❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ ❘■❖ ●❘❆◆❉❊ ❉❖ ◆❖❘❚❊ ❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❆ ❚❊❘❘❆
❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ❋❮❙■❈❆ ❚❊Ó❘■❈❆ ❊ ❊❳P❊❘■▼❊◆❚❆▲ P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ❋❮❙■❈❆
Pr♦♣r✐❡❞❛❞❡s ❊stát✐❝❛s ❡ ❉✐♥â♠✐❝❛s ❞❡
▼✉❧t✐❝❛♠❛❞❛s ▼❛❣♥ét✐❝❛s ❆❝♦♣❧❛❞❛s
◗✉❛s✐♣❡r✐♦❞✐❝❛♠❡♥t❡
▲❡♦♥❛r❞♦ ❉❛♥t❛s ▼❛❝❤❛❞♦
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❈❧❛✉❞✐♦♥♦r ●♦♠❡s ❇❡③❡rr❛
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋í✲ s✐❝❛ ❚❡ór✐❝❛ ❡ ❊①♣❡r✐♠❡♥t❛❧ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡✲ r❛❧ ❞♦ ❘✐♦ ●r❛♥❞❡ ❞♦ ◆♦rt❡ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ à ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❊❙❚❘❊ ❡♠ ❋❮❙■❈❆✳
❆❣r❛❞❡❝✐♠❡♥t♦s
Pr✐♠❡✐r❛♠❡♥t❡✱ ❣♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ❛♦ Pr♦❢✳ ❈❧❛✉❞✐♦♥♦r ●♦♠❡s ❇❡③❡rr❛✱ ♣♦r s✉❛ ♦r✐❡♥t❛çã♦ t❛♥t♦ ❞✉r❛♥t❡ ❛ ❣r❛❞✉❛çã♦ ❝♦♠♦ ❞✉r❛♥t❡ ♦ ♠❡s✲ tr❛❞♦✳ P♦r t✉❞♦ q✉❡ ❛♣r❡♥❞✐ ❞✉r❛♥t❡ ♦s ♠❛✐s ❞❡ ❝✐♥❝♦ ❛♥♦s ❡♠ q✉❡ ❢✉✐ s❡✉ ♦r✐❡♥t❛♥❞♦✳
❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❡ ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❉❋❚❊✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❛♦s ♣r♦❢❡ss♦r❡s ❏❛♥✐❧♦ ❙❛♥t♦s✱ ❘✉✐ ❚❡rt✉❧✐❛♥♦ ❡ P❛✉❧♦ ❋✉❧❝♦ ♣♦r s✉❛ ❛❥✉❞❛ ❞✉✲ r❛♥t❡ ❛ ❣r❛❞✉❛çã♦ ❡ ❛♦s ♣r♦❢❡ss♦r❡s ❊✉❞❡♥✐❧s♦♥ ▲✐♥s ❞❡ ❆❧❜✉q✉❡rq✉❡✱ ❆♥❛♥✐❛s ▼♦♥t❡✐r♦ ▼❛r✐③ ❡ ❈❛r❧♦s ❈❤❡s♠❛♥ ❞❡ ❆r❛ú❥♦ ❋❡✐t♦s❛ ♣♦r s✉❛ ❛❥✉❞❛ ❞✉r❛♥t❡ ♦ ♠❡str❛❞♦✳
❆♦ ❣r✉♣♦ ❞❡ ▼❛tér✐❛ ❈♦♥❞❡♥s❛❞❛ ♣❡❧♦ ✉s♦ ❞♦ ❈❧✉st❡r ❡ ❛♦ ❉r✳ ❏♦ã♦ ▼❡❞❡✐r♦s ❞❡ ❆r❛ú❥♦ ♣♦r s✉❛ ❛❥✉❞❛ ♥❛ ✐♠♣❧❛♥t❛çã♦ ❞❡ ❛❧❣♦r✐t♠♦s ✉s❛❞♦s ♥❡st❛ ❞✐ss❡rt❛çã♦✳
➚ ♠✐♥❤❛ ♠ã❡ ❚❡r❡s❛ ❡ ❛♦ ♠❡✉ ♣❛✐ ❙ér❣✐♦✱ q✉❡ s❡♠♣r❡ ♠❡ ❛♣♦✐❛r❛♠✳ ❆♦s ♠❡✉s ♣r✐♠♦s✱ t✐♦s ❡ ❛ t♦❞❛ ♠✐♥❤❛ ❢❛♠í❧✐❛✳ ●♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r t❛♠❜é♠ ❛ ❙✉❡❧✐✱ q✉❡ ♣♦r ♠♦r❛r ❝♦♥♦s❝♦ ❤á t❛♥t♦ t❡♠♣♦ t❛♠❜é♠ é ♣❛rt❡ ❞❛ ❢❛♠í❧✐❛✳
❆♦s ❛♥t✐❣♦s ❡ ❛t✉❛✐s ❝♦❧❡❣❛s ❞❛ s❛❧❛ ▼ár✐♦ ❙❝❤❡♥❜❡r❣✱ ♦♥❞❡ ♣❛ss❡✐
❣r❛♥❞❡ ♣❛rt❡ ❞❡st❡s ❞♦✐s ú❧t✐♠♦s ❛♥♦s✳
❆ t♦❞♦s ♠❡✉s ❛♠✐❣♦s✱ ❡♠ ♣❛rt✐❝✉❧❛r ❛ ❘❛❢❛❡❧✱ ❘✐❝❛r❞♦✱ P❛✉❧♦✱ ■r❡✲ ♥❛❧❞♦✱ ❋á❜✐♦ ❡ ❋r❛♥❝✐s❝♦ ❊❞✉❛r❞♦✳
❆♦ ❈◆Pq ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦ ✇❡ st✉❞②✱ ❢♦r t✇♦ ❞✐✛❡r❡♥t ❣r♦✇t❤ ❞✐r❡❝t✐♦♥s✱ ♠✉❧t✐❧❛②❡rs ♦❢ ♥❛♥♦♠❡tr✐❝ ♠❛❣♥❡t✐❝ ♠❡t❛❧❧✐❝ ✜❧♠s ❣r♦✇♥✱ ✉s✐♥❣ ❋✐❜♦♥❛❝❝✐ s❡q✉❡♥❝❡s✱ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ t❤✐❝❦♥❡ss ♦❢ t❤❡ ♥♦♥✲♠❛❣♥❡t✐❝ s♣❛❝❡r ♠❛② ✈❛r② ❢r♦♠ ❛ ♣❛✐r ♦❢ ✜❧♠s t♦ ❛♥♦t❤❡r✳ ❲❡ ❛♣♣❧✐❡❞ ❛ ♣❤❡♥♦♠❡♥♦❧♦❣✐❝❛❧ t❤❡♦r② t❤❛t ✉s❡s t❤❡ ♠❛❣♥❡t✐❝ ❡♥❡r❣② t♦ ❞❡s❝r✐❜❡ t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ s②st❡♠✳ ❆❢t❡r ✇❡ ❢♦✉♥❞ ♥✉♠❡r✐❝❛❧❧② t❤❡ ❣❧♦❜❛❧ ♠✐♥✐♠✉♠ ♦❢ t❤❡ t♦t❛❧ ❡♥❡r❣②✱ ✇❡ ✉s❡❞ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ❛♥❣❧❡s t♦ ♦❜t❛✐♥ ♠❛❣♥❡t✐③❛t✐♦♥ ❛♥❞ ♠❛❣♥❡t♦r❡s✐st❛♥❝❡ ❝✉r✈❡s✳ ◆❡①t✱ ✇❡ s♦❧✲ ✈❡❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ♠♦t✐♦♥ ♦❢ t❤❡ ♠✉❧t✐❧❛②❡rs t♦ ✜♥❞ t❤❡ ❞✐s♣❡rs✐♦♥ r❡❧❛t✐♦♥ ❢♦r t❤❡ s②st❡♠✳ ❚❤❡ r❡s✉❧ts s❤♦✇ t❤❛t✱ ✇❤❡♥ s♣❛❝❡rs ❛r❡ ✉s❡❞ ✇✐t❤ t❤✐❝❦✲ ♥❡ss s♦ t❤❛t t❤❡ ❜✐q✉❛❞r❛t✐❝ ❝♦✉♣❧✐♥❣ ✐s str♦♥❣ ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ ❜✐❧✐♥❡❛r ♦♥❡✱ ♥♦♥ ✉s✉❛❧ ❜❡❤❛✈✐♦rs ❢♦r ❜♦t❤ ♠❛❣♥❡t✐③❛t✐♦♥ ❛♥❞ ♠❛❣♥❡t♦r❡s✐st❛♥❝❡ ❛r❡ ♦❜s❡r✈❡❞✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ ❞❡♣❡♥❞❡♥❝❡ ♦♥ t❤❡ ♣❛r✐t② ♦❢ t❤❡ ❋✐❜♦♥❛❝❝✐ ❣❡♥❡r❛✲ t✐♦♥ ✉t✐❧✐③❡❞ ❢♦r ❝♦♥str✉❝t✐♥❣ t❤❡ s②st❡♠✱ ❛ ❧♦✇ ♠❛❣♥❡t♦r❡s✐st❛♥❝❡ st❡♣ ✐♥ ❧♦✇ ❡①t❡r♥❛❧ ♠❛❣♥❡t✐❝ ✜❡❧❞s ❛♥❞ r❡❣✐♦♥s t❤❛t s❤♦✇ ❤✐❣❤ s❡♥s✐❜✐❧✐t② t♦ s♠❛❧❧ ✈❛r✐✲ ❛t✐♦♥s ♦❢ t❤❡ ❛♣♣❧✐❡❞ ✜❡❧❞✳ ❚❤♦s❡ ❜❡❤❛✈✐♦rs ❛r❡ ♥♦t ♣r❡s❡♥t ✐♥ q✉❛s✐♣❡r✐♦❞✐❝ ♠❛❣♥❡t✐❝ ♠✉❧t✐❧❛②❡rs ✇✐t❤ ❝♦♥st❛♥t s♣❛❝❡r t❤✐❝❦♥❡ss✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦ ❡st✉❞❛♠♦s✱ ♣❛r❛ ❞✉❛s ❞✐r❡çõ❡s ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞✐st✐♥✲ t❛s✱ ♠✉❧t✐❝❛♠❛❞❛s ❞❡ ✜❧♠❡s ♥❛♥♦♠étr✐❝♦s ♠❡tá❧✐❝♦s ♠❛❣♥ét✐❝♦s ❝r❡s❝✐❞❛s✱ ✉s❛♥❞♦ s❡q✉ê♥❝✐❛s ❞❡ ❋✐❜♦♥❛❝❝✐✱ ❞❡ ♠♦❞♦ t❛❧ q✉❡ ❛ ❡s♣❡ss✉r❛ ❞♦s ❡s♣❛ç❛❞♦✲ r❡s ♥ã♦✲♠❛❣♥ét✐❝♦s ♣♦❞❡ ✈❛r✐❛r ❞❡ ✉♠ ♣❛r ❞❡ ✜❧♠❡s ♣❛r❛ ♦✉tr♦✳ ❯t✐❧✐③❛♠♦s ✉♠❛ t❡♦r✐❛ ❢❡♥♦♠❡♥♦❧ó❣✐❝❛ q✉❡ ✉s❛ ❛ ❡♥❡r❣✐❛ ♠❛❣♥ét✐❝❛ ♣❛r❛ ❞❡s❝r❡✈❡r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ s✐st❡♠❛✳ ❆♣ós ♠✐♥✐♠✐③❛r♠♦s ♥✉♠❡r✐❝❛♠❡♥t❡ ❛ ❡♥❡r❣✐❛ t♦t❛❧✱ ✉t✐❧✐③❛♠♦s ♦s â♥❣✉❧♦s ❞❡ ❡q✉✐❧í❜r✐♦ ♣❛r❛ ♦❜t❡r ❝✉r✈❛s ❞❡ ♠❛❣♥❡t✐③❛çã♦ ❡ ❞❡ ♠❛❣♥❡t♦r❡s✐stê♥❝✐❛✳ ❊♠ s❡❣✉✐❞❛✱ r❡s♦❧✈❡♠♦s ❛ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ ❞❛ ♠✉❧t✐❝❛♠❛❞❛ ♣❛r❛ ❡♥❝♦♥tr❛r♠♦s r❡❧❛çõ❡s ❞❡ ❞✐s♣❡rsã♦ ♣❛r❛ ♦ s✐st❡♠❛✳ ❖s r❡s✉❧t❛❞♦s ♠♦str❛♠ q✉❡✱ q✉❛♥❞♦ sã♦ ✉s❛❞♦s ❡s♣❛ç❛❞♦r❡s ❝♦♠ ❡s♣❡ss✉r❛ t❛❧ q✉❡ ♦ ❛❝♦♣❧❛♠❡♥t♦ ❜✐q✉❛❞rát✐❝♦ é ❢♦rt❡ ❡♠ ❝♦♠♣❛r❛çã♦ ❝♦♠ ♦ ❛❝♦♣❧❛♠❡♥t♦ ❜✐❧✐♥❡❛r✱ ♦❝♦rr❡♠ ❝♦♠♣♦rt❛♠❡♥t♦s ❞✐st✐♥t♦s ❞♦s ♦❜s❡r✈❛❞♦s ❡♠ ♠✉❧t✐❝❛♠❛❞❛s ♠❛❣♥ét✐❝❛s q✉❛s✐♣❡r✐ó❞✐❝❛s ❝♦♠ ❡s♣❛ç❛❞♦r❡s ❞❡ ❡s♣❡ss✉r❛ ❝♦♥st❛♥t❡✳ ❉❡♥✲ tr❡ ❡st❡s✱ ♣♦❞❡♠♦s ❝✐t❛r ✉♠❛ ❞❡♣❡♥❞ê♥❝✐❛ ❝♦♠ ❛ ♣❛r✐❞❛❞❡ ❞❛ ❣❡r❛çã♦ ❞❡ ❋✐❜♦♥❛❝❝✐ ✉t✐❧✐③❛❞❛✱ ✉♠ ♣❛t❛♠❛r ❞❡ ❜❛✐①❛ ♠❛❣♥❡t♦r❡s✐stê♥❝✐❛ ♣❛r❛ ❝❛♠♣♦s ❡①t❡r♥♦s ❜❛✐①♦s ❡ r❡❣✐õ❡s q✉❡ ❛♣r❡s❡♥t❛♠ ❛❧t❛ s❡♥s✐❜✐❧✐❞❛❞❡ ❛ ✈❛r✐❛çõ❡s ♣❡✲ q✉❡♥❛s ❞♦ ❝❛♠♣♦ ❛♣❧✐❝❛❞♦✳
❙✉♠ár✐♦
❆❣r❛❞❡❝✐♠❡♥t♦s ✐✐
❆❜str❛❝t ✐✈
❘❡s✉♠♦ ✈
■♥tr♦❞✉çã♦ ✶
✶ ❚❡♦r✐❛ ❢❡♥♦♠❡♥♦❧ó❣✐❝❛ ♣❛r❛ ✜❧♠❡s ♥❛♥♦♠étr✐❝♦s ✺ ✶✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ❊♥❡r❣✐❛ ♠❛❣♥ét✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷✳✶ ❊♥❡r❣✐❛ ❩❡❡♠❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷✳✷ ❆♥✐s♦tr♦♣✐❛s ❞❡ ❢♦r♠❛ ❡ s✉♣❡r❢í❝✐❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✷✳✸ ❉✐r❡çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✷✳✹ ❆♥✐s♦tr♦♣✐❛ ❝ú❜✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✷✳✺ ❆❝♦♣❧❛♠❡♥t♦ ❡♥tr❡ ❝❛♠❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✶✳✷✳✻ ❊♥❡r❣✐❛ ♠❛❣♥ét✐❝❛ t♦t❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✸ Pr♦♣r✐❡❞❛❞❡s ❡stát✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✳✹ Pr♦♣r✐❡❞❛❞❡s ❞✐♥â♠✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✷ ❚é❝♥✐❝❛s ❡①♣❡r✐♠❡♥t❛✐s ✹✺
✷✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷✳✷ ❊❢❡✐t♦ ❑❡rr ▼❛❣♥❡t♦✲Ó♣t✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✷✳✸ ▼❛❣♥❡t♦r❡s✐stê♥❝✐❛ ❉❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✷✳✹ ❘❡ss♦♥â♥❝✐❛ ❢❡rr♦♠❛❣♥ét✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷
✸ ❙❡q✉ê♥❝✐❛s q✉❛s✐♣❡r✐ó❞✐❝❛s ✺✽
✸✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✸✳✷ ❙❡q✉ê♥❝✐❛s s✉❜st✐t✉❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✸✳✸ ❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✸✳✹ ❖✉tr❛s s❡q✉ê♥❝✐❛s q✉❛s✐♣❡r✐ó❞✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
✹ ❋✐❧♠❡s ♥❛♥♦♠étr✐❝♦s ❛❝♦♣❧❛❞♦s q✉❛s✐♣❡r✐♦❞✐❝❛♠❡♥t❡ ✼✵ ✹✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✹✳✷ ❈✉r✈❛s ❞❡ ♠❛❣♥❡t✐③❛çã♦ ❡ ♠❛❣♥❡t♦r❡s✐stê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✹✳✷✳✶ ❉✐r❡çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❬✵✶✵❪ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✹✳✷✳✷ ❉✐r❡çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❬✶✶✵❪ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✻ ✹✳✸ ❘❡❧❛çõ❡s ❞❡ ❞✐s♣❡rsã♦ ♣❛r❛ ♦♥❞❛s ❞❡ s♣✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✻
✹✳✸✳✶ ❉✐r❡çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❬✵✶✵❪ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✾ ✹✳✸✳✷ ❉✐r❡çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❬✶✶✵❪ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✸
✺ ❈♦♥❝❧✉sõ❡s ❡ ♣❡rs♣❡❝t✐✈❛s ✶✸✵
❆ ▼ét♦❞♦s ♥✉♠ér✐❝♦s ✶✸✸
❆✳✶ ▼ét♦❞♦ ❞♦ ❣r❛❞✐❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✸ ❆✳✷ ▼ét♦❞♦ ❞❛ ❜✐ss❡❝çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✼
❇ ❈♦❡✜❝✐❡♥t❡s ❞❛ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ♣❛r❛ ❛ t❡r❝❡✐r❛ ❣❡r❛çã♦
❞❡ ❋✐❜♦♥❛❝❝✐ ✶✹✵
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✶✹✷
■♥tr♦❞✉çã♦
❖ tr❛❜❛❧❤♦ q✉❡ ❢♦r♥❡❝❡✉ ♦ ✐♠♣✉❧s♦ ✐♥✐❝✐❛❧ ♣❛r❛ ♦ ❡st✉❞♦ ❞❡ ♠✉❧t✐❝❛✲ ♠❛❞❛s ❞❡ ✜❧♠❡s ♥❛♥♦♠étr✐❝♦s ♠❡tá❧✐❝♦s ♠❛❣♥ét✐❝♦s ❢♦✐ ♦ ❛rt✐❣♦ ❞❡ ●rü♥❜❡r❣ ❡ ❝♦❧❛❜♦r❛❞♦r❡s ❬✶❪ q✉❡ ❡♠ ✶✾✽✻ ♠♦str♦✉✱ ❡♠ ✉♠❛ ❛♠♦str❛ ❝♦♠♣♦st❛ ♣♦r ❝❛✲ ♠❛❞❛s ❞❡ ❢❡rr♦ ❛❧t❡r♥❛❞❛s ♣♦r ❝❛♠❛❞❛s ❞❡ ❝r♦♠♦ ❞❡ ♣❡q✉❡♥❛ ❡s♣❡ss✉r❛ ✭❞❛ ♦r❞❡♠ ❞❡ ❛❧❣✉♥s ♥❛♥ô♠❡tr♦s✮✱ q✉❡ ❡♠❜♦r❛ ❛s ♠❛❣♥❡t✐③❛çõ❡s ♠❛❝r♦s❝ó♣✐❝❛s ❞❡ ✉♠ ✜❧♠❡ ❞❡ ❢❡rr♦ s❡ ❡♥❝♦♥tr❛ss❡♠ ❛❧✐♥❤❛❞❛s ❡♠ ❝❛❞❛ ❝❛♠❛❞❛✱ ❞❡ ✉♠❛ ❝❛♠❛❞❛ ♣❛r❛ ❛ s❡❣✉✐♥t❡ ❤❛✈✐❛ ✉♠❛ ✐♥✈❡rsã♦ ❞❡ s❡♥t✐❞♦✳ ❊st❡ ❛❝♦♣❧❛♠❡♥t♦✱ ❤♦❥❡ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❜✐❧✐♥❡❛r✱ ❞❡s♣❡rt♦✉ ❛t❡♥çã♦ ♣❛r❛ s✐st❡♠❛s ❞❡st❡ t✐♣♦✳
❊♠ ✶✾✽✽ ✉♠❛ ❞❡s❝♦❜❡rt❛ ❢❡✐t❛ ♣♦r ❇❛✐❜✐❝❤ ❡ ❝♦❧❛❜♦r❛❞♦r❡s ❬✷❪ ❛❞✐❝✐♦✲ ♥♦✉ ✉♠ ✐♥t❡r❡ss❡ t❡❝♥♦❧ó❣✐❝♦ ❛♦ ❡st✉❞♦ ❞❡ ♠✉❧t✐❝❛♠❛❞❛s ♠❛❣♥ét✐❝❛s✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ❢♦✐ ♦❜s❡r✈❛❞♦ q✉❡ ♥❡st❡s s✐st❡♠❛s ♦❝♦rr✐❛ ✉♠❛ ❣r❛♥❞❡ ✈❛r✐❛çã♦ ❞❛ r❡s✐stê♥❝✐❛ ❡❧étr✐❝❛ ❡♠ ❢✉♥çã♦ ❞♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦✱ s❡♥❞♦ ❡st❡ ❡❢❡✐t♦ ❝♦♥❤❡✲ ❝✐❞♦ ❝♦♠♦ ♠❛❣♥❡t♦r❡s✐stê♥❝✐❛ ❣✐❣❛♥t❡ ✭●▼❘✮✳ ❆t✉❛❧♠❡♥t❡✱ ♠✉❧t✐❝❛♠❛❞❛s ♠❛❣♥ét✐❝❛s ♣♦ss✉❡♠ ❛♣❧✐❝❛çõ❡s ❡♠ s❡♥s♦r❡s ❡ ♥❛ ár❡❛ ❞❡ ❣r❛✈❛çã♦ ♠❛❣♥ét✐❝❛ ❡ ❛r♠❛③❡♥❛♠❡♥t♦ ❞❡ ✐♥❢♦r♠❛çã♦ ✭s❡♥❞♦ ❡♥❝♦♥tr❛❞❛s✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ ❞✐s✲ ❝♦s rí❣✐❞♦s✮✳ P♦r s✉❛s ❝♦♥tr✐❜✉✐çõ❡s P❡t❡r ●rü♥❜❡r❣ ❡ ❆❧❜❡rt ❋❡rt ✭✉♠ ❞♦s ❛✉t♦r❡s ❞♦ ❛rt✐❣♦ ❬✷❪✮ r❡❝❡❜❡r❛♠✱ ❡♠ ✷✵✵✼✱ ♦ Prê♠✐♦ ◆♦❜❡❧ ❡♠ ❋ís✐❝❛✳
❖✉tr❛s ❞❡s❝♦❜❡rt❛s ✐♠♣♦rt❛♥t❡s ✐♥❝❧✉❡♠ ♦ tr❛❜❛❧❤♦ ❞❡ P❛r❦✐♥ ❡ ❝♦❧❛✲ ❜♦r❛❞♦r❡s ❬✸❪ q✉❡✱ ❡♠ ✶✾✾✵ ❛tr❛✈és ❞❡ ♠❡❞✐❞❛s ❞❡ ♠❛❣♥❡t♦r❡s✐stê♥❝✐❛✱ ♠♦s✲ tr❛r❛♠ q✉❡ ♦ ❛❝♦♣❧❛♠❡♥t♦ ❜✐❧✐♥❡❛r ❡♥tr❡ ✜❧♠❡s ❢❡rr♦♠❛❣♥ét✐❝♦s ♦s❝✐❧❛✈❛ ❡♥tr❡ ❢❡rr♦♠❛❣♥ét✐❝♦ ❡ ❛♥t✐❢❡rr♦♠❛❣♥ét✐❝♦ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❡s♣❡ss✉r❛ ❞♦ ❡s♣❛ç❛✲ ❞♦r ♥ã♦✲♠❛❣♥ét✐❝♦❀ ❡ ♦ tr❛❜❛❧❤♦ ❞❡ ❘ü❤r✐❣ ❡ ❝♦❧❛❜♦r❛❞♦r❡s ❬✹❪ q✉❡ ❡♠ ✶✾✾✶ r❡❧❛t❛r❛♠✱ ❛♦ ❡st✉❞❛r ✜❧♠❡s ❞❡ ❋❡✴❈r✱ q✉❡ ♣❛r❛ ❝❡rt❛s ❡s♣❡ss✉r❛s ❞♦ ❡s♣❛✲ ç❛❞♦r ♥ã♦✲♠❛❣♥ét✐❝♦ ♦❝♦rr✐❛ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ❞❡90◦ ❡♥tr❡ ❛s ♠❛❣♥❡t✐③❛çõ❡s✱
❛❧✐♥❤❛♠❡♥t♦ ❡st❡ ♣♦st❡r✐♦r♠❡♥t❡ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❜✐q✉❛❞rát✐❝♦✳
◆❛ ♠❡s♠❛ ❞é❝❛❞❛✱ ❛ ❞❡s❝♦❜❡rt❛ ❢❡✐t❛ ❡♠ ✶✾✽✹ ♣♦r ❙❤❡❝❤t♠❛♥ ❡ ❝♦❧❛✲ ❜♦r❛❞♦r❡s ❬✺❪✱ ❞❡ q✉❡ ❡♠ ❧✐❣❛s ❞❡ ❛❧✉♠í♥✐♦ ❡ ♠❛♥❣❛♥ês ✭♣r♦❞✉③✐❞❛s ❛tr❛✈és ❞♦ ♠ét♦❞♦ ❞❡ ✧♠❡❧t s♣✐♥♥✐♥❣✧✮ ♦❝♦rr✐❛ ✉♠❛ ❢❛s❡ q✉❛s✐❝r✐st❛❧✐♥❛✱ ❞❡✉ ♦r✐❣❡♠ ❛ ✉♠❛ ♥♦✈❛ ár❡❛ ❞❡ ♣❡sq✉✐s❛ ♥❛ ❢ís✐❝❛✱ ❛ ❞♦s s✐st❡♠❛s q✉❛s✐❝r✐st❛❧✐♥♦s✳ ❯♠ ❛♥♦ ❞❡♣♦✐s✱ ▼❡r❧✐♥ ❡ ❝♦❧❛❜♦r❛❞♦r❡s ❬✻❪✱ s❡❣✉✐♥❞♦ ❛s ✐❞❡✐❛s ❞❡ ▲❡✈✐♥❡ ❡ ❙t❡✐♥❤❛r❞t ❬✼❪✱ ❞❡s❡♥✈♦❧✈❡r❛♠ ❛ ♣r✐♠❡✐r❛ s✉♣❡r✲r❡❞❡ s❡❣✉✐♥❞♦ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✳ ◆❡st❛ ❞✐ss❡rt❛çã♦ ❡st✉❞❛r❡♠♦s ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✉♠❛ ♠✉❧t✐❝❛♠❛❞❛ ❞❡ ✜❧♠❡s ♥❛♥♦♠étr✐❝♦s ♠❛❣♥ét✐❝♦s ❝r❡s❝✐❞♦s✱ ✉t✐❧✐③❛♥❞♦ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐✲ ❜♦♥❛❝❝✐✱ ❞❡ ✉♠❛ ❢♦r♠❛ t❛❧ q✉❡ ❛ ❡s♣❡ss✉r❛ ❞♦ ❡s♣❛ç❛❞♦r ♣♦❞❡ ✈❛r✐❛r ❞❡ ✉♠ ✜❧♠❡ ♥ã♦✲♠❛❣♥ét✐❝♦ ♣❛r❛ ♦ ♦✉tr♦✳ ❖❜s❡r✈❛r❡♠♦s ❡♥tã♦ ❛ ✐♥✢✉ê♥❝✐❛ ❞❡st❛ ♠♦❞✐✜❝❛çã♦ ❡♠ s✉❛s ❝✉r✈❛s ❞❡ ♠❛❣♥❡t✐③❛çã♦✱ ♠❛❣♥❡t♦r❡s✐stê♥❝✐❛ ❡ ❡♠ s✉❛s r❡❧❛çõ❡s ❞❡ ❞✐s♣❡rsã♦✱ ♣❛r❛ ❞✉❛s ❞✐r❡çõ❡s ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞✐❢❡r❡♥t❡s✳ ❊st❡ tr❛✲ ❜❛❧❤♦ s❡ ❡♥❝♦♥tr❛ ❞✐✈✐❞✐❞♦ ❡♠ ❝✐♥❝♦ ❝❛♣ít✉❧♦s ❡ ❞♦✐s ❛♣ê♥❞✐❝❡s ❝♦♠♦ ❞❡s❝r✐t♦ ❛❜❛✐①♦✳
◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✱ ✈❡r❡♠♦s ❛ t❡♦r✐❛ ❢❡♥♦♠❡♥♦❧ó❣✐❝❛ ✉t✐❧✐③❛❞❛ ♣❛r❛ ❞❡s❝r❡✈❡r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ♠✉❧t✐❝❛♠❛❞❛s ❞❡ ✜❧♠❡s ♥❛♥♦♠étr✐❝♦s✳ ■♥✐❝✐✲
❛❧♠❡♥t❡✱ ❞❡s❝r❡✈❡r❡♠♦s ♦s t❡r♠♦s q✉❡ ❝♦♠♣õ❡♠ ❛ ❡♥❡r❣✐❛ ♠❛❣♥ét✐❝❛ ❡✱ ❡♠ s❡❣✉✐❞❛✱ ✉t✐❧✐③❛r❡♠♦s ❡st❛ ❡♥❡r❣✐❛ ♣❛r❛ ❞❡s❝r❡✈❡r ❛s ♣r♦♣r✐❡❞❛❞❡s ❡stát✐❝❛s ❞❡ ✉♠❛ ♠✉❧t✐❝❛♠❛❞❛ ❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞✐♥â♠✐❝❛s ♣❛r❛ ♦ ❝❛s♦ ❡s♣❡❝í✜❝♦ ❞❡ ✉♠❛ tr✐❝❛♠❛❞❛✳
◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ❡st✉❞❛r❡♠♦s ❛❧❣✉♠❛s té❝♥✐❝❛s ❡①♣❡r✐♠❡♥t❛✐s ♥♦r♠❛❧♠❡♥t❡ ✉s❛❞❛s ♥♦ ❡st✉❞♦ ❞❡ ♠✉❧t✐❝❛♠❛❞❛s✳ P❛r❛ ❝❛❞❛ t✐♣♦ ❞❡ ❝✉r✈❛ t❡ór✐❝❛ ♦❜t✐❞❛ ❞❡s❝r❡✈❡r❡♠♦s ✉♠ ♠ét♦❞♦ ❡①♣❡r✐♠❡♥t❛❧ q✉❡ ♣❡r♠✐t❡ ❝❤❡❝❛r ❛ ✈❛❧✐❞❛❞❡ ❞❡ ♥♦ss♦s r❡s✉❧t❛❞♦s✿ ♦ ❡❢❡✐t♦ ❑❡rr ♠❛❣♥❡t♦✲ó♣t✐❝♦ ♣❡r♠✐t❡ ♦❜t❡r ❝✉r✈❛s ❞❡ ♠❛❣♥❡t✐③❛çã♦✱ ❛ ♠❛❣♥❡t♦r❡s✐stê♥❝✐❛ ❉❈ ❝✉r✈❛s ❞❡ ♠❛❣♥❡t♦r❡s✐s✲ tê♥❝✐❛ ❡ ❛ r❡ss♦♥â♥❝✐❛ ❢❡rr♦♠❛❣♥ét✐❝❛✱ ❛ r❡❧❛çõ❡s ❞❡ ❞✐s♣❡rsã♦✳
◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ ❡st✉❞❛r❡♠♦s ❛❧❣✉♠❛s s❡q✉ê♥❝✐❛s q✉❛s✐♣❡r✐ó❞✐❝❛s✳ ■♥✐❝✐❛❧♠❡♥t❡ ❞❡✜♥✐r❡♠♦s ♦ q✉❡ é ✉♠❛ s❡q✉ê♥❝✐❛ s✉❜st✐t✉❝✐♦♥❛❧✱ ♣❛r❛ ❡♠ s❡✲ ❣✉✐❞❛ ❞❡s❝r❡✈❡r ❜r❡✈❡♠❡♥t❡ ❛❧❣✉♠❛s ❞❡st❛s s❡q✉ê♥❝✐❛s ❜❡♠ ❝♦♠♦ ❛❧❣✉♠❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s❀ tr❛t❛r❡♠♦s ❞❛s s❡q✉ê♥❝✐❛s ❞❡ ❋✐❜♦♥❛❝❝✐✱ ❞❡ ♣❡rí♦❞♦ ❞✉♣❧♦ ❡ ❞❡ ❚❤✉❡✲▼♦rs❡✳
◆♦ q✉❛rt♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s ♦s r❡s✉❧t❛❞♦s ❞❡st❛ ❞✐ss❡rt❛çã♦✳ Pr✐♠❡✐r❛♠❡♥t❡✱ ❞❡s❝r❡✈❡r❡♠♦s ♦ ♠ét♦❞♦ q✉❡ ✉t✐❧✐③❛♠♦s ♣❛r❛ ❝♦♥str✉✐r ♦ s✐s✲ t❡♠❛ ❛ s❡r ❡st✉❞❛❞♦✱ ♣❛r❛ ❞❡♣♦✐s ❞❡✜♥✐r ♦s ❝♦♥❥✉♥t♦s ❞❡ ♣❛râ♠❡tr♦s ♣♦r ♥ós ✉t✐❧✐③❛❞♦s✳ ❉✐s❝✉t✐r❡♠♦s ❡♥tã♦ ❛s ❝✉r✈❛s ❞❡ ♠❛❣♥❡t✐③❛çã♦✱ ♠❛❣♥❡t♦r❡s✐stê♥✲ ❝✐❛ ❡ ❞✐s♣❡rsã♦ ♦❜t✐❞❛s ♣❛r❛ ❛s ❞✐r❡çõ❡s ❞❡ ❝r❡s❝✐♠❡♥t♦ ❬✵✶✵❪ ❡ ❬✶✶✵❪✳
◆♦ q✉✐♥t♦ ❝❛♣ít✉❧♦ ❡①♣♦r❡♠♦s ❛s ❝♦♥❝❧✉sõ❡s ♦❜t✐❞❛s ❡ ❛s ♣❡rs♣❡❝t✐✈❛s ❞❡ ❡①t❡♥sã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
◆♦ ❛♣ê♥❞✐❝❡ ❆ ❞❡s❝r❡✈❡r❡♠♦s ♦s ♠ét♦❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❞♦ ❣r❛❞✐✲ ❡♥t❡ ❡ ❞❛ ❜✐ss❡❝çã♦✱ q✉❡ ❢♦r❛♠ ♣♦r ♥ós ✉t✐❧✐③❛❞♦s ♥❛ ♦❜t❡♥çã♦ ❞♦s r❡s✉❧t❛❞♦s
♥✉♠ér✐❝♦s✱ ❡♥q✉❛♥t♦ q✉❡ ♥♦ ❆♣ê♥❞✐❝❡ ❇ ❢♦r♥❡❝❡r❡♠♦s ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ r❡✲ ❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ♣❛r❛ ❛ t❡r❝❡✐r❛ ❣❡r❛çã♦ ❞❡ ❋✐❜♦♥❛❝❝✐✳
❈❛♣ít✉❧♦ ✶
❚❡♦r✐❛ ❢❡♥♦♠❡♥♦❧ó❣✐❝❛ ♣❛r❛ ✜❧♠❡s
♥❛♥♦♠étr✐❝♦s
✶✳✶ ■♥tr♦❞✉çã♦
❊st❡ ❝❛♣ít✉❧♦ ❞❡s❝r❡✈❡rá ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✜❧♠❡s ♥❛♥♦♠étr✐❝♦s ♠❡✲ tá❧✐❝♦s ❢❡rr♦♠❛❣♥ét✐❝♦s✱ ❛tr❛✈és ❞❡ ✉♠ ♠♦❞❡❧♦ ❢❡♥♦♠❡♥♦❧ó❣✐❝♦ q✉❡ ✉t✐❧✐③❛ ❛ ❡♥❡r❣✐❛ ♠❛❣♥ét✐❝❛✳ ❈♦♥s✐❞❡r❛♠♦s q✉❡ ❡st❡s ✜❧♠❡s ❡stã♦ ♠❛❣♥❡t✐③❛❞♦s ✉♥✐❢♦r✲ ♠❡♠❡♥t❡ ✭sã♦ ♠♦♥♦❞♦♠í♥✐♦s✮ ❡✱ ❞✉r❛♥t❡ ♦ ❡st✉❞♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❡stát✐❝❛s✱ q✉❡ ❡stã♦ ❡♠ ❡q✉✐❧í❜r✐♦ ✭♥ã♦ ❛♣r❡s❡♥t❛♠ ❡①❝✐t❛çõ❡s ❞✐♥â♠✐❝❛s✮✳ ❯♠ ❛rr❛♥❥♦ ❜❛st❛♥t❡ ❝♦♠✉♠ ❞❡st❡s ✜❧♠❡s é ❛ ❝❤❛♠❛❞❛ tr✐❝❛♠❛❞❛✱ q✉❡ s❡ ❡♥❝♦♥tr❛ ✐❧✉s✲ tr❛❞❛ ♥❛ ✜❣✉r❛ ✶✳✶✱ ❡♠ q✉❡ ❞♦✐s ✜❧♠❡s ❢❡rr♦♠❛❣♥ét✐❝♦s s❡ ❡♥❝♦♥tr❛♠ s❡♣❛✲ r❛❞♦s ♣♦r ✉♠ ♠❛t❡r✐❛❧ ♥ã♦✲♠❛❣♥ét✐❝♦ ❞❡♥♦♠✐♥❛❞♦ ❡s♣❛ç❛❞♦r✳ ❊st❡ ❛rr❛♥❥♦ é ✐♥t❡r❡ss❛♥t❡ ♣♦r ❛♣r❡s❡♥t❛r✱ ♣❛r❛ ❝❡rt❛s ❡s♣❡ss✉r❛s ❞♦ ❡s♣❛ç❛❞♦r✱ ♠❛❣♥❡✲ t♦r❡s✐stê♥❝✐❛ ❣✐❣❛♥t❡✳ P♦❞❡✲s❡ t❛♠❜é♠ ❝r❡s❝❡r ✈ár✐♦s ✜❧♠❡s ❢❡rr♦♠❛❣♥ét✐❝♦s
s❡♣❛r❛❞♦s ♣♦r ❡s♣❛ç❛❞♦r❡s ❡♠ s❡q✉ê♥❝✐❛✱ ❢♦r♠❛♥❞♦ ✉♠❛ ♠✉❧t✐❝❛♠❛❞❛✳
❋✐❣✉r❛ ✶✳✶✿ ❊sq✉❡♠❛ ❞❡ ✉♠❛ tr✐❝❛♠❛❞❛ ♠❛❣♥ét✐❝❛✳ ❆ é ✉♠ ✜❧♠❡ ♥❛♥♦♠é✲ tr✐❝♦ ♠❡tá❧✐❝♦ ❢❡rr♦♠❛❣♥ét✐❝♦ ✭♣♦r ❡①❡♠♣❧♦ ❢❡rr♦✮ ❡ ❇ ✉♠ ✜❧♠❡ ♥❛♥♦♠étr✐❝♦ ♠❡tá❧✐❝♦ ♥ã♦✲♠❛❣♥ét✐❝♦ ✭♣♦r ❡①❡♠♣❧♦ ❝r♦♠♦✮✳
❙❡❣✉✐♥❞♦ ♦ ♣r♦❝❡❞✐♠❡♥t♦ ❡♥❝♦♥tr❛❞♦ ❡♠ ❬✽❪✱ ✐♥✐❝✐❛❧♠❡♥t❡ ❞❡s❝r❡✈❡r❡✲ ♠♦s ♦s t❡r♠♦s q✉❡ ❝♦♠♣õ❡♠ ❛ ❡♥❡r❣✐❛ ♠❛❣♥ét✐❝❛✱ ❛ q✉❛❧ ♣♦❞❡ s❡r ♠✐♥✐♠✐③❛❞❛ ♣❛r❛ ❡♥❝♦♥tr❛r♠♦s ♦s â♥❣✉❧♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❞❛s ♠❛❣♥❡t✐③❛çõ❡s✳ ❊♠ s❡❣✉✐❞❛ ✉t✐❧✐③❛r❡♠♦s ❡st❡s â♥❣✉❧♦s ♣❛r❛ ❞❡s❝r❡✈❡r ♣r♦♣r✐❡❞❛❞❡s ❡stát✐❝❛s✳ ❊st❛♠♦s
✐♥t❡r❡ss❛❞♦s ♥❡st❛ ❞✐ss❡rt❛çã♦ ♥❛ ❝♦♠♣♦♥❡♥t❡ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ♣❛r❛❧❡❧❛ ❛♦ ❝❛♠♣♦ ❡①t❡r♥♦ ❡ ♥❛ ♠❛❣♥❡t♦r❡s✐stê♥❝✐❛ ❝✐t❛❞❛ ❛❝✐♠❛✳ ◆❛ ♣❛rt❡ ✜♥❛❧ ❞❡st❡ ❝❛♣ít✉❧♦✱ ❡st✉❞❛r❡♠♦s ❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ♦♥❞❛s ❡❧❡tr♦♠❛❣♥ét✐❝❛s ❡ tr✐❝❛♠❛❞❛s✳ ◆❡st❡ ❝❛s♦ ❛ ♠❛❣♥❡t✐③❛çã♦ s❛✐ ❞♦ ♣❧❛♥♦ ❞❡ s❡✉ r❡s♣❡❝t✐✈♦ ✜❧♠❡✱ ❡ ♣r❡❝❡ss✐♦♥❛ ❡♠ t♦r♥♦ ❞❡ s✉❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦✳
✶✳✷ ❊♥❡r❣✐❛ ♠❛❣♥ét✐❝❛
✶✳✷✳✶ ❊♥❡r❣✐❛ ❩❡❡♠❛♥
❈♦♠ ❛s ❝♦♥s✐❞❡r❛çõ❡s ❢❡✐t❛s ♥❛ ✐♥tr♦❞✉çã♦ ✭♦s ✜❧♠❡s sã♦ ♠♦♥♦❞♦♠í✲ ♥✐♦s s❡♠ ❡①❝✐t❛çõ❡s ❞✐♥â♠✐❝❛s✮✱ ♣♦❞❡♠♦s r❡♣r❡s❡♥t❛r ❛ ♠❛❣♥❡t✐③❛çã♦ ♣♦r ✉♠ ✈❡t♦r✳ ❆ ❡♥❡r❣✐❛ ❩❡❡♠❛♥ s✉r❣❡ q✉❛♥❞♦ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ é ❛♣❧✐❝❛❞♦ ♥♦ s✐st❡♠❛✱ ❞❡✈✐❞♦ à ✐♥t❡r❛çã♦ ❡♥tr❡ ♦ ✈❡t♦r ♠❛❣♥❡t✐③❛çã♦ ❡ ♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❡①t❡r♥♦✳ ❆ ❡♥❡r❣✐❛ ❩❡❡♠❛♥ t❡♥❞❡ ❛ ❛❧✐♥❤❛r ❛ ♠❛❣♥❡t✐③❛çã♦ ❝♦♠ ♦ ❝❛♠♣♦ ❡①t❡r♥♦✱ ❡ ♣♦❞❡♠♦s ❞❡s❝r❡✈❡r ❡st❛ ✐♥t❡r❛çã♦ ❛tr❛✈és ❞❡ ✉♠ ♣r♦❞✉t♦ ❡s❝❛❧❛r✳ P❛r❛ ♦ ❝❛s♦ ❣❡r❛❧ ❡♠ q✉❡ t❡♠♦s ✈ár✐♦s ✜❧♠❡s✱ ❝❛❞❛ ✉♠ ❝♦♠ ❡s♣❡ss✉r❛ di✱ ❛
❡♥❡r❣✐❛ ♠❛❣♥ét✐❝❛ ♣♦r ✉♥✐❞❛❞❡ ❞❡ ár❡❛ é ❞❛❞❛ ♣♦r
EZ =−
X
i
diM~i·H~0, ✭✶✳✶✮
♦♥❞❡ M~i é ❛ ♠❛❣♥❡t✐③❛çã♦ ❡ H~0 é ♦ ❝❛♠♣♦ ❡①t❡r♥♦✳ ▼✉✐t❛s ✈❡③❡s é ♠❛✐s
❝♦♥✈❡♥✐❡♥t❡ tr❛❜❛❧❤❛r ❡♠ ❝♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s✳ ◆❡st❡ ❝❛s♦✱ ♣❛r❛ ✉♠ ✜❧♠❡✱
~
Mi ❡ H~0 sã♦ ❞❛❞♦s ♣♦r
~
Mi =MS( s❡♥ θicosφi xˆ+ s❡♥ θi s❡♥ φi yˆ+ cosθi zˆ) ✭✶✳✷✮
❡
~
H0 =H0( s❡♥ θHcosφH xˆ+ s❡♥ θH s❡♥ φH yˆ+ cosθH zˆ), ✭✶✳✸✮
♦♥❞❡MS é ❛ ♠❛❣♥❡t✐③❛çã♦ ❞❡ s❛t✉r❛çã♦ ❞❡ ✉♠ ✜❧♠❡✳ ❯s❛♥❞♦ ❛s ❡q✉❛çõ❡s ✶✳✷
❡ ✶✳✸ ♦❜t❡♠♦s ❛ ❡①♣r❡ssã♦ ♣❛r❛ ❛ ❡♥❡r❣✐❛ ❩❡❡♠❛♥ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s✱
EZ =−
X
i
diMSH0[ s❡♥ θi s❡♥ θHcos(φi−φH) + cosθicosθH]. ✭✶✳✹✮
❊♥tr❡t❛♥t♦✱ q✉❛♥❞♦ ♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ é ❛♣❧✐❝❛❞♦ ♥♦ ♣❧❛♥♦ ❡♠ q✉❡ ♦ ✜❧♠❡
❋✐❣✉r❛ ✶✳✷✿ ❱❡t♦r❡s ♠❛❣♥❡t✐③❛çã♦ ❡ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❡①t❡r♥♦✳ ❊stã♦ ✐❧✉s✲ tr❛❞♦s ♦s â♥❣✉❧♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ❞❡s❝r❡✈ê✲❧♦s ❡♠ ❝♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s✳
s❡ ❡♥❝♦♥tr❛✱ ❞❡✈✐❞♦ ❛ ❛♥✐s♦tr♦♣✐❛ ♠❛❣♥ét✐❝❛ ❞❡ ❢♦r♠❛✱ q✉❡ s❡rá tr❛t❛❞❛ ♠❛✐s ❛❞✐❛♥t❡✱ ❛ ♠❛❣♥❡t✐③❛çã♦ ♣❡r♠❛♥❡❝❡ ♥❡st❡ ♣❧❛♥♦✳ ◆❡st❛ ❞✐ss❡rt❛çã♦ ❝♦♥s✐❞❡✲ r❛♠♦s q✉❡ ♦ ♣❧❛♥♦ ❡♠ q✉❡ ♦s ✜❧♠❡s s❡ ❡♥❝♦♥tr❛♠ é ♦ ♣❧❛♥♦ ①✲③✱ ❧♦❣♦ q✉❛♥❞♦ ♦ ❝❛♠♣♦ ❡①t❡r♥♦ é ❛♣❧✐❝❛❞♦ ♥❡st❡ ♣❧❛♥♦ t❡♠♦s φH = 0 ❡✱ ♣❡❧♦ ♠♦t✐✈♦ ❝✐t❛❞♦
❛❝✐♠❛✱ φi = 0✳ ◆❡st❡s ❝❛s♦s✱ ♣♦rt❛♥t♦✱ ❛ ❡①♣r❡ssã♦ ✶✳✹ s❡ r❡❞✉③ ❛
EZ =−
X
i
diMSH0cos(θi−θH). ✭✶✳✺✮
✶✳✷✳✷ ❆♥✐s♦tr♦♣✐❛s ❞❡ ❢♦r♠❛ ❡ s✉♣❡r❢í❝✐❡
❆♥✐s♦tr♦♣✐❛s ♠❛❣♥ét✐❝❛s sã♦ ❛ss♦❝✐❛❞❛s ❛ ❝♦♥✜❣✉r❛çõ❡s ❞♦s át♦♠♦s ♥♦s ❝r✐st❛✐s✱ ❡ ♣♦❞❡♠ s❡r ✐♥trí♥s❡❝❛s ✭r❡s✉❧t❛♥t❡s ❞❡ ✐♥t❡r❛çõ❡s q✉â♥t✐❝❛s ♦✉ ❡❧étr✐❝❛s✮ ♦✉ ❡①trí♥s❡❝❛s ✭❝✉❥❛ ♦r✐❣❡♠ ♣♦❞❡ s❡r ♦ ♠♦❞♦ ❝♦♠♦ é ❢❡✐t♦ ♦ ❛rr❛♥❥♦ ❝r✐st❛❧✐♥♦ ♦✉ ♦ ♠ét♦❞♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞❡st❡ ❝r✐st❛❧✮✳ ❙❡✉ ♣r✐♥❝✐♣❛❧ ❡❢❡✐t♦ é ♣r✐✈✐❧❡❣✐❛r ❛ ♠❛❣♥❡t✐③❛çã♦ ❡♠ ❝❡rt❛s ❞✐r❡çõ❡s ✭♦✉ ♣❧❛♥♦s✮✱ ♦✉ s❡❥❛✱ ❡♠ ❝❡rt❛s ❞✐r❡çõ❡s ✭♦✉ ♣❧❛♥♦s✮ ❛ ♠❛❣♥❡t✐③❛çã♦ ❡stá ♥✉♠ ❡st❛❞♦ ❞❡ ♠❡♥♦r ❡♥❡r❣✐❛ ✭❡✐①♦ ♦✉ ♣❧❛♥♦ ❢á❝✐❧✮ ❡♥q✉❛♥t♦ ❡♠ ♦✉tr❛s ♥✉♠ ❡st❛❞♦ ❞❡ ♠❛✐♦r ❡♥❡r❣✐❛ ✭❡✐①♦ ♦✉ ♣❧❛♥♦ ❞✉r♦✮✳
❈♦♥❢♦r♠❡ ❝✐t❛❞♦ ♥❛ s❡çã♦ ✶✳✷✳✶✱ ❛ ❛♥✐s♦tr♦♣✐❛ ❞❡ ❢♦r♠❛ ✭♦✉ ❞❡ ❞❡s✲ ♠❛❣♥❡t✐③❛çã♦✮ ❢❛③ ❝♦♠ q✉❡ ❛ ♠❛❣♥❡t✐③❛çã♦ ♣❡r♠❛♥❡ç❛ ♥♦ ♣❧❛♥♦ ❞♦ ✜❧♠❡ q✉❛♥❞♦ ♦ ❝❛♠♣♦ ❡①t❡r♥♦ é ❛♣❧✐❝❛❞♦ ♥❡st❡✳ ❊st❛ ❛♥✐s♦tr♦♣✐❛ s✉r❣❡ q✉❛♥❞♦ ✉♠ ♠❛t❡r✐❛❧ é ♠❛❣♥❡t✐③❛❞♦✱ ❞❛♥❞♦ ♦r✐❣❡♠ ❛ ❞✐♣♦❧♦s ♥ã♦ ❝♦♠♣❡♥s❛❞♦s✱ q✉❡ ❢❛③❡♠ ❝♦♠ q✉❡ ♦ ❝❛♠♣♦ ♥♦ ✐♥t❡r✐♦r ❞♦ ♠❛t❡r✐❛❧ s❡❥❛ ❞✐❢❡r❡♥t❡ ❞♦ ❝❛♠♣♦ ❡①✲ t❡r♥♦ ✭H~0✮✳ ❖✉tr♦ ❡❢❡✐t♦ ❝♦♥❤❡❝✐❞♦ ❡♠ ✜❧♠❡s ♥❛♥♦♠étr✐❝♦s é ❛ ❞✐♠✐♥✉✐çã♦
s♦tr♦♣✐❛ ❞❡ s✉♣❡r❢í❝✐❡ é ✉s❛❞❛ ♣❛r❛ ❡①♣❧✐❝❛r ❢❡♥♦♠❡♥♦❧♦❣✐❝❛♠❡♥t❡ ❡st❡ ❢❛t♦✱ ❡ s❡✉ ❡❢❡✐t♦ é t♦r♥❛r ❛ ♠❛❣♥❡t✐③❛çã♦ ✜♥❛❧ ❞♦ ✜❧♠❡ ♠❡♥♦r ❞♦ q✉❡ ❛ ♠❛❣✲ ♥❡t✐③❛çã♦ ❞❡ s❛t✉r❛çã♦ ❞❡ ✉♠❛ ❛♠♦str❛ ❣r♦ss❛✳ ❯♠❛ ❡①♣❧✐❝❛çã♦ ❞❡t❛❧❤❛❞❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ♥❛ r❡❢❡rê♥❝✐❛ ❬✾❪ ❛ r❡s♣❡✐t♦ ❞❛ ❛♥✐s♦tr♦♣✐❛ ❞❡ ❢♦r♠❛ ❡ ♥❛ r❡❢❡rê♥❝✐❛ ❬✶✵❪ ❛ r❡s♣❡✐t♦ ❞❛ ❛♥✐s♦tr♦♣✐❛ ❞❡ s✉♣❡r❢í❝✐❡✳ ▼❛t❡♠❛t✐❝❛♠❡♥t❡✱ ❛ ❛♥✐s♦tr♦♣✐❛ ❞❡ ❢♦r♠❛ é ❞❛❞❛✱ ♣❛r❛ ✉♠❛ ♠✉❧t✐❝❛♠❛❞❛✱ ♣♦r
ED = 2π
X
i
di
~ Mi·yˆ
2
, ✭✶✳✻✮
♦♥❞❡ yˆé ❛ ❞✐r❡çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞♦ ✜❧♠❡ ♠❛❣♥ét✐❝♦✳ P♦r s✉❛ ✈❡③✱ ❛ ❛♥✐s♦✲
tr♦♣✐❛ ❞❡ s✉♣❡r❢í❝✐❡ é ❞❛❞❛ ♣♦r
ES =−
X i kas M2 i ~ Mi·yˆ
2
. ✭✶✳✼✮
◆♦t❡ q✉❡ s❡ kas ❢♦r ♣♦s✐t✐✈♦ ❡st❡ t❡r♠♦ é ♠✐♥✐♠✐③❛❞♦ q✉❛♥❞♦ ❛ ♠❛❣♥❡t✐③❛çã♦
s❡ ❡♥❝♦♥tr❛ ♣❛r❛❧❡❧❛ ❛ ❞✐r❡çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦✳ ◆❡st❛ ❞✐ss❡rt❛çã♦ kas > 0✱
♠❛s ❝♦♥s✐❞❡r❛♠♦s ❡st❡ t❡r♠♦ ♠✉✐t♦ ♠❡♥♦r q✉❡ ♦ ❞❡ ❢♦r♠❛✱ ❞❡ ♠♦❞♦ q✉❡ ❛ ♠❛❣♥❡t✐③❛çã♦ ♣❡r♠❛♥❡❝❡ ♥♦ ♣❧❛♥♦ ♥♦ tr❛t❛♠❡♥t♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❡stát✐❝❛s✳ ◆❡st❡ ❝❛s♦✱ ♥ã♦ é ♥❡❝❡ssár✐♦ ✉t✐❧✐③❛r ❡①♣❧✐❝✐t❛♠❡♥t❡ ❡st❡s t❡r♠♦s✱ ✉♠❛ ✈❡③ q✉❡ ❞❡s❞❡ q✉❡ ❛ ♠❛❣♥❡t✐③❛çã♦ ♣❡r♠❛♥❡ç❛ ♥♦ ♣❧❛♥♦ ❞♦ ✜❧♠❡ M~i·yˆ= 0✳ ❆♣❡♥❛s
♥♦ tr❛t❛♠❡♥t♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞✐♥â♠✐❝❛s✱ ❡♠ q✉❡ ❛ ♠❛❣♥❡t✐③❛çã♦ ✐♥t❡r❛❣❡ ❝♦♠ ♦♥❞❛s ❡❧❡tr♦♠❛❣♥ét✐❝❛s ❡ s❛✐ ❞♦ ♣❧❛♥♦ ❞♦ ✜❧♠❡✱ é ♥❡❝❡ssár✐♦ ✐♥❝❧✉✐r ❡st❡s t❡r♠♦s ♥❛ ❡①♣r❡ssã♦ ❞❛ ❡♥❡r❣✐❛ t♦t❛❧✳
✶✳✷✳✸ ❉✐r❡çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦
❈♦♠♦ s❡rá ♠♦str❛❞♦ ❛❞✐❛♥t❡✱ ♦❜t✐✈❡♠♦s r❡s✉❧t❛❞♦s ♣❛r❛ ❞✉❛s ❞✐r❡çõ❡s ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞✐❢❡r❡♥t❡s✱ ❬✵✶✵❪ ❡ ❬✶✶✵❪✱ s❡♥❞♦ ♣♦rt❛♥t♦ ❝♦♥✈❡♥✐❡♥t❡ tr❛t❛r ❡st❡
t❡♠❛ ❝♦♠ ❛❧❣✉♠ ❞❡t❛❧❤❡✳ ❆ ❞✐r❡çã♦ ❬✵✶✵❪ é ❛ ❞✐r❡çã♦ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ♣❧❛♥♦ ❝r✐st❛❧✐♥♦ ✭✵✶✵✮✱ ❞❛ ♠❡s♠❛ ♠❛♥❡✐r❛ q✉❡ ❛ ❞✐r❡çã♦ ❬✶✶✵❪ é ❛ ❞✐r❡çã♦ ♣❡r♣❡♥❞✐✲ ❝✉❧❛r ❛♦ ♣❧❛♥♦ ❝r✐st❛❧✐♥♦ ✭✶✶✵✮✳ ◆❛ r❡❢❡rê♥❝✐❛ ❬✶✶❪ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ✉♠❛ ❞❡s❝r✐çã♦ ❞❡st❡s ♣❧❛♥♦s✳ P❛r❛ ❝r❡s❝❡r ✉♠ ✜❧♠❡✱ át♦♠♦s ❞❡ ✉♠ ♠❛t❡r✐❛❧ sã♦ ❧❛♥ç❛❞♦s ❡♠ ❞✐r❡çã♦ ❛♦ s✉❜str❛t♦✱ q✉❡ t❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ♦r❣❛♥✐③❛r ❡st❡s át♦♠♦s ❞❡ ♠❛♥❡✐r❛ ✐❣✉❛❧ ❛ ✉♠❛ ❞❛❞❛ ❞✐r❡çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦✱ ❛ss♦❝✐❛❞❛ ❛ ✉♠ ❞❛❞♦ ♣❧❛♥♦ ❝r✐st❛❧✐♥♦✱ ❞♦ ♠❛t❡r✐❛❧ ❛ s❡r ❝r❡s❝✐❞♦✳ ❆ ❞✐r❡çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ é✱ ♣♦rt❛♥t♦✱ ❞❡✜♥✐❞❛ ♣❡❧♦ s✉❜str❛t♦✱ q✉❡ ♣♦❞❡ s❡r ❛❞q✉✐r✐❞♦ ❝♦♠❡r❝✐❛❧♠❡♥t❡✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞✐r❡çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞❡s❡❥❛❞❛✳
❋✐❣✉r❛ ✶✳✸✿ ❊sq✉❡♠❛ ✐❧✉str❛♥❞♦ ♦ ❝r❡s❝✐♠❡♥t♦ ❞❡ ✉♠ ✜❧♠❡ ❞❡ ✉♠ ♠❛t❡r✐❛❧ ❝♦♠ ❡str✉t✉r❛ ❝ú❜✐❝❛ s✐♠♣❧❡s ♥❛ ❞✐r❡çã♦ ❬✵✶✵❪✳ P❧❛♥♦s ❞❡ át♦♠♦s ♣❡r♣❡♥❞✐✲ ❝✉❧❛r❡s ❛ ❡st❛ ❞✐r❡çã♦ ❞❡✈❡♠ s❡r ❝♦rt❛❞♦s ❡ ❡♠♣✐❧❤❛❞♦s ♣❛r❛ ❢♦r♠❛r ♦ ✜❧♠❡❀ ❛♣❡♥❛s ✉♠ ❞❡st❡s ♣❧❛♥♦s s❡ ❡♥❝♦♥tr❛ ✐❧✉str❛❞♦ ♥❛ ✜❣✉r❛✳
❉❡ ✉♠ ♣♦♥t♦ ❞❡ ✈✐st❛ t❡ór✐❝♦✱ ♣♦❞❡♠♦s ✈✐s✉❛❧✐③❛r ♦ ❝r❡s❝✐♠❡♥t♦ ❞❡ ✉♠ ✜❧♠❡ ❝♦♠♦ ✉♠ ♣r♦❝❡ss♦ ❡♠ q✉❡ ♣❧❛♥♦s ❞❡ át♦♠♦s✱ ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❛ ❞✐r❡çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞❡✜♥✐❞❛✱ sã♦ ❝♦rt❛❞♦s ❞❛ ❡str✉t✉r❛ ❝r✐st❛❧✐♥❛ ❡ ❡♠♣✐❧❤❛❞♦s✳
❋✐❣✉r❛ ✶✳✹✿ ❊sq✉❡♠❛ ✐❧✉str❛♥❞♦ ♦ ❝r❡s❝✐♠❡♥t♦ ❞❡ ✉♠ ✜❧♠❡ ❞❡ ✉♠ ♠❛t❡r✐❛❧ ❝♦♠ ❡str✉t✉r❛ ❝ú❜✐❝❛ s✐♠♣❧❡s ♥❛ ❞✐r❡çã♦ ❬✶✶✵❪✳
❋✐❣✉r❛ ✶✳✺✿ ❊sq✉❡♠❛ ✐❧✉str❛♥❞♦ ♦ ❝r❡s❝✐♠❡♥t♦ ❞❡ ✉♠ ✜❧♠❡ ❞❡ ✉♠ ♠❛t❡r✐❛❧ ❝♦♠ ❡str✉t✉r❛ ❝ú❜✐❝❛ s✐♠♣❧❡s ♥❛ ❞✐r❡çã♦ ❬✶✶✶❪✳
❆s ✜❣✉r❛s ✶✳✸✱ ✶✳✹ ❡ ✶✳✺ ✐❧✉str❛♠ ❡st❛ ✐❞❡✐❛ ♣❛r❛ três ❞✐r❡çõ❡s ❞❡ ❝r❡s❝✐♠❡♥t♦✿ ❬✵✶✵❪✱ ❬✶✶✵❪ ❡ ❬✶✶✶❪✳ ❆s ❧❡tr❛s ♥❛s ✜❣✉r❛s tê♠ ❛♣❡♥❛s ❝❛rát❡r ✐❧✉str❛t✐✈♦✳
✶✳✷✳✹ ❆♥✐s♦tr♦♣✐❛ ❝ú❜✐❝❛
❆♥✐s♦tr♦♣✐❛s ♠❛❣♥❡t♦✲❝r✐st❛❧✐♥❛s s✉r❣❡♠ ❞❛ ✐♥t❡r❛çã♦ ❞❛ ❡str✉t✉r❛ ❝r✐st❛❧✐♥❛ ❝♦♠ ♦s s♣✐♥s ❡❧❡trô♥✐❝♦s✱ s❡♥❞♦ ❛s ❛♥✐s♦tr♦♣✐❛s ❝ú❜✐❝❛ ❡ ✉♥✐❛①✐❛❧ s✉❛s ❢♦r♠❛s ♠❛✐s ❝♦♠✉♥s✳ ◆❛ ♦❜t❡♥çã♦ ❞♦s r❡s✉❧t❛❞♦s ❞❡st❡ tr❛❜❛❧❤♦✱ ❛♣❡♥❛s ❛ ❛♥✐s♦tr♦♣✐❛ ❝ú❜✐❝❛ ❢♦✐ ❝♦♥s✐❞❡r❛❞❛✳ ❉❡st❡ ♠♦❞♦✱ ♥ã♦ tr❛t❛r❡♠♦s ❞❛ ❛♥✐s♦✲ tr♦♣✐❛ ✉♥✐❛①✐❛❧✱ ♣♦❞❡♥❞♦ s❡r ❡♥❝♦♥tr❛❞♦ ♥❛ r❡❢❡rê♥❝✐❛ ❬✶✷❪ ✉♠ tr❛t❛♠❡♥t♦ ❛♣r♦♣r✐❛❞♦✳
❈♦♥❢♦r♠❡ s❡rá ✈✐st♦ ♥❛ s❡çã♦ ✶✳✸✱ ♣❛r❛ s❡ ♦❜t❡r ❝✉r✈❛s ❞❡ ♠❛❣♥❡t✐③❛✲ çã♦ ✉♠ ❝❛♠♣♦ ❡①t❡r♥♦ é ❛♣❧✐❝❛❞♦ ❡♠ ❝❡rt❛ ❞✐r❡çã♦ ❡ ♦ ♠ó❞✉❧♦ ❞❡st❡ ❝❛♠♣♦ é ✈❛r✐❛❞♦✱ ♠❡❞✐♥❞♦✲s❡ ❛ ❝♦♠♣♦♥❡♥t❡ ♣❛r❛❧❡❧❛ ❛♦ ❝❛♠♣♦ ❡①t❡r♥♦ ❞❛ ♠❛❣♥❡t✐③❛✲ çã♦✳ ❯♠❛ ♠❛♥❡✐r❛ ❞❡ s❡ ✐❞❡♥t✐✜❝❛r ❛ ♣r❡s❡♥ç❛ ❞❡ ❛♥✐s♦tr♦♣✐❛ ❝ú❜✐❝❛ ❡♠ ✉♠❛ ❞❛❞❛ ❛♠♦str❛ é ♦❜s❡r✈❛♥❞♦ q✉❡ ❡♠ ❝❛s♦s ❡♠ q✉❡ ❡st❛ s❡ ❡♥❝♦♥tr❛ ♣r❡s❡♥t❡✱ q✉❛♥❞♦ ♦ ❝❛♠♣♦ ❡①t❡r♥♦ é ❛♣❧✐❝❛❞♦ ❡♠ ❡✐①♦s ❝r✐st❛❧✐♥♦s ❡q✉✐✈❛❧❡♥t❡s✱ ❝♦♠♦ ❬✶✵✵❪ ❡ ❬✵✶✵❪ ♦✉ ❬✶✶✵❪ ❡ ❬✵✶✶❪✱ ❛s ❝✉r✈❛s ❞❡ ♠❛❣♥❡t✐③❛çã♦ ❞❡✈❡♠ s❡r s❡♠❡✲ ❧❤❛♥t❡s✳ ❉♦ ♣♦♥t♦ ❞❡ ✈✐st❛ t❡ór✐❝♦✱ ❞❡s❡❥❛♠♦s ♦❜t❡r ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ ❛ ❛♥✐s♦tr♦♣✐❛ ❝ú❜✐❝❛ ❡♠ t❡r♠♦s ❞♦s ❝♦ss❡♥♦s ❞✐r❡t♦r❡s ❞❛ ♠❛❣♥❡t✐③❛çã♦✳ P❛r❛ q✉❡ ❡st❛ ❡①♣r❡ssã♦ ♣♦ss✉❛ s✐♠❡tr✐❛ ❝ú❜✐❝❛ é ♥❡❝❡ssár✐♦ q✉❡✿
✶✳ ❙❡❥❛ ✐♥✈❛r✐❛♥t❡ q✉❛♥❞♦ ❞❛ ✐♥✈❡rsã♦ ❞♦ s❡♥t✐❞♦ ❞❛ ♠❛❣♥❡t✐③❛çã♦❀ ✷✳ ❙❡❥❛ ✐♥✈❛r✐❛♥t❡ q✉❛♥❞♦ ❞❛ tr♦❝❛ ❞❡ ❞♦✐s ❡✐①♦s ❡q✉✐✈❛❧❡♥t❡s✳
❉❡ ❛❝♦r❞♦ ❝♦♠ ❛s r❡❢❡rê♥❝✐❛s ❬✶✷❪ ❡ ❬✶✸❪✱ ❛ ❝♦♥tr✐❜✉✐çã♦ ❞❛ ❛♥✐s♦tr♦♣✐❛ ❝ú❜✐❝❛ ♣♦❞❡ s❡r ❞❡s❝r✐t❛✱ ♣❛r❛ ✉♠❛ ♠✉❧t✐❝❛♠❛❞❛ ❝♦♠ ♥ ✜❧♠❡s ❢❡rr♦♠❛❣♥ét✐❝♦s✱ ❝❛❞❛
✉♠ ❝♦♠ ❡s♣❡ss✉r❛ di✱ ♣♦r
Eac = n
X
i=1
Kac i di
|Mi|4
M2
ixMiy2 +Mix2Miz2 +Miy2Miz2
, ✭✶✳✽✮
♦♥❞❡ Kac
i é ❛ ❝♦♥st❛♥t❡ ❞❡ ❛♥✐s♦tr♦♣✐❛ ❝ú❜✐❝❛ ❝r✐st❛❧✐♥❛✳ ◆♦t❡ q✉❡ ❡st❛ ❡①✲
♣r❡ssã♦ s❛t✐s❢❛③ ♦s ❞♦✐s r❡q✉✐s✐t♦s ❡♥✉♠❡r❛❞♦s ❛❝✐♠❛✳
❋✐❣✉r❛ ✶✳✻✿ ■❧✉str❛çã♦ ❞♦s ❝♦ss❡♥♦s ❞✐r❡t♦r❡s ❞❛ ♠❛❣♥❡t✐③❛çã♦✳
❱❛♠♦s ❛❣♦r❛ r❡❡s❝r❡✈❡r ❡st❛ ❡①♣r❡ssã♦ ❡♠ t❡r♠♦s ❞♦s ❝♦ss❡♥♦s ❞✐r❡✲ t♦r❡s ❞❛ ♠❛❣♥❡t✐③❛çã♦✳ ❙❡❥❛♠ βi ♦ â♥❣✉❧♦ ❡♥tr❡ M~i ❡ ♦ ❡✐①♦ ①✱ γi ♦ â♥❣✉❧♦
❡♥tr❡ M~i ❡ ♦ ❡✐①♦ ② ❡ δi ♦ â♥❣✉❧♦ ❡♥tr❡ M~i ❡ ♦ ❡✐①♦ ③✳ P♦rt❛♥t♦✱ Mix = Micos(βi) = Miα1i✱ Miy = Micos(γi) = Miα2i ❡ Miz =Micos(δi) = Miα3i✱
❝♦♥❢♦r♠❡ ♣♦❞❡ s❡r ✈✐st♦ ♥❛ ✜❣✉r❛ ✶✳✻✳ ❙✉❜st✐t✉✐♥❞♦ ♥❛ ❡q✉❛çã♦ ✶✳✽✱ ♦❜t❡♠♦s
Eac = n
X
i=1
Kiacdi α21iα
2 2i +α
2 1iα
2 3i+α
2 2iα
2 3i
. ✭✶✳✾✮
❆ ❡①♣r❡ssã♦ ♦❜t✐❞❛ ❛❝✐♠❛ ❡♥❝♦♥tr❛✲s❡ ❡♠ ❢✉♥çã♦ ❞♦s ❡✐①♦s ❞❡ ✉♠ ❝r✐st❛❧✳ ◆♦ ❝❛s♦ ❬✵✶✵❪ ♦s ❡✐①♦s ❞❡ ✉♠ ✜❧♠❡ sã♦ ♦s ♠❡s♠♦s ❞♦ ❝r✐st❛❧✱ ♠❛s ♥♦s ❝❛s♦s ❬✶✶✵❪ ❡ ❬✶✶✶❪ ♦ s✐st❡♠❛ ❞❡ ❡✐①♦s ♠✉❞❛ ❡ ❞❡✈❡♠♦s r❡❡s❝r❡✈❡r ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ ❡♠ t❡r♠♦s ❞♦s ♥♦✈♦s ❝♦ss❡♥♦s ❞✐r❡t♦r❡s✳ ❱❛♠♦s ♦❜t❡r ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ ❝❛❞❛ ✉♠ ❞❡st❡s ❝❛s♦s ❡♠ ❝♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s✱ ♣♦r ❝♦♥✈❡♥✐ê♥❝✐❛✱ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ ♦ s✐st❡♠❛ é ❝♦♠♣♦st♦ ♣♦r ✉♠ ú♥✐❝♦ ✜❧♠❡✳ P❛r❛ ❣❡♥❡r❛❧✐③❛r ❡st❡s r❡s✉❧t❛✲ ❞♦s✱ ❜❛st❛ ❛❞✐❝✐♦♥❛r ✉♠ s♦♠❛tór✐♦ s♦❜r❡ t♦❞♦s ♦s ✜❧♠❡s ❡ ❛❞✐❝✐♦♥❛r ♦s í♥❞✐❝❡s ❛♣r♦♣r✐❛❞♦s ♥❛ ❡s♣❡ss✉r❛ ❡ ♥♦s â♥❣✉❧♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛ ❝❛❞❛ ✜❧♠❡✳
❉✐r❡çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❬✵✶✵❪
❋✐❣✉r❛ ✶✳✼✿ ❋✐❣✉r❛ ✐❧✉str❛♥❞♦ ♦ ♣❧❛♥♦ ✭✵✶✵✮ ❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s ✉t✐❧✐③❛❞♦✳
✶✳✾ ♥ã♦ é ♠♦❞✐✜❝❛❞❛✳ ❯s❛♥❞♦ α1 = s❡♥ θcosφ✱ α2 = s❡♥ θ s❡♥ φ ❡ α3 =
cosθ✱ ♦❜t❡♠♦s
Eac =dkac s❡♥ 4θcos2φ s❡♥ 2φ+ s❡♥ 2θcos2θcos2φ+ s❡♥ 2θcos2θ s❡♥ 2φ
.
✭✶✳✶✵✮ ❈♦♠ ❛ r❡❧❛çã♦ tr✐❣♦♥♦♠étr✐❝❛ ❛♣r♦♣r✐❛❞❛✱ ❡ ❧❡♠❜r❛♥❞♦ q✉❡ ❛ ♠❛❣♥❡t✐③❛çã♦ ♥ã♦ s❛✐ ❞♦ ♣❧❛♥♦ ❞♦ ✜❧♠❡ ✭φ = 0✮✱ ♦❜t❡♠♦s ❛ ❡①♣r❡ssã♦ ✜♥❛❧✱
Eac =
dkac
4 s❡♥
2(2θ). ✭✶✳✶✶✮
❆ ✜❣✉r❛ ✶✳✽ ✐❧✉str❛ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛♥❣✉❧❛r ❞❡st❡ t❡r♠♦ ❞❛ ❡♥❡r❣✐❛ ♠❛❣♥é✲
❋✐❣✉r❛ ✶✳✽✿ ❈♦♠♣♦rt❛♠❡♥t♦ ❛♥❣✉❧❛r ❞❡st❡ t❡r♠♦ ❞❛ ❡♥❡r❣✐❛ ♠❛❣♥ét✐❝❛ ♣❛r❛ ❛ ❞✐r❡çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❬✵✶✵❪✱ ❝♦♠ kac = 1 ❡r❣✴❝♠3 ✳ ❖ â♥❣✉❧♦ ♣❧❛♥❛r é
♠❡❞✐❞♦ ❛ ♣❛rt✐r ❞♦ ❡✐①♦ ③❀ θ= 90◦ ❝♦rr❡s♣♦♥❞❡ ❛♦ ❡✐①♦ ①✳
❛♦s â♥❣✉❧♦s θ = 0◦✱ θ = 90◦✱ θ = 180◦ ❡ θ = 270◦✱ ❡♥q✉❛♥t♦ ♦s ❡✐①♦s ❞✉r♦s
❝♦rr❡s♣♦♥❞❡♠ ❛ θ= 45◦✱θ = 135◦✱ θ= 225◦ ❡ θ = 315◦✳
❉✐r❡çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❬✶✶✵❪
❋✐❣✉r❛ ✶✳✾✿ ❋✐❣✉r❛ ✐❧✉str❛♥❞♦ ♦ ♣❧❛♥♦ ✭✶✶✵✮ ❡ ♦ ♥♦✈♦ ❝♦♥❥✉♥t♦ ❞❡ ❡✐①♦s ❝♦♦r✲ ❞❡♥❛❞♦s ✉t✐❧✐③❛❞♦✳
❙❡❥❛ {αi} ♦ ❝♦♥❥✉♥t♦ ❞❡ ❝♦ss❡♥♦s ❞✐r❡t♦r❡s ❞❛ ♠❛❣♥❡t✐③❛çã♦ ❝♦♠ r❡✲
❧❛çã♦ ❛♦s ❡✐①♦s ❞❡ ✉♠ ❝r✐st❛❧ ❡ {α′
i} ♦ ❝♦♥❥✉♥t♦ ❞❡ ❝♦ss❡♥♦s ❞✐r❡t♦r❡s ❝♦♠
r❡❧❛çã♦ ❛♦s ❡✐①♦s ❞♦s ✜❧♠❡s✳ ❖ ♦❜❥❡t✐✈♦ ❞❡st❛ s❡çã♦ é r❡❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ ✶✳✾✱ q✉❡ s❡ ❡♥❝♦♥tr❛ ❡♠ ❢✉♥çã♦ ❞♦ ❝♦♥❥✉♥t♦ {αi}✱ ❡♠ t❡r♠♦s ❞♦ ❝♦♥❥✉♥t♦
{α′
i}❀ ♣❛r❛ ✐st♦ é ♣r❡❝✐s♦ ❡♥❝♦♥tr❛r ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ♦s ❞♦✐s ❝♦♥❥✉♥t♦s✳
P♦❞❡♠♦s ❡s❝r❡✈❡r Mˆ✱ q✉❡ é ♦ ✈❡t♦r ✉♥✐tár✐♦ ❝✉❥❛ ❞✐r❡çã♦ é ❛q✉❡❧❛ ❞❛
♠❛❣♥❡t✐③❛çã♦✱ ❝♦♠♦ Mˆ =α1xˆ+α2yˆ+α3zˆ♦✉ ❝♦♠♦Mˆ =α′ 1xˆ′+α
′ 2yˆ′+α
′ 3zˆ′✳
◆♦t❡ q✉❡ t❛♥t♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ ✈❡t♦r❡sxˆ✱ yˆ❡zˆ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ ✈❡t♦r❡sxˆ′✱ ˆ
y′ ❡zˆ′ sã♦ ❝♦♥❥✉♥t♦s ♦rt♦❣♦♥❛✐s✳ ❊s❝r❡✈❡♥❞♦Mˆ ❡♠ t❡r♠♦s ❞♦ ❝♦♥❥✉♥t♦{α
i}✱
✉t✐❧✐③❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦ss❡♥♦ ❞✐r❡t♦r ❡ ❛s r❡❧❛çõ❡s ❝♦♥t✐❞❛s ♥❛ ✜❣✉r❛ ✶✳✾ ❡♥❝♦♥tr❛♠♦s
α′1 = ˆx′·Mˆ = ˆz·(α
1xˆ+α2yˆ+α3zˆ) = α3,
α2′ = ˆy′ ·Mˆ = √1
2(α1+α2) ✭✶✳✶✷✮
❡
α′3 = ˆz′ ·Mˆ = √1
2(−α1+α2).
■♥✈❡rt❡♥❞♦ ❛s r❡❧❛çõ❡s ❛❝✐♠❛ ❡ ❡s❝r❡✈❡♥❞♦α1✱α2 ❡α3 ❡♠ t❡r♠♦s ❞❡α′1✱ α′2 ❡
α′
3 ✈❛♠♦s ♦❜t❡r
α1 =
1
√
2(α ′
2−α′3),
α2 =
1
√
2(α ′
2+α′3) ✭✶✳✶✸✮
❡
α3 =α′1.
❯s❛♥❞♦ ❛s ❡q✉❛çõ❡s ✶✳✶✸ ♥❛ ❡q✉❛çã♦ ✶✳✾ ❡♥❝♦♥tr❛♠♦s ❛ ❡①♣r❡ssã♦ ❞❛ ❛♥✐s♦tr♦✲ ♣✐❛ ❝ú❜✐❝❛ ♣❛r❛ ❛ ❞✐r❡çã♦ ❬✶✶✵❪ ❡♠ t❡r♠♦s ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❝♦ss❡♥♦s ❞✐r❡t♦r❡s
{α′
i}✱
Eac =dkac
α′ 2 4 4 + α′ 3 4
4 +α
′ 1 2 α′ 2 2
+α′
1 2
α′
3 2
− α′2
2 α′ 3 2 2 . ✭✶✳✶✹✮
❊s❝r❡✈❡♥❞♦ ♦s ❝♦ss❡♥♦s ❞✐r❡t♦r❡s ❡①♣❧✐❝✐t❛♠❡♥t❡ ❡ ✉t✐❧✐③❛♥❞♦ ♦ ❢❛t♦ ❞❡ q✉❡ ❛ ♠❛❣♥❡t✐③❛çã♦ ♥ã♦ s❛✐ ❞♦ ♣❧❛♥♦ ❞♦ ✜❧♠❡ ✭φ′ = 0✮ ❝❤❡❣❛♠♦s ❛♦ r❡s✉❧t❛❞♦ ✜♥❛❧✿
Eac =
dkac
4 cos
4θ′+ s❡♥ 22θ′
. ✭✶✳✶✺✮
❋✐❣✉r❛ ✶✳✶✵✿ ❈♦♠♣♦rt❛♠❡♥t♦ ❛♥❣✉❧❛r ❞❡st❡ t❡r♠♦ ❞❛ ❡♥❡r❣✐❛ ♠❛❣♥ét✐❝❛ ♣❛r❛ ❛ ❞✐r❡çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❬✶✶✵❪✱ ♣❛r❛ kac = 1 ❡r❣✴❝♠3✳
❆ ✜❣✉r❛ ✶✳✶✵ ✐❧✉str❛ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛♥❣✉❧❛r ❞❡st❡ t❡r♠♦ ❞❛ ❡♥❡r❣✐❛ ♠❛❣✲ ♥ét✐❝❛✱ ♣❛r❛ ❡st❛ ❞✐r❡çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦✳ ◆❡st❡ ❝❛s♦ ❡①✐st❡♠ ❡✐①♦s ❢á❝❡✐s ✭θ = 90◦ ❡ θ = 270◦✮✱ ❡✐①♦s ✐♥t❡r♠❡❞✐ár✐♦s ✭θ = 0◦ ❡ θ = 180◦✮ ❡ ❡✐①♦s ❞✉r♦s
✭❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ θ ∼= 35◦ ❡ θ ∼= 215◦✮✳
❉✐r❡çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❬✶✶✶❪
◆❡st❛ ❞✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛♠♦s ❛♣❡♥❛s r❡s✉❧t❛❞♦s ♣❛r❛ ❛s ❞✐r❡çõ❡s ❞❡ ❝r❡s❝✐♠❡♥t♦ ❬✵✶✵❪ ❡ ❬✶✶✵❪✳ P♦r ❝♦♠♣❧❡t❡③❛✱ ❡♥tr❡t❛♥t♦✱ ✈❛♠♦s ✐♥❝❧✉✐r✱ ❞❡ ♠❛✲ ♥❡✐r❛ ❜r❡✈❡✱ ♦ ♣r♦❝❡❞✐♠❡♥t♦ ♣❛r❛ s❡ ♦❜t❡r ❛ ❡①♣r❡ssã♦ ❞❛ ❛♥✐s♦tr♦♣✐❛ ❝ú❜✐❝❛ ♣❛r❛ ❛ ❞✐r❡çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❬✶✶✶❪✳ ◆❛ ✜❣✉r❛ ✶✳✶✶✱ t❡♠♦s ♦s ✈❡t♦r❡s ✉♥✐tár✐♦s
❋✐❣✉r❛ ✶✳✶✶✿ ❋✐❣✉r❛ ✐❧✉str❛♥❞♦ ♦ ♣❧❛♥♦ ✭✶✶✶✮ ❡ ♦ ♥♦✈♦ ❝♦♥❥✉♥t♦ ❞❡ ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s ✉t✐❧✐③❛❞♦✳
q✉❡ r❡♣r❡s❡♥t❛♠ ♦ ♥♦✈♦ s✐st❡♠❛ ❞❡ ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s✳ ❖ ♣r♦❝❡❞✐♠❡♥t♦ ♣❛r❛ s❡ r❡❧❛❝✐♦♥❛r ♦s ❞♦✐s ❝♦♥❥✉♥t♦s ❞❡ ❝♦ss❡♥♦s ❞✐r❡t♦r❡s é ✐❞ê♥t✐❝♦ ❛♦ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❛ ❞✐r❡çã♦ ❬✶✶✵❪✳ ❊♠ s❡❣✉✐❞❛✱ ❞❡✈❡♠♦s ✐♥✈❡rt❡r ❛ r❡❧❛çã♦ ♣❛r❛ ♦❜t❡r♠♦s ♦ ❝♦♥❥✉♥t♦ {αi} ❡♠ ❢✉♥çã♦ ❞♦ ❝♦♥❥✉♥t♦ {α
′
i}❀ ♦ r❡s✉❧t❛❞♦ é
α1 =−
1 √ 2α ′ 1+ 1 √ 3α ′ 2+ 1 √ 6α ′ 3,
α2 =
1 √ 2α ′ 1+ 1 √ 3α ′ 2+ 1 √ 6α ′ 3 ✭✶✳✶✻✮ ❡
α3 =
❯t✐❧✐③❛♥❞♦ ✶✳✶✻ ❡♠ ✶✳✾ ♦❜t❡♠♦s✱
Eac =dkac
α′ 1 4 4 + α′ 2 4 3 + α′ 3 4 4 + 1 2α ′ 1 2 α′ 3 2
−√2α′
1 2
α′
2α3′ +
√
2 3 α
′
2α′3 3
!
.
✭✶✳✶✼✮ ❈♦♠♦ ❛ ♠❛❣♥❡t✐③❛çã♦ ♣❡r♠❛♥❡❝❡ ♥♦ ♣❧❛♥♦✱φ′ = 0✱α′
1 = s❡♥ θ′✱α′3 = cosθ′ ❡
α′
2 = 0✳ ❆♣ós ✉t✐❧✐③❛r♠♦s ❛ r❡❧❛çã♦ tr✐❣♦♥♦♠étr✐❝❛ ❛♣r♦♣r✐❛❞❛✱ ❡♥❝♦♥tr❛♠♦s
Eac =
dkac
4 . ✭✶✳✶✽✮
❉❡✈✐❞♦ à ❛❧t❛ s✐♠❡tr✐❛ ❞♦ ♣❧❛♥♦ ✭✶✶✶✮✱ t♦❞❛s ❛s ❞✐r❡çõ❡s ♥❡st❡ ♣❧❛♥♦ tê♠ ♦ ♠❡s♠♦ ✈❛❧♦r ❞❡ ❛♥✐s♦tr♦♣✐❛ ❝ú❜✐❝❛✱ s❡♥❞♦ ❡st❡ ♣❧❛♥♦ ✉♠ ♣❧❛♥♦ ♣r❡❢❡r❡♥❝✐❛❧ ♦✉ ❢á❝✐❧✳ ◆ã♦ ❡①✐st❡♠ ❡✐①♦s ❢á❝❡✐s✱ ✐♥t❡r♠❡❞✐ár✐♦s ♦✉ ❞✉r♦s✱ ✉♠❛ ✈❡③ q✉❡ ♥ã♦ ❡①✐st❡ ✉♠❛ ❞✐r❡çã♦ ♣r❡❢❡r❡♥❝✐❛❧ ♥♦ ♣❧❛♥♦✳
✶✳✷✳✺ ❆❝♦♣❧❛♠❡♥t♦ ❡♥tr❡ ❝❛♠❛❞❛s
◗✉❛♥❞♦ ❞♦✐s ✜❧♠❡s ♥❛♥♦♠étr✐❝♦s ♠❡tá❧✐❝♦s ♠❛❣♥ét✐❝♦s s❡ ❡♥❝♦♥tr❛♠ s❡♣❛r❛❞♦s ♣♦r ✉♠ ❡s♣❛ç❛❞♦r ♠❡tá❧✐❝♦ ♥ã♦✲♠❛❣♥ét✐❝♦✱ ❡st❡s ❛♣r❡s❡♥t❛♠ ✉♠ ❛❝♦♣❧❛♠❡♥t♦ ❡♥tr❡ s✐✳ ❈♦♥❢♦r♠❡ ❞✐t♦ ♥❛ s❡çã♦ ✶✳✶✱ ❡st❛ ❞✐ss❡rt❛çã♦ ✉t✐❧✐③❛ ✉♠❛ t❡♦r✐❛ ❢❡♥♦♠❡♥♦❧ó❣✐❝❛❀ ❧♦❣♦✱ ♥ã♦ tr❛t❛r❡♠♦s ❞❛ ♦r✐❣❡♠ ❞♦ ❛❝♦♣❧❛♠❡♥t♦✱ q✉❡ ♣♦❞❡ s❡r ❜✐❧✐♥❡❛r ♦✉ ❜✐q✉❛❞rát✐❝♦✳ ❚❡♦r✐❛s ❛ r❡s♣❡✐t♦ ❞❛ ♦r✐❣❡♠ ❞♦ ♣r✐✲ ♠❡✐r♦ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ♥❛s r❡❢❡rê♥❝✐❛s ❬✶✹❪✱ ❬✶✺❪ ❡ ❬✶✻❪✱ ❡♥q✉❛♥t♦ q✉❡ t❡♦r✐❛s ❛ r❡s♣❡✐t♦ ❞❛ ♦r✐❣❡♠ ❞♦ s❡❣✉♥❞♦ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ ❬✶✼❪✱ ❬✶✽❪✱ ❬✶✾❪✱ ❬✷✵❪✱ ❬✷✶❪ ❡ ✉♠❛ r❡✈✐sã♦ ❛ r❡s♣❡✐t♦ ❞❡st❛s t❡♦r✐❛s ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✷✷❪✳
❆❝♦♣❧❛♠❡♥t♦ ❜✐❧✐♥❡❛r
❖ ❛❝♦♣❧❛♠❡♥t♦ ❜✐❧✐♥❡❛r ♣♦❞❡ s❡r ❞❡ ❞♦✐s t✐♣♦s✿ ❢❡rr♦♠❛❣♥ét✐❝♦ ✭q✉❛♥❞♦ ❛s ♠❛❣♥❡t✐③❛çõ❡s ❞❡ ❞♦✐s ✜❧♠❡s ❡stã♦ ❛❧✐♥❤❛❞❛s ♣❛r❛❧❡❧❛♠❡♥t❡✮ ❡ ❛♥t✐❢❡rr♦✲ ♠❛❣♥ét✐❝♦ ✭❛❧✐♥❤❛❞❛s ❛♥t✐♣❛r❛❧❡❧❛♠❡♥t❡✮✳ ❋❡♥♦♠❡♥♦❧♦❣✐❝❛♠❡♥t❡✱ ♣♦❞❡♠♦s ♠♦❞❡❧❛r ♦ ❛❝♦♣❧❛♠❡♥t♦ ❜✐❧✐♥❡❛r ❛tr❛✈és ❞❡ ✉♠❛ ✐♥t❡r❛çã♦ ❛♥á❧♦❣❛ à ❞❡ tr♦❝❛ ✭Etr =−J ~S1 ·S~2✱ ♦♥❞❡ S~1 ❡ S~2 sã♦ s♣✐♥s ❞❡ í♦♥s ✈✐③✐♥❤♦s ❡ J é ❛ ❝♦♥st❛♥t❡
❞❡ tr♦❝❛✮✳ P❛r❛ ✐st♦✱ ♦s S~i sã♦ tr♦❝❛❞♦s ♣♦r M~i ❡ J é tr♦❝❛❞❛ ♣♦r Jbl✱ q✉❡
é ❝❤❛♠❛❞❛ ❞❡ ❝♦♥st❛♥t❡ ❞❡ ❛❝♦♣❧❛♠❡♥t♦ ❜✐❧✐♥❡❛r✳ ❖ ♠♦t✐✈♦ ❞❡ ✉t✐❧✐③❛r♠♦s ❡st❡ ♣r♦❝❡❞✐♠❡♥t♦ ♣♦r ❛♥❛❧♦❣✐❛ é q✉❡✱ ❛ss✐♠ ❝♦♠♦ ♦ ❛❝♦♣❧❛♠❡♥t♦ ❞❡ tr♦❝❛ t❡♥❞❡ ❛ ❛❧✐♥❤❛r s♣✐♥s ❞❡ át♦♠♦s ✭♦✉ ♠♦❧é❝✉❧❛s✮ ✈✐③✐♥❤♦s ♣❛r❛❧❡❧❛♠❡♥t❡ ✭❡♠ ❢❡rr♦♠❛❣♥❡t♦s✮ ♦✉ ❛♥t✐♣❛r❛❧❡❧❛♠❡♥t❡ ✭❡♠ ❛♥t✐❢❡rr♦♠❛❣♥❡t♦s✮✱ ♦ ❛❝♦♣❧❛♠❡♥t♦ ❜✐❧✐♥❡❛r t❡♥❞❡ ❛ ❛❧✐♥❤❛r ♠❛❣♥❡t✐③❛çõ❡s ❞❡ ✜❧♠❡s ✈✐③✐♥❤♦s ♣❛r❛❧❡❧❛♠❡♥t❡ ♦✉ ❛♥t✐♣❛r❛❧❡❧❛♠❡♥t❡❀ ♥♦t❡✱ ❡♥tr❡t❛♥t♦✱ q✉❡ ♦ ✈❛❧♦r ❞❡ Jbl é ♠✉✐t♦ ♠❡♥♦r q✉❡ ♦
❞❡ J✳ P❛r❛ ✉♠❛ ♠✉❧t✐❝❛♠❛❞❛ ❝♦♠ n ✜❧♠❡s ♠❡tá❧✐❝♦s ❡st❛ ❡♥❡r❣✐❛ ♣♦❞❡ s❡r
❡s❝r✐t❛ ❝♦♠♦
Ebl =− n−1
X
i=1
Jbl
~
Mi·M~i+1
|M~i||M~i+1|
. ✭✶✳✶✾✮
❆ ❡①♣r❡ssã♦ ✶✳✶✾ ♣♦❞❡ s❡r ✉s❛❞❛ ♣❛r❛ ♠♦❞❡❧❛r ♦s ❞♦✐s ❝❛s♦s ❞♦ ❛❝♦♣❧❛♠❡♥t♦s ❜✐❧✐♥❡❛r✿ s❡ Jbl > 0 ♦ ♠í♥✐♠♦ ❞❛ ❡♥❡r❣✐❛ ♦❝♦rr❡rá q✉❛♥❞♦ M~i ❡ M~i+1 ❢♦r❡♠
♣❛r❛❧❡❧♦s✱ ❡♥q✉❛♥t♦ q✉❡ s❡ Jbl < 0 ♦ ♠í♥✐♠♦ ❞❛ ❡♥❡r❣✐❛ ♦❝♦rr❡rá q✉❛♥❞♦
~
Mi ❡ M~i+1 ❡st✐✈❡r❡♠ ❛♥t✐♣❛r❛❧❡❧♦s✳ ❱❛♠♦s ❡♥❝♦♥tr❛r ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ ♦
❛❝♦♣❧❛♠❡♥t♦ ❡♠ t❡r♠♦s ❞♦s â♥❣✉❧♦s ❞❛s ♠❛❣♥❡t✐③❛çõ❡s✳ ❚❡♠♦s q✉❡ M~i = Mi( s❡♥ θicosφixˆ+ s❡♥ θi s❡♥ φiyˆ+ cosθizˆ)✳ P♦rt❛♥t♦✱
~
Mi·M~i+1
|M~i||M~i+1|
= s❡♥ θi s❡♥ θi+1(cosφicosφi+1+ s❡♥ φi s❡♥ φi+1)+cosθicosθi+1.
❯t✐❧✐③❛♥❞♦ ✉♠❛ r❡❧❛çã♦ tr✐❣♦♥♦♠étr✐❝❛ s✐♠♣❧❡s ♦❜t❡♠♦s✱
~
Mi·M~i+1
|M~i||M~i+1|
= s❡♥ θi s❡♥ θi+1cos(φi−φi+1) + cosθicosθi+1. ✭✶✳✷✵✮
❈♦♠♦ ❛s ♠❛❣♥❡t✐③❛çõ❡s ♣❡r♠❛♥❡❝❡♠ ♥♦ ♣❧❛♥♦ ❞❡ s❡✉s r❡s♣❡❝t✐✈♦s ✜❧♠❡s✱
φi = 0✳ ❖ r❡s✉❧t❛❞♦ ✜♥❛❧ é
Ebl =− n−1
X
i=1
Jblcos(θi−θi+1). ✭✶✳✷✶✮
◆♦t❡ q✉❡ ❡st❛ ❡♥❡r❣✐❛ ♥ã♦ ❛♣r❡s❡♥t❛ ♣r❡❢❡rê♥❝✐❛ ♣♦r q✉❛❧q✉❡r ❡✐①♦ ❝r✐st❛❧✐♥♦✱ s❡♥❞♦ ♣♦rt❛♥t♦ ✐s♦tró♣✐❝❛✱ só ❞❡♣❡♥❞❡♥❞♦ ❞❛ ♦r✐❡♥t❛çã♦ r❡❧❛t✐✈❛ ❞❛ ♠❛❣♥❡✲ t✐③❛çã♦ ❞♦s ❞♦✐s ✜❧♠❡s✳ ❆ ✜❣✉r❛ ✶✳✶✷ ✐❧✉str❛ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛♥❣✉❧❛r ❞❛ ❡♥❡r❣✐❛ ❞♦ ❛❝♦♣❧❛♠❡♥t♦ ❜✐❧✐♥❡❛r✱ ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ Jbl <0❀ ♥❛ ✜❣✉r❛✱ θ12 é
♦ â♥❣✉❧♦ ❢♦r♠❛❞♦ ❡♥tr❡ ❛ ♠❛❣♥❡t✐③❛çã♦ ❞❡ ❞♦✐s ✜❧♠❡s✳
❆❝♦♣❧❛♠❡♥t♦ ❜✐q✉❛❞rát✐❝♦
❖ t❡r♠♦ ❞❡ ❛❝♦♣❧❛♠❡♥t♦ ❜✐q✉❛❞rát✐❝♦ é ✉♠ t❡r♠♦ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❡♠ t❡r♠♦s ❞♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r ❞❛ ♠❛❣♥❡t✐③❛çã♦ ❞❡ ❞♦✐s ✜❧♠❡s✱ ❡ ♣♦❞❡ s❡r ❞❡s❝r✐t♦ ♣❡❧❛ s❡❣✉✐♥t❡ ❡①♣r❡ssã♦✱
Ebq = n−1
X
i=1
Jbq
~
Mi·M~i+1
|M~i||M~i+1|
!2
. ✭✶✳✷✷✮
❆ ❡①♣r❡ssã♦ ✶✳✷✷ é ✉s❛❞❛ ♣❛r❛ ❞❡s❝r❡✈❡r ♦ ❛❝♦♣❧❛♠❡♥t♦ ❜✐q✉❛❞rát✐❝♦ ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ✈✐st❛ ❢❡♥♦♠❡♥♦❧ó❣✐❝♦✱ s❡♥❞♦Jbq ❛ ❝♦♥st❛♥t❡ ❞❡ ❛❝♦♣❧❛♠❡♥t♦ ❜✐q✉❛✲
❞rát✐❝❛✱ q✉❡ é s❡♠♣r❡ ♣♦s✐t✐✈❛✳ ❊♠❜♦r❛ ❡st❡ t❡r♠♦ s❡❥❛ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✱ ♣❛r❛ ❝❡rt❛s ❡s♣❡ss✉r❛s ❞♦ ❡s♣❛ç❛❞♦r ♥ã♦✲♠❛❣♥ét✐❝♦✱ ♦ t❡r♠♦ ❜✐q✉❛❞rát✐❝♦ é
❋✐❣✉r❛ ✶✳✶✷✿ ❈♦♠♣♦rt❛♠❡♥t♦ ❛♥❣✉❧❛r ❞❛ ❡♥❡r❣✐❛ ❞♦ ❛❝♦♣❧❛♠❡♥t♦ ❜✐❧✐♥❡❛r✱ ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ Jbl <0✳ θ12 é ♦ â♥❣✉❧♦ ❢♦r♠❛❞♦ ❡♥tr❡ ❛ ♠❛❣♥❡t✐③❛çã♦ ❞❡
❞♦✐s ✜❧♠❡s✳
❞❛ ♠❡s♠❛ ♠❛❣♥✐t✉❞❡ ❞♦ ❜✐❧✐♥❡❛r ❬✷✸❪✳ ❊♠ t❡r♠♦s ❞♦s â♥❣✉❧♦s ❞❛s ♠❛❣♥❡t✐✲ ③❛çõ❡s ❞♦s ✜❧♠❡s✱ Ebq ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ✭❧❡♠❜r❛♥❞♦ q✉❡φi = 0✮
Ebq = n−1
X
i=1
Jbqcos2(θi−θi+1). ✭✶✳✷✸✮
❆ ✜❣✉r❛ ✶✳✶✸ ✐❧✉str❛ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛♥❣✉❧❛r ❞❛ ❡♥❡r❣✐❛ ❞♦ ❛❝♦♣❧❛♠❡♥t♦ ❜✐q✉❛❞rát✐❝♦❀ ♥♦ ❣rá✜❝♦✱ θ12 é ♥♦✈❛♠❡♥t❡ ♦ â♥❣✉❧♦ ❡♥tr❡ ❛ ♠❛❣♥❡t✐③❛çã♦ ❞❡
❞♦✐s ✜❧♠❡s✳ ❊st❡ t❡r♠♦ é ♠✐♥✐♠✐③❛❞♦ q✉❛♥❞♦θi−θi+1 = 90◦✱ ♦✉ s❡❥❛✱ q✉❛♥❞♦
❋✐❣✉r❛ ✶✳✶✸✿ ❈♦♠♣♦rt❛♠❡♥t♦ ❛♥❣✉❧❛r ❞❛ ❡♥❡r❣✐❛ ❞♦ ❛❝♦♣❧❛♠❡♥t♦ ❜✐q✉❛❞rá✲ t✐❝♦✳
❛s ♠❛❣♥❡t✐③❛çõ❡s ❡stã♦ ❛❧✐♥❤❛❞❛s ♣❡r♣❡♥❞✐❝✉❧❛r♠❡♥t❡✳ ◆♦t❡✱ ❡♥tr❡t❛♥t♦✱ q✉❡ ❛s ♠❛❣♥❡t✐③❛çõ❡s ♣❡r♠❛♥❡❝❡♠ ♥♦s ♣❧❛♥♦s ❞❡ s❡✉s r❡s♣❡❝t✐✈♦s ✜❧♠❡s✳
✶✳✷✳✻ ❊♥❡r❣✐❛ ♠❛❣♥ét✐❝❛ t♦t❛❧
❙♦♠❛♥❞♦ t♦❞❛s ❛s ❝♦♥tr✐❜✉✐çõ❡s ❛♥t❡r✐♦r❡s✱ ❛ ❡♥❡r❣✐❛ ♠❛❣♥ét✐❝❛ t♦t❛❧ é ❞❛❞❛ ♣♦r
ET =EZ +ED +ES+Eac+Ebl+Ebq. ✭✶✳✷✹✮
■♥✐❝✐❛❧♠❡♥t❡ ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ♠✐♥✐♠✐③❛r ❛ ❡♥❡r❣✐❛ ♠❛❣♥ét✐❝❛ ❡ ❡♥✲ ❝♦♥tr❛r ♦s â♥❣✉❧♦s ❞❡ ❡q✉✐❧í❜r✐♦✱ ❝❛s♦ ❡♠ q✉❡ ♠❛❣♥❡t✐③❛çã♦ ♣❡r♠❛♥❡❝❡ ♥♦ ♣❧❛♥♦✳ P♦rt❛♥t♦✱ ♦s t❡r♠♦s ED ❡ ES sã♦ ♥✉❧♦s✱ ❛♦ ♠❡♥♦s ❛té q✉❡ s❡ ❡st❡❥❛
❢❛③❡♥❞♦ ✉♠ ❡st✉❞♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞✐♥â♠✐❝❛s✳ P❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ ♦s ✜❧♠❡s ♠❛❣♥ét✐❝♦s tê♠ ❛ ♠❡s♠❛ ❡s♣❡ss✉r❛ ✭di =d✮ ❡ sã♦ ❢❡✐t♦s ❞♦ ♠❡s♠♦ ♠❛t❡r✐❛❧
✭Mi =MS✱ ♦♥❞❡ MS é ❛ ♠❛❣♥❡t✐③❛çã♦ ❞❡ s❛t✉r❛çã♦ ❞♦ ♠❛t❡r✐❛❧✮✱ ♣♦❞❡♠♦s
r❡❡s❝r❡✈❡r ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ ✉t✐❧✐③❛♥❞♦ ♦s ❝❛♠♣♦s ❞❛s ✐♥t❡r❛çõ❡s✱ ❞❡✜♥✐❞♦s ♣♦r ❬✷✹❪✱
Hac =
2kac
MS
, Hbl =
Jbl
dMS ❡
Hbq =
Jbq
dMS
.
❊s❝r❡✈❡♥❞♦ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❛ ❡♥❡r❣✐❛ ♠❛❣♥ét✐❝❛✱ ♣❛r❛ ❛ ❞✐r❡çã♦ ❞❡ ❝r❡s❝✐✲ ♠❡♥t♦ ❬✵✶✵❪✱ ✈❛♠♦s ♦❜t❡r
ET dMS = n X i=1
−H0cos(θi−θH) +
Hac
8 s❡♥
2(2θ
i)
+
n−1
X
i=1
−Hblcos(θi−θi+1) +Hbqcos2(θi−θi+1)
. ✭✶✳✷✺✮
P❛r❛ ❛ ❞✐r❡çã♦ ❬✶✶✵❪ ♦ r❡s✉❧t❛❞♦ é
ET dMS = n X i=1
−H0cos(θi−θH) +
Hac
8 cos
4θ
i+ s❡♥ 22θi
+
n−1
X
i=1
−Hblcos(θi−θi+1) +Hbqcos2(θi−θi+1)
. ✭✶✳✷✻✮
✶✳✸ Pr♦♣r✐❡❞❛❞❡s ❡stát✐❝❛s
❈♦♥❢♦r♠❡ ❞✐t♦ ♥❛ s❡çã♦ ✶✳✶✱ ❝♦♥❤❡❝✐❞❛ ❛ ❢♦r♠❛ ❞❛ ❡♥❡r❣✐❛ ♠❛❣♥ét✐❝❛ ♣♦❞❡♠♦s ♠✐♥✐♠✐③á✲❧❛ ♣❛r❛ ❡♥❝♦♥tr❛r ♦s â♥❣✉❧♦s ❞❡ ❡q✉✐❧í❜r✐♦✳ ❯♠❛ ♠❛♥❡✐r❛
❞❡ ♣r♦❝❡❞❡r ♣♦❞❡r✐❛ s❡r ❞❡r✐✈❛r ❛ ❡♥❡r❣✐❛ t♦t❛❧ ❝♦♠ r❡❧❛çã♦ ❛♦sθi ❡ ✐❣✉❛❧❛r ♦
r❡s✉❧t❛❞♦ ❛ ③❡r♦✳ ❊♥tr❡t❛♥t♦✱ ❡st❡ ♠ét♦❞♦ ❛♥❛❧ít✐❝♦ ✉s✉❛❧♠❡♥t❡ ❧❡✈❛ ❛ ❡q✉❛✲ çõ❡s q✉❡ sã♦ ♠✉✐t♦ ❞✐❢í❝❡✐s ❞❡ s❡r❡♠ r❡s♦❧✈✐❞❛s✳ ❖ ♣r♦❝❡❞✐♠❡♥t♦ ✉t✐❧✐③❛❞♦ ♥❡st❛ ❞✐ss❡rt❛çã♦ ❝♦♥s✐st❡ ❡♠ ♠✐♥✐♠✐③❛r ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡ ❛ ❡♥❡r❣✐❛ q✉❡✱ ❞❡♣❡♥❞❡♥❞♦ ❞❛ ❞✐r❡çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦✱ ♣♦❞❡ s❡r ❞❛❞❛ ♣♦r ✶✳✷✺ ♦✉ ✶✳✷✻✳ ❯t✐❧✐✲ ③❛♠♦s ♦ ❝❤❛♠❛❞♦ ♠ét♦❞♦ ❞♦ ❣r❛❞✐❡♥t❡ ♣❛r❛ ✐st♦✱ ✉♠❛ ✈❡③ q✉❡ ❡st❡ é ❡✜❝✐❡♥t❡ ❡♠ ❝❛s♦s ❡♠ q✉❡ ❛ ❢✉♥çã♦ ❛ s❡r ♠✐♥✐♠✐③❛❞❛ ❞❡♣❡♥❞❡ ❞❡ ❞✐✈❡rs❛s ✈❛r✐á✈❡✐s✱ ♦ q✉❡ ♦❝♦rr❡ q✉❛♥❞♦ tr❛❜❛❧❤❛♠♦s ❝♦♠ ♠✉❧t✐❝❛♠❛❞❛s ❬✷✺❪✳ ❯♠❛ ❞❡s❝r✐çã♦ ❞❡st❡ ♠ét♦❞♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ♥♦ ❛♣ê♥❞✐❝❡ ❆✳
❋✐❣✉r❛ ✶✳✶✹✿ ❋✐❣✉r❛ ✐❧✉str❛♥❞♦✱ ♣❛r❛ ✉♠❛ tr✐❝❛♠❛❞❛✱ ♦ ❝á❧❝✉❧♦ ❞❛ ❝♦♠♣♦♥❡♥t❡ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ♣❛r❛❧❡❧❛ ❛♦ ❝❛♠♣♦ ❡①t❡r♥♦✳
❊♥❝♦♥tr❛❞♦s ♦s â♥❣✉❧♦s ❞❡ ❡q✉✐❧í❜r✐♦✱ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ❛s ♣r♦♣r✐❡❞❛✲ ❞❡s ❡stát✐❝❛s✳ ❆ ♣r✐♠❡✐r❛ ❛ s❡r ❝❛❧❝✉❧❛❞❛ s❡rá ❛ ❝♦♠♣♦♥❡♥t❡ ❞❛ ♠❛❣♥❡t✐③❛çã♦
