❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ ❈❊❆❘➪ ❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ❋❮❙■❈❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ❋❮❙■❈❆
▼➪❘❈■❖ ❉❆ ❙■▲❱❆ ❚❆❱❆❘❊❙
❖◆❉❆❙ ❉❊ ❙P■◆ ❈❖▼ ■◆❚❊❘❆➬➹❖ ❉■P❖▲❆❘ ❊▼
❘❊❉❊❙ ❍❊❳❆●❖◆❆■❙ ▼❆●◆➱❚■❈❆❙✳
▼➪❘❈■❖ ❉❆ ❙■▲❱❆ ❚❆❱❆❘❊❙
❖◆❉❆❙ ❉❊ ❙P■◆ ❈❖▼ ■◆❚❊❘❆➬➹❖ ❉■P❖▲❆❘ ❊▼
❘❊❉❊❙ ❍❊❳❆●❖◆❆■❙ ▼❆●◆➱❚■❈❆❙✳
❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá ❝♦♠♦ ✉♠ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ❉♦✉t♦r ❡♠ ❋ís✐❝❛✳
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❘❛✐♠✉♥❞♦ ◆♦❣✉❡✐r❛ ❈♦st❛ ❋✐❧❤♦
▼ár❝✐♦ ❞❛ ❙✐❧✈❛ ❚❛✈❛r❡s
❖♥❞❛s ❙♣✐♥ ❝♦♠ ■♥t❡r❛çã♦ ❉✐♣♦❧❛r ❡♠ ❘❡❞❡s ❍❡①❛❣♦♥❛✐s ▼❛❣♥ét✐❝❛s✳ ✴ ▼ár❝✐♦ ❞❛ ❙✐❧✈❛ ❚❛✈❛r❡s✳ ✲ ✷✵✵✾
✶✻✹ ❢✳
❖r✐❡♥t❛❞♦r✿ ❘❛✐♠✉♥❞♦ ◆♦❣✉❡✐r❛ ❈♦st❛ ❋✐❧❤♦ ■♠♣r❡ss♦ ♣♦r ❝♦♠♣✉t❛❞♦r ✭❢♦t♦❝ó♣✐❛✮✳
❚❡s❡ ✭❉♦✉t♦r❛❞♦✮ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá✱ ❝✉rs♦ ❞❡ ❉♦✉t♦r❛❞♦ ❡♠ ❋ís✐❝❛✱ ❋♦rt❛❧❡③❛✱ ✷✵✶✹✳
✶✳ ❖♥❞❛s ❞❡ ❙♣✐♥✱ ✷✳■♥t❡r❛çã♦ ❉✐♣♦❧❛r✱ ✸✳ ▼❛❣♥❡t✐s♠♦ ❡♠ ❇❛✐①❛s ❉✐♠❡♥sõ❡s✳
■✳ ♦r✐❡♥t✳❘❛✐♠✉♥❞♦ ◆✳ ❈✳ ❋✐❧❤♦✱
▼➪❘❈■❖ ❉❆ ❙■▲❱❆ ❚❆❱❆❘❊❙
❖◆❉❆❙ ❉❊ ❙P■◆ ❈❖▼ ■◆❚❊❘❆➬➹❖ ❉■P❖▲❆❘ ❊▼
❘❊❉❊❙ ❍❊❳❆●❖◆❆■❙ ▼❆●◆➱❚■❈❆❙✳
❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡✲ r❛❧ ❞♦ ❈❡❛rá ❝♦♠♦ ✉♠ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ❉♦✉t♦r ❡♠ ❋ís✐❝❛✳
❆♣r♦✈❛❞❛ ❡♠ ✴ ✴
❇❆◆❈❆ ❊❳❆▼■◆❆❉❖❘❆
Pr♦❢✳ ❉r✳ ❘❛✐♠✉♥❞♦ ◆♦❣✉❡✐r❛ ❈♦st❛ ❋✐❧❤♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá
Pr♦❢✳ ❉r✳ ❲❛♥❞❡♠❜❡r❣ P❛✐✈❛ ❋❡rr❡✐r❛ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá
Pr♦❢✳ ❉r✳ ❆♥❞r❡② ❈❤❛✈❡s ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá
Pr♦❢✳ ❉r✳ ❉♦r② ❍é❧✐♦ ❆✐r❡s ❞❡ ▲✐♠❛ ❆♥s❡❧♠♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❘✐♦ ●r❛♥❞❡ ❞❡ ◆♦rt❡
Pr♦❢✳ ❉r✳ ❋á❜✐♦ ❍❡♥r✐q✉❡ ❙✐❧✈❛ ❙❛❧❡s ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛
❉❡❞✐❝❛tór✐❛
➚ ❉❊❯❙✳
➚ ♠✐♥❤❛ ❡s♣♦s❛ ❉❛②r✐s✳
❆♦s ♠❡✉s ♣❛✐s✳
➚ ♠✐♥❤❛ ✐r♠ã✳
➚ ♠✐♥❤❛ ♣❡ss♦❛✳
✧❆s ❞❡s❝♦❜❡rt❛s ❝✐❡♥tí✜❝❛s ❡✈✐❞❡♥❝✐❛♠ ♦ ❛❜s✉r❞♦ r❡❛❧✧✳ ▼ár❝✐♦ ❚❛✈❛r❡s
✧●r❛♥❞❡s ❧✉t❛s só sã♦ ❞❛❞❛s ❛ ❣r❛♥❞❡s ❣✉❡rr❡✐r♦s✧✳ ❆✉t♦r ❞❡s❝♦♥❤❡❝✐❞♦
❆❣r❛❞❡❝✐♠❡♥t♦s
➚ ❉❡✉s✱ ♣♦r t✉❞♦ q✉❡ r❡❛❧✐③♦✉ ♥❛ ♠✐♥❤❛ ✈✐❞❛✱ ♠❡ s✉st❡♥t❛♥❞♦ ❡ ❞❛♥❞♦ ❢♦rç❛s s❡♠♣r❡ q✉❡ ♣r❡❝✐s❡✐✳
➚ ♠✐♥❤❛ ❡s♣♦s❛ ♣♦r s✉❛ ❝♦♠♣r❡❡♥sã♦✱ ❞❡❞✐❝❛çã♦✱ ❛♣♦✐♦ ❡ ♣r❡s❡♥ç❛✳ ➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✳
❆♦s ♣r♦❢❡ss♦r❡s ❞♦ ❞❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛ ❞❛ ❯❋❈ ♣❡❧❛ ❢♦r♠❛çã♦✳ ❆♦ Pr♦❢✳ ❉r✳ ❘❛✐♠✉♥❞♦ ◆♦❣✉❡✐r❛ ❈✳ ❋✐❧❤♦ ♣❡❧❛ ♦r✐❡♥t❛çã♦✳
➚ ❈❆P❊❙✱ ♣❡❧❛s ❜♦❧s❛ ❝♦♥❝❡❞✐❞❛ ♣❛r❛ ❡①❡❝✉çã♦ ❞♦ tr❛❜❛❧❤♦ ❡ ✜♥❛♥❝✐❛♠❡♥t♦ ❞❡ ♣r♦❥❡t♦s✳
❘❡s✉♠♦
❊st❡ tr❛❜❛❧❤♦ ❢♦✐ ♠♦t✐✈❛❞♦ ♣❡❧❛s ♣r♦♣r✐❡❞❛❞❡s ❞✐✈❡rs❛s ❡ ♠✐rí❛❞❡ ❞❡ ❛♣❧✐❝❛çõ❡s ❞♦ ♠❛❣✲ ♥❡t✐s♠♦ ♥♦ ❝♦♥t❡①t♦ t❡❝♥♦❧ó❣✐❝♦ ❛t✉❛❧✱ ❞♦s q✉❛✐s ❡♥❢❛t✐③❛♠♦s ♦ ♣❛♣❡❧ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ❡ ♦♥❞❛s ❞❡ s♣✐♥ ❡♠ s✐st❡♠❛s ♠❛❣♥ét✐❝♦s ❜✐❞✐♠❡♥s✐♦♥❛✐s t❛✐s ❝♦♠ r❡❞❡s q✉❛❞r❛❞❛s ❡ ❤❡①❛❣♦✲ ♥❛✐s✳ ❆s ♦♥❞❛s ❞❡ s♣✐♥ t❡♠ tr❛③✐❞♦ ❛✈❛♥ç♦s ♦✉tr♦r❛ ♥ã♦ ✐♠❛❣✐♥❛❞♦s ❡ t❡♠ ❢❛✈♦r❡❝✐❞♦ ❡♠ ❞❡♠❛s✐❛ ❛ ❝♦♥str✉çã♦ ❞❡ ❞✐s♣♦s✐t✐✈♦s ♠❛✐s ❡✜❝✐❡♥t❡s ♥♦s r❛♠♦s ❞❛ ❡❧❡trô♥✐❝❛ ❡ tr❛♥s♣♦rt❡ ❞❡ ✐♥❢♦r♠❛çõ❡s✱ ❛❧é♠ ❞❡ ✜♥❝❛r ♥♦✈❛s ♣❡rs♣❡❝t✐✈❛s ❛✐♥❞❛ ❡①♣❧♦r❛❞❛s ❝♦♥❝❡✐t✉❛❧♠❡♥t❡✳ ❆s✲ s✐♠ ❡♥t❡♥❞❡✲s❡ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞♦ t❡♠❛ ❡ ♣r♦❝✉r❛♠♦s ❡st✉❞❛r ❛s ♦♥❞❛s ❞❡ s♣✐♥ ❡♠ ✉♠ ❢❡r✲ r♦♠❛❣♥❡t♦ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❡①♣♦st♦ ❛ ✐♥t❡r❛çõ❡s ♠❛❣♥ét✐❝❛s ❞❡ ❞✐❢❡r❡♥t❡s ♥❛t✉r❡③❛s ✭tr♦❝❛✱ ❛♥✐s♦tró♣✐❝❛✱ ❩❡❡♠❛♥ ❡ ❞✐♣♦❧❛r✮ ♥♦ ✐♥t✉✐t♦ ❞❡ ♠♦♥t❛r ❛ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❞♦s ♠♦❞❡❧♦s ♣r♦♣♦st♦s ❡ ❝♦♥str✉✐r ♦ ❡s♣❡❝tr♦ ❞❛ t❡♦r✐❛✱ ♦✉ s❡❥❛✱ ❛ r❡❧❛çã♦ ❡♥❡r❣✐❛✴✈❡t♦r ❞❡ ♦♥❞❛✳ ❆❞❡✲ ♠❛✐s✱ ♣❧♦t❛♠♦s ♦✉tr❛s r❡❧❛çõ❡s ❞❛ ❡♥❡r❣✐❛ ❝♦♠ ♣❛râ♠❡tr♦s ✐♠♣♦rt❛♥t❡s ♥♦ ❢♦❝♦ ❞❡ ♠❡❧❤♦r ❡♥t❡♥❞❡r ❛ ❡✈♦❧✉çã♦ ❞❛ ❡♥❡r❣✐❛ ❞♦ s✐st❡♠❛✱ ❛❧é♠ ❞❡ q✉❡ ✐♥✈❡st✐❣❛♠♦s ❛s ✐♥✢✉ê♥❝✐❛s ❞❛ ♦r✐❡♥t❛çã♦ ❞❛ r❡❞❡ ❢r❡♥t❡ à ♠❛❣♥❡t✐③❛çã♦ ❡①t❡r♥❛✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❖♥❞❛s ❞❡ ❙♣✐♥✱ ■♥t❡r❛çã♦ ❉✐♣♦❧❛r✱ ▼❛❣♥❡t✐s♠♦ ❡♠ ❇❛✐①❛ ❉✐♠❡♥sã♦✳
❆❜str❛❝t
❚❤✐s ✇♦r❦ ✇❛s ♠♦t✐✈❛t❡❞ ❜② t❤❡ ✈❛r✐♦✉s ♣r♦♣❡rt✐❡s ❛♥❞ ♠❛❣♥❡t✐s♠ ♠②r✐❛❞ ♦❢ ❛♣♣❧✐❝❛t✐✲ ♦♥s ✐♥ t❤❡ ❝✉rr❡♥t t❡❝❤♥♦❧♦❣✐❝❛❧ ❡♥✈✐r♦♥♠❡♥t✱ ✇❤✐❝❤ ❡♠♣❤❛s✐③❡ t❤❡ r♦❧❡ ♦❢ ♠❛❣♥❡t✐③❛t✐♦♥ ❛♥❞ s♣✐♥ ✇❛✈❡s ✐♥ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ♠❛❣♥❡t✐❝ s②st❡♠s ✇✐t❤ s✉❝❤ sq✉❛r❡ ❛♥❞ ❤❡①❛❣♦♥❛❧ ♥❡t✇♦r❦s✳ ❚❤❡ s♣✐♥ ✇❛✈❡s ❤❛✈❡ ❜r♦✉❣❤t ♣r❡✈✐♦✉s❧② ✉♥✐♠❛❣✐♥❡❞ ❛❞✈❛♥❝❡s ❛♥❞ ♦✈❡r✉s❡❞ t❤❛t ❤❛s ❢❛✈♦r❡❞ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ♠♦r❡ ❡✣❝✐❡♥t ❞❡✈✐❝❡s ✐♥ t❤❡ ✜❡❧❞s ♦❢ ❡❧❡❝tr♦♥✐❝s ❛♥❞ tr❛♥s♣♦rt❛t✐♦♥ ✐♥❢♦r♠❛t✐♦♥✱ ❛♥❞ ♣✉t ❞♦✇♥ ♥❡✇ ♣r♦s♣❡❝ts ②❡t ❡①♣❧♦r❡❞ ❝♦♥❝❡♣t✉❛❧❧②✳ ❙♦ ✇❡ ✉♥❞❡rst❛♥❞ t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤❡ t♦♣✐❝ ❛♥❞ tr②✐♥❣ t♦ st✉❞② t❤❡ s♣✐♥ ✇❛✈❡s ✐♥ ❛ ❧♦❝❛t✐♦♥ ❡①♣♦s❡❞ t♦ ♠❛❣♥❡t✐❝ ✐♥t❡r❛❝t✐♦♥s ♦❢ ❞✐✛❡r❡♥t ♥❛t✉r❡s ✭❡①❝❤❛♥❣❡✱ ❞✐♣♦❧❛r ❛♥❞ ❛♥✐s♦tr♦♣✐❝✮ ✐♥ ♦r❞❡r t♦ ❛ss❡♠❜❧❡ t❤❡ ❞✐s♣❡rs✐♦♥ r❡❧❛t✐♦♥ ♦❢ t❤❡ ♣r♦♣♦s❡❞ ♠♦❞❡❧s ❛♥❞ ❜✉✐❧❞ t❤❡ s♣❡❝tr✉♠ ♦❢ t❤❡ t❤❡♦r② ❞✐♠❡♥s✐♦♥❛❧ ❢❡rr♦♠❛❣♥❡t✱ t❤❛t ✐s✱ t❤❡ ❡♥❡r❣②✴✇❛✈❡ ✈❡❝t♦r✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ♣❧♦t t❤❡ ❡♥❡r❣② ✇✐t❤ ♦t❤❡r ✐♠♣♦rt❛♥t r❡❧❛t✐♦♥s❤✐♣s ✐♥ ❢♦❝✉s t♦ ❜❡tt❡r ✉♥❞❡rst❛♥❞ t❤❡ ❡✈♦✲ ❧✉t✐♦♥ ♦❢ t❤❡ ❡♥❡r❣② ♦❢ t❤❡ s②st❡♠ ♣❛r❛♠❡t❡rs✱ ❛♥❞ ✇❡ ✐♥✈❡st✐❣❛t❡ t❤❡ ✐♥✢✉❡♥❝❡ ♦❢ t❤❡ ♦r✐❡♥t❛t✐♦♥ ♦❢ t❤❡ ♠❛❣♥❡t✐③❛t✐♦♥ r❡❧❛t✐✈❡ t♦ t❤❡ ❡①t❡r♥❛❧ ♥❡t✇♦r❦✳
❑❡②✇♦r❞s✿ ❙♣✐♥ ❲❛✈❡s✱ ❉✐♣♦❧❡✲❉✐♣♦❧❡ ■♥t❡r❛❝t✐♦♥s✱ ▲♦✇ ❉✐♠❡♥s✐♦♥ ▼❛❣♥❡t✐s♠✳
❙✉♠ár✐♦
▲✐st❛ ❞❡ ❋✐❣✉r❛s ♣✳ ①
▲✐st❛ ❞❡ ❚❛❜❡❧❛s ♣✳ ①✈
✶ ■♥tr♦❞✉çã♦ ●❡r❛❧✳ ♣✳ ✶
✷ ▼❛❣♥❡t✐s♠♦✳ ♣✳ ✽
✷✳✶ ❆s♣❡❝t♦s ❍✐stór✐❝♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✽ ✷✳✶✳✶ ❆❜♦r❞❛❣❡♥s ▼❛❝r♦s❝ó♣✐❝❛ ❡ ▼✐❝r♦s❝ó♣✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✸ ✷✳✶✳✶✳✶ ▼❛❣♥❡t✐s♠♦ ▼❛❝r♦s❝ó♣✐❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✸ ✷✳✶✳✶✳✷ ▼❛❣♥❡t✐s♠♦ ▼✐❝r♦s❝ó♣✐❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✻ ✷✳✷ ❖ ▼♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✽
✸✳✶✳✺ ❖ ❍❛♠✐❧t♦♥✐❛♥♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✽ ✸✳✶✳✻ ❆♣r♦①✐♠❛çã♦ ❞❡ ❍♦❧st❡✐♥✲Pr✐♠❛❦♦✛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✾
✹ ❖♥❞❛s ❞❡ ❙♣✐♥ ❡♠ ✉♠❛ ❘❡❞❡ ◗✉❛❞r❛❞❛✳ ♣✳ ✹✺
✹✳✶ ❖♥❞❛s ❞❡ ❙♣✐♥ ❡♠ ❘❡❞❡s ◗✉❛❞r❛❞❛s ■♥✜♥✐t❛s s❡♠ ■♥t❡r❛çã♦ ❉✐♣♦❧❛r✳ ✳ ♣✳ ✹✺ ✹✳✷ ❘❡s✉❧t❛❞♦s ◆✉♠ér✐❝♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✼ ✹✳✸ ❖♥❞❛s ❞❡ ❙♣✐♥ ❡♠ ❘❡❞❡s ◗✉❛❞r❛❞❛s ■♥✜♥✐t❛s ❝♦♠ ■♥t❡r❛çã♦ ❉✐♣♦❧❛r✳ ✳ ♣✳ ✺✶ ✹✳✸✳✶ P❧❛♥♦ x−y✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✶
✹✳✸✳✷ P❧❛♥♦ x−z✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✼
✹✳✹ ❖♥❞❛s ❞❡ ❙♣✐♥ ❡♠ ❘❡❞❡ ◗✉❛❞r❛❞❛ ❋✐♥✐t❛ s❡♠ ■♥t❡r❛çã♦ ❉✐♣♦❧❛r ✳ ✳ ✳ ✳ ♣✳ ✻✶ ✹✳✺ ❖♥❞❛s ❞❡ ❙♣✐♥ ❡♠ ❘❡❞❡ ◗✉❛❞r❛❞❛ ❋✐♥✐t❛ ❝♦♠ ■♥t❡r❛çã♦ ❉✐♣♦❧❛r ✳ ✳ ✳ ✳ ♣✳ ✻✹ ✹✳✻ ❖♥❞❛s ❞❡ ❙♣✐♥ ❡♠ ✉♠❛ ❘❡❞❡ ❋❛✈♦ ❞❡ ▼❡❧ ■♥✜♥✐t❛ s❡♠ ■♥t❡r❛çã♦ ❉✐♣♦❧❛r ♣✳ ✼✵ ✹✳✼ ❘❡❞❡ ❋❛✈♦ ❞❡ ♠❡❧ ✐♥✜♥✐t❛ ❝♦♠ ✐♥t❡r❛çã♦ ❞✐♣♦❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✼✻ ✹✳✽ ❘❡❞❡ ❢❛✈♦ ❞❡ ♠❡❧ ✜♥✐t❛ s❡♠ ✐♥t❡r❛çã♦ ❞✐♣♦❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✽✺ ✹✳✾ ❘❡❞❡ ❢❛✈♦ ❞❡ ♠❡❧ ✜♥✐t❛ ❝♦♠ ✐♥t❡r❛çã♦ ❞✐♣♦❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✽✽
✺ ❈♦♥❝❧✉sõ❡s ❡ P❡rs♣❡❝t✐✈❛s✳ ♣✳ ✾✷
❆♣ê♥❞✐❝❡ ❆ ✲✲ ❈á❧❝✉❧♦ ❞♦s ❚❡r♠♦s ❉✐♣♦❧❛r❡s ❞❛ ❘❡❞❡ ◗✉❛❞r❛❞❛ ■♥✜♥✐t❛
♥♦ P❧❛♥♦ ✵①②✳ ♣✳ ✾✼
❆♣ê♥❞✐❝❡ ❇ ✲✲ ❈á❧❝✉❧♦ ❞♦s ❚❡r♠♦s ❉✐♣♦❧❛r❡s ❞❛ ❘❡❞❡ ◗✉❛❞r❛❞❛ ■♥✜♥✐t❛
♥♦ P❧❛♥♦ ✵①③✳ ♣✳ ✶✵✵
❆♣ê♥❞✐❝❡ ❈ ✲✲ ❈á❧❝✉❧♦ ❞♦s ❚❡r♠♦s ❉✐♣♦❧❛r❡s ❞❛ ❘❡❞❡ ◗✉❛❞r❛❞❛ ❋✐♥✐t❛
♥♦ P❧❛♥♦ ✵①②✳ ♣✳ ✶✵✷
❆♣ê♥❞✐❝❡ ❉ ✲✲ ❈á❧❝✉❧♦ ❞♦s ❚❡r♠♦s ❉✐♣♦❧❛r❡s ❞❛ ❘❡❞❡ ◗✉❛❞r❛❞❛ ❋✐♥✐t❛
♥♦ P❧❛♥♦ ✵①③✳ ♣✳ ✶✵✺
❆♣ê♥❞✐❝❡ ❊ ✲✲ ❈á❧❝✉❧♦ ❞♦s ❚❡r♠♦s ❉✐♣♦❧❛r❡s ❞❛ ❘❡❞❡ ❍❡①❛❣♦♥❛❧ ■♥✜♥✐t❛
♥♦ P❧❛♥♦ ✵①②✳ ♣✳ ✶✵✼
❆♣ê♥❞✐❝❡ ❋ ✲✲ ❈á❧❝✉❧♦ ❞♦s ❚❡r♠♦s ❉✐♣♦❧❛r❡s ❞❛ ❘❡❞❡ ❍❡①❛❣♦♥❛❧ ❋✐♥✐t❛
♥♦ P❧❛♥♦ ✵①②✳ ♣✳ ✶✶✶
❘❡❢❡rê♥❝✐❛s ♣✳ ✶✶✹
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶ ❈♦♠♣♦♥❡♥t❡ ❙♣✐♥trô♥✐❝♦ ❞❡ ❙✐❧í❝✐♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹ ✷ ❈♦❧❤❡r ❞❡ ▼❛❣♥❡t✐t❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✾ ✸ ❚❡rr❡❧❧❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✵ ✹ ❍✐st❡r❡s❡ ▼❛❣♥ét✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✺ ✺ ❋✉♥çã♦ ❞❡ ♦♥❞❛ ❛♥t✐ss✐♠étr✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✾ ✻ ❖♥❞❛s ❞❡ ❙♣✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✼ ✼ Pr❡❝❡ssã♦ ❞♦ ❙♣✐♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✽ ✽ ▼♦❞♦ ❯♥✐❢♦r♠❡ ❞❡ Pr♦♣❛❣❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✵ ✾ ▼♦❞♦ ♥ã♦ ❯♥✐❢♦r♠❡ ❞❡ Pr♦♣❛❣❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✵ ✶✵ ❙♣✐♥s ❆♥t✐♣❛r❛❧❡❧♦s ✭❛✮ ❡ ❙♣✐♥s P❛r❛❧❡❧♦s ✭❜✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✸ ✶✶ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ r❡❞❡ q✉❛❞r❛❞❛ ✐♥✜♥✐t❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✺ ✶✷ ❈✉r✈❛ ❞❡ ❝♦♥t♦r♥♦ ♣❛r❛ ❛s ❡♥❡r❣✐❛s ❞❡ ♦♥❞❛s ❞❡ s♣✐♥✳ ❆ ❡♥❡r❣✐❛ ❝r❡s❝❡
❞♦ ❛③✉❧ ❛té ♦ ✈❡r♠❡❧❤♦ 0< E/J1 <9✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✽
✶✸ ❆ ③♦♥❛ ❞❡ ❇r✐❧❧♦✉✐♥ ♣❛r❛ ✉♠❛ r❡❞❡ q✉❛❞r❛❞❛ ♠♦str❛♥❞♦ ♦s ♣♦♥t♦sΓ✱ X✱
❡ M✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✽
✶✹ ❘❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❞❛s ♦♥❞❛s ❞❡ s♣✐♥ ❛♦ ❧♦♥❣♦ ❞♦s ❝❛♠✐♥❤♦s∆✱ Z✱ ❡Σ✳ ♣✳ ✹✾
✶✺ ❈✉r✈❛ ❞❡ ❝♦♥t♦r♥♦ ♣❛r❛ ❛s ❡♥❡r❣✐❛s ❞❡ ♦♥❞❛s ❞❡ s♣✐♥✳ ❆ ❡♥❡r❣✐❛ ❝r❡s❝❡ ❞♦ ❛③✉❧ ❛té ♦ ✈❡r♠❡❧❤♦ 0 < E/J1 < 9✳ ❆q✉✐ ❝♦♥s✐❞❡r❛♠♦s s❡❣✉♥❞♦s
✈✐③✐♥❤♦s✿❛✮J2/J1 = 0,1✱ ❜✮J2/J1 = 0,2✱ ❝✮J2/J1 = 0,3✱ ❞✮J2/J1 = 0,4 ✳ ♣✳ ✹✾
✶✻ ❈✉r✈❛ ❞❡ ❝♦♥t♦r♥♦ ♣❛r❛ ❛s ❡♥❡r❣✐❛s ❞❡ ♦♥❞❛s ❞❡ s♣✐♥✳ ❆ ❡♥❡r❣✐❛ ❝r❡s❝❡ ❞♦ ❛③✉❧ ❛té ♦ ✈❡r♠❡❧❤♦ 0 < E/J1 < 14✳ ❆q✉✐ ❝♦♥s✐❞❡r❛♠♦s qy = 0 ♥♦
✶✼ ❘❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❞❛s ♦♥❞❛s ❞❡ s♣✐♥ ❝♦♠ ✐♥t❡r❛çã♦ ❞❡ s❡❣✉♥❞♦s ✈✐③✐✲ ♥❤♦s ❛♦ ❧♦♥❣♦ ❞♦s ❝❛♠✐♥❤♦s ∆✱ Z✱ ❡ Σ✳ ❆ ❧✐♥❤❛ s♦❧✐❞❛ J2 = 0✱ ❛ ❧✐♥❤❛
♣♦♥t✐❧❤❛❞❛ é ♣❛r❛ J2 = 0.1J1✱ ♦ ❝❛s♦ ♦♥❞❡ J2 = 0.2 é r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛
tr❛❝❡❥❛❞❛✱ ❛ ❧✐♥❤❛ tr❛❝❡❥❛❞❛ ❧♦♥❣❛ ❝♦rr❡s♣♦♥❞❡ ❛ J2 = 0.3J1✱ ❡ ❛ ❧✐♥❤❛
tr❛❝❡❥❛❞❛ ♣♦♥t✐❧❤❛❞❛ ✐♥❞✐❝❛ ♦ ❝❛s♦ ♦♥❞❡ J2 = 0.4J1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✵
✶✽ ❆ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ❞❛s ♦♥❞❛s ❞❡ s♣✐♥ ❡♠ ✉♠❛ r❡❞❡ q✉❛❞r❛❞❛✳ ❆ ❝✉r✈❛ só❧✐❞❛ é ♣❛r❛ ✉♠ ❡s♣❡❝tr♦ s❡♠ s❡❣✉♥❞♦s ✈✐③✐♥❤♦s✱ ❛ ❝✉r✈❛ ♣♦♥t✐❧❤❛❞❛ r❡♣r❡s❡♥t❛ ♦ ❝❛s♦ ♦♥❞❡ J2 = 0,1J1✱ ❡ ❛ ❝✉r✈❛ tr❛❝❡❥❛❞❛ é ♣❛r❛ ♦ ❝❛s♦
♦♥❞❡ Js = 0,4J1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✶ ✶✾ ❘❡❞❡ q✉❛❞r❛❞❛ ■♥✜♥✐t❛ ❡♠ ❆♠❜❛s ❛s ❉✐r❡çõ❡s ♥♦ ♣❧❛♥♦ x−y✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✷
✷✵ ❈✉r✈❛ ❞❡ ❝♦♥t♦r♥♦ ♣❛r❛ ❛s ❡♥❡r❣✐❛s ❞❡ ♦♥❞❛s ❞❡ s♣✐♥ ♥✉♠❛ r❡❞❡ q✉❛❞r❛❞❛ ❝♦♠ ✐♥t❡r❛çã♦ ❞✐♣♦❧❛r✳ ❆ ❡♥❡r❣✐❛ ❝r❡s❝❡ ❞♦ ❛③✉❧ ❛té ♦ ✈❡r♠❡❧❤♦ 0 <
E/J1 <10✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✺
✷✶ ❈✉r✈❛ ❞❡ ❝♦♥t♦r♥♦ ♣❛r❛ ❛s ❡♥❡r❣✐❛s ❞❡ ♦♥❞❛s ❞❡ s♣✐♥ ♥✉♠❛ r❡❞❡ q✉❛❞r❛❞❛ ❝♦♠ ✐♥t❡r❛çã♦ ❞✐♣♦❧❛r✳ ❆ ❡♥❡r❣✐❛ ❝r❡s❝❡ ❞♦ ❛③✉❧ ❛té ♦ ✈❡r♠❡❧❤♦ 0 <
E/J1 <9.5✳ ❖ ♣❛✐♥❡❧ ❞❛ ❞✐r❡✐t❛ qy = 0 ❡ qy =π/a♥♦ ♣❛✐♥❡❧ ❞❛ ❞✐r❡✐t❛✳ ♣✳ ✺✺ ✷✷ ❘❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❞❛s ♦♥❞❛s ❞❡ s♣✐♥ ❝♦♠ ✐♥t❡r❛çã♦ ❞❡ s❡❣✉♥❞♦s ✈✐✲
③✐♥❤♦s ❛♦ ❧♦♥❣♦ ❞♦s ❝❛♠✐♥❤♦s ∆✱ Z✱ ❡ Σ✳ ❆ ❧✐♥❤❛ s♦❧✐❞❛ 4πgµBM = 0✱ ❛ ❧✐♥❤❛ ♣♦♥t✐❧❤❛❞❛ é ♣❛r❛ 4πgµBM = 1✱ ♦ ❝❛s♦ ♦♥❞❡ 4πgµBM = 2 é r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ tr❛❝❡❥❛❞❛ ❧♦♥❣❛✱ ❡ ❛ ❧✐♥❤❛ tr❛❝❡❥❛❞❛ ♣♦♥t✐❧❤❛❞❛ ✐♥❞✐❝❛ ♦ ❝❛s♦ ♦♥❞❡ 4πgµBM = 3✳ ❆ ❞✐r❡✐t❛ t❡♠♦s ❛ ❞✐❢❡r❡♥ç❛ ❞❡ ❡♥❡r❣✐❛ ❞❛s ♦♥❞❛s ❞❡ s♣✐♥ ♣❛r❛4πµBM = 0 ❡4πµBM = 1❛♦ ❧♦♥❣♦ ❞♦s ❝❛♠✐♥❤♦s ∆✱
Z✱ ❡Σ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✻
✷✸ ❆ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ❞❛s ♦♥❞❛s ❞❡ s♣✐♥ ❡♠ ✉♠❛ r❡❞❡ q✉❛❞r❛❞❛✳ ❆ ❝✉r✈❛ só❧✐❞❛ é ♣❛r❛ ✉♠ ❡s♣❡❝tr♦ s❡♠ ✐♥t❡r❛çã♦ ❞✐♣♦❧❛r✱ ❛ ❝✉r✈❛ ♣♦♥t✐❧❤❛❞❛ r❡♣r❡s❡♥t❛ ♦ ❝❛s♦ ♦♥❞❡ 4πµBM = 1✱❛ ❝✉r✈❛ tr❛❝❡❥❛❞❛ é ♣❛r❛ ♦ ❝❛s♦ ♦♥❞❡
4πµBM = 2✱ ❛ ❝✉r✈❛ tr❛❝❡❥❛❞❛✲♣♦♥t✐❧❤❛❞❛ é ♣❛r❛ ♦ ❝❛s♦ 4πµBM = 3✳ ✳ ♣✳ ✺✼ ✷✹ ❈✉r✈❛ ❞❡ ❝♦♥t♦r♥♦ ♣❛r❛ ❛s ❡♥❡r❣✐❛s ❞❡ ♦♥❞❛s ❞❡ s♣✐♥ ♥✉♠❛ r❡❞❡ q✉❛❞r❛❞❛
❝♦♠ ✐♥t❡r❛çã♦ ❞✐♣♦❧❛r ♥♦ ♣❧❛♥♦ x−z✳ ❆ ❡♥❡r❣✐❛ ❝r❡s❝❡ ❞♦ ❛③✉❧ ❛té ♦
✈❡r♠❡❧❤♦ 0< E/J1 <10✿ ❛✮ 4πµBM = 0✱ ❜✮ 4πµBM = 1✱ ❝✮4πµBM =
2✱ ❞✮4πµBM = 3✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✾
✷✺ ❈✉r✈❛ ❞❡ ❝♦♥t♦r♥♦ ♣❛r❛ ❛s ❡♥❡r❣✐❛s ❞❡ ♦♥❞❛s ❞❡ s♣✐♥ ♥✉♠❛ r❡❞❡ q✉❛✲ ❞r❛❞❛ ❝♦♠ ✐♥t❡r❛çã♦ ❞✐♣♦❧❛r✳ ❆ ❡♥❡r❣✐❛ ❝r❡s❝❡ ❞♦ ❛③✉❧ ❛té ♦ ✈❡r♠❡❧❤♦ ♥♦ ✐♥t❡r✈❛❧♦ 0< E/J1 <10✳ ❖ ♣❛✐♥❡❧ ❞❛ ❡sq✉❡r❞❛qx = 0 ❡ qx =πa ♥♦
♣❛✐♥❡❧ ❞❛ ❞✐r❡✐t❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✾ ✷✻ ❘❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❞❛s ♦♥❞❛s ❞❡ s♣✐♥ ❝♦♠ ✐♥t❡r❛çã♦ ❞❡ s❡❣✉♥❞♦s ✈✐③✐✲
♥❤♦s ❡ ✐♥t❡r❛çã♦ ❞✐♣♦❧❛r ❛♦ ❧♦♥❣♦ ❞♦s ❝❛♠✐♥❤♦s∆✱Z✱ ❡Σ✳ ❆ ❧✐♥❤❛ s♦❧✐❞❛ 4πgµBM = 0✱ ❛ ❧✐♥❤❛ ♣♦♥t✐❧❤❛❞❛ é ♣❛r❛ 4πgµBM = 1✱ ♦ ❝❛s♦ ♦♥❞❡
4πgµBM = 2 é r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ tr❛❝❡❥❛❞❛ ❧♦♥❣❛✱ ❡ ❛ ❧✐♥❤❛ tr❛❝❡❥❛❞❛✲ ♣♦♥t✐❧❤❛❞❛ ✐♥❞✐❝❛ ♦ ❝❛s♦ ♦♥❞❡4πgµBM = 3✳ ❆ ❞✐r❡✐t❛ ❛ t❡♠♦s ❛ ❞✐❢❡r❡♥ç❛ ❞❡ ❡♥❡r❣✐❛ ❡♥tr❡ ❛s ♦♥❞❛s ❞❡ s♣✐♥ ♣❛r❛ 4πµBM = 0 ❡ 4πµBM = 1 ❛♦
❧♦♥❣♦ ❞♦s ❝❛♠✐♥❤♦s ∆✱ Z✱ ❡ Σ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✵
✷✼ ❆ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ❞❛s ♦♥❞❛s ❞❡ s♣✐♥ ❡♠ ✉♠❛ r❡❞❡ q✉❛❞r❛❞❛✳ ❆ ❝✉r✈❛ só❧✐❞❛ é ♣❛r❛ ✉♠ ❡s♣❡❝tr♦ s❡♠ ✐♥t❡r❛çã♦ ❞✐♣♦❧❛r✱ ❛ ❝✉r✈❛ ♣♦♥t✐❧❤❛❞❛ r❡♣r❡s❡♥t❛ ♦ ❝❛s♦ ♦♥❞❡ 4πµBM = 1✱❛ ❝✉r✈❛ tr❛❝❡❥❛❞❛ é ♣❛r❛ ♦ ❝❛s♦ ♦♥❞❡
4πµBM = 2✱ ❛ ❝✉r✈❛ tr❛❝❡❥❛❞❛✲♣♦♥t✐❧❤❛❞❛ é ♣❛r❛ ♦ ❝❛s♦ 4πµBM = 3✳ ✳ ♣✳ ✻✵ ✷✽ ❋✐t❛ ▼❛❣♥ét✐❝❛ ❋✐♥✐t❛ ❡♠ ✉♠❛ ❉✐r❡çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✶ ✷✾ ❆ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❞❡ ♦♥❞❛s ❞❡ s♣✐♥ ♣❛r❛ q✉❛tr♦ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s
❞❛ ✐♥t❡r❛çã♦ ❞❡ tr♦❝❛ ❞❡ s❡❣✉♥❞♦s ✈✐③✐♥❤♦s✿ ❛✮J2/J1 = 0✱ ❜✮J2/J1 = 0,1✱
❝✮J2/J1 = 0,2✱ ❡ J2/J1 = 0,3✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✸
✸✵ ❘❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❞❡ ♦♥❞❛s ❞❡ s♣✐♥ ❝♦♥tr❛ ❛ ✐♥t❡r❛çã♦ ❞❡ tr♦❝❛ ❞❡ s❡❣✉♥❞♦s ✈✐③✐♥❤♦s J2 ♣❛r❛ ✉♠❛ ✜t❛ q✉❛❞r❛❞❛ ❝♦♠ ✶✵ ❧✐♥❤❛s ❛tô♠✐❝❛s
✭◆❂✶✵✮✿ ❛✮qx = 0✱ ❜✮qx = 0,5πa✱ ❡ ❝✮qx =πa✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✹ ✸✶ ❆ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❞❡ ♦♥❞❛s ❞❡ s♣✐♥ ♣❛r❛ q✉❛tr♦ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❛
✐♥t❡♥s✐❞❛❞❡ ❞❛ ✐♥t❡r❛çã♦ ❞✐♣♦❧❛r✿ ❛✮4πM = 0✱ ❜✮4πM = 1✱ ❝✮4πM = 2✱
❞✮4πM = 3✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✻
✸✷ ❘❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❞❡ ♦♥❞❛s ❞❡ s♣✐♥ ❝♦♥tr❛ ❛ ✐♥t❡r❛çã♦ ❞❡ tr♦❝❛ ❞❡ s❡❣✉♥❞♦s ✈✐③✐♥❤♦s J2 ♣❛r❛ ✉♠❛ ✜t❛ q✉❛❞r❛❞❛ ❝♦♠ ✶✵ ❧✐♥❤❛s ❛tô♠✐❝❛s
✭◆❂✶✵✮✿ ❛✮qx = 0✱ ❜✮qx = 0,5πa✱ ❡ ❝✮qx =πa✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✻ ✸✸ ❘❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❞❡ ♦♥❞❛s ❞❡ s♣✐♥ ❝♦♥tr❛ ❛ ✐♥t❡r❛çã♦ ❞❡ tr♦❝❛ ❞❡
s❡❣✉♥❞♦s ✈✐③✐♥❤♦s J2 ♣❛r❛ ✉♠❛ ✜t❛ q✉❛❞r❛❞❛ ❝♦♠ ✶✵ ❧✐♥❤❛s ❛tô♠✐❝❛s
✭◆❂✶✵✮✿ ❛✮qx = 0✱ ❜✮qx = 0,5πa✱ ❡ ❝✮qx =πa✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✼
✸✹ ❆ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❞❡ ♦♥❞❛s ❞❡ s♣✐♥ ♣❛r❛ q✉❛tr♦ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❛ ✐♥t❡♥s✐❞❛❞❡ ❞❛ ✐♥t❡r❛çã♦ ❞✐♣♦❧❛r ♣❛r❛ ♦ s✐st❡♠❛ ♥♦ ♣❧❛♥♦x−z✿ ❛✮4πM =
0✱ ❜✮4πM = 1✱ ❝✮4πM = 2✱ ❞✮4πM = 3✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✾
✸✺ ❘❡❧❛çã♦ ❡♥tr❡ ❡♥❡r❣✐❛ ❡ ✐♥t❡r❛çã♦ ❞✐♣♦❧❛r ♣❛r❛ ✉♠❛ ✜t❛ q✉❛❞r❛❞❛ ❝♦♠ ✶✵ ❧✐♥❤❛s ❛tô♠✐❝❛s ✭◆❂✶✵✮ ♥♦ ♣❧❛♥♦ x−z✿ ❛✮qz = 0✱ ❜✮qz = 0,5πa✱ ❡ ❝✮
qz =πa✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✾ ✸✻ ❖s ♠♦❞♦s ❞❡ ♠❛✐s ❜❛✐①❛ ❡♥❡r❣✐❛ ♣❛r❛ ✉♠ s✐st❡♠❛ ♥♦ ♣❧❛♥♦ x−y ❧✐♥❤❛
só❧✐❞❛ ❡ ♥♦ ♣❧❛♥♦ x−z ❧✐♥❤❛ tr❛❝❡❥❛❞❛✳ ❆ ❝✉r✈❛ ♣❛r❛ ♦ ♣❧❛♥♦x−z ♣♦❞❡
♣♦❞❡ s❡r ✜t❛❞❛ ♣♦r ✉♠❛ ❢✉♥çã♦ ❧✐♥❡❛r ❝♦♠♦ ♠♦str❛ ❛ ❧❡❣❡♥❞❛✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✼✵ ✸✼ ❋✐t❛ ▼❛❣♥ét✐❝❛ ❋✐♥✐t❛ ❡♠ ✉♠❛ ❉✐r❡çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✼✵ ✸✽ ❈✉r✈❛ ❞❡ ❝♦♥t♦r♥♦ ♣❛r❛ ❛s ❡♥❡r❣✐❛s ❞❡ ♦♥❞❛s ❞❡ s♣✐♥ ❡♠ ✉♠❛ r❡❞❡ ❢❛✈♦ ❞❡
♠❡❧ ♣❛r❛ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❛ ✐♥t❡r❛çã♦ ❞❡ s❡❣✉♥❞♦s ✈✐③✐♥❤♦s ❛✮J2 = 0✱
❜✮J2 = 0,1J1✱ ❝✮ J2 = 0,2J1✱ ❡ ❞✮ J2 = 0,3J1✳ ❆ ❡♥❡r❣✐❛ ❝r❡s❝❡ ❞♦ ❛③✉❧
❛té ♦ ✈❡r♠❡❧❤♦ 2< E/J1 <6✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✼✹
✸✾ ❈✉r✈❛ ❞❡ ❝♦♥t♦r♥♦ ♣❛r❛ ❛s ❡♥❡r❣✐❛s ❞❡ ♦♥❞❛s ❞❡ s♣✐♥ ♥✉♠❛ r❡❞❡ ❢❛✈♦ ❞❡ ♠❡❧ s❡♠ ✐♥t❡r❛çã♦ ❞✐♣♦❧❛r✳ ❆ ❡♥❡r❣✐❛ ❝r❡s❝❡ ❞♦ ❛③✉❧ ❛té ♦ ✈❡r♠❡❧❤♦
2< E/J1 <6✳ ❖ ♣❛✐♥❡❧ ❞❛ ❞✐r❡✐t❛ qy = 0 ❡ qy =π/a♥♦ ♣❛✐♥❡❧ ❞❛ ❞✐r❡✐t❛✳ ♣✳ ✼✹ ✹✵ ❆ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ♣❛r❛ ❛s ♦♥❞❛s ❞❡ s♣✐♥ ❡♠ ✉♠❛ r❡❞❡ ❢❛✈♦ ❞❡ ♠❡❧
❛♦ ❧♦♥❣♦ ❞♦s ❝❛♠✐♥❤♦s Γ−K✱ K −M✱ M −Γ ♣❛r❛ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s
❞❛ ✐♥t❡r❡❛çã♦ ❞❡ s❡❣✉♥❞♦s ✈✐③✐♥❤♦s J2✳ ◆♦ ♣❛✐♥❡❧ ❞❛ ❞✐r❡✐t❛ t❡♠♦s ❛
❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦s ♠♦❞♦s ❝♦♠ J2 = 0 ❡ J2 = 0,1J1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✼✺
✹✶ ❆ ❞❡♥s✐❞❛❞❡ ❞❡ ❡st❛❞♦s σ ♣❛r❛ ❛ r❡❞❡ ❢❛✈♦ ❞❡ ♠❡❧ ✐♥✜♥✐t❛ s❡♠ ✐♥t❡r❛çã♦
❞✐♣♦❧❛r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✼✺ ✹✷ ❈✉r✈❛ ❞❡ ❝♦♥t♦r♥♦ ♣❛r❛ ❛s ❡♥❡r❣✐❛s ❞❡ ♦♥❞❛s ❞❡ s♣✐♥ ♣❛r❛ ✉♠❛ r❡❞❡ ❢❛✈♦
❞❡ ♠❡❧✿ ❛✮ 4πM = 0✱ ❜✮4πM = 1✱ ❝✮4πM = 2✱ ❞✮ 4πM = 3✳ ❆ ❡♥❡r❣✐❛
❝r❡s❝❡ ❞♦ ❛③✉❧ ❛té ♦ ✈❡r♠❡❧❤♦ 2< E/J1 <6✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✽✸
✹✸ ❈✉r✈❛ ❞❡ ❝♦♥t♦r♥♦ ♣❛r❛ ❛s ❡♥❡r❣✐❛s ❞❡ ♦♥❞❛s ❞❡ s♣✐♥ ♣❛r❛ ✈❛❧♦r❡s ✜①♦s ❞❡
qy✳ ❖ ♣❛✐♥❡❧ ❞❛ ❞✐r❡✐t❛ é ♣❛r❛qy = 0✱ ❡♥q✉❛♥t♦ q✉❡ ♦ ♣❛✐♥❡❧ ❞❛ ❡sq✉❡r❞❛
é ♣❛r❛ qy = 2π/3a✳ ❆ ❡♥❡r❣✐❛ ❝r❡s❝❡ ❞♦ ❛③✉❧ ❛té ♦ ✈❡r♠❡❧❤♦ 2< E/J1 <6✳ ♣✳ ✽✸
✹✹ ❆ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ♣❛r❛ ❛s ♦♥❞❛s ❞❡ s♣✐♥ ❡♠ ✉♠❛ r❡❞❡ ❢❛✈♦ ❞❡ ♠❡❧ ❛♦ ❧♦♥❣♦ ❞♦s ❝❛♠✐♥❤♦s Γ−K✱ K−M✱ M−Γ ♣❛r❛ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❛
✐♥t❡r❡❛çã♦ ❞✐♣♦❧❛r MS✳ ◆♦ ♣❛✐♥❡❧ ❞❛ ❞✐r❡✐t❛ t❡♠♦s ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦s
♠♦❞♦s ❝♦♠ MS = 0 ❡ MS = 1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✽✹ ✹✺ ❆ ❞❡♥s✐❞❛❞❡ ❞❡ ❡st❛❞♦sσ ♣❛r❛ ❛ r❡❞❡ ❢❛✈♦ ❞❡ ♠❡❧ ✐♥✜♥✐t❛ ❝♦♠ ✐♥t❡r❛çã♦
❞✐♣♦❧❛r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✽✹ ✹✻ ❆ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❞❡ ♦♥❞❛s ❞❡ s♣✐♥ ♣❛r❛ q✉❛tr♦ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s
❞❛ ❞❡ s❡❣✉♥❞♦s ✈✐③✐♥❤♦s ♣❛r❛ ✉♠❛ ✜t❛ ❝♦♠ N = 10 ❧✐♥❤❛s ❛tô♠✐❝❛s✿
❛✮J2 = 0✱ ❜✮J2 = 0,1J1✱ ❝✮J2 = 0,2J1✱ ❞✮J2 = 0,3J1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✽✻
✹✼ ❆ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❞❡ ♦♥❞❛s ❞❡ s♣✐♥ ♣❛r❛ q✉❛tr♦ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❛ ❞❡ s❡❣✉♥❞♦s ✈✐③✐♥❤♦s ♣❛r❛ ✉♠❛ ✜t❛ ❝♦♠ N = 11 ❧✐♥❤❛s ❛tô♠✐❝❛s✿
❛✮J2 = 0✱ ❜✮J2 = 0,1J1✱ ❝✮J2 = 0,2J1✱ ❞✮J2 = 0,3J1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✽✼
✹✽ ❆ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❞❡ ♦♥❞❛s ❞❡ s♣✐♥ ✈❛r✐❛♥❞♦ ❛ ✐♥t❡♥s✐❞❛❞❡ ❞❛ ✐♥t❡✲ r❛çã♦ ❞❡ tr♦❝❛ ❞❡ s❡❣✉♥❞♦s ✈✐③✐♥❤♦s✿ ❛✮N = 10 ❡ qx = 0✱ ❜✮N = 11 ❡
qx0✱ ❝✮N = 10 ❡ qx = 2π/3a✱ ❞✮N = 11 ❡ qx = 2π/3a✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✽✼ ✹✾ ❆ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❞❡ ♦♥❞❛s ❞❡ s♣✐♥ ♣❛r❛ q✉❛tr♦ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s
❞❛ ✐♥t❡♥s✐❞❛❞❡ ❞❛ ✐♥t❡r❛çã♦ ❞✐♣♦❧❛r✿ ❛✮MS = 0✱ ❜✮MS = 1✱ ❝✮MS = 2✱
❞✮MS = 3✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✾✵ ✺✵ ❆ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❞❡ ♦♥❞❛s ❞❡ s♣✐♥ ♣❛r❛ q✉❛tr♦ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s
❞❛ ✐♥t❡♥s✐❞❛❞❡ ❞❛ ✐♥t❡r❛çã♦ ❞✐♣♦❧❛r✿ ❛✮MS = 0✱ ❜✮MS = 1✱ ❝✮MS = 2✱
❞✮MS = 3✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✾✵ ✺✶ ❆ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❞❡ ♦♥❞❛s ❞❡ s♣✐♥ ♣❛r❛ q✉❛tr♦ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s
❞❛ ✐♥t❡♥s✐❞❛❞❡ ❞❛ ✐♥t❡r❛çã♦ ❞✐♣♦❧❛r✿ ❛✮N = 10❡ qx = 0✱ ❜✮N = 11 ❡qx0✱
❝✮N = 10 ❡ qx = 2π/3a✱ ❞✮N = 11 ❡ qx = 2π/3a✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✾✶
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✶ ❈♦♠♣♦♥❡♥t❡s ❡ ❆✉t♦❡st❛❞♦s ❞❡ ❙♣✐♥✲1
✶
✶ ■♥tr♦❞✉çã♦ ●❡r❛❧✳
❆ ❝✐ê♥❝✐❛ é r❡s♣♦♥sá✈❡❧ ♣♦r ❣r❛♥❞❡ ♣❛rt❡ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦ ❤♦♠❡♠ ❛ ❝❡r❝❛ ❞❛ ♥❛✲ t✉r❡③❛✱ ❡ ❡ss❛ ❝♦♥q✉✐st❛ é ❢r✉t♦ ❞❡ sé❝✉❧♦s ❞❡ ♣❡sq✉✐s❛ ♦ q✉❡ ♥♦s r❡♠❡t❡ às ❝✐✈✐❧✐③❛çõ❡s ❛♥t✐❣❛s✳ ❚♦❞♦ ❡ss❡ ❡♥t❡♥❞✐♠❡♥t♦ ❤♦❥❡ ❝♦♠♣õ❡ q✉❛tr♦ ❣r❛♥❞❡s ár❡❛s ❞❛ ❢ís✐❝❛✿ ❊❧❡tr♦♠❛❣✲ ♥❡t✐s♠♦✱ ●r❛✈✐t❛❝✐♦♥❛❧✱ ◆✉❝❧❡❛r ❋♦rt❡ ❡ ❋r❛❝❛✳ ❍á ❛❧❣✉♠ t❡♠♣♦ tê♠✲s❡ t❡♥t❛❞♦ ❡❧❛❜♦r❛r ✉♠❛ ✧t❡♦r✐❛ ❞❡ t✉❞♦✧q✉❡ s❡r✐❛ r❡❣✐❞❛ ♣♦r ✉♠❛ s✉♣❡r✲❢♦rç❛ q✉❡ ❣♦✈❡r♥❛r✐❛ ❛ ♥❛t✉r❡③❛✳ ❊st❛s t❡♥t❛t✐✈❛s ❛té ♦s ❞✐❛s ❛t✉❛✐s ❛✐♥❞❛ ♥ã♦ ❧♦❣r❛r❛♠ ê①✐t♦ ❞❡✈✐❞♦ ❛ ❛❧❣✉♠❛s q✉❡stõ❡s ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❛s ❝♦♠♣❧✐❝❛çõ❡s ♥❛ ❡❧❛❜♦r❛çã♦ ❞❡ ✉♠❛ t❡♦r✐❛ q✉â♥t✐❝❛ ❣r❛✈✐t❛❝✐♦♥❛❧✳
❊♠❜♦r❛ ❛✐♥❞❛ ♥ã♦ t❡♥❤❛♠♦s ♦❜t✐❞♦ s✉❝❡ss♦ ♥❛ ❢♦r♠✉❧❛çã♦ ❞❡ t❛❧ t❡♦r✐❛✱ ♦s ❛✈❛♥ç♦s ♦❜t✐❞♦s t❡♠ ❡❧❡✈❛❞♦ ♦ ❤♦♠❡♠ ❛ ♣❛t❛♠❛r❡s ❝❛❞❛ ✈❡③ ♠❛✐s ♣r♦❢✉♥❞♦s ♥♦ t♦❝❛♥t❡ à ❝♦♠♣r❡✲ ❡♥sã♦ ❞♦ ♠✉♥❞♦ ♠❛❝r♦ ❡ ♠✐❝r♦ q✉❡ ♥♦s r♦❞❡✐❛❀ r❡s✉❧t❛♥❞♦ ❡♠ ❛✈❛♥ç♦s t❡❝♥♦❧ó❣✐❝♦s q✉❡ ✈❡♠ ❛ ♣❛ss♦s ❧❛r❣♦s ♦t✐♠✐③❛♥❞♦ ❛ ✈✐❞❛ s♦❝✐❛❧✳ ❆ ❡①❡♠♣❧♦ ❞✐ss♦ ♣♦❞❡♠♦s ❝✐t❛r ♦ ♠❛❣♥❡✲ t✐s♠♦✱ ♦ q✉❛❧ ♣♦ss✉✐ ♣♦s✐çã♦ ❞❡ ❞❡st❛q✉❡ ♥♦ ❝❡♥ár✐♦ t❡❝♥♦❧ó❣✐❝♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥❛ ❡r❛ ❞❛ ✐♥❢♦r♠❛t✐③❛çã♦ ❡♠ q✉❡ ✈✐✈❡♠♦s✳
❖ ♠❛❣♥❡t✐s♠♦ ❢♦✐ ♦❜s❡r✈❛❞♦ ❛ sé❝✉❧♦s ❛♥t❡s ❞❡ ❈r✐st♦✱ ❡ t❡♠ ❞❡s❞❡ ❡♥tã♦ s✐❞♦ ❢♦❝♦ ❞❡ ♠✉✐t❛s ❞❡s❝♦❜❡rt❛s r❡s♣♦♥sá✈❡✐s ♣♦r ✉♠❛ sér✐❡ ❞❡ ❝♦♥q✉✐st❛s ❡♠ ❞✐✈❡rs♦s r❛♠♦s ❝♦♠♦ ♥❛ ♠❡❞✐❝✐♥❛✱ ✐♥❢♦r♠át✐❝❛✱ t❡❧❡❝♦♠✉♥✐❝❛çõ❡s✱ ❡t❝✳ ❊ss❡ ❣r❛♥❞❡ ❧❡q✉❡ ❞❡ ❛♣❧✐❝❛çõ❡s s❡ ❞❡✈❡ ❛♦s ♠❛t❡r✐❛✐s ♠❛❣♥ét✐❝♦s q✉❡ ❛♣r❡s❡♥t❛♠ ♣r♦♣r✐❡❞❛❞❡s ✐♠♣♦rt❛♥t❡s ❡ ❝♦♠ ✜♥❛❧✐❞❛❞❡s ❜❡♠ ❞❡✜♥✐❞❛s✳
❚♦❞❛ ♠❛tér✐❛ é ❝♦♠♣♦st❛ ♣♦r át♦♠♦s q✉❡ ♣♦ss✉❡♠ ♣❛rtí❝✉❧❛s ❢✉♥❞❛♠❡♥t❛✐s ❞❡t❡♥t♦r❛s ❞❡ ♣r♦♣r✐❡❞❛❞❡s ✐♥trí♥s❡❝❛s ❝♦♠♦ ❝❛r❣❛✱ ♠❛ss❛ ❡ s♣✐♥✳ ❆❧❣✉♥s ❞❡ss❡s át♦♠♦s ❛♣r❡s❡♥t❛♠ ❝❛r❛❝t❡ríst✐❝❛s ♠❛❣♥ét✐❝❛s ♣r♦✈❡♥✐❡♥t❡s ❞❛ ♣r❡s❡♥ç❛ ❞❡ ♠♦♠❡♥t♦s ❞❡ ❞✐♣♦❧♦ ♠❛❣♥ét✐❝♦s✱ ♦s q✉❛✐s sã♦ r❡s♣♦♥sá✈❡✐s ♣❡❧❛ ❢♦r♠❛çã♦ ❞❡ ❝❛♠♣♦s ♠❛❣♥ét✐❝♦s✳ ❆ r❡s♣♦st❛ ❞❡ss❡s ♠❛t❡r✐❛✐s ❛♦s ❝❛♠♣♦s ♠❛❣♥ét✐❝♦s ❡①t❡r♥♦s é ❢r✉t♦ ❞❛ ♣r❡s❡♥ç❛ ❞❡ss❡s ❞✐♣♦❧♦s ♠❛❣♥ét✐❝♦s ❡ ❞❡ s✉❛s ✐♥t❡r❛çõ❡s❀ ❡ ❝♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❛ ✈❛st❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❞❡ss❡s ♠❛t❡r✐❛✐s s❡ ❞❡✈❡ ❛♦ t✐♣♦ ❞❡ r❡❛çã♦ ❛ ❡ss❡ ❝❛♠♣♦✱ ❛❧é♠ ❞❡ ♦✉tr♦s ❢❛t♦r❡s ♣♦st❡r✐♦r♠❡♥t❡ ❞✐s❝✉t✐❞♦s✳
✶ ■♥tr♦❞✉çã♦ ●❡r❛❧✳ ✷
♠❛t❡r✐❛✐s ❡♠ ♣❛r❛♠❛❣♥ét✐❝♦s✱ q✉❛♥❞♦ ♦s ♠♦♠❡♥t♦s✱ ❛ ♣r✐♦r✐✱ ❡stã♦ ❞❡s♦r✐❡♥t❛❞♦s ❡ s❡ ❛❧✐♥❤❛♠ ❝♦♠ ♦ ❝❛♠♣♦ ❡①t❡r♥♦✱ ❢♦rç❛♥❞♦ ♦ ✈❡t♦r ♠❛❣♥❡t✐③❛çã♦ ❛ t❡r ♠❡s♠❛ ♦r✐❡♥t❛çã♦✳ ❖s ❞✐❛♠❛❣♥ét✐❝♦s✱ q✉❛♥❞♦ ♦s ❞✐♣♦❧♦s s❡ ❛❧✐♥❤❛♠ ❞❡ ❢♦r♠❛ ❛♥t✐♣❛r❛❧❡❧❛ ❛♦ ❝❛♠♣♦ ❡ sã♦ ❢r❛❝❛♠❡♥t❡ ❛tr❛í❞♦s ♣♦r ❡❧❡✳ ❆❧é♠ ❞✐ss♦✱ ❤á ❛q✉❡❧❡s ❡♠ q✉❡ ♦s ❞✐♣♦❧♦s ♠❛❣♥ét✐❝♦s ❛♣r❡✲ s❡♥t❛♠ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ✧❝♦♥❣❡❧❛♠❡♥t♦✧♥❛ ❛✉sê♥❝✐❛ ❞♦ ❝❛♠♣♦ q✉❡ ♦s ❛❧✐♥❤♦✉✱ ❡st❡s ❞❡♥♦♠✐♥❛❞♦s ❢❡rr♦♠❛❣♥ét✐❝♦s✳
❖s ❛♥t✐❢❡rr♦♠❛❣♥❡t♦s ❝♦rr❡s♣♦♥❞❡♠ ❛ ♦✉tr♦ ❡①❡♠♣❧♦ ❞❡ ♠❛t❡r✐❛✐s ♠❛❣♥ét✐❝♦s ❡♠ q✉❡ ♦ ❛❧✐♥❤❛♠❡♥t♦ ❞♦s ❞✐♣♦❧♦s ♦❝♦rr❡ ❞❡ ❢♦r♠❛ ❛♥t✐♣❛r❛❧❡❧❛✱ ♦✉ s❡❥❛✱ ♦s ♠♦♠❡♥t♦s s❡ ❛❧✐♥❤❛♠ ♥❛ ♠❡s♠❛ ❞✐r❡çã♦ ❡ s❡♥t✐❞♦s ♦♣♦st♦s✱ ❡ ❝♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞✐ss♦ ❛ ♠❛❣♥❡t✐③❛çã♦ ♠❛❝r♦s✲ ❝ó♣✐❝❛ é ♥✉❧❛✳ ❙✐♠✐❧❛r♠❡♥t❡ t❡♠♦s ✉♠ ♠❛t❡r✐❛❧ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❢❡rr✐♠❛❣♥❡t♦s✱ ♦s q✉❛✐s ❛♣r❡s❡♥t❛♠ ❞♦✐s t✐♣♦s ❞✐❢❡r❡♥t❡s ❞❡ í♦♥s ♠❛❣♥ét✐❝♦s ✭♣♦r ❡①❡♠♣❧♦ N i++ ❡ F e+++✮ ♥❡✲
❧❡s ❡①✐st✐♥❞♦ ✐♥t❡r❛çõ❡s ❡❧❡tr♦♠❛❣♥ét✐❝❛s s✐♠✐❧❛r❡s ❛♦ ❛♥t✐❢❡rr♦♠❛❣♥❡t✐s♠♦✱ ♣♦ré♠✱ ♣❡❧❛ ♣r❡s❡♥ç❛ ❞♦s í♦♥s ❝♦♠ ♠ó❞✉❧♦s ❞✐❢❡r❡♥t❡s ♥♦ ♠♦♠❡♥t♦ ❞✐♣♦❧♦ ♠❛❣♥ét✐❝♦✱ ❛ ♠❛❣♥❡t✐③❛✲ çã♦ ♠❛❝r♦s❝ó♣✐❝❛ ♥ã♦ é ♥✉❧❛✳ ❆❧é♠ ❞❡st❡s✱ ♣♦❞❡♠♦s ❝✐t❛r ❛✐♥❞❛ ♦s ♠❡t❛♠❛❣♥❡t♦s q✉❡ ❝♦rr❡s♣♦♥❞❡♠ ❛ ❝♦♠♣♦st♦s q✉❡ ♦r❛ ❛♣r❡s❡♥t❛♠ ❝♦♠♣♦rt❛♠❡♥t♦ ❢❡rr♦♠❛❣♥ét✐❝♦ ♦r❛ ❛♥t✐✲ ❢❡rr♦♠❛❣♥ét✐❝♦✳
❖ ✈❡t♦r ♠❛❣♥❡t✐③❛çã♦ q✉❡ ❢♦r❛ ❝✐t❛❞♦ ❛ss✉♠❡ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ♥♦ ❡❧❡tr♦♠❛❣♥❡t✐s♠♦ ❝❧áss✐❝♦✱ ❡①♣r❡ss❛♥❞♦ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ♠♦♠❡♥t♦s ❞❡ ❞✐♣♦❧♦ ♠❛❣♥ét✐❝♦s ♣❡r♠❛♥❡♥t❡s ♦✉ ✐♥❞✉③✐❞♦s ❡♠ ✉♠ ♠❛t❡r✐❛❧ ♠❛❣♥ét✐❝♦✳ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞✐ss♦✱ ❛ ♠❛❣♥❡t✐③❛çã♦ ❞❡s❝r❡✈❡ ❝♦♠♦ ✉♠ ♠❛t❡r✐❛❧ r❡s♣♦♥❞❡ ❛ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦✱ ❛❧é♠ ❞❡ q✉❡ ♣♦❞❡ s❡r ✉s❛❞❛ ♣❛r❛ ❝❛❧❝✉❧❛r ❛s ❢♦rç❛s q✉❡ r❡s✉❧t❛♠ ❞❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ♠❛t❡r✐❛❧ ❡ ❝❛♠♣♦ ❡①t❡r♥♦ ❛♣❧✐❝❛❞♦✳ ❉❡ss❛ ❢♦r♠❛ ♣♦❞❡♠♦s ❛ss♦❝✐❛r ❛ ♠❛❣♥❡t✐③❛çã♦ ❛♦ s❡✉ ❛♥á❧♦❣♦ ❡❧étr✐❝♦✱ ❛ ♣♦❧❛r✐③❛çã♦ q✉❡ é ❛ ♠❡❞✐❞❛ ❞❛ r❡❛çã♦ ❞❡ ✉♠ ♠❛t❡r✐❛❧ ❝♦rr❡s♣♦♥❞❡♥t❡ à ❛♣❧✐❝❛çã♦ ❞❡ ✉♠ ❝❛♠♣♦ ❡❧étr✐❝♦ ❡♠ ❡❧❡tr♦stát✐❝❛✳ ▼❛t❡♠❛t✐❝❛♠❡♥t❡ ♣♦❞❡♠♦s ❡s❝r❡✈❡r
M = lim
∆V→0 N m
∆V ,
♦♥❞❡ ◆ é ♦ ♥ú♠❡r♦ ❞❡ ♠♦♠❡♥t♦s ♠❛❣♥ét✐❝♦s ✭m✮ ♥❛ ❛♠♦str❛ ❡ ❱ ♦ ✈♦❧✉♠❡✳
✶ ■♥tr♦❞✉çã♦ ●❡r❛❧✳ ✸
✐♥t❡r❡ss❛❞♦s ❡♠ ❝♦♠♣r❡❡♥❞❡r ♦ ♣♦rq✉❡ ❞♦ ❞❡s❞♦❜r❛♠❡♥t♦ ❞❛s ❧✐♥❤❛s ❡s♣❡❝tr❛✐s ♥♦ át♦♠♦ ❞❡ ❤✐❞r♦❣ê♥✐♦ ❡ át♦♠♦s ❛❧❝❛❧✐♥♦s✱ ♦ q✉❡ ❡r❛ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❡str✉t✉r❛ ✜♥❛✳
❊ss❛ ❡str✉t✉r❛ ✜♥❛ ❤❛✈✐❛ s✐❞♦ tr❛t❛❞❛ ♣♦r ❙♦♠♠❡r❢❡❧❞✱ ♦ q✉❛❧ ✉t✐❧✐③❛♥❞♦ ♦ ♠♦❞❡❧♦ ❞❡ ❇♦❤r ❝❤❡❣♦✉ à ❝♦♥❝❧✉sã♦ ❞❡st❛ s❡r ♦ ❞❡s❞♦❜r❛♠❡♥t♦ ❞♦s ♥í✈❡✐s ❞❡ ❡♥❡r❣✐❛ ❛tô♠✐❝♦s r❡s✉❧✲ t❛♥t❡s ❞❡ ✉♠❛ ♣❡q✉❡♥❛ ❝♦♥tr✐❜✉✐çã♦✭10−4✮ à ❡♥❡r❣✐❛ t♦t❛❧✱ ❛❞✈✐♥❞❛ ❞❛ ✈❛r✐❛çã♦ r❡❧❛t✐✈❛
❞❛ ♠❛ss❛ ❞♦ ❡❧étr♦♥ ❞❡✈✐❞❛ à s✉❛ ✈❡❧♦❝✐❞❛❞❡ r❡❧❛t✐✈íst✐❝❛ ❡♠ t♦r♥♦ ❞♦ ♥ú❝❧❡♦ ❞♦ át♦♠♦ ❞❡ ❍✐❞r♦❣ê♥✐♦✳ P♦ré♠✱ t❛❧ t❡♦r✐❛ ♥ã♦ ❧♦❣r♦✉ ê①✐t♦ ♥❛ ❡①♣❧✐❝❛çã♦ ❞❡ t❛✐s ❡❢❡✐t♦s ❡♠ át♦♠♦s ❛❧❝❛❧✐♥♦s✱ r❡s✉❧t❛♥❞♦ ❡♠ s✉s♣❡✐t❛s q✉❛♥t♦ à ✈❛❧✐❞❛❞❡ ❞❡ s✉❛ t❡♦r✐❛ ♣❛r❛ ❛ ♦r✐❣❡♠ ❞❡ss❛ ❡str✉t✉r❛ ✜♥❛✳
◆❡ss❡ í♥t❡r✐♠✱ ❞❡♥tr❡ ♦✉tr❛s t❡♥t❛t✐✈❛s✱ ●♦✉❞s♠✐t ❡ ❯❤❧❡♥❜❡❝❦ ♣r♦♣✉s❡r❛♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❡ ♠♦♠❡♥t♦ ❞❡ ❞✐♣♦❧♦ ♠❛❣♥ét✐❝♦✱ ❝✉❥❛s ❝♦♠♣♦♥❡♥t❡s z s❡r✐❛♠
❡s♣❡❝✐✜❝❛❞❛s ♣♦r ✉♠ q✉❛rt♦ ♥ú♠❡r♦ q✉â♥t✐❝♦ ms✱ q✉❡ ❛ss✉♠✐r✐❛ ❞♦✐s ✈❛❧♦r❡s ±12✳ ❊st❡
❡st❛r✐❛ ❛ss♦❝✐❛❞♦ ♥ã♦ ❛♦ ❝♦♥❤❡❝✐❞♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ♦r❜✐t❛❧✱ ♠❛s s✐♠✱ ❛ ✉♠ ♥♦✈♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❝❤❛♠❛❞♦ s♣✐♥✱ q✉❡ ❤❛✈❡r✐❛ ❞❡ s❡r ❝♦♠♣r♦✈❛❞♦ ❡♠♣✐r✐❝❛♠❡♥t❡ ❡♠ ✶✾✷✼ ♣♦r P❤✐♣♣s ❡ ❚❛②❧♦r q✉❛♥❞♦ ❛tr❛✈és ❞❛ té❝♥✐❝❛ ❞❡ ❙t❡r♥✲●❡r❧❛❝❤ ✉t✐❧✐③❛♥❞♦ át♦♠♦s ❞❡ ❤✐❞r♦❣ê♥✐♦ ✭♥♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧✮ ✈❡r✐✜❝❛r❛♠ s❡r ❡st❡ ✉♠ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r r❡❛❧ ✐♥trí♥s❡❝♦ ❛♦ ❡❧étr♦♥✳ ❈♦♠♦ ♣♦❞❡♠♦s ✈❡r✱ ❡st❛s ❝♦♥❝❧✉sõ❡s ❢♦r❛♠ ❝♦♥t❡♠♣♦râ♥❡❛s à ❞❡s❝♦❜❡rt❛ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ ❡ ❛ ✐❞❡✐❛ ❞♦ s♣✐♥ tr❛r✐❛ ❡❢❡✐t♦s s❡♠ ❛♥á❧♦❣♦ ❝❧áss✐❝♦✳ P♦rt❛♥t♦ ❛s ✐♥✢✉ê♥❝✐❛s ❞♦ s♣✐♥ ♣♦❞❡♠ s❡r ❞❡s❝r✐t❛s ❛tr❛✈és ❞❛s t❡♦r✐❛s ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ❝♦♠♦ ❛ ❞❡ ❙❝❤rö❞✐♥❣❡r✱ q✉❡ ♣❡❧❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r✲P❛✉❧✐✱ [ 1
2m(σ·(p−qA))
2 +qφ]|ψi = i~∂
t|ψi✱ r❡✈❡❧❛ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✉♠ s✐st❡♠❛ ❝♦♠♣♦st♦ ♣♦r ♣❛rtí❝✉❧❛s ❞❡ s♣✐♥ ✭1
2✮ ✐♥t❡r❛❣✐♥❞♦ ❝♦♠ ♦
❝❛♠♣♦ ❡❧❡tr♦♠❛❣♥ét✐❝♦ ❡♠ ✉♠ r❡❣✐♠❡ ♥ã♦ r❡❧❛t✐✈íst✐❝♦✳ ❏á ❡♠ ❡s❝❛❧❛s ❞❡ ❛❧t❛s ❡♥❡r❣✐❛s ❛ t❡♦r✐❛ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ r❡❧❛t✐✈íst✐❝❛✱ ❡❧❛❜♦r❛❞❛ ♣♦r ❉✐r❛❝ ❡♠ 1928✱ ❛ss✉♠❡ s❡r ✉♠❛
❢♦r♠✐❞á✈❡❧ ❢❡rr❛♠❡♥t❛ ❞❡ ♣❡sq✉✐s❛✳
P❡❧♦ q✉❡ ❢♦✐ ❞❡s❝r✐t♦ ❛❝✐♠❛✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ❛ ✐♥tr♦❞✉çã♦ ❞❡ss❡ ♥♦✈♦ ♥ú♠❡r♦ q✉â♥t✐❝♦ ❛♣r❡s❡♥t❛ ❡❢❡✐t♦s q✉❡ ❞❡ ♠❛♥❡✐r❛ ♥❡♥❤✉♠❛ sã♦ s✉t✐s✱ ❝♦♠♦ ❡①❡♠♣❧♦ ❛ ♣ró♣r✐❛ ♠❛❣♥❡t✐③❛çã♦ q✉❡ é ❢r✉t♦ ❞❛ ♣r❡❝❡ssã♦ ❡♠ ♠❡s♠❛ ❢r❡q✉ê♥❝✐❛ ❡ ❢❛s❡ ❞♦s ♠♦♠❡♥t♦s ✐♥❞✐✈✐✲ ❞✉❛✐s✱ ♦✉ s❡❥❛✱ ❞♦s s♣✐♥s✳
✶ ■♥tr♦❞✉çã♦ ●❡r❛❧✳ ✹
♠❛t❡r✐❛✐s ♠❛❣♥ét✐❝♦s sã♦ ✉t✐❧✐③❛❞♦s❀ ❞❛í ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ s❡ ❝♦♥❤❡❝❡r ♠❛✐s s♦❜r❡ ❡ss❛s ❡①❝✐t❛çõ❡s ❝♦❧❡t✐✈❛s✱ ♦ q✉❡ é ✉♠ ❞♦s ❢♦❝♦s ❞❡ ❡st✉❞♦ ❞❛ s♣✐♥trô♥✐❝❛✳
❆ s♣✐♥trô♥✐❝❛ é ✉♠❛ t❡❝♥♦❧♦❣✐❛ q✉❡ ✈❡♠ ❡①♣❧♦r❛♥❞♦ ♦ ♣❛♣❡❧ ❞♦ s♣✐♥ ❡❧❡trô♥✐❝♦ ❡✴♦✉ ♥✉❝❧❡❛r ♥❛ ❢ís✐❝❛ ❞♦ ❡st❛❞♦ só❧✐❞♦✱ ❡♠ ♦✉tr❛s ♣❛❧❛✈r❛s ♣♦❞❡♠♦s ❞✐③❡r q✉❡ é ✉♠❛ ❡❧❡trô♥✐❝❛ ❞♦s s♣✐♥s✱ q✉❡ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❝♦♥tr♦❧❛r ♦ ✢✉①♦ ❞❡ ❝♦rr❡♥t❡ ❡♠ ✉♠ ❞✐s♣♦s✐t✐✈♦ ♥ã♦ s♦♠❡♥t❡ ♣❡❧❛ ❝❛r❣❛ ❞♦s ❡❧étr♦♥s✱ ♠❛s t❛♠❜é♠ ♣❡❧♦ ♦ s♣✐♥✳ ❆ ❝❛r❣❛ ❞♦ ❡❧étr♦♥ é ❛❢❡t❛❞❛ ♣❡❧❛ ❛çã♦ ❞❡ ❝❛♠♣♦s ❡❧étr✐❝♦s✱ ♠❛s ♦ s♣✐♥ é ♣❡rt✉r❜❛❞♦ ♣♦r ❝❛♠♣♦s ♠❛❣♥ét✐❝♦s✳ ❊ss❛ ♥♦✈❛ ♣r♦♣♦st❛ ♣♦❞❡rá ♣r♦❞✉③✐r ❞✐s♣♦s✐t✐✈♦s ♠❛✐s rá♣✐❞♦s ❡ q✉❡ ❞✐ss✐♣❡♠ ♠❡♥♦s ❡♥❡r❣✐❛✳
❆ ❡①❡♠♣❧♦ ♣♦❞❡♠♦s ❝✐t❛r ❛ ❡❧❛❜♦r❛çã♦ ❞❡ ❞✐s♣♦s✐t✐✈♦ s♣✐♥trô♥✐❝♦ ❜❛s❡❛❞♦ ♥♦ ❙✐❧í❝✐♦ ✭✸✮✱ ✈✐❞❡ ❋✐❣✉r❛ ✶✱ ♦ q✉❛❧ ❥á t❡♠ s✐❞♦ ✉t✐❧✐③❛❞♦ ♥❛ ❡❧❛❜♦r❛çã♦ ❞❡ ❝♦♠♣✉t❛❞♦r❡s ❡♠ q✉❡ ❛s ✐♥❢♦r♠❛çõ❡s sã♦ ❛r♠❛③❡♥❛❞❛s ♥♦ s♣✐♥✳ ❆❧é♠ ❞✐ss♦ ♣❡sq✉✐s❛s r❡❝❡♥t❡s ✭✹✮ ✈❡♠ ❡①♣❧♦r❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ tr❛♥s♣♦rt❡ ❞❛s ♦♥❞❛s ❞❡ s♣✐♥ ❡♠ ❝r✐st❛✐s✱ ✈✐s❛♥❞♦ ❛♣❧✐❝❛çõ❡s ❝♦♠♦ ❛ r❡t✐r❛❞❛ ❞❡ ❝❛❧♦r ❞♦ ✐♥t❡r✐♦r ❞♦s ♣r♦❝❡ss❛❞♦r❡s ❡❧❡trô♥✐❝♦s ❝♦♥✈❡♥❝✐♦♥❛✐s✱ ❡st✉❞♦ ❡st❡ ❢♦❝♦ ❞❛ ❝❛❧♦r✐trô♥✐❝❛✱ r❛♠♦ ❡♠❡r❣❡♥t❡ ❞❛ s♣✐♥trô♥✐❝❛✳
❋✐❣✉r❛ ✶✿ ❈♦♠♣♦♥❡♥t❡ ❙♣✐♥trô♥✐❝♦ ❞❡ ❙✐❧í❝✐♦✳
❆ ❞❡s❝♦❜❡rt❛ ❞❡ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ✐♥✉s✐t❛❞❛ ❝♦♠♦ ♦ s♣✐♥ ❧❡✈♦✉ ❛ ✉♠❛ ♠❡❧❤♦r ❝♦♠♣r❡✲ ❡♥sã♦ ❞♦s ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s ❡ ❡st❡s ♣❡r♠✐t✐r❛♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ♥♦✈❛s ❢♦r♠❛s ❞❡ ♣r♦❝❡ss❛r✱ ♠❛♥✐♣✉❧❛r ❡ ❛r♠❛③❡♥❛r ✐♥❢♦r♠❛çõ❡s✳ ❊♠ t❡❧❡❝♦♠✉♥✐❝❛çõ❡s ♣♦❞❡♠♦s ❝✐t❛r ❛ ❝r✐❛çã♦ ❞❡ ❞✐s♣♦s✐t✐✈♦s ❞❡ ❡♠✐ssã♦ ❞❡ ♦♥❞❛s ❡❧❡tr♦♠❛❣♥ét✐❝❛s ❡ ♠❛✐s r❡❝❡♥t❡♠❡♥t❡ ❛ ✉t✐❧✐③❛çã♦ ❞♦ s♣✐♥ ♥♦ ♣r♦❝❡ss❛♠❡♥t♦ ❞❡ ✐♥❢♦r♠❛çõ❡s q✉❡ ♣❡r♠✐t✐rá ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ♥♦✈♦s ❛❧❣♦r✐t♠♦s ❞❡ ❝♦♠♣✉t❛çã♦✱ ♦s q✉❛✐s ✉t✐❧✐③❛rã♦ ❛s ♣r♦♣r✐❡❞❛❞❡s q✉â♥t✐❝❛s ❞♦ s♣✐♥✳ ❊ss❡ ♥♦✈♦ r❛♠♦ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ ❝❤❛♠❛✲s❡ ❞❡ ❝♦♠♣✉t❛çã♦ q✉â♥t✐❝❛✳
✶ ■♥tr♦❞✉çã♦ ●❡r❛❧✳ ✺
❯♠ ✜❧♠❡ ✜♥♦ é ❡♥t❡♥❞✐❞♦ ❝♦♠♦ ✉♠❛ ❡str✉t✉r❛ ♦♥❞❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ✐♥t❡r❡ss❡ sã♦ ✐♥✢✉❡♥❝✐❛❞❛s ♣♦r ❡❢❡✐t♦s ❞❡ ✐♥t❡r❢❛❝❡✳ ❙✉❛ ❡s♣❡ss✉r❛ ♥ã♦ é ❜❡♠ ❞❡✜♥✐❞❛✱ s❡♥❞♦ ❡st❛ ❡st❛❜❡❧❡❝✐❞❛ ♣❡❧♦s t✐♣♦s ❞❡ ♠❛t❡r✐❛✐s ❝♦♠♣♦♥❡♥t❡s ❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❡♠ q✉❡stã♦✳ ❯♠❛ ❜✐❝❛♠❛❞❛ ❞❡ ✜❧♠❡s ✜♥♦s é ❝♦♠♣♦st❛ ♣♦r ❞✉❛s ❝❛♠❛❞❛s ❞❡ ✜❧♠❡s s♦❜r❡♣♦st❛s✱ s❡♥❞♦ ❝❧❛ss✐✜❝❛❞❛ ❞❡ ♠✉❧t✐❝❛♠❛❞❛ ❛ ♣❛rt✐r ❞❡ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✐❣✉❛❧ ♦✉ s✉♣❡r✐♦r ❛ três ❝❛♠❛❞❛s ❞❡ ✜❧♠❡s ✭✻✮✳
◆♦s ❞✐❛s ❛t✉❛✐s ❝♦❧❤❡✲s❡ ♦s ❢r✉t♦s ❞❡ té❝♥✐❝❛s ♠✉✐t♦ ❡✜❝✐❡♥t❡s ♥♦ t♦❝❛♥t❡ ❛♦ ❝r❡s❝✐♠❡♥t♦ ❞❡ ♠❛t❡r✐❛✐s ♠❛❣♥ét✐❝♦s✱ ♣♦ss✐❜✐❧✐t❛♥❞♦ ❛ss✐♠ ❛ ❢❛❜r✐❝❛çã♦ ❞❡ ✜❧♠❡s ✜♥♦s ❛❞❡q✉❛❞♦s às ❛♣❧✐❝❛çõ❡s ♥♦ ♥✐❝❤♦ ❞❛ ❡❧❡trô♥✐❝❛✳ ❈♦♠♦ ❡①❡♠♣❧♦s ❞❡ss❛s té❝♥✐❝❛s ♣♦❞❡♠♦s ❝✐t❛r ❛ ❊♣✐t❛✲ ①✐❛ ♣♦r ❋❡✐①❡ ▼♦❧❡❝✉❧❛r ✭▼❇❊✮ ❡ ❛ ❉❡❝♦♠♣♦s✐çã♦ ♣♦r ❱❛♣♦r ◗✉í♠✐❝♦ ❞❡ ▼❡t❛✐s ❖r❣â♥✐❝♦s ✭▼❖❈❱❉✮✱ ♣r♦♣♦r❝✐♦♥❛♥❞♦ ❛ ❢❛❜r✐❝❛çã♦ ❞❡ s✐st❡♠❛s ♠✉❧t✐❝❛♠❛❞❛s ❝♦♠ ❡s♣❛ç❛♠❡♥t♦s ❞❡ ❛❧t❛ q✉❛❧✐❞❛❞❡ ❣♦③❛♥❞♦ ❞❡ ❞✐♠❡♥sõ❡s ♥❛ ♦r❞❡♠ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ ❞❡ ❞❡ ❇r♦❣❧✐❡ ♦✉ ♣ró①✐♠♦ ❞♦ ❧✐✈r❡ ❝❛♠✐♥❤♦ ♠é❞✐♦ ❡❧❡trô♥✐❝♦ ✭✼✮✳
❆ ✈❛st❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ t❡❝♥♦❧ó❣✐❝❛ ❥✉st✐✜❝❛✲s❡ ♥❛ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❞♦s ✜❧✲ ♠❡s ✜♥♦s ♠❛❣♥ét✐❝♦s✱ t❛✐s ❝♦♠♦ ❛ ♠❛❣♥❡t♦rr❡s✐stê♥❝✐❛ ❣✐❣❛♥t❡ ✭▼❘●✮ ❞❡ ❝✉❥♦s ❡❢❡✐t♦s ❢♦r❛♠ ♣✐♦♥❡✐r❛♠❡♥t❡ ♦❜s❡r✈❛❞♦s ♣♦r ❇❛✐❜✐❝❤✐ ❡t ❛❧✳✭✽✮✱ ❡♠ ✶✾✽✽ ❡♠ ♠✉❧t✐❝❛♠❛❞❛s ♠❛❣♥é✲ t✐❝❛s ❞♦ t✐♣♦ ❋▼✴◆▼✴❋▼ ✭❋❡✴❈r✴❋❡✮ ❛❝♦♣❧❛❞❛s ❛♥t✐❢❡rr♦♠❛❣♥❡t✐❝❛♠❡♥t❡ ❡ ♣r♦❞✉③✐❞❛s ♣♦r ❡♣✐t❛①✐❛ ❞❡ ❢❡✐①❡ ♠♦❧❡❝✉❧❛r✱ ❡ ♣❡❧❛ ❛❞❡q✉❛çã♦ ❞❛ ❛♥✐s♦tr♦♣✐❛ ♠❛❣♥ét✐❝❛ ✭✾✮✳ ❊ss❛s ❛♣❧✐❝❛çõ❡s s❡ ❡st❡♥❞❡♠ ❞❡s❞❡ ♦ ❝✉♥❤♦ ♣rát✐❝♦ ❝♦♠ ❛ ❡❧❛❜♦r❛çã♦ ❞❡ ❞✐s❝♦s rí❣✐❞♦s ❡ ✢❡①í✈❡✐s ♣❛r❛ ❣r❛✈❛çã♦ ❞❡ ❞❛❞♦s ✭✶✵✮✱ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ✉♠❛ ♥♦✈❛ ❡ ♣r♦♠✐ss♦r❛ ❡❧❡trô♥✐❝❛ ❞♦s s♣✐♥s ✭✶✶✮✱ ❛té ♦ ❝✉♥❤♦ t❡ór✐❝♦ q✉❡ ❛♣❡s❛r ❞❛s ♣r✐♠❡✐r❛s t❡♦r✐❛s ❞❡ tr❛♥s♣♦rt❡ ❡❧❡✲ trô♥✐❝♦ ❡♠ ✜❧♠❡s ✜♥♦s ♠❡tá❧✐❝♦s ✈✐r ❞❡ ❧♦♥❣❛s ❞❛t❛s ❝♦♠ tr❛❜❛❧❤♦s ❝❧áss✐❝♦s ❞❡ ❋✉❝❤s ✭✶✷✮✱ ❈❤❛♠❜❡rs ✭✶✸✮ ❡ ❙♦♥❞❤❡✐♠❡r ✭✶✹✮✱ ♦ t❡♠❛ r❡❝❡♥t❡♠❡♥t❡ t❡♠ ✐♥❝♦r♣♦r❛❞♦ ♥♦✈♦s ❡♥❢♦q✉❡s ❞❡ ✐♥✈❡st✐❣❛çã♦✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❡❢❡✐t♦s q✉â♥t✐❝♦s ✭✶✺✮✱ ✐♥t❡rr❡❧❛❝✐♦♥❛♠❡♥t♦ ❝♦♠ ♣r♦♣r✐❡❞❛❞❡s ♠❛❣♥ét✐❝❛s ✭✶✻✮✱ ❢❡♥ô♠❡♥♦s ❞❡ ❧♦❝❛❧✐③❛çã♦ ✭✶✼✮✱ ❡t❝✳
❊♠ s✉♠❛✱ ❡ss❡s ♠❛t❡r✐❛✐s ❛❜r❡♠ ♣♦rt❛s ♣❛r❛ ♥♦✈❛s ♣❡rs♣❡❝t✐✈❛s ❡ tr❛t❛♠ ❞❡ ♦t✐♠✐✲ ③❛r ❛s ❞✐♠❡♥sõ❡s ❞♦s ❞✐s♣♦s✐t✐✈♦s✱ ❛✉♠❡♥t❛♥❞♦ ❛ ✈❡❧♦❝✐❞❛❞❡ ❡ ❜❛✐①❛♥❞♦ ❛ ✈♦❧t❛❣❡♠ ❞❡ ♦♣❡r❛çã♦✳ ❆t✉❛❧♠❡♥t❡✱ ❡st❡s ✜❧♠❡s ✈❡♠ s❡♥❞♦ tr❛t❛❞♦s ❡♠ ❞✐♠❡♥sõ❡s ❝❛❞❛ ✈❡③ ♠❛✐s r❡✲ ❞✉③✐❞❛s✱ ❛t✐♥❣✐♥❞♦ ♦ ♣♦♥t♦ ❞❡ s❡r❡♠ ❡❧❛❜♦r❛❞❛s ❡♠2D✭q✉❛s❡✲❜✐❞✐♠❡♥s✐♦♥❛✐s✮ ❡1D ✭✜♦s
♠❛❣♥ét✐❝♦s✮✳
✶ ■♥tr♦❞✉çã♦ ●❡r❛❧✳ ✻
♣♦r ✉♠ tr❛t♦ ♠❛t❡♠át✐❝♦ ❝♦♠♣❧✐❝❛❞♦✱ t❡♠ s❡ t♦r♥❛❞♦ ❝❛❞❛ ♠❛✐s ❢r❡q✉❡♥t❡s ♥❛s ♣❡sq✉✐s❛s✱ ❞❡✈✐❞♦ ❛♦s ♥♦tór✐♦s ❛✈❛♥ç♦s ❛ ✉♠❛ ❞❡s❝r✐çã♦ ♠❛✐s r❡❛❧íst✐❝❛ ❞♦s s✐st❡♠❛s ❢ís✐❝♦s✳
❉❡ ♠❛♥❡✐r❛ ❛ ❛❣r❡❣❛r ✈❛❧♦r ❛♦ ❡st✉❞♦ ❡♠ ❜❛✐①❛s ❞✐♠❡♥sõ❡s✱ ❛❧é♠ ❞♦ ✐♥t❡r❡ss❡ ❡♠ ♣❛rtí❝✉❧❛s ✐s♦❧❛❞❛s✱ ♦s ❡st✉❞♦s ❞❡ r❡❞❡s r❡❣✉❧❛r❡s ♣❧❛♥❛r❡s✱ t❛♠❜é♠ t❡♠ r❡❝❡❜✐❞♦ ♠❛✐s ❛t❡♥çã♦ ♥♦s ú❧t✐♠♦s ❛♥♦s✱ ❡ ❝♦♠♣r❡❡♥❞❡r ❝♦♠♦ ♦ ❡s♣❛ç❛♠❡♥t♦ ❡ ❛ ✐♥t❡r❛çã♦ ❞✐♣♦❧❛r ❡♥tr❡ ❛s ♣❛rtí❝✉❧❛s ❛❢❡t❛ ❛s ♣r♦♣r✐❡❞❛❞❡s ♠❛❣♥ét✐❝❛s ❞♦s s✐st❡♠❛s é r❡❧❡✈❛♥t❡ ♥♦ ❡♥t❡♥❞✐♠❡♥t♦ ❡ ✐♥t❡r♣r❡t❛çã♦ ❞♦s r❡s✉❧t❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s✳ ❖✉tr♦ ❢❛t♦r ❞❡ s✉♠❛ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ❛ r❡s✲ ♣♦st❛ ❢ís✐❝❛ ❞♦ s✐st❡♠❛ ❛ ❡stí♠✉❧♦s ❡①t❡r♥♦s é r❡✢❡t✐❞♦ ♣❡❧❛ ❣❡♦♠❡tr✐❛ ❞❛ r❡❞❡ ♠❛❣♥ét✐❝❛✱ ❛ q✉❛❧ é ❡①♣❧♦r❛❞❛ ♥❡ss❡ tr❛❜❛❧❤♦✳
❋✐♥❛❧♠❡♥t❡✱ ♣❡r❝❡❜❡✲s❡ q✉❡ ♥❛ ✈✐tr✐♥❡ ❞❛s ♣❡sq✉✐s❛s ❛t✉❛✐s ♦s s✐st❡♠❛s ❞❡ ❜❛✐①❛ ❞✐✲ ♠❡♥s✐♦♥❛❧✐❞❛❞❡ ❣❛♥❤❛♠ ❢♦rç❛✱ ❡ ♥❛ ♣❡rs♣❡❝t✐✈❛ ❞❛ ♠✐♥✐❛t✉r✐③❛çã♦ ❞♦s ❡q✉✐♣❛♠❡♥t♦s ❡ ❞✐s♣♦s✐t✐✈♦s ✉t✐❧✐③❛❞♦s ❡♠ ❝✐r❝✉✐t♦s ❡❧❡trô♥✐❝♦s ❝♦♠ ❝❛♣❛❝✐❞❛❞❡ ❝❛❞❛ ✈❡③ ♠❛✐♦r❡s✱ ❛ ❜✉s❝❛ ♣♦r ♠❛t❡r✐❛✐s ♥♦✈♦s ✈❡♠ ❣❛♥❤❛♥❞♦ ♥♦t♦r✐❡❞❛❞❡✳ ◆❡ss❡ ♣❛❧❝♦ ♦s ♣r✐♥❝✐♣❛✐s ♣r♦t❛❣♦♥✐st❛s t❡♠ s✐❞♦ ♦s ✜❧♠❡s ✜♥♦s ♠❛❣♥ét✐❝♦s ❡ ❛❣♦r❛ ♦ ❣r❛❢❡♥♦✱ ❞♦s q✉❛✐s ❝♦♠ ♦ ❝♦♥tr♦❧❡ ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ♣♦❞❡♠♦s ❡st❡♥❞❡r ❛ ♥♦✈❛s ❞✐r❡çõ❡s✱ ♣❡r♠✐t✐♥❞♦ ❛ ❝r✐❛çã♦ ❞❡ s✐st❡♠❛s ❝♦♠ r♦❜✉st❛ ❝❛r❛❝t❡ríst✐❝❛s s✉♣❡r❝♦♥❞✉t♦r❛s ❡ ♠❛❣♥ét✐❝❛s ✭✶✾✮✳
❊♠❜♦r❛ ❡ss❡ ❝❛♠♣♦ ❞❡ ❡st✉❞♦ ❡st❡❥❛ ❛♣❡♥❛s ♥♦ ❝♦♠❡ç♦✱ ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s t❡❝♥♦❧ó❣✐❝❛s ❡ ❝✐❡♥tí✜❝❛s ♣❛r❡❝❡♠ s❡r ✐♠❡♥s❛s✳ ❖ ❡♥t❡♥❞✐♠❡♥t♦ ❡ ♦ ❝♦♥tr♦❧❡ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ss❡s ♠❛t❡r✐❛✐s ♣♦❞❡♠ ❛❜r✐r ♣♦rt❛s ♣❛r❛ ✉♠❛ ♥♦✈❛ ❢r♦♥t❡✐r❛ ♥♦ s❡t♦r ❡❧❡trô♥✐❝♦✳
❆ ❡str✉t✉r❛ ❞❡ss❡ tr❛❜❛❧❤♦ ❡stá ♦r❣❛♥✐③❛❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ◆♦ ❝❛♣ít✉❧♦ ✷ ❢❛r❡♠♦s ✉♠❛ ❛❜♦r❞❛❣❡♠ ❤✐stór✐❝❛ s♦❜r❡ ♦ ♠❛❣♥❡t✐s♠♦✱ r❡ss❛❧t❛♥❞♦ s❡✉s ❛s♣❡❝t♦s ♠❛❝r♦ ❡ ♠✐❝r♦s✲ ❝ó♣✐❝♦s✳ ❊♥❢❛t✐③❛♠♦s ♦ ❝❛rát❡r q✉â♥t✐❝♦ ❞♦ ♠❛❣♥❡t✐s♠♦ ❛tr❛✈és ❞♦ ♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣✱ ♠♦❞❡❧♦ q✉❡ s❡rá ❜❛s❡ ♣❛r❛ ❛s ❛♥á❧✐s❡s ♥❡st❡ tr❛❜❛❧❤♦✳
◆♦ ❝❛♣ít✉❧♦ q✉❡ s❡ s❡❣✉❡ ❡st✉❞❛♠♦s ❛s ♦♥❞❛s ❞❡ s♣✐♥ ❡♠ r❡❞❡s ♠❛❣♥ét✐❝❛s✱ ♦♥❞❡ ❛❜♦r✲ ❞❛♠♦s ♦s ❛s♣❡❝t♦s ❤✐stór✐❝♦s ❛ss♦❝✐❛❞♦s ❛ ❡ss❡ ❝♦♥❝❡✐t♦ ❡ ❛♥❛❧✐s❛♠♦s ❛s ♣❡rs♣❡❝t✐✈❛s s❡✲ ♠✐❝❧áss✐❝❛ ❡ q✉â♥t✐❝❛✳ ❉✐s❝✉t✐♠♦s t❡♠❛s ❝♦rr❡❧❛❝✐♦♥❛❞♦s ❞❡ ✐♥t❡r❡ss❡ ❝♦♠♦ ❛s ❞✐❢❡r❡♥t❡s ✐♥t❡r❛çõ❡s ♥♦s s✐st❡♠❛s ♠❛❣♥ét✐❝♦s t✐♣♦ ❚r♦❝❛✱ ❩❡❡♠❛♥ ❡ ❉✐♣♦❧❛r✳ ❆❝r❡s❝❡♥t❛♠♦s ✉♠❛ ❞✐s❝✉ssã♦ ♠❛✐s ❞❡t❛❧❤❛❞❛ s♦❜r❡ ♦ ♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣✱ ♦ q✉❛❧ ❡st❛rá s✉❥❡✐t♦ à ❢❛♠♦s❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❍♦❧st❡✐♥✲Pr✐♠❛❦♦✛✱ ♥❛ q✉❛❧ ♦s ♦♣❡r❛❞♦r❡s ❞❡ s♣✐♥ sã♦ ❡s❝r✐t♦s ❡♠ t❡r✲ ♠♦s ❞♦s ♦♣❡r❛❞♦r❡s ❞❡ ❝r✐❛çã♦ ❡ ❛♥✐q✉✐❧❛çã♦ ❞❡ ♠❛❣♥♦♥s✱ ❣❛r❛♥t✐♥❞♦ ❛ ✈✐s✉❛❧✐③❛çã♦ ❞♦s ♠♦❞♦s ❞❛s ♦♥❞❛s ❞❡ s♣✐♥ ❝♦♠♦ ✉♠ ❣ás ❜♦sô♥✐❝♦✳
✶ ■♥tr♦❞✉çã♦ ●❡r❛❧✳ ✼
✽
✷ ▼❛❣♥❡t✐s♠♦✳
✷✳✶ ❆s♣❡❝t♦s ❍✐stór✐❝♦s✳
❖ ❝♦♥❤❡❝✐♠❡♥t♦ s♦❜r❡ ♦ ♠❛❣♥❡t✐s♠♦ é ❞❡ ♠✉✐t♦ t❡♠♣♦ ❛trás✱ sé❝✉❧♦s ❛♥t❡s ❞❡ ❈r✐st♦✱ ♠❛s q✉❡ só ✈❡✐♦ ❛t✐♥❣✐r ✉♠ ♣❛t❛♠❛r ❞❡ ❝♦♠♣r❡❡♥sã♦ ♠❛✐s ♠♦❞❡r♥♦ ♥♦ ✜♥❛❧ ❞♦ sé❝✉❧♦ ❳■❳✱ ❝♦♠♦ r❡✢❡①♦ ❞♦ tr❛❜❛❧❤♦ ❞❡ ❛❧❣✉♥s ♥♦tá✈❡✐s✳
◆❛ ❡✈♦❧✉çã♦ ❞♦s ❢❛t♦s✱ ❛ ❤✐stór✐❛ ❝♦♥t❛ q✉❡ ♦ ♠❛❣♥❡t✐s♠♦ ❢♦✐ ♣✐♦♥❡✐r❛♠❡♥t❡ ♦❜s❡r✈❛❞♦ ♣♦r ✉♠ ♣❛st♦r ❝r❡t❡♥s❡ ♣♦r ♥♦♠❡ ▼❛❣♥❡s ✭✷✵✮✱ q✉❡ ❛♦ ♣❛st♦r❡❛r ❝♦♠ ✉♠ ❝❛❥❛❞♦ ❞❡ ♣♦♥t❛ ❞❡ ❢❡rr♦✱ t♦❝♦✉ ❡♠ ✉♠❛ r♦❝❤❛✱ ❛ q✉❛❧ ♣♦r ❛❧❣✉♥s ✐♥st❛♥t❡s ♠❛♥✐❢❡st♦✉ ✉♠❛ ❢♦rç❛ ❛tr❛t✐✈❛✳ ❊st❡ ❢❛t♦ ❝❤❡❣♦✉ ❛♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦ ✜❧ós♦❢♦ ❚❤❛❧❡s q✉❡ ❝♦♥❝❧✉✐✉ s❡r ❡st❛ ✉♠❛ r♦❝❤❛ ❝♦♠ ♦r✐❣❡♠ ❞✐✈✐♥❛✱ ❜❛s❡❛❞♦ ♥❛ s✉♣♦s✐çã♦ ❞❡❧❛ ♣♦ss✉✐r ❛❧♠❛✳
◆❛ t❡♥t❛t✐✈❛ ❞❡ ❡①♣❧✐❝❛r ❛ s✐t✉❛çã♦ ❚❤❛❧❡s s❡ r❡♣♦rt♦✉ à ❜✉s❝❛ ♣♦r ♦✉tr♦s ♠❛t❡r✐❛✐s ♦✉ ❡①❡♠♣❧❛r❡s q✉❡ ♣♦ss✉íss❡♠ ♦ ♠❡s♠♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ♠❛❣♥❡t✐t❛✱ ❡ ❞❡♥tr❡ s❡✉s ❛❝❤❛❞♦s r❡❧❛t❛✲s❡ ✉♠❛ r❡s✐♥❛ ✈❡❣❡t❛❧ ❛♠❛r❡❧❛❞❛ ❡ tr❛♥s❧ú❝✐❞❛ ✭♦ â♠❜❛r✮ q✉❡ ❛♣ós s❡r ❛tr✐t❛❞❛ ❡①♣❡❧✐❛ ❛r♦♠❛ ❡ ♠♦str❛✈❛✲s❡ ❝♦♠ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❛tr❛✐r ♣❡❞❛ç♦s ❞❡ ♣❛♣❡❧✱ ❢❡❧♣♦✱ ❜♦❧❛s ❞❡ ❝♦rt✐ç❛✱ ❡t❝❀ ❞❡♥tr❡ ♦s q✉❛✐s ♥ã♦ s❡ ✐♥❝❧✉í❛ ♦ ❢❡rr♦✳ ◆♦ í♥t❡r✐♠ ❞❛s ❞❡s❝♦❜❡rt❛s✱ ❚❤❛❧❡s ♥ã♦ ❝❤❡❣♦✉ ❛ ✉♠❛ ❝♦♠♣r❡❡♥sã♦ ♣r❡❝✐s❛ ❡ ❝♦♥✈✐♥❝❡♥t❡ ❛❝❡r❝❛ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ♠❛❣♥❡t✐t❛✱ ❡ ❛❣♦r❛ t❛♠❜é♠ ❞❡♣❛r❛✈❛✲s❡ ❝♦♠ ❛ ✐♥❝❡rt❡③❛ s♦❜r❡ ❛ s✐♠✐❧❛r✐❞❛❞❡ ❝♦♠ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ â♠❜❛r✳
P♦st❡r✐♦r♠❡♥t❡✱ ♦s ✜❧ós♦❢♦s ♠❡❝❛♥✐❝✐st❛s ♦✉ ❛t♦♠✐st❛s✱ t❡♥t❛r❛♠ ❛tr❛✈és ❞❡ ❞✐✈❡rs❛s ❝♦♥❝❡♣çõ❡s ♠❡t❛❢ís✐❝❛s ❝♦♠♣r❡❡♥❞❡r ❡st❡s ❡ ♦✉tr♦s ❢❡♥ô♠❡♥♦s ♥❛t✉r❛✐s✳ ◆♦ ❝❡♥ár✐♦ ❞♦ q✉❡ ❢♦r❛ ❞✐s❝✉t✐❞♦✱ ❛tr✐❜✉ír❛♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠✐❞❛❞❡ ♥♦ ❢❡rr♦ ❞❛ q✉❛❧ ❛ ♠❛❣♥❡t✐t❛ s❡ ❛❧✐♠❡♥t❛✈❛✳ ➱ ✐♥t❡r❡ss❛♥t❡ ♥♦t❛r q✉❡ ♠❡s♠♦ s❡ tr❛t❛♥❞♦ ❞❡ ♣❡♥s❛♠❡♥t♦s s❡♠ ❝♦♥st❛t❛çõ❡s ❡①♣❡r✐♠❡♥t❛✐s✱ ❜❛s❡❛❞❛s ❛♣❡♥❛s ❡♠ ❛✈❛❧✐❛çõ❡s s✉❜❥❡t✐✈❛s✱ ❥á ❤❛✈✐❛ ❛ ✐❞❡✐❛ ❞♦s ❝♦♥trár✐♦s✱ ♦✉ s❡❥❛✱ ❡❧❡♠❡♥t♦s ♦♣♦st♦s q✉❡ s❡ ❝♦♠♣❧❡t❛✈❛♠✳
✷✳✶ ❆s♣❡❝t♦s ❍✐stór✐❝♦s✳ ✾
❡r❛♠ ❢✉♥❞❛♠❡♥t❛❞❛s s♦♠❡♥t❡ ❛♦ ❝❛♠♣♦ ❞❛s ✐❞❡✐❛s ✭✷✶✮✳ ■st♦ só ✈✐r✐❛ ❛ s❡r ❝♦♥❝❡❜✐❞♦ ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ ♣♦r ◆✐❝❝♦❧♦ ❈❛❜❡♦ ❥á ♥♦ sé❝✉❧♦ ❳❱■✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥♦ q✉❡ t❛♥❣❡ ❛♦s ❡❢❡✐t♦s ❡❧étr✐❝♦s✳ ❈❛❜❡♦ ❝♦♥❝❡❜❡✉ ♦ ❝♦♥❝❡✐t♦ ❞❛ ✧❡s❢❡r❛ ❞❡ ✐♥✢✉ê♥❝✐❛✧✱ q✉❡ t❡r✐❛ ❝❛rát❡r ✐♠❛t❡r✐❛❧ ❡ ♦❝✉♣❛r✐❛ ♦ ❡s♣❛ç♦ ❛♦ r❡❞♦r ❞♦ ✐♠ã✱ ♠❛♥✐❢❡st❛♥❞♦ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❧❛t❡♥t❡ ❝❛♣❛③ ❞❡ ❛tr❛✐r ♦ ❢❡rr♦✱ q✉❛♥❞♦ ❡st❡ s❡ ❛♣r♦①✐♠❛✈❛✳ ❊st❡ ❝♦♥❝❡✐t♦ s❡r✐❛ ♠✉✐t♦ s✐♠✐❧❛r ❛♦ ❞♦ ❡✤✉✈✐✉♠✱ q✉❡ ❝♦rr❡s♣♦♥❞❡r✐❛ ❛ ✉♠❛ ❡s♣é❝✐❡ ❞❡ ✧❝❛♠♣♦ ♠❡❝â♥✐❝♦ ♠❛t❡r✐❛❧✧ q✉❡ ❛♦ s❡ ♣r♦♣❛❣❛r tr❛r✐❛✱ à ❞✐stâ♥❝✐❛✱ ❛ ❝♦♥❡①ã♦ ❡♥tr❡ ❝♦r♣♦s ❛tr✐t❛❞♦s✱ ❢❛③❡♥❞♦ ❝♦♠ q✉❡ ❛♠❜♦s s❡ ❛tr❛íss❡♠✳
❊st❡s ❢❛t♦s ❝✐t❛❞♦s ❢♦r❛♠ ❛ ❣ê♥❡s✐s ❞❛ ♣❡r❝❡♣çã♦ ❞♦ ❤♦♠❡♠ ❛❝❡r❝❛ ❞♦s ❢❡♥ô♠❡♥♦s ♠❛❣✲ ♥ét✐❝♦s ❡ ❡❧étr✐❝♦s✳ ❖s ♠❛❣♥ét✐❝♦s ✈✐♥❝✉❧❛❞♦s às ✐♥t❡r❛çõ❡s ❡①♣❧í❝✐t❛s ♣❡❧❛ ♠❛❣♥❡t✐t❛ ❡ ♦s ❡❧étr✐❝♦s ❞❡✈✐❞♦ ❛♦ â♠❜❛r ✭❡❧é❦tr♦♥✱ ❡♠ ❣r❡❣♦✮✳ ❍á ✐♥❞í❝✐♦s ❞❡ q✉❡ ♦s ❝❤✐♥❡s❡s ❝♦♥❤❡❝✐❛♠ ♦ ♠❛❣♥❡t✐s♠♦ ❤á ♠❛✐s t❡♠♣♦ q✉❡ ♦s ❣r❡❣♦s✱ ❡ ❥á s❡ ✉t✐❧✐③❛✈❛♠ ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳ ◆♦ ✐♥í❝✐♦ ❞❛ ❡r❛ ❈r✐stã✱ ♦s ❝❤✐♥❡s❡s ❢❛③✐❛♠ ✉s♦ ❞❛ ❝♦❧❤❡r ❞❡ ♠❛❣♥❡t✐t❛ ❋✐❣✉r❛ ✷✱ q✉❡ ❛♣♦✐❛❞❛ ❡♠ ✉♠ ♣✐♥♦ ❛♣♦♥t❛✈❛ ❝♦♠ s❡✉ ❝❛❜♦ ♣❛r❛ ♦ s✉❧ ❣❡♦❣rá✜❝♦✳ ❉❛í t✐♥❤❛✲s❡ ❛ ♣r✐♠❡✐r❛ ❝♦♥✲ ❝❡♣çã♦ ❞❛ ❜úss♦❧❛ ✭✉t✐❧✐③❛❞❛ ❡♠ ♥❛✈❡❣❛çõ❡s ❛ ♣❛rt✐r ❞♦ sé❝✉❧♦ ❳ ♦✉ ❳■✮ q✉❡ ❛tr❡❧❛✈❛✲s❡ ❛♦ ❝♦♥❝❡✐t♦ ❞♦ ✐♠ã✱ ♣❛❧❛✈r❛ q✉❡ s✉r❣✐r✐❛ ♠❛✐s t❛r❞❡ ❞❡r✐✈❛❞❛ ❞♦ t❡r♠♦ ❢r❛♥❝ês ✧❛✐♠❛♥t✧✱ ❛♠❛♥t❡ ✭✷✷✮✳
❋✐❣✉r❛ ✷✿ ❈♦❧❤❡r ❞❡ ▼❛❣♥❡t✐t❛✳
