥❛✐s ❡♠ q✉❡ ♦s át♦♠♦s ❞❡ ❜♦r♦ ❡ ♥✐tr♦❣ê♥✐♦ ❡stã♦ ❧✐❣❛❞♦s ❛tr❛✈és ❞❡ ❧✐❣❛çõ❡s s♣2 _{❡♠ ✉♠❛ ❡str✉t✉r❛ t✐♣♦}

❝♦❧♠é✐❛ ❞❡ ❛❜❡❧❤❛ ❬✹✱✺❪✱ ❝♦♠ ❝❛❞❛ át♦♠♦ ❞❡ ❜♦r♦ ❧✐❣❛❞♦ ❛ três át♦♠♦s ❞❡ ♥✐tr♦❣ê♥✐♦✱ ❡ ✈✐❝❡ ❡ ✈❡rs❛✳ ❉❡♥tr♦ ❞❡ ❝❛❞❛ ❝❛♠❛❞❛✱ ♦s át♦♠♦s ❡stã♦ ❧✐❣❛❞♦s ❝♦♠ ❢♦rt❡s ❧✐❣❛çõ❡s ❝♦✈❛❧❡♥t❡s✱ ❡♥q✉❛♥t♦ ❛s ❧✐❣❛çõ❡s ✐♥t❡r♣❧❛♥❛r❡s sã♦ ❜❛s✲ t❛♥t❡ ❢r❛❝❛s ✭✜❣✉r❛ ✶✳✶✮✳ ❆ ❡str✉t✉r❛ ❣❡r❛❧ ❡ ♦s ❡s♣❛ç❛♠❡♥t♦s ❛tô♠✐❝♦s ❞♦ ❤✲❇◆ sã♦ ♠✉✐t♦ s❡♠❡❧❤❛♥t❡s ❛♦s ❞♦ ❣r❛✜t❡✳ ◆♦ ❤✲❇◆✱ ♥♦ ❡♥t❛♥t♦✱ ♦s át♦♠♦s ❞❡ ❜♦r♦ ❡ ♥✐tr♦❣ê♥✐♦ ❡stã♦ ❡♠♣✐❧❤❛❞♦s ❞✐r❡t❛♠❡♥t❡ ❡♠ ❝✐♠❛ ✉♥s ❞♦s ♦✉tr♦s✱ r❡s✉❧t❛♥❞♦ ❡♠ ✉♠ ❡♠♣✐❧❤❛♠❡♥t♦ t✐♣♦ ❆❆❆ ❬✻✱✼❪✱ ❡♥q✉❛♥t♦ ♦ ❣r❛✜t❡ ♣♦ss✉✐ ✉♠ ❡♠♣✐❧❤❛♠❡♥t♦ ❞♦ t✐♣♦ ❇❡r♥❛❧ ✭❆❇❆✮ ❬✹❪✳ ❆ ❝❛♠❛❞❛ ú♥✐❝❛ ❞❡ ❤✲❇◆ é ❝♦♥s✐❞❡r❛❞❛ ♦ ❝r✐st❛❧ ✷❉ ♠❛✐s ✜♥♦ ♣♦ssí✈❡❧ ❝♦♠ ❧✐❣❛çõ❡s ✐ô♥✐❝❛s ❢r❛❝❛s✳ ❊ss❛ ❝❛r❛❝t❡ríst✐❝❛ ❢❛③ ❞♦ ❤✲❇◆ ✉♠ s✐st❡♠❛ ♠♦❞❡❧♦ ✐❞❡❛❧ ♥♦ q✉❛❧ é ♣♦ssí✈❡❧ ❡st✉✲ ❞❛r ❝♦♥✜❣✉r❛çõ❡s ❛tô♠✐❝❛s✱ ✐♥❝❧✉✐♥❞♦ ❞❡❢❡✐t♦s✱ ❜♦r❞❛s ❡ ✈❛❝â♥❝✐❛s ❡♠ ❝r✐st❛✐s ✐ô♥✐❝♦s ❜✐❞✐♠❡♥s✐♦♥❛✐s✳

❆♦ ❝♦♥trár✐♦ ❞♦ ❣r❛❢❡♥♦✱ ♦ ♥✐tr❡t♦ ❞❡ ❜♦r♦ ❤❡①❛❣♦♥❛❧ é ✉♠ ✐s♦❧❛♥t❡✱ t♦r✲ ♥❛♥❞♦ ♦ ❤✲❇◆ ✉♠ ❡①❝❡❧❡♥t❡ s✉❜str❛t♦ ♣❛r❛ ❞✐s♣♦s✐t✐✈♦s ❜❛s❡❛❞♦s ❡♠ ❣r❛✲ ❢❡♥♦ ❬✽❪✳ ❋♦✐ ♠♦str❛❞♦ ♥❛ ❧✐t❡r❛t✉r❛ q✉❡ ❡ss❡s ❞✐s♣♦s✐t✐✈♦s ♣♦ss✉❡♠ ♠♦❜✐❧✐✲ ❞❛❞❡ ❞❡ ♣♦rt❛❞♦r❡s q✉❡ sã♦ ❝❡r❝❛ ❞❡ ✉♠❛ ♦r❞❡♠ ❞❡ ❣r❛♥❞❡③❛ ♠❛✐♦r❡s q✉❡ ♦s ❞✐s♣♦s✐t✐✈♦s s♦❜r❡ ❙✐❖x✱ ❛❧é♠ ❞❡ ♣♦ss✉✐r❡♠ ✉♠❛ r✉❣♦s✐❞❛❞❡ ❡ r❡❛t✐✈✐❞❛❞❡

❈❆P❮❚❯▲❖ ✶ ✸

❋✐❣✉r❛ ✶✳✶✿ ◆✐tr❡t♦ ❞❡ ❇♦r♦ ❍❡①❛❣♦♥❛❧✳ ✭❛✮ ❊sq✉❡♠❛ ❞❛ r❡❞❡ ❤❡①❛❣♦♥❛❧✱ ✭❜✮ ♦ ♣ó ❝♦♠❡r❝✐❛❧ ❞❡ ❤✲❇◆ ❡ ✭❝✮ ✉♠❛ ✐♠❛❣❡♠ ❞❡ t♦♣♦❣r❛✜❛ ❞❡ ✉♠ ✢♦❝♦ ❞❡ ❤✲❇◆✳

❆❧é♠ ❞✐ss♦✱ ♦ ❤✲❇◆ ♣♦ss✉✐ ❞✐✈❡rs❛s ♦✉tr❛s ♣r♦♣r✐❡❞❛❞❡s ❬✾✕✶✶❪✱ t❛✐s ❝♦♠♦ ❛ ❝♦♥❞✉t✐✈✐❞❛❞❡ tér♠✐❝❛ ❡❧❡✈❛❞❛✱ ❜❛✐①❛ ❡①♣❛♥sã♦ tér♠✐❝❛✱ ❛❧t❛ r❡s✐stê♥❝✐❛ ♠❡❝â♥✐❝❛✱ ❞✉r❡③❛ ❡❧❡✈❛❞❛✱ r❡s✐stê♥❝✐❛ à ❝♦rr♦sã♦✱ ♣♦♥t♦ ❞❡ ❢✉sã♦ ♣ró①✐♠♦ ❛ ✷✻✵✵◦ ❈✳ ❆❧é♠ ❞✐ss♦✱ ♣♦ss✉✐ ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❧✉❜r✐✜❝❛r ❡♠ ✉♠ ❧♦♥❣♦ ✐♥t❡r✲

✈❛❧♦ ❞❡ t❡♠♣❡r❛t✉r❛s✱ ❥á q✉❡ s❡✉ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛tr✐t♦ é ♠❛♥t✐❞♦ ❛té ✾✵✵◦

❈✱ ❡♥q✉❛♥t♦ ♦✉tr♦s ❧✉❜r✐✜❝❛♥t❡s só❧✐❞♦s✱ ❝♦♠♦ ♦ ❞✐ss✉❧❢❡t♦ ❞❡ ♠♦❧✐❜❞ê♥✐♦ ❡ ❣r❛✜t❡ q✉❡✐♠❛♠ ❥á ❡♠ t❡♠♣❡r❛t✉r❛s ♠❛✐s ❜❛✐①❛s✳ ❉❛❞❛s t♦❞❛s ❡ss❛s ♣r♦♣r✐✲ ❡❞❛❞❡s✱ ♦ ♥✐tr❡t♦ ❞❡ ❜♦r♦ é ✉s❛❞♦ ❡♠ ❞✐❢❡r❡♥t❡s s❡t♦r❡s✱ ❝♦♠♦ ❧✉❜r✐✜❝❛♥t❡s ❡ r❡✈❡st✐♠❡♥t♦ ❝♦♥tr❛ ❝♦rr♦sã♦✱ ✐s♦❧❛♥t❡s tér♠✐❝♦s ❞❡ ❝♦♠♣✉t❛❞♦r❡s✱ ❛❧é♠ ❞❡ s❡r ❜❛st❛♥t❡ ❡♠♣r❡❣❛❞♦ ♥❛ ✐♥❞✉str✐❛ ❞❡ ❝♦s♠ét✐❝♦s∗ ❬✶✷❪✳

❖ ❝❛♣ít✉❧♦ ✷ ❛♣r❡s❡♥t❛ ❛s ❝❛r❛❝t❡ríst✐❝❛s ❡str✉t✉r❛✐s ❡ ❡❧❡trô♥✐❝❛s ❞♦ ♥✐✲ tr❡t♦ ❞❡ ❜♦r♦ ❤❡①❛❣♦♥❛❧ ❡✱ ♥♦ ❝❛♣ít✉❧♦ ✸✱ ❛♣r❡s❡♥t❛♠♦s ❛s té❝♥✐❝❛s ❡①♣❡r✐✲ ♠❡♥t❛✐s ❡ ♠❡t♦❞♦❧♦❣✐❛s t❡ór✐❝❛s q✉❡ ❢♦r❛♠ ✉t✐❧✐③❛❞❛s ♥❡ss❛ t❡s❡✳ ❖ ❝❛♣ít✉❧♦ ✹ ❛♣r❡s❡♥t❛ ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦✱ ✈✐❛ ▼✐❝r♦s❝♦♣✐❛ ❞❡ ❋♦rç❛ ❊❧étr✐❝❛✱ ❞❛s ❝❛♠❛✲ ❞❛s ❞❡ ♥✐tr❡t♦ ❞❡ ❜♦r♦ ❤❡①❛❣♦♥❛❧✳ ❆ ♣❛rt✐r ❞❡ss❛ ❝❛r❛❝t❡r✐③❛çã♦✱ ❡st✉❞❛♠♦s ✉♠ ❝♦♠♣♦rt❛♠❡♥t♦ ❛♥ô♠❛❧♦ ❞❛ ❝♦♥st❛♥t❡ ❞✐❡❧étr✐❝❛ ❞❛s ❝❛♠❛❞❛s ❞❡ ❤✲❇◆ ❝♦♠ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ❛♣❧✐❝❛❞♦✳ ❏á ♥♦ ❝❛♣ít✉❧♦ ✺✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❢♦r♠❛çã♦ ❞❡ ♥❛♥♦❡str✉t✉r❛s ❝r✐st❛❧♦❣r❛✜❝❛♠❡♥t❡ ♦r❣❛♥✐③❛❞❛s s♦❜r❡ ❝❛♠❛❞❛s ❞❡ ♥✐tr❡t♦ ❞❡ ❜♦r♦ ❤❡①❛❣♦♥❛❧✳ ❊ss❛s ❡str✉t✉r❛s sã♦ ❢♦r♠❛❞❛s q✉❛♥❞♦ ❛s ❛♠♦str❛s ❞❡ ❤✲❇◆✱ ❝♦♥t❡♥❞♦ ♥❛♥♦♣❛rtí❝✉❧❛s ♠❡tá❧✐❝❛s✱ sã♦ ❧❡✈❛❞❛s ❛ ✉♠ ❢♦r♥♦ ❡♠ ✉♠ ♣r♦❝❡ss♦

∗❊♠ ✶✾✹✵✱ ♦ ♥✐tr❡t♦ ❞❡ ❜♦r♦ ❤❡①❛❣♦♥❛❧ ❢♦✐ ✉s❛❞♦ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ❡♠ ❢♦r♠✉❧❛çõ❡s

❈❆P❮❚❯▲❖ ✶ ✹

✺

❈❛♣ít✉❧♦ ✷

▼❛t❡r✐❛❧ ❊st✉❞❛❞♦

◆❡ss❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞♦ ♠❛t❡r✐❛❧ ❡st✉❞❛❞♦✿ ♦ ♥✐tr❡t♦ ❞❡ ❜♦r♦ ❤❡①❛❣♦♥❛❧✳ ❆ s✉❛ ❡str✉t✉r❛ ❝r✐st❛❧✐♥❛ é ❞✐s❝✉t✐❞❛✱

♠♦str❛♥❞♦ ❛ s✉❛ r❡❞❡ ❞✐r❡t❛ ❡ r❡❝í♣r♦❝❛✳ ❆❧é♠ ❞✐ss♦✱ ✉♠ ♠♦❞❡❧♦ t✐❣❤t✲❜✐♥❞✐♥❣ ❜❛st❛♥t❡ s✐♠♣❧❡s é ❛♣r❡s❡♥t❛❞♦ ♣❛r❛ ❞❡s❝r❡✈❡r ❛ ❡str✉t✉r❛ ❞❡

❈❆P❮❚❯▲❖ ✷ ✻

✷✳✶ ◆✐tr❡t♦ ❞❡ ❇♦r♦ ❍❡①❛❣♦♥❛❧

✷✳✶✳✶ Pr♦♣r✐❡❞❛❞❡s ❊str✉t✉r❛✐s

❖ ❝r✐st❛❧ ❞❡ ♥✐tr❡t♦ ❞❡ ❜♦r♦ ❤❡①❛❣♦♥❛❧ ✭❤✲❇◆✮ ❝♦♥s✐st❡ ❞❡ ❝❛♠❛❞❛s ❜✐❞✐✲ ♠❡♥s✐♦♥❛✐s ❝♦♠ ♦s át♦♠♦s ❞❡ ❜♦r♦ ❡ ♥✐tr♦❣ê♥✐♦ ♦❝✉♣❛♥❞♦ ✈ért✐❝❡s ❞❡ ❤❡①á❣♦✲ ♥♦s✳ ❖s át♦♠♦s ❞❡ ❜♦r♦ ❡ ♥✐tr♦❣ê♥✐♦ s❡ ❧✐❣❛♠ ❛tr❛✈és ❞❡ ❧✐❣❛çõ❡s ❝♦✈❛❧❡♥t❡s✱ ❝❤❛♠❛❞❛s ❧✐❣❛çõ❡s σ✱ ❤í❜r✐❞❛s ♥❛ ❝♦♥✜❣✉r❛çã♦ sp2_{✳ ❖ ♦✉tr♦ ♦r❜✐t❛❧ ✷♣}

z é

♣❡r♣❡♥❞✐❝✉❧❛r ❛ ❢♦❧❤❛ ❡ ❢♦r♠❛ ❛s ❧✐❣❛çõ❡s π✳ ◆♦ ❝r✐st❛❧ ✸❉ ❞❡ ♥✐tr❡t♦ ❞❡

❜♦r♦ ❤❡①❛❣♦♥❛❧✱ ❛s ❧✐❣❛çõ❡s ✐♥t❡r♣❧❛♥❛r❡s sã♦ ❜❛st❛♥t❡ ❢r❛❝❛s ❡ sã♦✱ ♣r♦✈❛✲ ✈❡❧♠❡♥t❡✱ ✉♠❛ ♠✐st✉r❛ ❞❡ ❛tr❛çã♦ ✐ô♥✐❝❛ ❡♥tr❡ í♦♥s ❞❡ ❝❛r❣❛s ♦♣♦st❛s ♥♦s ♣❧❛♥♦s ❛❞❥❛❝❡♥t❡s ❡ ❧✐❣❛çõ❡s ❞❡ ✈❛♥ ❞❡r ❲❛❛❧s ❝♦♠♦ ♦❝♦rr❡ ♥♦ ❣r❛✜t❡ ❬✶✸❪✳

❆ r❡❞❡ ❞♦ ♥✐tr❡t♦ ❞❡ ❜♦r♦ ❤❡①❛❣♦♥❛❧ é tr✐❛♥❣✉❧❛r✱ ❝♦♠ ✉♠❛ ❜❛s❡ q✉❡ ❝♦♥té♠ ✉♠ át♦♠♦ ❞❡ ❜♦r♦ ❡ ♦✉tr♦ ❞❡ ♥✐tr♦❣ê♥✐♦ ♣♦r ❝é❧✉❧❛ ✉♥✐tár✐❛✳ ❈❛❞❛ át♦♠♦ ❞❡ ❜♦r♦ s❡ ❧✐❣❛ ❛ três át♦♠♦s ❞❡ ♥✐tr♦❣ê♥✐♦ ❡ ✈✐❝❡✲✈❡rs❛✱ ❝♦♥❢♦r♠❡ ♠♦str❛ ❛ ✜❣✉r❛ ✷✳✶✭❛✮✳

❋✐❣✉r❛ ✷✳✶✿ ✭❛✮ ❘❡❞❡ ❞✐r❡t❛ ❞♦ ♥✐tr❡t♦ ❞❡ ❜♦r♦ ❤❡①❛❣♦♥❛❧✱ ❝♦♠ ❛ ❝é❧✉❧❛ ✉♥✐✲ tár✐❛ ❞❡st❛❝❛❞❛✳ ✭❜✮ ❘❡❞❡ r❡❝í♣r♦❝❛✱ ❞❡st❛❝❛♥❞♦ ❛ ♣r✐♠❡✐r❛ ③♦♥❛ ❞❡ ❇r✐❧❧♦✉✐♥✳ ❖s ✈❡t♦r❡s ✉♥✐tár✐♦s ♥♦ ❡s♣❛ç♦ ❞✐r❡t♦ ❡ ♥♦ ❡s♣❛ç♦ r❡❝í♣r♦❝♦ t❛♠❜é♠ ❡stã♦ ♠♦str❛❞♦s✳

❖s ✈❡t♦r❡s ♣r✐♠✐t✐✈♦sa1~ ❡a2~ ♣♦❞❡♠ s❡r ❡s❝r✐t♦s✱ ❡♠ t❡r♠♦s ❞♦ ♣❛râ♠❡tr♦

❈❆P❮❚❯▲❖ ✷ ✼

~ a1 = a

2( √

3ˆx+ ˆy) a2~ = a 2(−

√

3ˆx+ ˆy).

❖ ✈❛❧♦r ❞♦ ♣❛râ♠❡tr♦ ❞❡ r❡❞❡ é ❛ ❂ ✷✱✺✶➴ ❬✶✹❪✳ ❆ r❡❞❡ r❡❝í♣r♦❝❛ ❡stá ❛♣r❡s❡♥t❛❞❛ ♥❛ ✜❣✉r❛ ✷✳✶✭❜✮✱ ❝♦♠ ❛ ♣r✐♠❡✐r❛ ③♦♥❛ ❞❡ ❇r✐❧❧♦✉✐♥ ❞❡st❛❝❛❞❛ ❡ ♦s ✈❡t♦r❡s _b1~ _{❡ b2}~ _{♣♦❞❡♠ s❡r ❡s❝r✐t♦s ❝♦♠♦✿}

~ b1 = 2π

a

√3

3 kˆx+ ˆky

~ b2 = 2π

a

−

√ 3

3 kˆx+ ˆky

.

✷✳✶✳✷ Pr♦♣r✐❡❞❛❞❡s ❊❧❡trô♥✐❝❛s

❆ ♠♦♥♦❝❛♠❛❞❛ ❞❡ ♥✐tr❡t♦ ❞❡ ❜♦r♦ ❤❡①❛❣♦♥❛❧ ♣♦ss✉✐ ❞♦✐s át♦♠♦s ♣♦r ❝é✲ ❧✉❧❛ ✉♥✐tár✐❛✱ ✉♠ ❞❡ ❜♦r♦ ❡ ✉♠ ❞❡ ♥✐tr♦❣ê♥✐♦✳ ■ss♦ s✐❣♥✐✜❝❛ ❞♦✐s ❝♦♥❥✉♥t♦s ❞❡ ❡st❛❞♦s ✷s ❡ ✷♣✱ ❡♠ ✉♠ t♦t❛❧ ❞❡ ✽ ❡❧étr♦♥s ❞❡ ✈❛❧ê♥❝✐❛ ♣♦r ❝é❧✉❧❛ ✉♥✐tár✐❛ ❬✶✸❪✳ ❆ss✐♠✱ ✻ ❞♦s ❡❧étr♦♥s ❡st❛rã♦ ❡♥✈♦❧✈✐❞♦s ♥❛s ❧✐❣❛çõ❡s ♣❧❛♥❛r❡sσ✱ ❡♥q✉❛♥t♦ ♦s

♦✉tr♦s ✷ ❡stã♦ ❡♥✈♦❧✈✐❞♦s ♥❛s ❧✐❣❛çõ❡sπ✳ ❈♦♠♦ ❛s ❤✐❜r✐❞✐③❛çõ❡s ❡♥tr❡ ♦r❜✐t❛✐s

♣❡r♣❡♥❞✐❝✉❧❛r❡s sã♦ ♥✉❧❛s✱ ♣♦❞❡♠♦s tr❛t❛r ♦s ♦r❜✐t❛✐s π s❡♣❛r❛❞❛♠❡♥t❡ ❞♦s

♦r❜✐t❛✐s σ✳

◆❡ss❛ s❡çã♦✱ ❢❛r❡♠♦s ✉♠ ♠♦❞❡❧♦ t✐❣❤t✲❜✐♥❞✐♥❣ ❜❛st❛♥t❡ s✐♠♣❧✐✜❝❛❞♦ q✉❡ s❡rá ❝❛♣❛③ ❞❡ ❞❡s❝r❡✈❡r✱ ♠❡s♠♦ q✉❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡✱ ❛s ❜❛♥❞❛sπ❞♦ ❤✲❇◆✱

✐♥❝❧✉s✐✈❡ ♣r❡✈❡♥❞♦ ❛ ♣r❡s❡♥ç❛ ❞♦ ❣❛♣ ❞❡ ❡♥❡r❣✐❛ ❡♥tr❡ ❛s ❜❛♥❞❛s ❞❡ ✈❛❧ê♥❝✐❛ ❡ ❞❡ ❝♦♥❞✉çã♦✳ ❖ ♠ét♦❞♦ t✐❣❤t✲❜✐♥❞✐♥❣ ❞❡♥♦t❛ ✉♠ ♠ét♦❞♦ q✉❡ s❡ ❛♣❧✐❝❛r✐❛✱ ❡♠ s✉❛ ✜❧♦s♦✜❛✱ ❛ s✐st❡♠❛s ❝♦♠ ❡❧étr♦♥s ❢♦rt❡♠❡♥t❡ ❧✐❣❛❞♦s ❛ s❡✉s ♥ú❝❧❡♦s✱ ❝♦♠ s✉♣❡r♣♦s✐çã♦ ♣❡q✉❡♥❛ ❞❡ ♦r❜✐t❛✐s ❧♦❝❛❧✐③❛❞♦s ❡♠ sít✐♦s ✈✐③✐♥❤♦s ❡♠ ✉♠❛ r❡❞❡✳

❉❡♥tr♦ ❞♦ ♠ét♦❞♦ t✐❣❤t✲❜✐♥❞✐♥❣✱ ♦s ❛✉t♦✲✈❡t♦r❡s s❡rã♦ r❡♣r❡s❡♥t❛❞♦s ♣❡❧♦s ♦r❜✐t❛✐s ❛tô♠✐❝♦s ❡ ♦ ♣♦t❡♥❝✐❛❧ ❝r✐st❛❧✐♥♦ s❡rá tr❛t❛❞♦ ❝♦♠♦ ✉♠❛ ♣❡rt✉r❜❛çã♦✳ ❆ss✐♠✱ ♦s ❡st❛❞♦s ❡❧❡trô♥✐❝♦s ❝r✐st❛❧✐♥♦s✱ ♣❛r❛ ♦s ❞♦✐s sít✐♦s ♥ã♦ ❡q✉✐✈❛❧❡♥t❡s ❆ ❡ ❇ s❡rã♦ r❡♣r❡s❡♥t❛❞♦s ♣♦r ❡st❛❞♦s ❞❡ ❇❧♦❝❤ ✭ΦA ❡ ΦB✮✱ ❝♦♥str✉í❞♦s ❛

♣❛rt✐r ❞♦s ♦r❜✐t❛✐s ❛tô♠✐❝♦spz✭ϕ✮✱ ❢♦r♥❡❝❡♥❞♦ ❛s ❢✉♥çõ❡s ❜❛s❡ ♣❛r❛ ❞❡s❝r❡✈❡r

❈❆P❮❚❯▲❖ ✷ ✽

ΦA,B ∝

X

R

ei~k·R~_ϕ

A,B(~r−R~)

❆ ❡q✉❛çã♦ s❡❝✉❧❛r é ❞❡r✐✈❛❞❛ ❞❡ ✉♠ ❍❛♠✐❧t♦♥✐❛♥♦ ✷×✷✱ q✉❡ ♣♦ss✉✐ ❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

Hi,j =hΦi |H |Φji H =

HAA HAB

HBA HBB

.

◗✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s s♦♠❡♥t❡ ✐♥t❡r❛çõ❡s ❡♥tr❡ ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s✱ ♣♦❞❡✲ ♠♦s ❝❛❧❝✉❧❛r ♦s t❡r♠♦s ❞❛ ❞✐❛❣♦♥❛❧ ❞❛ ♠❛tr✐③ ❍ ✿

HAA =

1 2

X

RA

X

RA′

hϕA(~r−R~A)|H |ϕA′(~r−R~_A′)i=ǫ_A

HBB =

1 2

X

RB

X

RB′

hϕB(~r−R~B)|H |ϕB′(~r−R~_B′)i=ǫ_B,

❡♠ q✉❡ ǫA❡ǫB sã♦ ❛s ❡♥❡r❣✐❛s ♣❛r❛ ♦ ♥í✈❡❧ ✷♣ ❞❡ ❝❛❞❛ át♦♠♦ ✐s♦❧❛❞♦ ❞♦ sít✐♦

❆ ❡ ❞♦ sít✐♦ ❇✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ P❛r❛ ♦s t❡r♠♦s ❢♦r❛ ❞❛ ❞✐❛❣♦♥❛❧✱ ❍AB ❡

❍BA✱ ❞❡✈❡♠♦s ❝♦♥s✐❞❡r❛r ♦s três ♣r✐♠❡✐r♦s át♦♠♦s ❞❡ ◆ ♣ró①✐♠♦s ❞♦ át♦♠♦

❞❡ ❇✱ q✉❡ sã♦ ❞❡♥♦t❛❞♦s ♣❡❧♦s ✈❡t♦r❡s _δ1~_{✱ δ2}~ _{❡ δ3}~_{✱ ❝♦♥❡❝t❛♥❞♦ ♦ át♦♠♦ ❞❡}

❜♦r♦ ❛♦s át♦♠♦s ❞❡ ♥✐tr♦❣ê♥✐♦✿

HAB =

1 2 X RA X RB

ei~k·(R~A−R~B)_hϕ

A(~r−R~A)|H |ϕB(~r−R~B)i

= 1

2hϕA(~r−R~A)|H |ϕB(~r−R~B)i(e

−i~k·~δ1

+e−i~k·~δ2

+e−i~k·~δ3

)

HBA =

1 2 X RB X RA

ei~k·(R~B−R~A)_hϕ

B(~r−R~B)|H |ϕA(~r−R~A)i

= 1

2hϕB(~r−R~B)|H |ϕA(~r−R~A)i(e

i~k·~δ1

+ei~k·~δ2

+ei~k·~δ3

)

❉❡ ❢♦r♠❛ s✐♠♣❧✐✜❝❛❞❛✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ♦s t❡r♠♦s ❢♦r❛ ❞❛ ❞✐❛❣♦♥❛❧ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

2HAB =t(ei~k· ~ δ1

+ei~k·~δ2

+ei~k·~δ3

❈❆P❮❚❯▲❖ ✷ ✾

❡♠ q✉❡ t é ❛ ✐♥t❡❣r❛❧ ❞❡ tr❛♥s❢❡rê♥❝✐❛ ❡♥tr❡ ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s ✭hϕA | H |

ϕBi✮✱ q✉❡ é ♥♦r♠❛❧♠❡♥t❡ ❝❤❛♠❛❞❛ ❞❡ −γ0 ✭t❂−γ0✮✱ ❡♠ q✉❡ γ0 é ✉♠ ✈❛❧♦r

♣♦s✐t✐✈♦✳ ❆ ❢✉♥çã♦ ❢✭❦✮ é ✉♠❛ s♦♠❛ ❞♦s ❢❛t♦r❡s ❞❡ ❢❛s❡ ei~k·R~j ✭❥❂✶✱✷✱✸✮✳

❯s❛♥❞♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s✱ t❡♠♦s q✉❡✿

f(k) = eikxa/

√

3_{+ 2e}−ikxa/2

√

3_coskya

2

.

❆ss✐♠✱ ❝♦♠♦ ❢✭❦✮ é ✉♠❛ ❢✉♥çã♦ ❝♦♠♣❧❡①❛ ❡ ♦ ❍❛♠✐❧t♦♥✐❛♥♦ ❢♦r♠❛ ✉♠❛ ♠❛tr✐③ ❤❡r♠✐t✐❛♥❛✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✿

HAB = tf(k)

HBA = HAB∗ =tf∗(K) H =

ǫA tf(k)

tf∗_{(k) ǫ}

B .

❉❛❞❛ ❛ ❢♦r♠❛ ♠❛tr✐❝✐❛❧ ♣❛r❛ ♦ ❍❛♠✐❧t♦♥✐❛♥♦ ❞♦ s✐st❡♠❛✱ ♣♦❞❡♠♦s r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ s❡❝✉❧❛r ♣❛r❛ ♦❜t❡r♠♦s ♦s ❛✉t♦✈❛❧♦r❡s ❊✭❦✮✱ ❝♦♠ ❦❂✭kx, ky✮✿

det(H₋EU) = 0 H=

ǫA−E(k) tf(k)

tf∗_{(k) ǫ}

B−E(k)

= 0.

❆ss✐♠✱ t❡♠♦s✿

E = (ǫA+ǫB)± p

(ǫA−ǫB)2−4t2 |f(k)|2

2 . ❙❡ ✜③❡r♠♦s ❛s s❡❣✉✐♥t❡s ❞❡✜♥✐çõ❡s✱

E0 = (ǫA+ǫB)

2 Eg =ǫA−ǫB ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ♦s ❛✉t♦✈❛❧♦r❡s ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

E =E0_±

p

(Eg)2−4t2 |f(k)|2

2

❡♠ q✉❡ ❊0 é ❛ ❡♥❡r❣✐❛ ♥♦ ♠❡✐♦ ❞♦ ❣❛♣ ❡ ❊g é ❛ ❡♥❡r❣✐❛ ❞♦ ❣❛♣ ❡ ♦s s✐♥❛✐s ±

r❡♣r❡s❡♥t❛♥❞♦ ❛s ❝♦♠❜✐♥❛çõ❡s s✐♠étr✐❝❛ ❡ ❛♥t✐✲s✐♠étr✐❝❛ ❞❡ ΦA ❡ ΦB ❬✶✸❪✳

❈❆P❮❚❯▲❖ ✷ ✶✵

✷✳✷✮✱ ❥á q✉❡ ❛ q✉❡❜r❛ ❞❡ s✐♠❡tr✐❛ ❡♥tr❡ ❛s s✉❜r❡❞❡s ❆ ❡ ❇ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✐♠♣❡❞❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❝♦♥❡s ❞❡ ❉✐r❛❝ ♥❛s ❡①tr❡♠✐❞❛❞❡s ❞❛ ③♦♥❛ ❞❡ ❇r✐❧❧♦✉✐♥✱ ❝r✐❛♥❞♦ ✉♠ ❣❛♣ ❞❡ ❡♥❡r❣✐❛✱ q✉❡ é ♠❛✐♦r q✉❡ ✺ ❡❱✳ ❊①❝❡t♦ ♣♦r ❡ss❛ ❞✐❢❡r❡♥ç❛✱ ❛ ❡str✉t✉r❛ ❞❡ ❜❛♥❞❛s ❡❧❡trô♥✐❝❛s ❞♦ ❇◆ ❛ss❡♠❡❧❤❛✲s❡ ❜❛st❛♥t❡ ❛ ❞♦ ❣r❛❢❡♥♦✳ ❖ ❣❛♣ ❡stá ♥♦ ♣♦♥t♦ ❑ ❡ ♥ã♦ ♥♦ ♣♦♥t♦ Γ✭❝♦♠♦ ✉s✉❛❧♠❡♥t❡ ♦❝♦rr❡ ♥♦s s❡♠✐✲ ❝♦♥❞✉t♦r❡s✮ ❡✱ ❡①❝❡t♦ ♣♦r ❡ss❛ r❡❣✐ã♦ ♥♦ ❡s♣❛ç♦ r❡❝í♣r♦❝♦✱ ❛ ❢♦r♠❛ ❞❛s ❜❛♥❞❛s sã♦ ❜❛st❛♥t❡ s✐♠✐❧❛r❡s ❬✶✹❪✳

❋✐❣✉r❛ ✷✳✷✿ ❈♦♠♣❛r❛çã♦ ❡♥tr❡ ❛s ❡str✉t✉r❛s ❞❡ ❜❛♥❞❛ ❞♦ ❤✲❇◆ ✭❛③✉❧✮ ❡ ❞♦ ❣r❛❢❡♥♦ ✭♣♦♥t✐❧❤❛❞♦ ✈❡r♠❡❧❤♦✮✳ ➱ ♣♦ssí✈❡❧ ♥♦t❛r q✉❡ ❛ ❡str✉t✉r❛ ❞❡ ❜❛♥❞❛s ❞♦ ❞♦✐s ♠❛t❡r✐❛✐s é ❜❛st❛♥t❡ ♣❛r❡❝✐❞❛✱ ❡①❝❡t♦ ♣❡❧❛ r❡❣✐ã♦ ❞♦s ♣♦♥t♦s ❑ ❡ ❑′✳

✶✶

❈❛♣ít✉❧♦ ✸

▼❡t♦❞♦❧♦❣✐❛s ❊①♣❡r✐♠❡♥t❛✐s ❡

❚❡ór✐❝❛s

◆❡ss❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❛s ♠❡t♦❞♦❧♦❣✐❛s ✉t✐❧✐③❛❞❛s ♥❡ss❛ t❡s❡✱ t❛♥t♦ ❡①♣❡r✐♠❡♥t❛✐s q✉❛♥t♦ t❡ór✐❝❛s✳ ❖ ♣r✐♥❝í♣✐♦ ❞❡ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞❛s té❝♥✐❝❛s

❞❡ ❙P▼ é ❞❡s❝r✐t♦✱ ❞❛♥❞♦ ê♥❢❛s❡ às té❝♥✐❝❛s q✉❡ ❢♦r❛♠ ✉t✐❧✐③❛❞❛s ♥❡st❡ tr❛❜❛❧❤♦✿ ❛ ▼✐❝r♦s❝♦♣✐❛ ❞❡ ❋♦rç❛ ❆tô♠✐❝❛✱ ▼✐❝r♦s❝♦♣✐❛ ❞❡ ❋♦rç❛ ❊❧étr✐❝❛ ❡ ♦ ▼♦❞♦ ❞❡ ❈❛♣❛❝✐tâ♥❝✐❛✳ ❆❧é♠ ❞✐ss♦✱ ❛ ❚❡♦r✐❛ ❞♦ ❋✉♥❝✐♦♥❛❧ ❞❛ ❉❡♥s✐❞❛❞❡ é

❈❆P❮❚❯▲❖ ✸ ✶✷

✸✳✶ ❚é❝♥✐❝❛s ❊①♣❡r✐♠❡♥t❛✐s

✸✳✶✳✶ ■♥tr♦❞✉çã♦

❆s té❝♥✐❝❛s ❞❡ ▼✐❝r♦s❝♦♣✐❛ ❞❡ ❱❛rr❡❞✉r❛ ♣♦r ❙♦♥❞❛ ✭❞♦ ✐♥❣❧ês ❙P▼✿ ❙❝❛♥✲ ♥✐♥❣ Pr♦❜❡ ▼✐❝r♦s❝♦♣②✮ s✉r❣✐r❛♠ ❛ ♣❛rt✐r ❞❡ ❞é❝❛❞❛ ❞❡ 1980❡ ♣♦ss✐❜✐❧✐t❛r❛♠ ❛ ♦❜t❡♥çã♦ ❞❛s ♣r✐♠❡✐r❛s ✐♠❛❣❡♥s ❞❡ r❡s♦❧✉çã♦ ❛tô♠✐❝❛ ♥♦ ❡s♣❛ç♦ r❡❛❧✳ ❖ ♣r✐♥❝í♣✐♦ ❜ás✐❝♦ ❞❡ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞❡ss❛ ❢❛♠í❧✐❛ ❞❡ té❝♥✐❝❛s ❝♦♥s✐st❡ ❡♠ ✉♠❛ s♦♥❞❛ ♠❡❝â♥✐❝❛ ✐♥t❡r❛❣✐♥❞♦ ❧♦❝❛❧♠❡♥t❡ ❝♦♠ ❛ s✉♣❡r❢í❝✐❡ ❞❡ ✉♠❛ ❛♠♦str❛✱ r❡✈❡❧❛♥❞♦ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ♠♦r❢♦❧ó❣✐❝❛s ❡ ❡str✉t✉r❛✐s✱ ❛❧é♠ ❞❡ ♣❡r♠✐t✐r ❛ ♠❛♥✐♣✉❧❛çã♦ ❞❛ ♠❛tér✐❛ ❡♠ ❡s❝❛❧❛ ♥❛♥♦♠étr✐❝❛✳

❯♠❛ ❣r❛♥❞❡ ✈❛r✐❡❞❛❞❡ ❞❡ té❝♥✐❝❛s ❞❡ ❙P▼ ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞❛s ❜❛s❡❛❞❛s ♥❛ ❞❡t❡❝çã♦ ❞❛s ❢♦rç❛s ❡♥tr❡ ❛ s♦♥❞❛ ❡ ❛ ❛♠♦str❛ ❬✶✺❪✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ✐♥t❡r❛çõ❡s s♦♥❞❛✲❛♠♦str❛ ♣♦❞❡♠ s❡r ♠❡❞✐❞❛s ♣♦r ✉♠ ♠✐❝r♦s❝ó♣✐♦ ❞❡ ✈❛rr❡✲ ❞✉r❛ ♣♦r s♦♥❞❛ ❞❡♣❡♥❞❡♥❞♦ ❞❛ s❡♣❛r❛çã♦ ❡♥tr❡ ❛ s♦♥❞❛ ❡ ❛ ❛♠♦str❛✳ ❆ss✐♠✱ ♦ t✐♣♦ ❞❡ ✐♥t❡r❛çã♦ ♠♦♥✐t♦r❛❞♦ ❡♥tr❡ ❛ s♦♥❞❛ ❡ ❛ ❛♠♦str❛ ❞❡t❡r♠✐♥❛ ♦ ♠♦❞♦ ❞❡ ♦♣❡r❛çã♦ ❞♦ ♠✐❝r♦s❝ó♣✐♦✳ ❉✉r❛♥t❡ ♦ ❝♦♥t❛t♦ ❝♦♠ ❛ ❛♠♦str❛ ❛ s♦♥❞❛ ❡①✲ ♣❡r✐♠❡♥t❛ ♣r❡❞♦♠✐♥❛♥t❡♠❡♥t❡ ❢♦rç❛s r❡♣✉❧s✐✈❛s✳ ◗✉❛♥❞♦ ❡ss❛ é ❛❢❛st❛❞❛ ❞❛ s✉♣❡r❢í❝✐❡✱ ❡♥tr❡t❛♥t♦✱ ❢♦rç❛s ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡✱ ❝♦♠♦ ❛ ❡❧étr✐❝❛ ❡ ❛ ♠❛❣♥ét✐❝❛✱ ♣♦❞❡♠ ❞♦♠✐♥❛r ❛ ✐♥t❡r❛çã♦ ❝♦♠ ❛ s♦♥❞❛ ❬✶✺❪✳ ❉❡✈❡✲s❡ t❡r ❡♠ ♠❡♥t❡ q✉❡ t♦❞❛s ❛s té❝♥✐❝❛s ❞❛ ❢❛♠í❧✐❛ ❙P▼ sã♦ ❞❡ s✉♣❡r❢í❝✐❡✱ ✉♠❛ ✈❡③ q✉❡ ❛ s♦♥❞❛ ❢❛③ ❝♦♥t❛t♦ ❢ís✐❝♦ ❛♣❡♥❛s ❝♦♠ ❛ ❝❛♠❛❞❛ ♠❛✐s s✉♣❡r✜❝✐❛❧ ❡ ♣♦❞❡ ❡①♣❡r✐♠❡♥t❛r ❢♦rç❛s ♣r♦✈❡♥✐❡♥t❡s ❞❡ ✉♠❛ ✜♥❛ ❝❛♠❛❞❛✳

❈❆P❮❚❯▲❖ ✸ ✶✸

♠❡♥t❛çã♦ ❧❛t❡r❛❧ ❞❛ ❛♠♦str❛ ❡♠ r❡❧❛çã♦ à s♦♥❞❛✱ ❞❡s❝r❡✈❡♥❞♦ ✉♠ ♣❛❞rã♦ ❞❡ ✈❛rr❡❞✉r❛✳ ❆tr❛✈és ❞♦ ♠❡❝❛♥✐s♠♦ ❞❡ ♠♦♥✐t♦r❛çã♦ ✭❈✮✱ ❞❡t❡❝t❛✲s❡ ❛ ✈❛r✐❛çã♦ ❞❛ ✐♥t❡r❛çã♦ s♦♥❞❛✲❛♠♦str❛ ❞✉r❛♥t❡ ❛ ✈❛rr❡❞✉r❛ ❡ ❡st❛ ✐♥❢♦r♠❛çã♦ é ♣❛ss❛❞❛ ❛ ✉♠ s✐st❡♠❛ ❞❡ r❡❛❧✐♠❡♥t❛çã♦ q✉❡ ❝♦♥tr♦❧❛ ❛ ♣♦s✐çã♦ ✈❡rt✐❝❛❧ ❞❛ s♦♥❞❛✳ ❆ ♣♦s✐çã♦ ✈❡rt✐❝❛❧ ❞❛ s♦♥❞❛ s♦❜r❡ ❛ s✉♣❡r❢í❝✐❡ ❞❛ ❛♠♦str❛ t❛♠❜é♠ é ❞❡✜♥✐❞❛ ♣❡❧♦ s❝❛♥♥❡r✱ q✉❡ é✱ ♣♦rt❛♥t♦✱ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ♠♦✈✐♠❡♥t❛çã♦ r❡❧❛t✐✈❛ s♦♥❞❛✲ ❛♠♦str❛ ♥❛s três ❞✐r❡çõ❡s ❡s♣❛❝✐❛✐s✳ ❚♦❞♦ ❡ss❡ ♣r♦❝❡ss♦ é ❝♦♥tr♦❧❛❞♦ ♣♦r ✉♠ ❝♦♠♣✉t❛❞♦r ✭❊✮✱ q✉❡ ❝♦♥tr♦❧❛ ❛ ♠♦✈✐♠❡♥t❛çã♦ ❞♦ s❝❛♥♥❡r✱ r❡❝❡❜❡ ♦s ❞❛❞♦s ❞❛ ✐♥t❡r❛çã♦ s♦♥❞❛✲❛♠♦str❛ ❡ ❝♦♥✈❡rt❡ t♦❞❛ ❛ ✐♥❢♦r♠❛çã♦ ♦❜t✐❞❛✱ ❢♦r♠❛♥❞♦ ❛ ✐♠❛❣❡♠✳ ❈♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ s❡ ♣♦s✐❝✐♦♥❛r ❛ s♦♥❞❛ s♦❜r❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ r❡❣✐ã♦ ❞❛ ❛♠♦str❛✱ ♦ s✐st❡♠❛ ♣♦ss✉✐ ❛✐♥❞❛ ✉♠ ♠❡❝❛♥✐s♠♦ ❞❡ ❛♣r♦①✐♠❛çã♦ ✭❉✮ q✉❡ ❝♦❧♦❝❛ ❛ s♦♥❞❛ ♥❛s ♣r♦①✐♠✐❞❛❞❡s ❞❛ ❧♦❝❛❧✐③❛çã♦ ❞❡s❡❥❛❞❛✳ P❛r❛ t❛♥t♦✱ ✉♠❛ ❧✉♣❛ ♦✉ ✉♠ ♠✐❝r♦s❝ó♣✐♦ ó♣t✐❝♦ ♥♦r♠❛❧♠❡♥t❡ ❢❛③❡♠ ♣❛rt❡ ❞❡st❡ ♠❡❝❛✲ ♥✐s♠♦ ❞❡ ❛♣r♦①✐♠❛çã♦✳ ❉❡♥tr❡ ❛s ❞✐✈❡rs❛s té❝♥✐❝❛s ❞❡ss❛ ❢❛♠í❧✐❛✱ ❛❧❣✉♠❛s ❢♦r❛♠ ✉t✐❧✐③❛❞❛s ♥❡ss❡ tr❛❜❛❧❤♦ ❡ s❡rã♦ ❞❡s❝r✐t❛s ❛ s❡❣✉✐r✳

❈❆P❮❚❯▲❖ ✸ ✶✹

✸✳✶✳✷ ▼✐❝r♦s❝♦♣✐❛ ❞❡ ❋♦rç❛ ❆tô♠✐❝❛

❆ ▼✐❝r♦s❝♦♣✐❛ ❞❡ ❋♦rç❛ ❆tô♠✐❝❛ ✭❆❋▼ ✲ ❆t♦♠✐❝ ❋♦r❝❡ ▼✐❝r♦s❝♦♣②✮ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞❛ ❡♠ ✶✾✽✻ ♣♦r ❇✐♥♥✐❣✱ ◗✉❛t❡ ❡ ●❡r❜❡r✱ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❡st✉✲ ❞❛r s✉♣❡r❢í❝✐❡ ❞❡ ♠❛t❡r✐❛✐s ✐s♦❧❛♥t❡s ❡♠ ❡s❝❛❧❛ ❛tô♠✐❝❛ ❬✶✼❪✳ ❖ ♣r✐♥❝í♣✐♦ ❞❡ ❢✉♥❝✐♦♥❛♠❡♥t♦ é ❛ ♠❡❞✐❞❛ ❞❛ ❢♦rç❛ ❞❡ ✐♥t❡r❛çã♦ ❡♥tr❡ ✉♠❛ s♦♥❞❛ ♠❡❝â♥✐❝❛ ❡ ❛ ❛♠♦str❛✱ ❛tr❛✈és ❞❛ ❞❡✢❡①ã♦ ❞❡ ✉♠❛ ❛❧❛✈❛♥❝❛ q✉❡ s❡ ❝✉r✈❛ q✉❛♥❞♦ ❛ s♦♥❞❛ ✐♥t❡r❛❣❡ ❝♦♠ ❛ s✉♣❡r❢í❝✐❡✳

❆s ❢♦rç❛s ❞❡ ✐♥t❡r❛çã♦ ♠❡❞✐❞❛s ♣❡❧♦ ❆❋▼ ♣♦❞❡♠ s❡r q✉❛❧✐t❛t✐✈❛♠❡♥t❡ ❡①♣❧✐❝❛❞❛s ❝♦♥s✐❞❡r❛♥❞♦ ❛s ❢♦rç❛s ❞❡ ✈❛♥ ❞❡r ❲❛❛❧s ❞❡ ❞♦✐s át♦♠♦s ❧♦❝❛❧✐③❛❞♦s ❛ ✉♠❛ ❞✐stâ♥❝✐❛ r✱ ❝✉❥♦ ♣♦t❡♥❝✐❛❧ é ❛♣r♦①✐♠❛❞♦ ♣❡❧♦ ♣♦t❡♥❝✐❛❧ ❞❡ ▲❡♥♥❛r❞✲

❏♦♥❡s✱ ♦✉ s❡❥❛✱ ❛ s♦♥❞❛ é ❛tr❛í❞❛ ♣❡❧❛ ❛♠♦str❛ ❛ ❧♦♥❣❛s ❞✐stâ♥❝✐❛s ❡ é r❡♣❡❧✐❞❛ ♣❛r❛ ❝✉rt❛s ❞✐stâ♥❝✐❛s ❬✶✽❪✳ P❛r❛ ✈❛❧♦r❡s ❣r❛♥❞❡s ❞❡ s❡♣❛r❛çã♦ s♦♥❞❛✲❛♠♦str❛✱ ♥ã♦ ❤á ♥❡♥❤✉♠❛ ❢♦rç❛ ❡♥tr❡ ❡❧❛s✳ ❆ ♠❡❞✐❞❛ q✉❡ ♦❝♦rr❡ ❛ ❛♣r♦①✐♠❛çã♦✱ ❢♦rç❛s ❛tr❛t✐✈❛s ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡ ❝♦♠❡ç❛♠ ❛ ❛♣❛r❡❝❡r✳ ❆ ❢♦rç❛ ❛t✐♥❣❡ ✉♠ ♠á①✐♠♦ ❡ ❞❡❝❛✐✱ ❛té s❡ t♦r♥❛r r❡♣✉❧s✐✈❛✳ ■ss♦ s❡ ❞á ♣♦r ❝❛✉s❛ ❞❛ r❡♣✉❧sã♦ ❞❛s ♥✉✈❡♥s ❡❧❡trô♥✐❝❛s ❞♦s át♦♠♦s ♠❛✐s ❡①tr❡♠♦s ❞❛ s♦♥❞❛ ❡ ❞❛ ❛♠♦str❛✳ ◆❡st❛ r❡❣✐ã♦✱ ❞✐③✲s❡ q✉❡ ❛ s♦♥❞❛ ❡stá ❡♠ ❝♦♥t❛t♦ ❢ís✐❝♦ ❝♦♠ ❛ s✉♣❡r❢í❝✐❡ ❞❛ ❛♠♦str❛✳

➱ ♣♦ssí✈❡❧ ❞✐st✐♥❣✉✐r ❡♥tr❡ ❞♦✐s ♠♦❞♦s ❞❡ ♦♣❡r❛çã♦ ❞♦ ❆❋▼✳ ❖ ♣r✐♠❡✐r♦ ❝❛s♦ é ❝❤❛♠❛❞♦ ♠♦❞♦ ❝♦♥t❛t♦✱ ♥♦ q✉❛❧ é ♠♦♥✐t♦r❛❞❛ ❛ ❞❡✢❡①ã♦ ❡stát✐❝❛ ❞❛ ❛❧❛✈❛♥❝❛✱ ❡♥q✉❛♥t♦ ♦ s❡❣✉♥❞♦ ♣♦ss✉✐ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ♥♦♠❡s✱ ❡♥tr❡ ♦s q✉❛✐s s❡ ♣♦❞❡ ❛♣♦♥t❛r ♠♦❞♦ t❛♣♣✐♥❣ ♦✉ ♠♦❞♦ ❝♦♥t❛t♦ ✐♥t❡r♠✐t❡♥t❡✳ ◆❡ss❡ ❝❛s♦✱ ❛ ❛❧❛✈❛♥❝❛ ♦s❝✐❧❛ ❝♦♠ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❛♠♣❧✐t✉❞❡ ✭❡①❝✐t❛❞❛ ❡♠ ✉♠❛ ❢r❡q✉ê♥✲ ❝✐❛ω♣ró①✐♠❛ ❛ ❢r❡q✉ê♥❝✐❛ ♥❛t✉r❛❧ ❞❛ ❛❧❛✈❛♥❝❛✮ ❡ ♦ s✐st❡♠❛ ❞❡ r❡tr♦❛❧✐♠❡♥t❛✲

❈❆P❮❚❯▲❖ ✸ ✶✺

s✉♣❡r❢í❝✐❡ ❞❛ ❛♠♦str❛ ❡ ❞❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ s❡ ❡stá ✐♥t❡r❡ss❛❞♦ ❡♠ ❛♥❛❧✐s❛r✳ ❆s r❡❣✐õ❡s ❞❡ ✐♥t❡r❛çã♦ ❞❡st❡s ♠♦❞♦s ♣♦❞❡♠ s❡r ✐❞❡♥t✐✜❝❛❞❛s ♥❛ ✜❣✉r❛ ✸✳✷✳

❋✐❣✉r❛ ✸✳✷✿ ❈✉r✈❛ ❡sq✉❡♠át✐❝❛ ♠♦str❛♥❞♦ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞❛ ❢♦rç❛ ❞❡ ✐♥✲ t❡r❛çã♦ s♦♥❞❛✲❛♠♦str❛ ❡♠ ❢✉♥çã♦ ❞❛ s❡♣❛r❛çã♦ ❡♥tr❡ ❡❧❛s ❡ s✉❛ r❡❧❛çã♦ ❝♦♠ ♦s ♠♦❞♦s ❞❡ ♦♣❡r❛çã♦ ❞♦ ❆❋▼✿ ❝♦♥t❛t♦✱ ♥ã♦✲❝♦♥t❛t♦ ❡ ❝♦♥t❛t♦ ✐♥t❡r♠✐t❡♥t❡✳ ❋✐❣✉r❛ ❛❞❛♣t❛❞❛ ❞❡ ❬✶✾❪✳

▼♦❞♦ ❈♦♥t❛t♦

◆♦ ♠♦❞♦ ❝♦♥t❛t♦✱ ❛ s♦♥❞❛ é ❝♦❧♦❝❛❞❛ ❡♠ ❝♦♥t❛t♦ ❢ís✐❝♦ ❝♦♠ ❛ s✉♣❡r❢í❝✐❡ ❡ ❛ ❢♦rç❛ ❛❣✐♥❞♦ ❡♥tr❡ ♦s át♦♠♦s ❞❛ s♦♥❞❛ ❡ ❞❛ ❛♠♦str❛ é ❝♦♥tr❛❜❛❧❛♥❝❡❛❞❛ ♣❡❧❛ ❢♦rç❛ ❡❧ást✐❝❛ ♣r♦❞✉③✐❞❛ ♣❛r❛ ❞❡✢❡t✐r ❛ ❛❧❛✈❛♥❝❛ ❬✶✽❪ ❡ ❡ss❛ ❞❡✢❡①ã♦ ❞❛ ❛❧❛✈❛♥❝❛ é ♠❛♥t✐❞❛ ❝♦♥st❛♥t❡ ❞✉r❛♥t❡ ❛ ✈❛rr❡❞✉r❛ ♣❡❧♦ s✐st❡♠❛ ❞❡ r❡tr♦❛❧✐✲ ♠❡♥t❛çã♦ ❬✶✾❪✳ ❆ss✐♠✱ ❛ ✐♥❢♦r♠❛çã♦ t♦♣♦❣rá✜❝❛ ❞❛ s✉♣❡r❢í❝✐❡ é ♦❜t✐❞❛ ❛tr❛✈és ❞♦ ♠♦♥✐t♦r❛♠❡♥t♦ ❞❡ss❛ ❞❡✢❡①ã♦ q✉❡ ♦❝♦rr❡ ❝♦♠♦ r❡s✉❧t❛❞♦ ❞❛ ✈❛r✐❛çã♦ ❞❛ ❢♦rç❛ ❝❛✉s❛❞❛ ♣❡❧❛s ♠✉❞❛♥ç❛s ♥❛ t♦♣♦❣r❛✜❛ ❞❛ ❛♠♦str❛ ❬✷✵❪ ✭✜❣✉r❛ ✸✳✸✮✳

❈❆P❮❚❯▲❖ ✸ ✶✻

❋✐❣✉r❛ ✸✳✸✿ ❉❡s❡♥❤♦ ❡sq✉❡♠át✐❝♦ ❞❛s ❞❡✢❡①õ❡s ❞❛ ❛❧❛✈❛♥❝❛ ❡ ❞❡❢♦r♠❛çõ❡s ❞♦ s❝❛♥♥❡r ❡♠ ❢✉♥çã♦ ❞❛ ❢♦rç❛ ❞❡ ✐♥t❡r❛çã♦ s♦♥❞❛✲❛♠♦str❛ ♥♦ ♠♦❞♦ ❞❡ ♦♣❡r❛çã♦ ❝♦♥t❛t♦✳ ❊♠ ✭❛✮ tê♠✲s❡ ❛ ❢♦rç❛ ❞❡ r❡❢❡rê♥❝✐❛ ❞❡✜♥✐❞❛ ♣❡❧♦ ♦♣❡r❛❞♦r✱ ❡♠ ✭❜✮ ❛ ❢♦rç❛ ❞❡ ✐♥t❡r❛çã♦ ♠❛✐♦r q✉❡ ❛ ❞❡ r❡❢❡rê♥❝✐❛ ❡ ❡♠ ✭❝✮ ❛ ❢♦rç❛ ❞❡ ✐♥t❡r❛çã♦ ♠❡♥♦r q✉❡ ❛ ❞❡ r❡❢❡rê♥❝✐❛✳

♠♦❞♦ ❢♦rç❛ ❝♦♥st❛♥t❡✱ ♦ s✐st❡♠❛ ❞❡ r❡tr♦❛❧✐♠❡♥t❛çã♦ ♠❛♥té♠ ✉♠ ✈❛❧♦r ❝♦♥s✲ t❛♥t❡ ♣❛r❛ ❛ ❞❡✢❡①ã♦ ❞❛ ❛❧❛✈❛♥❝❛✱ ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❞❛ ❢♦rç❛ ❞❡ ✐♥t❡r❛çã♦✳ ❆ss✐♠✱ ❛ r❡s♣♦st❛ ✉s❛❞❛ ♥♦ s✐st❡♠❛ ❞❡ r❡tr♦❛❧✐♠❡♥t❛çã♦ é ♣r♦♣♦r❝✐♦♥❛❧ ❛ t♦✲ ♣♦❣r❛✜❛ ❞❛ ❛♠♦str❛✳ ❖ ♠♦❞♦ ❞❡ ❞✐stâ♥❝✐❛ ♠é❞✐❛ ❝♦♥st❛♥t❡ ❡♥tr❡ ❛ s♦♥❞❛ ❡ ❛ ❛♠♦str❛ é ❢r❡q✉❡♥t❡♠❡♥t❡ ✉s❛❞♦ ❡♠ ❛♠♦str❛s ❞❡ ❜❛✐①❛ r✉❣♦s✐❞❛❞❡✳ ◆❡ss❡ ♠♦❞♦✱ ❛ s♦♥❞❛ s❡ ♠♦✈❡ ❛ ✉♠❛ ❞✐stâ♥❝✐❛ ③ ❞❛ ❛♠♦str❛ ❡ ❛ ❞❡✢❡①ã♦ ∆z ❞❛

❛❧❛✈❛♥❝❛ ✭♣r♦♣♦r❝✐♦♥❛❧ ❛ ❢♦rç❛✮ é ♠♦♥✐t♦r❛❞♦ ❡♠ ❝❛❞❛ ♣♦♥t♦✳ ❆ ✐♠❛❣❡♠ ❞❡ ❆❋▼ ♥❡ss❡ ❝❛s♦ ❞❡s❝r❡✈❡ ❛ ❞✐str✐❜✉✐çã♦ ❞❛ ❢♦rç❛ ❞❡ ✐♥t❡r❛çã♦✳

❈♦♠♦ ♥♦ ♠♦❞♦ ❝♦♥t❛t♦ ❛ ❛♠♦str❛ ❡①♣❡r✐♠❡♥t❛ t❛♥t♦ ❢♦rç❛s ❝♦♠♣r❡ss✐✈❛s q✉❛♥t♦ ❢♦rç❛s ❞❡ ❝✐s❛❧❤❛♠❡♥t♦✱ q✉❡ sã♦ ❝❛✉s❛❞❛s ♣❡❧♦ ♠♦✈✐♠❡♥t♦ ❞❡ ✈❛r✲ r❡❞✉r❛ ❞❛ s♦♥❞❛✱ ❛ ♣r✐♥❝✐♣❛❧ ❞❡s✈❛♥t❛❣❡♠ ❞❡ss❡ ♠♦❞♦ é q✉❡ ❛♠❜❛s ❢♦rç❛s ♣♦❞❡♠ ✐♥❞✉③✐r ❞❡❢♦r♠❛çõ❡s ♥❛ ❛♠♦str❛✳

▼♦❞♦ ◆ã♦✲❈♦♥t❛t♦

❈❆P❮❚❯▲❖ ✸ ✶✼

✐♥❞✉③❡♠ ✉♠ ❞❡s❧♦❝❛♠❡♥t♦ ♥❛ ❢r❡q✉ê♥❝✐❛ ❞❡ r❡ss♦♥â♥❝✐❛ ❞❛ ❛❧❛✈❛♥❝❛ ❬✶✾❪✳ ❆s ✐♠❛❣❡♥s sã♦ ♦❜t✐❞❛s ♠❛♥t❡♥❞♦ ♦ ❞❡s❧♦❝❛♠❡♥t♦ ❡♠ ❢r❡q✉ê♥❝✐❛ ❝♦♥st❛♥t❡ ❞✉r❛♥t❡ ❛ ✈❛rr❡❞✉r❛✱ ❡ ♥♦r♠❛❧♠❡♥t❡ ✐ss♦ é ❢❡✐t♦ ♠♦♥✐t♦r❛♥❞♦ ❛ ❛♠♣❧✐t✉❞❡ ❞❡ ♦s❝✐❧❛çã♦ ❞❛ ❛❧❛✈❛♥❝❛ ♣❛r❛ ✉♠❛ ❢r❡q✉ê♥❝✐❛ ✜①❛✱ r❡♣❛ss❛♥❞♦ ❡ss❡ ✈❛❧♦r ♣❛r❛ ♦ ❝♦♥tr♦❧❡ ❞❡ r❡tr♦❛❧✐♠❡♥t❛çã♦✱ ❛ss✐♠ ❝♦♠♦ ❛❝♦♥t❡❝❡ ♥♦ ♠♦❞♦ ❝♦♥t❛t♦✳ ❆s ✐♥t❡r❛çõ❡s s♦♥❞❛✲❛♠♦str❛ sã♦ ❜❛st❛♥t❡ ♣❡q✉❡♥❛s ♥♦ ♠♦❞♦ ♥ã♦✲❝♦♥t❛t♦ ❡ ✉♠❛ ❜♦❛ r❡s♦❧✉çã♦ ✈❡rt✐❝❛❧ ♣♦❞❡ s❡r ❛❧❝❛♥ç❛❞❛✳ ❊①✐st❡ s♦♠❡♥t❡ ✉♠❛ ♣❡q✉❡♥❛ r❡❣✐ã♦ ♥❛ q✉❛❧ ❛ ❛♠♣❧✐t✉❞❡ ❞❡ ♦s❝✐❧❛çã♦ é ❛❢❡t❛❞❛ ♣❡❧❛ ✐♥t❡r❛çã♦ ❞❡ ✈❛♥ ❞❡r ❲❛❛❧s ❞❡ ❝✉rt♦ ❛❧❝❛♥❝❡ ❛♥t❡s ❞❛ s♦♥❞❛ s❡r ❝❛♣t✉r❛❞❛ ♣❡❧❛ ❝❛♠❛❞❛ ❞❡ ❝♦♥t❛♠✐♥❛çã♦ ❬✷✵❪✳ ❊ s❡ ❛ ❝❛♠❛❞❛ ❞❡ ❝♦♥t❛♠✐♥❛çã♦ ❢♦r ❡s♣❡ss❛ ♦✉ s❡ ❤á á❣✉❛ ♣r❡s❡♥t❡✱ ❛ s♦♥❞❛ ♣♦❞❡ ✜❝❛r ♣r❡s❛✱ ♥ã♦ ♣♦ss✉✐♥❞♦ ❡♥❡r❣✐❛ s✉✜❝✐❡♥t❡ ♣❛r❛ s❡ s♦❧t❛r ♣♦r ❝♦♥t❛ ❞❛ s✉❛ ❜❛✐①❛ ❛♠♣❧✐t✉❞❡ ❞❡ ♦s❝✐❧❛çã♦✳

▼♦❞♦ ❈♦♥t❛t♦ ■♥t❡r♠✐t❡♥t❡

❆ ♠♦t✐✈❛çã♦ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ♠♦❞♦ ❝♦♥t❛t♦ ✐♥t❡r♠✐t❡♥t❡ ✭t❛♠✲ ❜é♠ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♠♦❞♦ t❛♣♣✐♥❣✮ é s✉♣❡r❛r ❛s ❞✐✜❝✉❧❞❛❞❡s ❞❡ ♦♣❡r❛çã♦ ❞♦ ♠♦❞♦ ♥ã♦✲❝♦♥t❛t♦ ❡ ♠✐♥✐♠✐③❛r ♦s ❞❛♥♦s ❝❛✉s❛❞♦s ♣❡❧♦ ♠♦❞♦ ❝♦♥t❛t♦ ❬✷✵❪✳ ❖ ♠♦❞♦ ❝♦♥t❛t♦ ✐♥t❡r♠✐t❡♥t❡ ❞✐❢❡r❡ ❞♦ ♠♦❞♦ ♥ã♦✲❝♦♥t❛t♦ ❥á q✉❡ ❛ ❛❧❛✈❛♥❝❛ ♦s✲ ❝✐❧❛ ♣❡rt♦ ❞❡ s✉❛ ❢r❡q✉ê♥❝✐❛ ❞❡ r❡ss♦♥â♥❝✐❛ ❡ ❛ s♦♥❞❛ t♦❝❛ ❛ ❛♠♦str❛ ❡♠ ❝❛❞❛ ❝✐❝❧♦ ❞❡ ♦s❝✐❧❛çã♦✳ ❆ ❛♠♣❧✐t✉❞❡ ❞❡ ♦s❝✐❧❛çã♦ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❛ ❡♥❡r❣✐❛ ❛ss♦s❝✐❛❞❛ ❛ ♦s❝✐❧❛çã♦✱ é s✉✜❝✐❡♥t❡ ♣❛r❛ ✈❡♥❝❡r ❛ ✈✐s❝♦s✐❞❛❞❡ ❞❛ ❛♠♦str❛✳ ■ss♦ r❡s✉❧t❛ ❡♠ ✉♠❛ ❛♠♣❧✐t✉❞❡ ❞❡ ♦s❝✐❧❛çã♦ q✉❡ é t✐♣✐❝❛♠❡♥t❡ ❞❛ ♦r❞❡♠ ❞❡ ✷✵ ❛ ✶✵✵ ♥♠✱ ♠✉✐t♦ ♠❛✐♦r q✉❡ ❛s ❛♠♣❧✐t✉❞❡s ❞♦ ♠♦❞♦ ♥ã♦✲❝♦♥t❛t♦✳ ❖ s✐st❡♠❛ ❞❡ r❡tr♦❛❧✐♠❡♥t❛çã♦ ❞❡t❡❝t❛ ❛ ♣❡rt✉r❜❛çã♦ ♥❛ ❛♠♣❧✐t✉❞❡ ❞❡ ♦s❝✐❧❛çã♦ ❝❛✉✲ s❛❞❛ ♣❡❧♦ ❝♦♥t❛t♦ ✐♥t❡r♠✐t❡♥t❡ ❝♦♠ ❛ s✉♣❡r❢í❝✐❡ ✭✜❣✉r❛ ✸✳✹✮ ❡ ❛ ✐♠❛❣❡♠ ❞❛ ❛♠♦str❛ é ❢❡✐t❛ ❡♥q✉❛♥t♦ ♦ s✐st❡♠❛ ❞❡ r❡tr♦❛❧✐♠❡♥t❛çã♦ ❛❥✉st❛ ❛ s❡♣❛r❛çã♦ s♦♥❞❛✲❛♠♦str❛ ♣❛r❛ ♠❛♥t❡r ❛ ❛♠♣❧✐t✉❞❡ ❞❡ ♦s❝✐❧❛çã♦ ❝♦♥st❛♥t❡ ❬✷✶❪✳ ❉✉r❛♥t❡ ✉♠❛ ♦s❝✐❧❛çã♦✱ ❛s ❢♦rç❛s ❛❣✐♥❞♦ ♥❛ s♦♥❞❛ sã♦ ❞♦ t✐♣♦ ✈❛♥ ❞❡r ❲❛❛❧s ♦✉✱ ♠❛✐s ❣❡r❛❧♠❡♥t❡✱ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❞❡ ❢♦rç❛s ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡ ✭❛tr❛t✐✈❛s✮ ❡ ❝✉rt♦ ❛❧❝❛♥❝❡ ✭r❡♣✉❧s✐✈❛s✮ ❬✷✷❪✳

❈❆P❮❚❯▲❖ ✸ ✶✽

❋✐❣✉r❛ ✸✳✹✿ ❉❡s❡♥❤♦ ❡sq✉❡♠át✐❝♦ ❞❛s ❛♠♣❧✐t✉❞❡s ❞❡ ♦s❝✐❧❛çã♦ ❞❛ s♦♥❞❛ ♥♦ ♠♦❞♦ ❞❡ ♦♣❡r❛çã♦ ❝♦♥t❛t♦ ✐♥t❡r♠✐t❡♥t❡✳ ❊♠ ✭❛✮ ❛ s♦♥❞❛ ♦s❝✐❧❛ ❧♦♥❣❡ ❞❛ s✉♣❡r❢í❝✐❡✱ ❡ ❡♠ ✭❜✮ ❛ s♦♥❞❛ ♦s❝✐❧❛ ♣ró①✐♠❛ à s✉♣❡r❢í❝✐❡✳

❛q✉✐s✐çã♦ ❞❡ ✈ár✐❛s ✐♥❢♦r♠❛çõ❡s✱ ❝❛❞❛ ✉♠❛ ❞❡❧❛s r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ✉♠❛ ♣r♦✲ ♣r✐❡❞❛❞❡ ❬✷✸❪✳ ❆❞✐❝✐♦♥❛❧♠❡♥t❡✱ ♦ ♠♦❞♦ ❝♦♥t❛t♦ ✐♥t❡r♠✐t❡♥t❡ ❡❧✐♠✐♥❛ q✉❛s❡ t♦❞♦ ♦ ❝✐s❛❧❤❛♠❡♥t♦ ❧❛t❡r❛❧ ❡①❡r❝✐❞♦ ♥❛ ❛♠♦str❛✱ ✈✐st♦ q✉❡ ❛ s♦♥❞❛ ♥ã♦ ♠❛✐s é ❛rr❛st❛❞❛ s♦❜r❡ ❛ s✉♣❡r❢í❝✐❡✳ ■ss♦ t❡♠ ❛ ✈❛♥t❛❣❡♠ ❞❡ ❡①❝❧✉✐r ❛ ✈❛r✐❛çã♦ ❞❡ ❛tr✐t♦ ❞✉r❛♥t❡ ❛ ♠❡❞✐❞❛ ❞❡ t♦♣♦❣r❛✜❛ ❬✷✵❪✳

✸✳✶✳✸ ▼✐❝r♦s❝♦♣✐❛ ❞❡ ❋♦rç❛ ❊❧étr✐❝❛

❆❧é♠ ❞❡ ♠❡❞✐❞❛s ❞❡ ❢♦rç❛ ❞❡ ❝✉rt♦ ❛❧❝❛♥❝❡✱ ❝♦♠♦ ✐♠♣❧❡♠❡♥t❛❞♦ ♥♦ ❆❋▼✱ ✉♠ ♠✐❝r♦s❝ó♣✐♦ ❞❡ ❙P▼ t❛♠❜é♠ ♣❡r♠✐t❡ ♠❛♣❡❛r ❢♦rç❛s ❞❡ ♦✉tr❛s ♥❛t✉r❡✲ ③❛s✱ ❝♦♠♦ ❛s ❢♦rç❛s ❛tr❛t✐✈❛s ❞❡ ✈❛♥ ❞❡r ❲❛❛❧s✱ ❢♦rç❛s ♠❛❣♥ét✐❝❛s ❡ ❡❧étr✐❝❛s✳ ❊ss❛s ✐♥t❡r❛çõ❡s ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡ ♣♦❞❡♠ s❡r ♠❡❞✐❞❛s ♥❛s té❝♥✐❝❛s ❞❡ ❙P▼ ✈❛r✐❛♥❞♦ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❛ s✉♣❡r❢í❝✐❡ ❡ ❛ s♦♥❞❛✳ P❛r❛ ♣❡q✉❡♥❛s s❡♣❛r❛çõ❡s s♦♥❞❛✲❛♠♦str❛✱ ✈✐♠♦s q✉❡ ❛s ❢♦rç❛s ❞❡ ✈❛♥ ❞❡r ❲❛❛❧s ❞❡ ❝✉rt♦ ❛❧❝❛♥❝❡ ♣r❡✈❛✲ ❧❡❝❡♠✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ s❡♣❛r❛çõ❡s s♦♥❞❛✲❛♠♦str❛ ♠❛✐♦r❡s✱ ❛s ✐♥t❡r❛çõ❡s ❡❧étr✐❝❛s ❡ ♠❛❣♥ét✐❝❛s ✐rã♦ ❞♦♠✐♥❛r ❞❡✈✐❞♦ ❛♦ rá♣✐❞♦ ❞❡❝❛✐♠❡♥t♦ ❞❛s ❢♦rç❛s ❞❡ ✈❛♥ ❞❡r ❲❛❛❧s✳ ❯♠ ♣❛ss♦ ✐♠♣♦rt❛♥t❡ ♥❛ ❞❡t❡❝çã♦ ❞❡ ❢♦rç❛s ❞❡ ❧♦♥❣♦ ❛❧✲ ❝❛♥❝❡ ❢♦✐ ❛ ✐♥✈❡♥çã♦ ❞♦s ♠♦❞♦s ❞❡ ♥ã♦✲❝♦♥t❛t♦✳ ❆ss✐♠✱ ❛ s♦♥❞❛ ♦❜té♠ ❛s ✐♠❛❣❡♥s ❞❡ t♦♣♦❣r❛✜❛ ❜❡♠ ♣ró①✐♠❛ ❛ s✉♣❡r❢í❝✐❡✱ ❡♥tã♦ é r❡tr❛í❞❛ ❛ ✉♠❛ ❝❡rt❛ ❞✐stâ♥❝✐❛ ❛❝✐♠❛ ❞❛ s✉♣❡r❢í❝✐❡ ♣❛r❛ ♠❡❞✐r ❡ss❛s ♦✉tr❛s ✐♥t❡r❛çõ❡s✳

❈❆P❮❚❯▲❖ ✸ ✶✾

♣❛r❛ s❡ ♦❜t❡r ❞✐✈❡rs❛s ♣r♦♣r✐❡❞❛❞❡s ❡❧étr✐❝❛s t❛✐s ❝♦♠♦✿ ❝❛r❣❛✱ ❞✐str✐❜✉✐çã♦ ❞❡ ♣♦t❡♥❝✐❛❧ ❡❧étr✐❝♦✱ ❝♦♥st❛♥t❡ ❞✐❡❧étr✐❝❛✱ ❝♦♥❞✉t✐✈✐❞❛❞❡✱ ❡t❝✳

❇❛s❡❛❞♦ ♥❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❛ s♦♥❞❛ ❡ ❛ ❛♠♦str❛✱ ❛ té❝♥✐❝❛ ❞❡ ❊❋▼ ♣♦❞❡ s❡r ❞✐✈✐❞✐❞❛ ❡♠ três r❡❣✐♠❡s✿ ❧♦♥❣♦ ❛❧❝❛♥❝❡✱ ✐♥t❡r♠❡❞✐ár✐♦ ❡ ❝✉rt♦ ❛❧❝❛♥❝❡ ❬✶✺❪✳ ◆♦ r❡❣✐♠❡ ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡✱ ❛ ❞✐stâ♥❝✐❛ s♦♥❞❛✲❛♠♦str❛ é ③ ❃ ✶✵✲✺✵ ♥♠ ❡ s♦♠❡♥t❡ ❢♦rç❛s ❡❧❡tr♦stát✐❝❛s ❡♥tr❡ ❛ s♦♥❞❛ ❡ ❛ s✉♣❡r❢í❝✐❡ sã♦ s✐❣♥✐✜❝❛♥t✐✲ ✈❛s✳ ❆ ❛♠♣❧✐t✉❞❡ ❞❡ ♦s❝✐❧❛çã♦ é✱ ❣❡r❛❧♠❡♥t❡✱ ♠✉✐t♦ ♠❡♥♦r q✉❡ ❛ s❡♣❛r❛çã♦ s♦♥❞❛✲❛♠♦str❛✱ ❞❡ ♠♦❞♦ q✉❡ é ♣♦ssí✈❡❧ ✉♠❛ ❡①tr❛çã♦ r❡❧❛t✐✈❛♠❡♥t❡ ❢á❝✐❧ ❞♦ ❣r❛❞✐❡♥t❡ ❞❡ ❢♦rç❛✳ P❛r❛ t❛❧✱ é ♥❡❝❡ssár✐♦ ❛♣❡♥❛s q✉❡ s❡❥❛♠ ❝♦♥❤❡❝✐❞❛s ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❛❧❛✈❛♥❝❛ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❝♦♥st❛♥t❡ ❡❧ást✐❝❛✳ ❖ s❡❣✉♥❞♦ r❡❣✐♠❡ é ❝❛r❛❝t❡r✐③❛❞♦ ♣♦r ✉♠❛ ❞✐stâ♥❝✐❛ s♦♥❞❛ ❛♠♦str❛ r❡❧❛t✐✈❛♠❡♥t❡ ♣❡✲ q✉❡♥❛ ✭③ ∼ ✶✵✲✺✵ ♥♠✮✱ ♣♦ré♠ ❛ ❝♦♥tr✐❜✉✐çã♦ ❞❡ ❢♦rç❛s ❡❧❡tr♦stát✐❝❛s ❛✐♥❞❛ ❞♦♠✐♥❛ s♦❜r❡ ❛s ✐♥t❡r❛çõ❡s ❞❡ ✈❛♥ ❞❡r ❲❛❛❧s✳ ❉✐❢❡r❡♥t❡♠❡♥t❡ ❞♦ r❡❣✐♠❡ ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡✱ ❛ ❛♠♣❧✐t✉❞❡ ❞❡ ♦s❝✐❧❛çã♦ é ❝♦♠♣❛rá✈❡❧ à s❡♣❛r❛çã♦ s♦♥❞❛✲ ❛♠♦str❛ ❡ ❛ s♦♥❞❛ ♣♦❞❡ t♦❝❛r ❛ s✉♣❡r❢í❝✐❡✱ t♦r♥❛♥❞♦ ❛ ❡①tr❛çã♦ ❞♦s ❞❛❞♦s ♠❛✐s ❞✐❢í❝✐❧✳ ◆❡st❡ ❝❛s♦ tr❛♥s❢❡rê♥❝✐❛s ❞❡ ❝❛r❣❛ ❡♥tr❡ ❛ s♦♥❞❛ ❡ ❛ ❛♠♦str❛ ❡ ❢♦rç❛s ❡❧❡tr♦stát✐❝❛s ❝♦♥tr✐❜✉❡♠ ♣❛r❛ ❞❡t❡❝çã♦ ❞♦ s✐♥❛❧✳ ◆♦ ú❧t✐♠♦ ❝❛s♦✱ ❛ ♣♦♥t❛ ❡stá ❡♠ ❝♦♥t❛t♦ ❝♦♠ ❛ s✉♣❡r❢í❝✐❡✳ ❈♦♠♦ ❛s ✐♥t❡r❛çõ❡s ❞❡ ✈❛♥ ❞❡r ❲❛❛❧s ❞♦♠✐♥❛♠ s♦❜r❡ ❛s ❢♦rç❛s ❡❧❡tr♦stát✐❝❛s✱ ❛♣❡♥❛s ❞❡t❡❝çã♦ ♥♦ ▼♦❞♦ ❈♦♥t❛t♦ é ♣♦ssí✈❡❧✳ ❈♦♥t✉❞♦✱ ♥❡st❡ ❝❛s♦✱ ❛ ♣♦♥t❛ ♣♦❞❡ s❡r ✉s❛❞❛ ❝♦♠♦ ✉♠ ♠❡❞✐❞♦r ❧♦❝❛❧ ❞❡ ❝❛♣❛❝✐tâ♥❝✐❛ ✭❙❈▼ ✲ ❙❝❛♥♥✐♥❣ ❈❛♣❛❝✐t❛♥❝❡ ▼✐❝r♦s❝♦♣②✮ ♦✉ r❡s✐st✐✈✐❞❛❞❡ ✭❙❙❘▼ ✲ ❙❝❛♥♥✐♥❣ ❙♣r❡❛❞✐♥❣ ❘❡s✐st❛♥❝❡ ▼✐❝r♦s❝♦♣②✮✱ ♣♦r ❡①❡♠♣❧♦✳

◆♦ s❡♥t✐❞♦ ❞❡ r❡❛❧✐③❛r ❛ ✈❛rr❡❞✉r❛ ❛ ✉♠❛ ❞✐stâ♥❝✐❛ ❝♦♥st❛♥t❡✱ ✉♠ ❡s✲ q✉❡♠❛ ❞❡ ❞✉❛s ♣❛ss❛❣❡♥s é ✉t✐❧✐③❛❞♦ ❬✶✽❪✳ ❊♠ ❝❛❞❛ ✈❛rr❡❞✉r❛✱ ♦ s❡❣✉✐♥t❡ ❡sq✉❡♠❛ é r❡❛❧✐③❛❞♦✿ ♥❛ ♣r✐♠❡✐r❛ ♣❛ss❛❣❡♠✱ ❛ s♦♥❞❛ ❢❛③ ✉♠❛ ✈❛rr❡❞✉r❛ ♥♦ ♠♦❞♦ ❈♦♥t❛t♦ ■♥t❡r♠✐t❡♥t❡ ♦✉ ♥♦ ♠♦❞♦ ❈♦♥t❛t♦✱ ♦❜t❡♥❞♦ ❛ss✐♠ ❛ ✐♠❛❣❡♠ ❞❛ t♦♣♦❣r❛✜❛ ❞❛ ❛♠♦str❛✳ ❉❡♣♦✐s✱ ❛ s♦♥❞❛ é s✉s♣❡♥s❛ ❞❡ ✉♠❛ ❞✐stâ♥❝✐❛ ③✱ ❡ r❡♣❡t❡ ❛ ✈❛rr❡❞✉r❛ ❞❛ t♦♣♦❣r❛✜❛ r❡❛❧✐③❛❞❛ ♥❛ ♣r✐♠❡✐r❛ ♣❛ss❛❣❡♠✱ ♦s❝✐❧❛♥❞♦ ❡♠ s✉❛ ❢r❡q✉ê♥❝✐❛ ❞❡ r❡ss♦♥â♥❝✐❛ ❡ ✉♠❛ t❡♥sã♦ ❉❈ ♦✉ ❆❈ ✭♦✉ ❛♠❜❛s✮ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞❛ ❡♥tr❡ ❛ s♦♥❞❛ ❡ ❛ s✉♣❡r❢í❝✐❡✳ ❉✉r❛♥t❡ ❡ss❛ s❡❣✉♥❞❛ ♣❛ss❛❣❡♠✱ ❛❧❣✉♠ t✐♣♦ ❞❡ ✐♥t❡r❛çã♦ ❡❧étr✐❝❛ ♣♦❞❡ ❛♣❛r❡❝❡r ❡♥tr❡ s♦♥❞❛✲❛♠♦str❛ ❡ ❡st❛ ✐♥t❡r❛çã♦ s❡rá r❡s♣♦♥á✈❡❧ ♣❡❧❛ ✈❛r✐❛çã♦ ❞❛ ❢r❡q✉ê♥❝✐❛ ♦✉ ❢❛s❡ ❞❛ ♦s❝✐❧❛çã♦ ❞❛ s♦♥❞❛ ✭✜❣✉r❛ ✸✳✺✮✳

❈❆P❮❚❯▲❖ ✸ ✷✵

❋✐❣✉r❛ ✸✳✺✿ ❉❡s❡♥❤♦ ❞♦ ❡sq✉❡♠❛ ❞❡ ❞✉❛s ♣❛ss❛❣❡♥s ✉t✐❧✐③❛❞♦ ♥❛s ♠❡❞✐❞❛s ❞❡ ❊❋▼✳ ❊♠ ✭❛✮ ❛ s♦♥❞❛ ❢❛③ ❛ ✈❛rr❡❞✉r❛ ♥♦ ♠♦❞♦ ❝♦♥t❛t♦ ✐♥t❡r♠✐t❡♥t❡✱ ❡ ❡♠ ✭❜✮ ❛ s♦♥❞❛ é r❡tr❛í❞❛ ❞❡ ✉♠❛ ❞✐stâ♥❝✐❛ ③ ❞❛ s✉♣❡r❢í❝✐❡ ❞❛ ❛♠♦str❛ ❡ ❛s ✐♥t❡r❛çõ❡s ❡❧étr✐❝❛s sã♦ ♠❡❞✐❞❛s ❛tr❛✈és ❞❡ ✉♠ ❞❡s❧♦❝❛♠❡♥t♦ ❞❛ ❢r❡q✉ê♥❝✐❛ ❞❡ r❡ss♦♥â♥❝✐❛✳

✉♠❛ ❢r❡q✉ê♥❝✐❛ ω0 s♦❜ ❛ ❛çã♦ ❞❡ ✉♠❛ ❢♦rç❛ ❡①t❡r♥❛ ❡ ✈❛r✐á✈❡❧ ❋✱ ♣♦❞❡♠♦s

❡s❝r❡✈❡r ❞❡ ❢♦r♠❛ ❣❡r❛❧✿

ω0 =c(m)√k′_,

❡♠ q✉❡ ❝✭♠✮ é ✉♠❛ ❣r❛♥❞❡③❛ q✉❡ ❞❡♣❡♥❞❡ ❞❛ ♠❛ss❛ ❡ ❞❛ ❣❡♦♠❡tr✐❛ ❞♦ ❝♦r♣♦✳ ❙❡ ❝♦♥s✐❞❡r❛r♠♦s ✉♠ ❡❢❡✐t♦ ❞❡ ❣r❛❞✐❡♥t❡ ❞❡ ❢♦rç❛✱∂Fz/∂z✱ ❛t✉❛♥❞♦ ♥❛ ❞✐r❡çã♦

❞❡ ♦s❝✐❧❛çã♦✱ ♦ ✈❛❧♦r ❡❢❡t✐✈♦ ❞❡ k′ ✜❝❛ r❡❞✉③✐❞♦ ❡ ♦❝♦rr❡ ✉♠❛ ♠✉❞❛♥ç❛ ♥❛

❢r❡q✉ê♥❝✐❛ ❞❡ ♦s❝✐❧❛çã♦ ❞❛ s♦♥❞❛✿

k′ _=k− ∂Fz

∂z ⇒ ωF =c(m)

r

k₋∂Fz ∂z .

❆ss✐♠✱ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ❛ ✈❛r✐❛çã♦ ♣r♦❞✉③✐❞❛ ♥❛ ❢r❡q✉ê♥❝✐❛ ❞❡ ♦s❝✐❧❛çã♦ ❞❛ s♦♥❞❛ ♣❡❧❛ ❢♦rç❛ ✈❛r✐á✈❡❧ F✿

∆ω = ωF −ω0

= c(m) r

k₋ ∂Fz ∂z

−√k

= c(m)√k

r

1₋ 1

k ∂Fz

∂z

❈❆P❮❚❯▲❖ ✸ ✷✶

❙❡∂Fz/∂z ≪k✱ ♣♦❞❡♠♦s ✉s❛r ❛ ❡①♣❛♥sã♦ ❡♠ sér✐❡ ❞❡ ❚❛②❧♦r ✭

p

(1₋x)∼= 1₋x/2✱ ♣❛r❛x_≪1✮✿

r

1₋ 1

k ∂Fz

∂z

∼

= 1− 1 2k

∂Fz

∂z ⇒ ∆ω=ω0

1₋ 1

2k ∂Fz

∂z −1

.

❉❡ss❡ ♠♦❞♦ ♦❜t❡♠♦s✿

∆ω=₋ω0 2k

∂Fz

∂z , ✭✸✳✶✮

q✉❡ é ❛ ❡q✉❛çã♦ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ❛ ✈❛r✐❛çã♦ ❞❡ ❢r❡q✉ê♥❝✐❛ ❞❡ ♦s❝✐❧❛çã♦ ❞❡ ✉♠❛ ❛❧❛✈❛♥❝❛✱ ♦s❝✐❧❛♥❞♦ s♦❜ ❛ ❛çã♦ ❞❡ ✉♠❛ ❢♦rç❛ ✈❛r✐á✈❡❧✳ ❖✉✱ ❡♠ t❡r♠♦s ❞❛ ✈❛r✐❛çã♦ ❞❛ ❢❛s❡✱ ∆ϕ✱ ❞❛ ❛❧❛✈❛♥❝❛ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✿

∆ϕ= Q

k ∂Fz

∂z ✭✸✳✷✮

❡♠ q✉❡ ◗ é ♦ ❢❛t♦r ❞❡ q✉❛❧✐❞❛❞❡ ❞❛ ❛❧❛✈❛♥❝❛✳

◆❛ ❛✉sê♥❝✐❛ ❞❡ ❝❛r❣❛s ♣❡r♠❛♥❡♥t❡s ♥❛ s✉♣❡r❢í❝✐❡ ❞❛ ❛♠♦str❛ ❡ tr❛t❛♥❞♦ ♦ s✐st❡♠❛ s♦♥❞❛✲❛♠♦str❛ ❝♦♠♦ ✉♠ ❝❛♣❛❝✐t♦r✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✿

F = 1

2V 2∂C ∂z ⇒ ∂F ∂z = 1 2V

2∂2C ∂z2,

❝♦♠ ❈ s❡♥❞♦ ❛ ❝❛♣❛❝✐tâ♥❝✐❛ ❞♦ s✐st❡♠❛ s♦♥❞❛✲❛♠♦str❛ ❡ ❱ ❛ t❡♥sã♦ ❛♣❧✐❝❛❞❛✳ ❙❡ ❤♦✉✈❡r ❝❛r❣❛s ♣❡r♠❛♥❡♥t❡s ♥❛ s✉♣❡r❢í❝✐❡✱ ❛ ❢♦rç❛ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦✿

F = qqs

4πǫ0z2 −

1 2V

2∂C ∂z,

❡♠ q✉❡ qs = −q +CV é ❛ ❝❛r❣❛ ♥❛ s♦♥❞❛✱ ✐♥❞✉③✐❞❛ ♣♦r q✱ s✉♣♦♥❞♦ ✉♠❛

❛♣r♦①✐♠❛çã♦ ♣❧❛♥♦✲❡s❢❡r❛ ❡ s❡♥❞♦ ❛ s♦♥❞❛ ❝♦♥❞✉t♦r❛✳ ▲❡✈❛♥❞♦ ❛ ❡①♣r❡ssã♦ ♣❛r❛ qs ♥❛ ❢♦rç❛✱ t❡♠♦s✿

F = q(−q+CV) 4πǫ0z2 −

1 2V

❈❆P❮❚❯▲❖ ✸ ✷✷

❆ss✐♠✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✿

F = q

2

4πǫ0z2 + qCV

4πǫ0z2 −

1 2V 2∂C ∂z ∂F ∂z = q2

4πǫ0z3

+ qV 4πǫ0

∂C/∂z

z2 −

2C z3

−1₂V2∂ 2_C

∂z2

❆ss✐♠✱ s✉❜st✐t✉✐♥❞♦ ❡♠ ∆ω✱ t❡♠♦s✿

∆ω=₋ω0 2k

q2

4πǫ0z3 + qV

4πǫ0

∂C/∂z

z2 −

2C z3

− 1₂V2∂ 2_C

∂z2

✭✸✳✸✮

P♦❞❡♠♦s ♣❡r❝❡❜❡r q✉❡ ❡①✐st❡ ✉♠❛ r❡❧❛çã♦ ❞✐r❡t❛ ❡♥tr❡ ♦ s✐♥❛❧ ♠❡❞✐❞♦ ❛ ♣❛rt✐r ❞❛ ✐♠❛❣❡♠ ❞❡ ❊❋▼ ✭∆ω✮✱ ❛ ❝❛r❣❛ ❡①✐st❡♥t❡ ♥❛ s✉♣❡r✜❝✐❡ ❡✴♦✉ ❛

❝❛♣❛❝✐tâ♥❝✐❛ ❞♦ s✐st❡♠❛ s♦♥❞❛ ❛♠♦str❛✳

✸✳✶✳✹ ▼✐❝r♦s❝♦♣✐❛ ❞❡ ❱❛rr❡❞✉r❛ ❞❡ ❈❛♣❛❝✐tâ♥❝✐❛

❈♦♥s✐❞❡r❡ q✉❡ ✉♠❛ t❡♥sã♦ ❝♦♥st❛♥t❡V0 ❡ ❛ t❡♥sã♦ ✈❛r✐á✈❡❧V′ _=V1sin(ωt)

sã♦ ❛♣❧✐❝❛❞❛s ❡♥tr❡ ❛ s♦♥❞❛ ❡ ❛ ❛♠♦str❛✳ ❙❡ ✉♠❛ ♣❡q✉❡♥❛ ❝❛♠❛❞❛ ♥♦ s✉❜s✲ tr❛t♦ ❝♦♥❞✉t♦r é ❞❡ ✉♠ ♠❛t❡r✐❛❧ s❡♠✐❝♦♥❞✉t♦r ♦✉ ✐s♦❧❛♥t❡✱ ❛ ❝❛r❣❛ ♥❛ s✉♣❡r✲ ❢í❝✐❡ ♣r♦❞✉③ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♣♦t❡♥❝✐❛❧ φ(x, y) ♥❛ s✉♣❡r❢í❝✐❡ ❞❛ ❛♠♦str❛✳ ❆ t❡♥sã♦ ❡♥tr❡ ❛ s♦♥❞❛ ❡ ❛ ❛♠♦str❛ é✿

V =V0+V1sin(ωt)₋φ(x, y)

❡ ❛ ❝❛♣❛❝✐tâ♥❝✐❛ ❞♦ s✐st❡♠❛ s♦♥❞❛✲❛♠♦str❛✱ s♦❜ ❡ss❛ t❡♥sã♦ ❱✱ ❛r♠❛③❡♥❛ ❛ ❡♥❡r❣✐❛✿

U = CV

2