❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ ❞❡ ▼❛r✐♥❣á
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛
Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛
❘❡♥❛t♦ ❋❡rr❡✐r❛ ❞❡ ❙♦✉③❛
■♥✢✉ê♥❝✐❛ ❞❛ ❱✐s❝♦s✐❞❛❞❡ ❞❡ ❙✉♣❡r❢í❝✐❡✱
❊♥❡r❣✐❛ ❞❡ ❆♥❝♦r❛♠❡♥t♦✱ ❊s♣❡ss✉r❛ ❡ ➶♥❣✉❧♦
♥❛ ❙✉♣❡r❢í❝✐❡ ♥♦ ❚❡♠♣♦ ❞❡ ❘❡❧❛①❛♠❡♥t♦ ❞❡
❈r✐st❛✐s ▲íq✉✐❞♦s ◆❡♠át✐❝♦s
❘❡♥❛t♦ ❋❡rr❡✐r❛ ❞❡ ❙♦✉③❛
■♥✢✉ê♥❝✐❛ ❞❛ ❱✐s❝♦s✐❞❛❞❡ ❞❡ ❙✉♣❡r❢í❝✐❡✱
❊♥❡r❣✐❛ ❞❡ ❆♥❝♦r❛♠❡♥t♦✱ ❊s♣❡ss✉r❛ ❡ ➶♥❣✉❧♦
♥❛ ❙✉♣❡r❢í❝✐❡ ♥♦ ❚❡♠♣♦ ❞❡ ❘❡❧❛①❛♠❡♥t♦ ❞❡
❈r✐st❛✐s ▲íq✉✐❞♦s ◆❡♠át✐❝♦s✳
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▲✉✐③ ❘♦❜❡rt♦ ❊✈❛♥❣❡❧✐st❛
❈♦♦r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❘❛❢❛❡❧ ❙♦❛r❡s ❩♦❧❛
❉✐ss❡rt❛çã♦ s✉❜♠❡t✐❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡
Pós✲●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡
❊st❛❞✉❛❧ ❞❡ ▼❛r✐♥❣á ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tí✲
t✉❧♦ ❞❡ ▼❡str❡ ❡♠ ❋ís✐❝❛✳
Dados Internacionais de Catalogação na Publicação (CIP) (Biblioteca Central - UEM, Maringá, PR, Brasil)
Souza, Renato Ferreira de
S729i Influência da viscosidade de superfície, energia de ancoramento, espessura e ângulo na superfície no tempo de relaxamento de cristais líquidos nemáticos / Renato Ferreira de Souza. -- Maringá, 2013. 99 f. : il. (algumas color.), figs., tabs.
Orientador: Prof. Dr. Luiz Roberto Evangelista. Coorientador: Prof. Dr. Rafael Soares Zola. Dissertação (mestrado) - Universidade Estadual de
Maringá, Centro de Ciências Exatas, Departamento de Física, Programa de Pós-Graduação em Física, 2013.
1. Cristal líquido. 2. Cristal líquido -Viscosidade de superfície. 3. Cristal líquido - Tempo de resposta. I. Evangelista, Luiz Roberto, orient. II. Zola, Rafael Soares, coorient. III. Universidade Estadual de Maringá. Centro de Ciências Exatas. Departamento de Física. Programa de Pós-Graduação em Física. IV. Título.
❆❣r❛❞❡❝✐♠❡♥t♦s
❆♦s ♠❡✉s ♣❛✐s✱ ♣♦r t♦❞♦ ❛♠♦r ❡ ❛♣♦✐♦ ❡ss❡♥❝✐❛✐s ❡♠ t♦❞♦s ❡ss❡s ❧♦♥❣♦s ❛♥♦s✳ ❆♦ ♣r♦❢❡ss♦r ❉r✳ ❘❛❢❛❡❧ ❙♦❛r❡s ❩♦❧❛✱ ♣❡❧❛ ❛t❡♥çã♦✱ ❞✐s♣♦s✐çã♦ ❡ ♣❛❝✐ê♥❝✐❛✳ ❆♦ ♣r♦❢❡ss♦r ❉r✳ ▲✉✐③ ❘♦❜❡rt♦ ❊✈❛♥❣❡❧✐st❛✱ ♣♦r t♦❞♦ ❛♣♦✐♦ ❡ ✐♥❝❡♥t✐✈♦✳
❆ t♦❞♦s ♦s ❛♠✐❣♦s q✉❡ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
➚ ❈❆P❊❙✱ ♣❡❧♦ s✉♣♦rt❡ ✜♥❛♥❝❡✐r♦✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦✱ ❡st✉❞❛♠♦s ♥✉♠ér✐❝❛ ❡ ❛♥❛❧✐t✐❝❛♠❡♥t❡ ❛ ✐♥✢✉ê♥❝✐❛ ❞❛ ✈✐s❝♦s✐❞❛❞❡ ❞❡ s✉♣❡r❢í❝✐❡ ♥♦s t❡♠♣♦s ❝❛r❛❝t❡ríst✐❝♦s ❞❡ r❡❧❛①❛çã♦ ❞❡ ✉♠❛ ❝é❧✉❧❛ ❞❡ ❝r✐st❛❧ ❧íq✉✐❞♦ ♥❡♠á✲ t✐❝♦✳ ❆ ❡q✉❛çã♦ q✉❡ ❞❡s❝r❡✈❡ ♦ r❡❧❛①❛♠❡♥t♦ ❢♦✐ ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❛ ❡q✉❛çã♦ ❞❛ ❡♥❡r❣✐❛ ❧✐✈r❡ ❞❡ ❋r❛♥❦ ❡ ❞♦ ♠♦❞❡❧♦ ❞❡ ❉❡r③❤❛♥s❦✐ ❡ P❡tr♦✈ ♣❛r❛ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦✳ ❊ss❛s ❡q✉❛çõ❡s ❢♦r❛♠ r❡s♦❧✈✐❞❛s ❛♥❛❧✐t✐❝❛♠❡♥t❡ ♣❛r❛ ♦ ❝❛s♦ ❞❡ ♣❡q✉❡♥❛s ❞❡❢♦r♠❛çõ❡s ❡ ❞❡ ❝♦♥s✲ t❛♥t❡s ❡❧ást✐❝❛s ❝♦♠ ✈❛❧♦r❡s ♠✉✐t♦ ♣ró①✐♠♦s✳ P❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞♦ ❡st✉❞♦✱ ❡s❝♦❧❤❡♠♦s ❛ ❣❡♦♠❡tr✐❛ ❞❡ ❛❧✐♥❤❛♠❡♥t♦ ✈❡rt✐❝❛❧ ❡ ✉♠ ❝r✐st❛❧ ❧íq✉✐❞♦ ❝♦♠ ❛♥✐s♦tr♦♣✐❛ ❞✐❡❧étr✐❝❛ ♥❡❣❛t✐✈❛✳ ❚❛♠❜é♠✱ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦ ✉♠ ❛❧❣♦rít✐♠♦ ❝❛♣❛③ ❞❡ r❡s♦❧✈❡r ❡ss❛s ❡q✉❛çõ❡s ♥✉♠❡r✐❝❛♠❡♥t❡ s❡♠ ♥❡♥❤✉♠ t✐♣♦ ❞❡ ❛♣r♦①✐♠❛çã♦✳ ❯t✐❧✐③❛♥❞♦ ❡ss❡ ❛❧❣♦rít✐♠♦✱ ♦❜t✈❡♠♦s ❛ ❞✐str✐❜✉✐çã♦ ❞♦ ❞✐r❡t♦r ❡♠ ❢✉♥çã♦ ❞♦ t❡♠♣♦ ❛♦ ❧♦♥❣♦ ❞❛ ❝é❧✉❧❛✳ ❆ ♣❛rt✐r ❞❡ss❡ r❡s✉❧t❛❞♦✱ ❢♦r❛♠ ♦❜t✐❞♦s ♦s t❡♠♣♦s ❞❡ r❡❧❛①❛çã♦ ❞♦ s✐st❡♠❛✱ ❝♦♠ ♦s q✉❛✐s ❛♥❛❧✐s❛♠♦s ❝♦♠♦ ❛ ✈❛r✐❛çã♦ ❞❡ ❛❧❣✉♥s ♣❛râ♠❡tr♦s ✐♥✢✉❡♥❝✐❛♠ ❝❛❞❛ ✉♠ ❞♦s t❡♠♣♦s ❝❛r❛❝t❡ríst✐❝♦s ❞❡❧❡s✱ ❡♠ ❡s♣❡❝✐❛❧✱ ❛ ✈✐s❝♦s✐✲ ❞❛❞❡ ❞❡ s✉♣❡r❢í❝✐❡✳ ❈♦♠ ✐ss♦✱ ✉♠❛ ❝♦♠♣❛r❛çã♦ ❝♦♠ ♦s r❡s✉❧t❛❞♦s ❛♥❛❧ít✐❝♦s ❢♦✐ r❡❛❧✐③❛❞❛✳ ❖s r❡s✉❧t❛❞♦s ✐♥❞✐❝❛♠ q✉❡✱ s♦❜ ❝❡rt❛s ❝✐r❝✉♥stâ♥❝✐❛s✱ ❛ ✈✐s❝♦s✐❞❛❞❡ ❞❡ s✉♣❡r❢í❝✐❡ é ❝r✉❝✐❛❧ ♣❛r❛ ❛ ❞❡t❡r♠✐♥❛çã♦ ❝♦rr❡t❛ ❞♦s t❡♠♣♦s ❞❡ r❡❧❛①❛çã♦ ❞♦ s✐st❡♠❛✳ ❊ss❛s s✐t✉❛çõ❡s sã♦ ✐♠♣♦rt❛♥t❡s ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ✐♥❞✉str✐❛❧✳ ❊①♣❡r✐♠❡♥t❛❧♠❡♥t❡✱ ❛✐♥❞❛ ♥ã♦ ❢♦r❛♠ ❞❡s❡♥✈♦❧✲ ✈✐❞❛s té❝♥✐❝❛s ❝❛♣❛③❡s ❞❡ ✐❞❡♥t✐✜❝❛r ♦ ✈❛❧♦r ❞❛ ✈✐s❝♦s✐❞❛❞❡ ❞❡ s✉♣❡r❢í❝✐❡ ♣❛r❛ ❛ ❣❡♦♠❡tr✐❛ ❡st✉❞❛❞❛✳ ❉❡ss❛ ❢♦r♠❛✱ ❡s♣❡r❛♠♦s q✉❡ ❡st❡ tr❛❜❛❧❤♦ ❛♣♦♥t❡ ❛♣♦♥t❡ ♣❛r❛ s✐t✉❛çõ❡s ❡♠ q✉❡ ❡ss❡ ♣❛râ♠❡tr♦ ♣♦ss❛ s❡r ♠❡❞✐❞♦ ❝♦♠ ♠❛✐s ❢❛❝✐❧✐❞❛❞❡✳
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦ ✇❡ st✉❞② t❤❡ r♦❧❡ ♦❢ t❤❡ s✉r❢❛❝❡ ✈✐s❝♦s✐t② ♦♥ t❤❡ r❡❧❛①❛t✐♦♥ t✐♠❡s ♦❢ ❛ ♥❡♠❛t✐❝ ❧✐q✉✐❞ ❝r②st❛❧ ❝❡❧❧✳ ❚❤❡ ❡q✉❛t✐♦♥s ✇❡r❡ ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ❋r❛♥❦ ❢r❡❡ ❡♥❡r❣② ❛♥❞ t❤❡ ❉❡r③❤❛♥s❦✐ ❛♥❞ P❡tr♦✈ ♠♦❞❡❧ ❢♦r t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❚❤❡s❡ ❡q✉❛t✐♦♥ ✇❡r❡ s♦❧✈❡❞ ✇❤❡♥ t❤❡ ❞✐r❡❝t♦r ❛♥❣❧❡s ❛r❡ s♠❛❧❧ ❛♥❞ t❤❡ ❡❧❛st✐❝ ❝♦♥st❛♥ts ❛r❡ ❛♣♣r♦①✐♠❛t❡❧② t❤❡ s❛♠❡✳ ❚♦ r❡❛❧✐③❡ t❤✐s ✇♦r❦ t❤❡ ✈❡rt✐❝❛❧ ❛❧✐❣♥♠❡♥t ❣❡♦♠❡tr② ✇❛s ❝❤♦s❡♥✳ ❆ ♥✉♠❡r✐❝❛❧ ❛❧❣♦r✐t❤♠ ✇❛s ❛❧s♦ ❞❡s✐❣♥❡❞ t♦ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥s ✇✐t❤♦✉t ❛♥② ❦✐♥❞ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥✳ ❲✐t❤ t❤❡ s♦❧✉t✐♦♥ ✇❡ ♦❜t❛✐♥❡❞ t❤❡ r❡❧❛①❛t✐♦♥ t✐♠❡s ♦❢ t❤❡ s②st❡♠ ❛♥❞ ❛♥❛❧②③❡❞ t❤❡ r♦❧❡ ♦❢ s♦♠❡ ♣❛r❛♠❡t❡rs ♦♥ t❤❡ r❡❧❛①❛t✐♦♥ t✐♠❡ ♦❢ t❤❡ s②st❡♠✱ ♠❛✐♥❧② t❤❡ s✉r❢❛❝❡ ✈✐s❝♦s✐t②✳ ❚❤❡ r❡s✉❧ts ✐♥❞✐❝❛t❡ t❤❛t ✉♥❞❡rs ❝❡rt❛✐♥ ❝✐r❝✉♥st❛♥❝❡s t❤❡ s✉r❢❛❝❡ ✈✐s❝♦s✐t② ✐s ❝r✉❝✐❛❧ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ r❡❧❛①❛t✐♦♥ t✐♠❡ ♦❢ t❤❡ s②st❡♠✳ ❚❤❡ s✉r❢❛❝❡ ✈✐s❝♦s✐t② st✐❧❧ ❤❛✈❡ ♥♦t ❜❡❡♥ ♠❡❛s✉r❡❞ ❢♦r t❤❡ st✉❞✐❡❞ ❣❡♦♠❡tr②✱ ❤❡♥❝❡ ✇❡ ❤♦♣❡ t❤❛t t❤✐s ✇♦r❦ ✐♥❞✐❝❛t❡ ♣♦ss✐❜❧❡ ✇❛②s ✇❤❡r❡ t❤✐s ♣❛r❛♠❡t❡r ❝♦✉❧❞ ❜❡ ♠♦r❡ ❡❛s✐❧② ♦❜t❛✐♥❡❞✳
❙✉♠ár✐♦
❘❡s✉♠♦ ✸
❆❜str❛❝t ✹
■♥tr♦❞✉çã♦ ✼
✶ ❈❧❛ss✐✜❝❛çã♦ ❡ Pr♦♣r✐❡❞❛❞❡s ❋ís✐❝❛s ❞♦s ❈r✐st❛✐s ▲íq✉✐❞♦s ✾ ✶✳✶ ❈❧❛ss✐✜❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✷ Pr♦♣r✐❡❞❛❞❡s ❡❧ást✐❝❛s ❞❡ ❝r✐st❛✐s ❧íq✉✐❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✸ ❊♥❡r❣✐❛ ❞❡ ❛♥❝♦r❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✹ ❘❡s♣♦st❛ ❡❧étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✹✳✶ ❘❡s♣♦st❛ ❡❧étr✐❝❛ ❛ ❝❛♠♣♦ ❡❧étr✐❝♦ ❝♦♥st❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✹✳✷ ❘❡s♣♦st❛ ❡❧étr✐❝❛ à ❞✐❢❡r❡♥ç❛ ❞❡ ♣♦t❡♥❝✐❛❧ ❝♦♥st❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✺ Pr♦♣r✐❡❞❛❞❡s ót✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✺✳✶ ❚r❛♥s♠✐tâ♥❝✐❛ ót✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✳✻ ❱✐s❝♦s✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✷ ❚r❛♥s✐çõ❡s ❞❡ ❋ré❡❞❡r✐❝❦s③ ✷✺
✷✳✶ ❊stát✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✶✳✶ ❆♥❝♦r❛♠❡♥t♦ ❢♦rt❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✶✳✷ ❆♥❝♦r❛♠❡♥t♦ ❢r❛❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✶✳✸ ❆♥❝♦r❛♠❡♥t♦ ❢♦rt❡ ❝♦♠ ♣r❡t✐❧t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✷ ❉✐♥â♠✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✷✳✶ ❆♥❝♦r❛♠❡♥t♦ ❢♦rt❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✷✳✷ ❆♥❝♦r❛♠❡♥t♦ ❢r❛❝♦ ❝♦♠ ♣r❡t✐❧t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✷✳✸ ❙♦❧✉çã♦ ❞❛s ❡q✉❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✸ ❚❡♠♣♦s ❝❛r❛❝t❡ríst✐❝♦s ❞❡ r❡❧❛①❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✹ ❉✐s♣❧❛②s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✸ ❋♦r♠✉❧❛çã♦ ◆✉♠ér✐❝❛ ✹✶
✸✳✶ ❉✐❢❡r❡♥ç❛s ✜♥✐t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✷ ❊♥❝♦♥tr❛♥❞♦ ❛ ❞✐str✐❜✉✐çã♦ ❞♦ ❞✐r❡t♦r ♥✉♠❡r✐❝❛♠❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✷✳✶ ❚❡st❛♥❞♦ ♦ ❛❧❣♦r✐t♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✸ ❈❛❧❝✉❧❛♥❞♦ ❛ tr❛♥s♠✐tâ♥❝✐❛ ót✐❝❛ ❡ ❛ ❞✐❢❡r❡♥ç❛ ❞❡ ❢❛s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✸✳✶ ▼ét♦❞♦ ❞❡ ❏♦♥❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✸✳✸✳✷ ▼ét♦❞♦ ❞❡ ❏♦♥❡s ❡♠ ♠❡✐♦s ❝♦♠ ♣❛râ♠❡tr♦s ✈❛r✐á✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✹ ❈❛❧❝✉❧❛♥❞♦ ♦s t❡♠♣♦s ❞❡ r❡s♣♦st❛ ❞♦ s✐st❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
✹ ❆♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s ✺✾ ✹✳✶ ❚❡♠♣♦ ❞❡ r❡❧❛①❛♠❡♥t♦ ❞♦ ❞✐r❡t♦rτ0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾
✹✳✶✳✶ ❈♦♠♣♦rt❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ à ❡s♣❡ss✉r❛ ❡ ❛♦ ❛♥❝♦r❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✺✾ ✹✳✶✳✷ ❈♦♠♣♦rt❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛♦ ♣r❡t✐❧t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✹✳✶✳✸ ❈♦♠♣♦rt❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ à ❞✐❢❡r❡♥ç❛ ❞❡ ♣♦t❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✹✳✷ ❚❡♠♣♦s ót✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✹✳✷✳✶ ❈♦♠♣♦rt❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ à ❡s♣❡ss✉r❛ ❞❛ ❝é❧✉❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✹✳✷✳✷ ❈♦♠♣♦rt❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛♦ ♣r❡t✐❧t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✹✳✷✳✸ ❈♦♠♣♦rt❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ à ❞✐❢❡r❡♥ç❛ ❞❡ ♣♦t❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✹✳✸ ❚❡♠♣♦s ❞❡ r❡❧❛①❛çã♦ ❞❡ ❢❛s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✹✳✸✳✶ ❈♦♠♣♦rt❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ à ❡s♣❡ss✉r❛ ❡ ❛♦ ❛♥❝♦r❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✼✽ ✹✳✸✳✷ ❈♦♠♣♦rt❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛♦ ♣r❡t✐❧t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ✹✳✸✳✸ ❈♦♠♣♦rt❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ à ❞✐❢❡r❡♥ç❛ ❞❡ ♣♦t❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷
❈♦♥❝❧✉sõ❡s ✽✻
❆♣ê♥❞✐❝❡s ✽✼
❆ ❆❧❣♦r✐t♠♦s ❞❡s❡♥✈♦❧✈✐❞♦s ✽✽
❆✳✶ ❉✐str✐❜✉✐çã♦ ❞♦ ❞✐r❡t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✽ ❆✳✷ ❚❡♠♣♦s ❞❡ r❡❧❛①❛♠❡♥t♦ ót✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸ ❆✳✸ ❚❡♠♣♦s ❞❡ r❡❧❛①❛♠❡♥t♦ ❞♦ ❞✐r❡t♦r ❡ ❞❡ ❢❛s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✾✽
■♥tr♦❞✉çã♦
❈r✐st❛s ❧íq✉✐❞♦s sã♦ ♠❛t❡r✐❛✐s q✉❡ ✢✉❡♠✱ ♠❛s q✉❡ ❛♣r❡s❡♥t❛♠ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ só❧✐❞♦s✱ ❝♦♠♦ ❛ ❜✐rr❡❢r✐♥❣ê♥❝✐❛ ót✐❝❛✳ ❖ ✈❛❧♦r ❞❡ss❛ ❜✐rr❡❢r✐♥❣ê♥❝✐❛ ♣♦❞❡ s❡r ❛❧t❡r❛❞♦ ♣♦r ❢❛t♦r❡s ❡①t❡r♥♦s✱ ❝♦♠♦ ❝❛♠♣♦s ❡❧étr✐❝♦s ❡ ♠❛❣♥ét✐❝♦s✳
❊st❛ ú❧t✐♠❛ ♣r♦♣r✐❡❞❛❞❡✱ ♣♦r s✉❛ ✈❡③✱ é ❝❤❛♠❛❞❛ ❞❡ ❜✐rr❡❢r✐♥❣ê♥❝✐❛ ❝♦♥tr♦❧❛❞❛✳ ❉❡✈✐❞♦ ❛ ❡❧❛✱ ♦s ❝r✐st❛✐s ❧íq✉✐❞♦s t♦r♥❛r❛♠✲s❡ ♠❛t❡r✐❛✐s ❝♦♠ ✉♠❛ ❣r❛♥❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ ❛♣❧✐❝❛çõ❡s ót✐❝❛s✱ ❛ ❞❡ ♠❛✐♦r s✉❝❡ss♦ ❢♦✐ s✉❛ ✉t✐❧✐③❛çã♦ ❡♠ ♠♦str❛❞♦r❡s ❞✐❣✐t❛✐s ✭❞✐s♣❧❛②s✮ ❬✶✱✷❪✳
P❛r❛ q✉❡ ✉♠ ❝r✐st❛❧ ❧íq✉✐❞♦ ♣♦ss❛ s❡r ✉t✐❧✐③❛❞♦ ❡♠ ✉♠ ❞✐s♣❧❛②✱ é ♥❡❝❡ssár✐♦ q✉❡ ❡❧❡ ♣♦ss✉❛ ✉♠ ❣r✉♣♦ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❞❡s❡❥❛❞❛s✳ ❊♥tr❡ ❡❧❛s✱ é ✐♠♣♦rt❛♥t❡ q✉❡ ♦ t❡♠♣♦ ❞❡ r❡s♣♦st❛ ❡♠ r❡❧❛çã♦ ❛♦ ❝❛♠♣♦ ❡❧étr✐❝♦ s❡❥❛ ♠❡♥♦r q✉❡ ❛ s❡♥s✐❜✐❧✐❞❛❞❡ ❞♦ ♦❧❤♦ ❤✉♠❛♥♦ ✭❞❛ ♦r❞❡♠ ❞❡ ♠✐❧✐ss❡❣✉♥❞♦s✮✳
❖s t❡♠♣♦s ❞❡ r❡s♣♦st❛ sã♦ ❝♦♥tr♦❧❛❞♦s ♣♦r ♣❛râ♠❡tr♦s ❞♦ ❝r✐st❛❧ ❧íq✉✐❞♦ ❡ ❞❛s ♣❛r❡❞❡s q✉❡ ♦ ❝♦♥✜♥❛♠✱ ❞❡♥tr❡ ❡❧❡s✱ ♣♦❞❡♠♦s ❝✐t❛r ♦ ❛♥❝♦r❛♠❡♥t♦✱ ❛ ❡s♣❡ss✉r❛ ❞❛ ❝é❧✉❧❛ ❡ ❛ ✈✐s❝♦✲ s✐❞❛❞❡ ❞♦ ❝r✐st❛❧ ❧íq✉✐❞♦✳ ◗✉❛♥❞♦ ♦ ❛♥❝♦r❛♠❡♥t♦ é ❞♦ t✐♣♦ ❢r❛❝♦✱ t❛♠❜é♠ ❤á ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✐♥❝❧✉sã♦ ❞❛ ✈✐s❝♦s✐❞❛❞❡ ❞❡ s✉♣❡r❢í❝✐❡✱ ❝♦♠♦ ♠♦str❛r❡♠♦s ♥❡st❡ tr❛❜❛❧❤♦✳
❖r✐❣✐♥❛❞❛ ❞❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ♦ ❝r✐st❛❧ ❧íq✉✐❞♦ ❡ ♦ s✉❜str❛t♦ ❝♦♥✜♥❛❞♦r✱ ❛ ✈✐s❝♦s✐❞❛❞❡ ❞❡ s✉♣❡r❢í❝✐❡ ❣♦✈❡r♥❛ ❛ ❞✐♥â♠✐❝❛ ❞❡ ♠♦✈✐♠❡♥t♦s ❞❡ ♠♦❧é❝✉❧❛s ♣ró①✐♠❛s à ✐♥t❡r❢❛❝❡✳ ❆ ❢♦r♠❛ ❞❡ tr❛t❛r ♦ ♣r♦❜❧❡♠❛ ❛✐♥❞❛ é ❛❧✈♦ ❞❡ ❞✐s❝✉ssõ❡s✳ ◆❛ ❧✐t❡r❛t✉r❛✱ ❡♥❝♦♥tr❛♠♦s ❞♦✐s ♠♦❞♦s ❞❡ ❛❜♦r❞á✲❧♦✳
◆♦ ♠♦❞❡❧♦ ❞❡s❧♦❝❛❧✐③❛❞♦✱ ❛ ✈✐s❝♦s✐❞❛❞❡ ♣♦ss✉✐ ✉♠ ✈❛❧♦r ♥❛ s✉♣❡r❢í❝✐❡ q✉❡ s❡ ❛♣r♦①✐♠❛ ❝♦♥t✐♥✉❛♠❡♥t❡ ❞♦ ✈❛❧♦r ❞❛ ✈✐s❝♦s✐❞❛❞❡ ❞❡ ✈♦❧✉♠❡ ❝♦♥❢♦r♠❡ ❛ ❞✐stâ♥❝✐❛ ❡♠ r❡❧❛çã♦ ❛♦ s✉❜str❛t♦ ❛✉♠❡♥t❛ ❬✸✕✺❪✳
◆♦ ♠♦❞❡❧♦ ❧♦❝❛❧✐③❛❞♦✱ ♣r♦♣♦st♦ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ♣♦r ❉❡r③❤❛♥s❦✐ ❡ P❡tr♦✈ ❡ ♣♦r ❑❧é♠❛♥ ❡ P✐❝❦✐♥ ❬✺❪✱ ❛❞♠✐t❡✲s❡ q✉❡ ❛ r❡❣✐ã♦ ❞❡ ✐♥t❡r❢❛❝❡ é tã♦ ♣❡q✉❡♥❛ q✉❡ ❛ ✈✐s❝♦s✐❞❛❞❡ ❞❡ s✉♣❡r❢í❝✐❡ ✈❛r✐❛ ❞❡s❝♦♥t✐♥✉❛♠❡♥t❡ ❞♦ ✈❛❧♦r ❞❛ s✉♣❡r❢í❝✐❡ ♣❛r❛ ♦ ❞♦ ✈♦❧✉♠❡ ❬✸✱✻❪✳
◆❡st❡ tr❛❜❛❧❤♦✱ ✉t✐❧✐③❛r❡♠♦s ♦ ♠♦❞❡❧♦ ❞❡ ✈✐s❝♦s✐❞❛❞❡ ❞❡ s✉♣❡r❢í❝✐❡ ♣r♦♣♦st♦ ♣♦r ❉❡r③❤❛♥s❦✐ ❡ P❡tr♦✈ ♣❛r❛ ❡st✉❞❛r ❝♦♠♦ ❡ss❡ ♣❛râ♠❡tr♦ ✐♥✢✉❡♥❝✐❛ ♦s t❡♠♣♦s ❞❡ r❡❧❛①❛♠❡♥t♦ ❞❡ ✉♠ ❝r✐st❛❧ ❧íq✉✐❞♦ ❡♠ ✉♠❛ ❝é❧✉❧❛ ♥❛ ❣❡♦♠❡tr✐❛ ❞❡ s❧❛❜✳ P❛r❛ t❛♥t♦✱ ❞✐✈✐❞✐♠♦s ♦ tr❛❜❛❧❤♦ ♥❛ s❡❣✉✐♥t❡ s❡q✉ê♥❝✐❛✳
◆♦ ❝❛♣ít✉❧♦ ✶✱ ❢❛r❡♠♦s ✉♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ s♦❜r❡ ❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❛s ❢❛s❡s ❧íq✉✐❞♦✲ ❝r✐st❛❧✐♥❛s ❡ s✉❛s ❝❧❛ss✐✜❝❛çõ❡s✳ ❊♠ s❡❣✉✐❞❛✱ ❛❜♦r❞❛r❡♠♦s ❛s ♣r♦♣r✐❡❞❛❞❡s ❡❧ást✐❝❛s✱ ❡❧é✲ tr✐❝❛s ❡ ót✐❝❛s ❞❡ss❡s ♠❛t❡r✐❛✐s✳
◆♦ ❝❛♣ít✉❧♦ ✷✱ ✉t✐❧✐③❛r❡♠♦s ❡st❡s ❝♦♥❝❡✐t♦s ♣❛r❛ ♦ ❡st✉❞♦ ❛♥❛❧ít✐❝♦ ❞❛s tr❛♥s✐çõ❡s ❞❡ ❋ré❡❞❡r✐❝❦③✳ ■♥✐❝✐❛r❡♠♦s ❝♦♠ ♦ ❝❛s♦ ♠❛✐s s✐♠♣❧❡s✱ ❡♠ q✉❡ ♦ ❛♥❝♦r❛♠❡♥t♦ ♥❛ s✉♣❡r❢í❝✐❡ é ❞♦ t✐♣♦ ❢♦rt❡ ❡ ♥ã♦ ❤á ♣r❡t✐❧t✳ ❊♠ s❡❣✉✐❞❛✱ ❛❜♦r❞❛r❡♠♦s ❝❛s♦s ♠❛✐s ❝♦♠♣❧❡①♦s ❡✱ ❞❡✲ ♣♦✐s✱ ❛❜♦r❞❛r❡♠♦s ♦ ❝❛s♦ ♠❛✐s ❣❡r❛❧✱ ❡♠ q✉❡ ❛ ❝é❧✉❧❛ é s✉❜♠❡t✐❞❛ ❛ ✉♠ ❛♥❝♦r❛♠❡♥t♦ ❞♦ t✐♣♦ ❢r❛❝♦ ♥❛ ♣r❡s❡♥ç❛ ❞❡ ♣r❡t✐❧t✱ ❡ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ✈✐s❝♦s✐❞❛❞❡ ❞❡ s✉♣❡r❢í❝✐❡✳ P♦r ✜♠✱ ❞❡✜♥✐r❡♠♦s ♦s t❡♠♣♦s ❞❡ r❡❧❛①❛çã♦ ❡ ❞✐s❝✉t✐r❡♠♦s ❜r❡✈❡♠❡♥t❡ ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞❡ ✉♠ ❞✐s♣❧❛②✳
◆♦ ❝❛♣ít✉❧♦ ✸✱ ❡st✉❞❛r❡♠♦s ♥✉♠❡r✐❝❛♠❡♥t❡ ♦ ❝❛s♦ ♠❛✐s ❣❡r❛❧ ❞❛ tr❛♥s✐çã♦ ❞❡ ❋ré❡✲
❞r✐❝❦③s✳ ❆❧é♠ ❞♦s t❡♠♣♦s ❞❡ r❡❧❛①❛♠❡♥t♦ ❞♦ ❞✐r❡t♦r✱ ❛♣r❡s❡♥t❛r❡♠♦s t❛♠❜é♠ ♥❡st❡ ❝❛♣ít✉❧♦✱ ❢♦r♠❛s ❞❡ ♦❜t❡r♠♦s ♦s t❡♠♣♦s ❞❡ r❡❧❛①❛♠❡♥t♦ ót✐❝♦ ❡ ❞❡ r❡❧❛①❛♠❡♥t♦ ❞❡ ❢❛s❡✳
◆♦ ❝❛♣ít✉❧♦ ✹✱ s❡rã♦ ❝♦♠♣❛r❛r❡♠♦s ♦s r❡s✉❧t❛❞♦s ♥✉♠ér✐❝♦s ❡ ♦s t❡ór✐❝♦s ♣❛r❛ ♦s t❡♠♣♦s ❞❡ r❡❧❛①❛♠❡♥t♦ ❞♦ ❞✐r❡t♦r✳ ❊♠ s❡❣✉✐❞❛✱ ❛♥❛❧✐s❛r❡♠♦s ♦ ❡❢❡✐t♦ ❞❛ ✈✐s❝♦s✐❞❛❞❡ ❞❡ s✉♣❡r❢í❝✐❡ ♥♦s t❡♠♣♦s ❞❡ r❡❧❛①❛çã♦✳
P♦r ✜♠✱ ✐r❡♠♦s s✉♠❛r✐③❛r ❛s ❝♦♥❝❧✉sõ❡s ❞♦s ❡st✉❞♦s ♥♦ ❝❛♣ít✉❧♦ s❡❣✉✐♥t❡✳ ❊✱ ♥♦ ❛♣ê♥❞✐❝❡ ❆✱ ✐r❡♠♦s ❡♥❝♦♥tr❛r ♦s ❛❧❣♦rít✐♠♦s ✉t✐❧✐③❛❞♦s ♥❡st❡ tr❛❜❛❧❤♦✳
❈❛♣ít✉❧♦ ✶
❈❧❛ss✐✜❝❛çã♦ ❡ Pr♦♣r✐❡❞❛❞❡s ❋ís✐❝❛s ❞♦s
❈r✐st❛✐s ▲íq✉✐❞♦s
❯s✉❛❧♠❡♥t❡✱ ❞✐✈✐❞✐♠♦s ❛ ♠❛tér✐❛ ❡♠ três ❢❛s❡s✿ só❧✐❞❛✱ ❧íq✉✐❞❛ ❡ ❣❛s♦s❛✳ ◆♦ ❡♥t❛♥t♦✱ ❡①✐st❡♠ s✉❜stâ♥❝✐❛s q✉❡✱ ❡♠ ❝❡rt❛s ❝♦♥❞✐çõ❡s✱ ❛♣r❡s❡♥t❛♠ ❝❛r❛❝t❡ríst✐❝❛s q✉❡ ♣♦❞❡♠ ♣❡r✲ t❡♥❝❡r ❛ ♠❛✐s ❞❡ ✉♠❛ ❢❛s❡✳ ◗✉❛♥❞♦ ✐ss♦ ♦❝♦rr❡✱ ❞✐③❡♠♦s q✉❡ ❡ss❛ s✉❜stâ♥❝✐❛ s❡ ❡♥❝♦♥tr❛ ❡♠ ✉♠❛ ❢❛s❡ ✐♥t❡r♠❡❞✐ár✐❛✱ ❝❤❛♠❛❞❛ ❞❡ ♠❡s♦❢❛s❡s✳ ❖s ❝r✐st❛✐s ❧íq✉✐❞♦s sã♦ ♠❡s♦❢❛s❡s ✭❤á ❞✐✈❡rs❛s ❢❛s❡s ❞❡ ❝r✐st❛✐s ❧íq✉✐❞♦s✮ q✉❡ ❣❡r❛❧♠❡♥t❡ s❡ ❡♥❝♦♥tr❛♠ ❡♥tr❡ ❛ ❢❛s❡ ❧íq✉✐❞❛✲ ✐s♦tró♣✐❝❛ ❡ ❛ ❢❛s❡ só❧✐❞❛✲❝r✐st❛❧✐♥❛✳ ❚r❛t❛✲s❡ ❞❡ s✉❜stâ♥❝✐❛s q✉❡ ✢✉❡♠✱ ♠❛s q✉❡ t❛♠❜é♠ ❛♣r❡s❡♥t❛♠ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ só❧✐❞♦s ❝r✐st❛❧✐♥♦s✱ ❝♦♠♦ ❛♥✐s♦tr♦♣✐❛ ❡❧étr✐❝❛ ❡ ♦r❞❡♠ ♦r✐❡♥t❛❝✐♦♥❛❧✳
❖ ❢❛t♦ ❞❡ ✉♠ ♠❛t❡r✐❛❧ ♣♦ss✉✐r ♦r❞❡♠ ♦r✐❡♥t❛❝✐♦♥❛❧ ✐♠♣❧✐❝❛ ❡①✐stê♥❝✐❛ ❞❡ ❝♦rr❡❧❛çã♦ ♥❛ ♦r✐❡♥t❛çã♦ ❡♥tr❡ s❡✉s ❡❧❡♠❡♥t♦s ❝♦♥st✐t✉✐♥t❡s✳ ❊♠ ✉♠ ❧íq✉✐❞♦✱ ♦♥❞❡ ♥ã♦ ❤á ❝♦rr❡❧❛çã♦✱ ❛s ♠♦❧é❝✉❧❛s sã♦ ❞✐str✐❜✉í❞❛s ❛❧❡❛t♦r✐❛♠❡♥t❡ ❡ sã♦ ♦r✐❡♥t❛❞❛s ❛♦ ❛❝❛s♦ ✭♥❡♥❤✉♠❛ ♦r❞❡♠✮✱ ♠❛s✱ ❡♠ ✉♠ ❝r✐st❛❧ ❧íq✉✐❞♦✱ ♦s ❡✐①♦s ❧♦♥❣♦s ❞❡ ♠♦❧é❝✉❧❛s ♣ró①✐♠❛s t❡♥❞❡♠ ❛ s❡ ❛❧✐♥❤❛r ❡♠ ✉♠❛ ❞✐r❡çã♦ ❝♦♠✉♠ ❬✼❪✳ ❯♠❛ ❝♦♠♣❛r❛çã♦ ❡♥tr❡ ❛s ❞✐str✐❜✉✐çõ❡s ♠♦❧❡❝✉❧❛r❡s ❞❛s ❢❛s❡s só❧✐❞❛✲❝r✐st❛❧✐♥❛✱ ❧íq✉✐❞♦✲✐s♦tró♣✐❝❛ ❡ ❧íq✉✐❞❛✲❝r✐st❛❧✐♥❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ♥❛ ✜❣✉r❛ ✶✳✶
❖s ❝r✐st❛✐s ❧íq✉✐❞♦s sã♦ ❝♦♠♣♦st♦s ❞❡ ♠♦❧é❝✉❧❛s ❛♥✐s♦♠étr✐❝❛s✱ s❡♥❞♦ ♦ ❢♦r♠❛t♦ ❞❡ ❜❛stã♦ ♦ ♠❛✐s ❢❛♠♦s♦ ✭❝❤❛♠❛❞♦ ❞❡ ❝❛❧❛♠ít✐❝♦✮✳ ❚❛♠❜é♠ ❡♥❝♦♥tr❛♠♦s ❝r✐st❛✐s ❧íq✉✐❞♦s ❝♦♠ ♦ ❢♦r♠❛t♦ ❞❡ ❞✐s❝♦ ✭❝❤❛♠❛❞♦s ❞✐s❝ót✐❝♦s✮ ❡ ♦✉tr❛s ❢♦r♠❛s ♠❡♥♦s ❝♦♠✉♥s✱ ❝♦♠♦ ♦ ❢♦r♠❛t♦ ❞❡ ❜❛♥❛♥❛✳
✭❛✮ ✭❜✮ ✭❝✮
❋✐❣✉r❛ ✶✳✶✿ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ❢❛s❡✿ ❛✮ ❙ó❧✐❞❛✲❝r✐st❛❧✐♥❛✳ ❜✮ ▲íq✉✐❞❛✲❝r✐st❛❧✐♥❛✳ ❝✮ ▲íq✉✐❞♦✲ ✐s♦tró♣✐❝❛✳
✶✳✶ ❈❧❛ss✐✜❝❛çã♦
P♦❞❡♠♦s ❝❧❛ss✐✜❝❛r ❛s ❢❛s❡s ❧íq✉✐❞❛✲❝r✐st❛❧✐♥❛s ❡♠ ❞✉❛s ❣r❛♥❞❡s ❝❛t❡❣♦r✐❛s✿ ❧✐♦tró♣✐❝♦s ❡ t❡r♠♦tró♣✐❝♦s ❬✽❪✳
❈r✐st❛✐s ❧íq✉✐❞♦s ❧✐♦tró♣✐❝♦s sã♦ ❢♦r♠❛❞♦s ♣♦r ♠♦❧é❝✉❧❛s ❛♥✜❢í❧✐❝❛s q✉❡✱ ♣♦r s✉❛ ✈❡③✱ ❛♣r❡s❡♥t❛♠ ❞✉❛s ❛✜♥✐❞❛❞❡s✱ ✉♠❛ ♣❛rt❡ ❤✐❞r♦❢í❧✐❝❛ ❡ ♦✉tr❛ ❤✐❞r♦❢ó❜✐❝❛✳ ❉❡ss❛ ❢♦r♠❛✱ ❞❡♣❡♥❞❡♥❞♦ ❞❛ ♣r♦♣♦rçã♦ ❝♦♠ q✉❡ ❡ss❛ s✉❜stâ♥❝✐❛ é ❞✐ss♦❧✈✐❞❛ ❡♠ á❣✉❛✱ ♣♦❞❡ ♦❝♦rr❡r ❛ ❢♦r♠❛çã♦ ❞❡ ❛❣❧♦♠❡r❛❞♦s ❞❡ ♠♦❧é❝✉❧❛s ❞❡♥♦♠✐♥❛❞♦s ❞❡ ♠✐❝❡❧❛s✱ q✉❡ ♣♦❞❡♠ ♣♦ss✉✐r ❝♦♠♣♦rt❛♠❡♥t♦ ❧íq✉✐❞♦ ❝r✐st❛❧✐♥♦✳ ❈r✐st❛✐s ❧íq✉✐❞♦s ❧✐♦tró♣✐❝♦s ♣♦❞❡♠ s❡r ❢❛❝✐❧♠❡♥t❡ ❡♥✲ ❝♦♥tr❛❞♦s ♥♦ ❞✐❛ ❛ ❞✐❛✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ ❞❡t❡r❣❡♥t❡s ♦✉ s❛❜õ❡s ✭❛♠❜♦s ❞✐ss♦❧✈✐❞♦s ❡♠ á❣✉❛✮ ❡ ❡♠ ✈ár✐♦s s✐st❡♠❛s ❜✐♦❧ó❣✐❝♦s ❬✽❪✳
❈r✐st❛✐s ❧íq✉✐❞♦s t❡r♠♦tró♣✐❝♦s sã♦ ❝♦♠♣♦st♦s ❞❡ ♠♦❧é❝✉❧❛s q✉❡ ♣♦ss✉❡♠ ♣r♦♣♦rçõ❡s ❝❛r❛❝t❡ríst✐❝❛s ✭❡✐①♦ ❧♦♥❣♦ é ❝❡r❝❛ ❞❡ ✺ ✈❡③❡s ♠❛✐♦r q✉❡ ♦s ❞❡♠❛✐s ❡✐①♦s✮ ❬✷❪✳ ❙✉❛ ❡str✉t✉r❛ ♠♦❧❡❝✉❧❛r é ❝♦♠♣♦st❛ ❞❡ ✉♠❛ ♣❛rt❡ rí❣✐❞❛ ❡ ❞❡ ♦✉tr❛ ✢❡①í✈❡❧✱ s❡♥❞♦ q✉❡ ❛s ❞✉❛s sã♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❛ ❢♦r♠❛çã♦ ❞❛ ❢❛s❡ ❧íq✉✐❞♦✲❝r✐st❛❧✐♥❛✳ ❆ ♣❛rt❡ rí❣✐❞❛ ❢❛✈♦r❡❝❡ t❛♥t♦ ❛ ♦r❣❛♥✐③❛çã♦ ❡s♣❛❝✐❛❧ q✉❛♥t♦ ❛ ♦r✐❡♥t❛❝✐♦♥❛❧✱ ❡♥q✉❛♥t♦ q✉❡ ❛ ♣❛rt❡ ✢❡①í✈❡❧ ♥ã♦ ❢❛✈♦r❡❝❡ ♥❡♥❤✉♠❛ ❞❡❧❛s✳ ❈❛s♦ ❛ ♠♦❧é❝✉❧❛ ❢♦ss❡ ❝♦♠♣♦st❛ ❛♣❡♥❛s ❞❛ ♣❛rt❡ rí❣✐❞❛✱ ❡❧❛ ♣❛ss❛r✐❛ ❞❛ ❢❛s❡ ❝r✐st❛❧✐♥❛ ❞✐r❡t❛♠❡♥t❡ ♣❛r❛ ❛ ❧íq✉✐❞♦✲✐s♦tró♣✐❝❛✱ ❡ s❡ ❡❧❛ ❢♦ss❡ ❝♦♠♣❧❡t❛♠❡♥t❡ ✢❡①í✈❡❧✱ ♥ã♦ ❛♣r❡s❡♥t❛r✐❛ ♦r❞❡♠ ♦r✐❡♥t❛❝✐♦♥❛❧✳ ❖ ❡q✉✐❧í❜r✐♦ ❡♥tr❡ ❛s ❞✉❛s ❝❛r❛❝t❡ríst✐❝❛s é ♦ q✉❡ t♦r♥❛ ♣♦ssí✈❡❧ ❛ ❢♦r♠❛çã♦ ❞❛ ❢❛s❡ ❬✷❪✳
❚❛♥t♦ ❝r✐st❛✐s ❧íq✉✐❞♦s ❧✐♦tró♣✐❝♦s ❝♦♠♦ t❡r♠♦tró♣✐❝♦s ❛♣r❡s❡♥t❛♠ ❞✐✈❡rs❛s ❢❛s❡s✱ q✉❡ ♣♦❞❡♠ s❡r ❝❧❛ss✐✜❝❛❞❛s ♣❡❧❛ ❝♦rr❡❧❛çã♦ ♣♦s✐❝✐♦♥❛❧ ❡ ♦r✐❡♥t❛❝✐♦♥❛❧✱ ♣♦r ❡①❡♠♣❧♦✳ ❉❡♥tr♦ ❞❡ss❡s ❝r✐tér✐♦s✱ ♦s ❝r✐st❛✐s ❧íq✉✐❞♦s t❡r♠♦tró♣✐❝♦s ♣♦❞❡♠ s❡r ❞✐✈✐❞✐❞♦s ❡♠ três ❣r✉♣♦s✿ ♥❡♠át✐❝♦✱ ❡s♠ét✐❝♦ ❡ ❝♦❧❡stér✐❝♦✳
◆❡♠át✐❝♦s✿ ♥♦s ❝r✐st❛✐s ❧íq✉✐❞♦s ♥❡♠át✐❝♦s✱ ♦s ❝❡♥tr♦s ❞❡ ♠❛ss❛ ❞❛s ♠♦❧é❝✉❧❛s sã♦ ❞✐str✐❜✉í❞♦s ❛❧❡❛t♦r✐❛♠❡♥t❡ ✭s❡♠ ❝♦rr❡❧❛çã♦✮✱ ♣♦ré♠✱ ❛s ❞✐r❡çõ❡s ❞♦s ❡✐①♦s ❧♦♥❣♦s sã♦ ❞✐str✐❜✉í❞❛s ♣r❡❢❡r❡♥❝✐❛❧♠❡♥t❡ ❡♠ ✉♠❛ ú♥✐❝❛ ❞✐r❡çã♦✳ ❊ss❛ ❢❛s❡ é ❣❡r❛❧♠❡♥t❡ ❢♦r♠❛❞❛ ♣♦r ♠♦❧é❝✉❧❛s ❛❧♦♥❣❛❞❛s ♥♦ ❢♦r♠❛t♦ ❞❡ ❜❛stã♦✱ ♠❛s t❛♠❜é♠ é ♣♦ssí✈❡❧ ❡♥❝♦♥trá✲❧❛s ❡♠ ♦✉tr♦s ❢♦r♠❛t♦s✱ ❝♦♠♦ ♣♦❞❡ s❡r ✈✐st♦ ♥❛ ✜❣✉r❛ ✶✳✷
P❛r❛ ❞❡s❝r❡✈❡r ❛ ❝♦rr❡❧❛çã♦ ♦r✐❡♥t❛❝✐♦♥❛❧✱ é ✉t✐❧✐③❛❞♦ ✉♠ ✈❡t♦r ❝❤❛♠❛❞♦ ❞❡ ❞✐r❡t♦r✱ r❡♣r❡s❡♥t❛❞♦ ♣♦r ~n✳ ❊♠ ❝r✐st❛✐s ❧íq✉✐❞♦s ❝❛❧❛♠ít✐❝♦s✱ ❡❧❡ r❡♣r❡s❡♥t❛ ❛ ❞✐r❡çã♦ ♠é❞✐❛ ❞♦
❡✐①♦ ❧♦♥❣♦ ♠♦❧❡❝✉❧❛r ❡♠ ✉♠❛ r❡❣✐ã♦✳ P❛r❛ ♦s ♥❡♠át✐❝♦s ❞✐s❝ót✐❝♦s✱ ❡ss❡ ✈❡t♦r r❡♣r❡s❡♥t❛ ❛ ❞✐r❡çã♦ ♠é❞✐❛ ❞♦s ✈❡t♦r❡s ♥♦r♠❛✐s ❡♠ r❡❧❛çã♦ ❛♦ ♣❧❛♥♦ q✉❡ ❝♦♥té♠ ❛ ♠♦❧é❝✉❧❛✳
❆ ❢❛s❡ ♥❡♠át✐❝❛ ♥ã♦ ❛♣r❡s❡♥t❛ ♣♦❧❛r✐③❛çã♦ ❡s♣♦♥tâ♥❡❛ q✉❛♥❞♦ ❝♦♠♣♦st❛ ♣♦r ♠♦❧é❝✉❧❛s
✭❛✮ ✭❜✮ ✭❝✮
❋✐❣✉r❛ ✶✳✷✿ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ❢❛s❡ ♥❡♠át✐❝❛✿ ❛✮ ❈❛❧❛♠ít✐❝❛ ✉♥✐❛①✐❛❧✳ ❜✮ ❇✐❛①✐❛❧✳ ❝✮ ❉✐s❝ót✐❝❛✳
♥♦ ❢♦r♠❛t♦ ❝❛❧❛♠ít✐❝♦✳ ▼❡s♠♦ q✉❛♥❞♦ ♣♦ss✉❡♠ ♠♦♠❡♥t♦s ❞❡ ❞✐♣♦❧♦ ♣❡r♠❛♥❡♥t❡✱ ❛s ✐♥t❡r❛çõ❡s ♠♦❧❡❝✉❧❛r❡s ❢❛③❡♠ ❝♦♠ q✉❡ ❛s ♠❧é❝✉❧❛s s❡ ♦r❣❛♥✐③❡♠ ❞❡ ♠♦❞♦ q✉❡ ❛ ♣♦❧❛r✐③❛çã♦ r❡s✉❧t❛♥t❡ s❡❥❛ ♥✉❧❛✳ ❆ ♣❛rt✐r ❞✐ss♦✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡✱ ♣❛r❛ ❡ss❛ ❢❛s❡✱ ~n ❡ −~n sã♦
❡q✉✐✈❛❧❡♥t❡s ❬✷❪✳
●❡r❛❧♠❡♥t❡✱ ♠♦❧é❝✉❧❛s q✉❡ ❝♦♠♣õ❡♠ ❛ ❢❛s❡ ♥❡♠át✐❝❛ ♣♦ss✉❡♠ s✐♠❡tr✐❛ ❛①✐❛❧✱ s❡♥❞♦ ♥❡❝❡ssár✐♦ ❛♣❡♥❛s ✉♠ ❞✐r❡t♦r ♣❛r❛ ❛ s✉❛ ❝❛r❛❝t❡r✐③❛çã♦✳ ▼✉✐t❛s ♠♦❧é❝✉❧❛s✱ t♦❞❛✈✐❛✱ ♥ã♦ ❛♣r❡s❡♥t❛♠ ❡ss❛ s✐♠❡tr✐❛✱ ♠❛s ❛ ✈❡❧♦❝✐❞❛❞❡ ❝♦♠ q✉❡ ❡❧❛s ❡①❡❝✉t❛♠ r♦t❛çõ❡s ❡♠ t♦r♥♦ ❞♦ ❡✐①♦ ❧♦♥❣♦ ✭❝❡r❝❛ ❞❡ 1Ghz✮ ❢❛③ ❝♦♠ q✉❡ ❡❧❛s s❡ ❝♦♠♣♦rt❡♠ ❝♦♠♦ s❡ ❛ ♣♦ss✉íss❡♠ ❬✷❪✳
❈r✐st❛✐s ❧íq✉✐❞♦s ❝♦♠♣♦st♦s ♣♦r ❡ss❡ t✐♣♦ ❞❡ ♠♦❧é❝✉❧❛s sã♦ ❝❤❛♠❛❞♦s ❞❡ ✉♥✐❛①✐❛✐s✳
P❛r❛ ♦s ❝r✐st❛✐s ❧íq✉✐❞♦s q✉❡ ♥ã♦ ❛♣r❡s❡♥t❛♠ s✐♠❡tr✐❛ ✉♥✐❛①✐❛❧✱ é ♣r❡❝✐s♦ ❞♦✐s ❞✐r❡t♦r❡s ♣❛r❛ ❝❛r❛❝t❡r✐③❛r s✉❛ ♦r✐❡♥t❛çã♦ ♠é❞✐❛✳ ❊ss❡ t✐♣♦ ❞❡ ❝r✐st❛❧ ❧íq✉✐❞♦ é ❝❤❛♠❛❞♦ ❞❡ ❜✐❛①✐❛❧✳ ❊ss❡s ú❧t✐♠♦s ❢♦r❛♠ ♦❜s❡r✈❛❞♦s ❛♣❡♥❛s ❡♥tr❡ ❝r✐st❛✐s ❧íq✉✐❞♦s ❧✐♦tró♣✐❝♦s✱ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❝r✐st❛✐s ❧íq✉✐❞♦s ❜✐❛①✐❛✐s t❡r♠♦tró♣✐❝♦s ❛✐♥❞❛ é ♦❜❥❡t♦ ❞❡ ❞✐s❝✉ssõ❡s ❬✾❪✳
P❛r❛ ❢❛③❡r♠♦s ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❝♦♠♣❧❡t❛ ❞♦ ❝r✐st❛❧ ❧íq✉✐❞♦ ♥❡♠át✐❝♦✱ ♣r❡❝✐s❛♠♦s ✐♥tr♦❞✉③✐r ♦✉tr♦ ♣❛râ♠❡tr♦ ❝♦♠♣❧❡♠❡♥t❛r ❛♦ ❞✐r❡t♦r✳ ❊♠❜♦r❛ ♦ ❞✐r❡t♦r ❞❡♠♦♥str❡ ❛ ♦r✐❡♥t❛çã♦ ♠é❞✐❛ ❡♠ ✉♠❛ r❡❣✐ã♦✱ ❡❧❡ ♥ã♦ ❢♦r♥❡❝❡ ✐♥❢♦r♠❛çã♦ ❞❡ q✉ã♦ ❜❡♠ ♦r✐❡♥t❛❞♦ ♦ ❝r✐st❛❧ ❧íq✉✐❞♦ s❡ ❡♥❝♦♥tr❛✱ ♣♦r ❡①❡♠♣❧♦✿ ❛s ✜❣✉r❛s ✶✳✸❛ ❡ ✶✳✸❜ ♠♦str❛♠ ❞✉❛s ❞✐str✐❜✉✐çõ❡s ♠♦❧❡❝✉❧❛r❡s q✉❡ ♣♦ss✉❡♠ ♦ ♠❡s♠♦ ❞✐r❡t♦r✱ ♠❛s ❡❧❛s ❛♣r❡s❡♥t❛♠ ❡st❛❞♦ ❞❡ ♦r❞❡♥❛♠❡♥t♦ ❞✐st✐♥t♦s✳
P❛r❛ ❝♦♥t♦r♥❛r ❡ss❡ ♣r♦❜❧❡♠❛✱ é ❞❡✜♥✐❞♦ ✉♠ ♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠ ❡s❝❛❧❛r q✉❡ ✐♥❞✐❝❛ ♦ ❣r❛✉ ❞❡ ♦r✐❡♥t❛çã♦ ❡♠ q✉❡ ♦ ♠❡✐♦ s❡ ❡♥❝♦♥tr❛✳ ❯♠❛ ♣r✐♠❡✐r❛ t❡♥t❛t✐✈❛ s❡r✐❛ ❞❡✜♥✐r ❡ss❡ ♣❛râ♠❡tr♦ ❝♦♠♦ ♦ ✈❛❧♦r ♠é❞✐♦ ❞♦ ❞❡s✈✐♦ ❞♦s ❡✐①♦s ❧♦♥❣♦s ❡♠ r❡❧❛çã♦ ❛♦ ❞✐r❡t♦r✳ ❊ss❡ ♣❛râ♠❡tr♦ ♣♦❞❡r✐❛ s❡r ❡①♣r❡ss♦ ♣❡❧❛ ♠é❞✐❛ ❞❡ ✉♠ ♣r♦❞✉t♦ ❡s❝❛❧❛r ❡♥tr❡ ♦ ✈❡t♦r ~a✱ q✉❡
r❡♣r❡s❡♥t❛ ❛ ♦r✐❡♥t❛çã♦ ❞❡ ✉♠❛ ♠♦❧é❝✉❧❛✱ ❡ ♦ ❞✐r❡t♦r ~n✱ ✐❧✉str❛❞♦ ♥❛ ✜❣✉r❛ ✶✳✹✳ ❉❡ss❛
❢♦r♠❛✱ ♦ ✈❛❧♦r ❞♦ ♣❛râ♠❡tr♦ s❡r✐❛1s❡ ♦ ♠❡✐♦ ❡st✐✈❡ss❡ ♣❡r❢❡✐t❛♠❡♥t❡ ♦r❞❡♥❛❞♦ ❡0♥❛ ❢❛s❡
✐s♦tró♣✐❝❛✳ ▼❛s ❡ss❛ ❞❡✜♥✐çã♦ ♥ã♦ ❧❡✈❛ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ~n ❡ −~n✱ ❛
q✉❛❧ ❢❛r✐❛ ❝♦♠ q✉❡ ❛ ♠é❞✐❛ ❢♦ss❡ s❡♠♣r❡ 0✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❞♦ ❡st❛❞♦ ❞❡ ♦r✐❡♥t❛çã♦✳
❯♠❛ ❢♦r♠❛ ♠❛✐s ❝♦♥✈❡♥✐❡♥t❡ ❝♦♥s✐st❡ ❡♠ ❞❡✜♥✐r ♦ ♣❛râ♠❡tr♦ ❝♦♠♦ S = 1/2 < (3 cos2θ −1) >✱ ❡♠ q✉❡ θ é ♦ â♥❣✉❧♦ ❡♥tr❡ ~a ❡ ~n✳ ◆❡ss❛ ❞❡✜♥✐çã♦✱ ♦ ♣❛râ♠❡tr♦ ❙ t❡♠
✈❛❧♦r 1 q✉❛♥❞♦ ♦ ♥❡♠át✐❝♦ ❡stá ♣❡r❢❡✐t❛♠❡♥t❡ ♦r✐❡♥t❛❞♦ ❡ 0 ♥❛ ❢❛s❡ ✐s♦tró♣✐❝❛✳ P♦❞❡♠♦s
❛✐♥❞❛✱ ✉t✐❧✐③❛r S ❡ ♦ ❞✐r❡t♦r ♣❛r❛ ❞❡✜♥✐r ✉♠ ♣❛râ♠❡tr♦ q✉❡ ❡①♣r❡ss❡ t♦❞❛s ❛s s✐♠❡tr✐❛s
❞❛ ❢❛s❡✳ ■ss♦ é ❢❡✐t♦ ✐♥tr♦❞✉③✐♥❞♦ ✉♠ ♣❛râ♠❡tr♦ t❡♥s♦r✐❛❧ Q ❝✉❥❛s ❝♦♠♣♦♥❡♥t❡s ♣♦ss✉❡♠
n
✭❛✮
n
✭❜✮
❋✐❣✉r❛ ✶✳✸✿ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ❞♦✐s ❡st❛❞♦s ❞❡ ♦r❞❡♥❛♠❡♥t♦ q✉❡ ❛♣r❡s❡♥t❛♠ ❛ ♠❡s♠❛ ❞✐r❡çã♦ ♠é❞✐❛✳
â
ϕ θ
n
❋✐❣✉r❛ ✶✳✹✿ ❘❡♣r❡s❡♥t❛çã♦ ❞♦s â♥❣✉❧♦s ❡♥tr❡ ✉♠❛ ♠♦❧é❝✉❧❛ ❞❡ ❝r✐st❛❧ ❧íq✉✐❞♦ ❡ ♦ ❞✐r❡t♦r✳
❛ ❢♦r♠❛ ❬✷❪✿
Qij =
3 2S
ninj −
δij
3
, ✭✶✳✶✮
❡♠ q✉❡ ni ❡ nj sã♦ ❛s ❝♦♠♣♦♥❡♥t❡s ❞♦ ❞✐r❡t♦r✱ ❙ é ♦ ♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠ ❡s❝❛❧❛r ❡ δij é
❛ ❞❡❧t❛ ❞❡ ❑r♦♥❡❝❦❡r✳
❊s♠ét✐❝♦s✿ ❆❧❣✉♠❛s ❢❛s❡s ❞❡ ❝r✐st❛✐s ❧íq✉✐❞♦s✱ ❛❧é♠ ❞❡ ❛♣r❡s❡♥t❛r ❝♦rr❡❧❛çã♦ ♦r✐❡♥✲ t❛❝✐♦♥❛❧✱ t❛♠❜é♠ ❛♣r❡s❡♥t❛♠ ❝❡rt❛ ♦r❞❡♠ ♣♦s✐❝✐♦♥❛❧✳ ❊ss❡ é ♦ ❝❛s♦ ❞❛ ❢❛s❡ ❡s♠ét✐❝❛✱ ❡♥❝♦♥tr❛❞❛ ❡♥tr❡ ❛ ❢❛s❡ só❧✐❞♦✲❝r✐st❛❧✐♥❛ ❡ ❛ ♥❡♠át✐❝❛✳ ◆❡ss❛ ♠❡s♦❢❛s❡✱ ♦s ❝r✐st❛✐s ❧íq✉✐❞♦s sã♦ ♦r❣❛♥✐③❛❞♦s ❡♠ ❝❛♠❛❞❛s✱ ❝✉❥❛s ❞✐stâ♥❝✐❛s sã♦ ✜①❛s ❡ ❤á ✈ár✐❛s ❢♦r♠❛s ❞❡ ♦r❣❛♥✐③❛çã♦ ❡♥tr❡ ❛s ❝❛♠❛❞❛s ❡ ♦s ❞✐r❡t♦r❡s✱ s❡♥❞♦ q✉❡ ❝❛❞❛ t✐♣♦ ❞❡ ♦r❣❛♥✐③❛çã♦ ❝❛r❛❝t❡r✐③❛ ✉♠❛ ❢❛s❡ ❡s♠ét✐❝❛ ❞✐❢❡r❡♥t❡✳
❆s ❢❛s❡s ❡s♠ét✐❝❛s ♠❛✐s ❝♦♥❤❡❝✐❞❛s sã♦ ❛ ❊s♠ét✐❝❛ ❆ ❡ ❛ ❊s♠ét✐❝❛ ❈✳ ❊♠ ❛♠❜❛s✱ ♦s ❝♦♠♣♦♥❡♥t❡s ♣r❡s❡♥t❡s ♥❛ ♠❡s♠❛ ❝❛♠❛❞❛ ❛♣r❡s❡♥t❛♠ ❢♦rt❡ ♦r❞❡♠ ♦r✐❡♥t❛❝✐♦♥❛❧ ❡ ❛s ❞✉❛s ♠❡s♦❢❛s❡s s❡ ❞✐❢❡r❡♠ ♣❡❧❛ ❞✐r❡çã♦ ❞♦ ❞✐r❡t♦r ❡♠ r❡❧❛çã♦ às ❝❛♠❛❞❛s✳ ◆❛ ❢❛s❡ ❆✱ ♦ ❞✐r❡t♦r s❡ ❡♥❝♦♥tr❛ ♥♦r♠❛❧ ❛ ❡❧❛✱ ❡♥q✉❛♥t♦ q✉❡✱ ♥❛ ❢❛s❡ ❈✱ ❡❧❡ s❡ ❡♥❝♦♥tr❛ ✐♥❝❧✐♥❛❞♦✳ ❊ss❛s ❞✉❛s ❢❛s❡s ♣♦❞❡♠ s❡r ✈✐st❛s ♥❛ ✜❣✉r❛ ✶✳✺✳
❈♦❧❡stér✐❝♦s✿ P♦❞❡♠♦s ❡♥❝♦♥tr❛r ♥❛ ♥❛t✉r❡③❛ ❝r✐st❛✐s ❧íq✉✐❞♦s ♥❡♠át✐❝♦s ❝♦♠ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❛ ♠❛✐s✱ ❛ q✉✐r❛❧✐❞❛❞❡✳ ❈r✐st❛✐s ❧íq✉✐❞♦s ♥❡♠át✐❝♦s q✉✐r❛✐s sã♦ ❝❤❛♠❛❞♦s ❞❡ ❝♦❧❡stér✐❝♦s✳ ❆ q✉✐r❛❧✐❞❛❞❡ ❢❛③ ❝♦♠ q✉❡ ♦ ❞✐r❡t♦r ❞❡ ✉♠ ♥❡♠át✐❝♦ s♦❢r❛ ✉♠❛ ❧❡✈❡ t♦rçã♦
✭❛✮ ✭❜✮
❋✐❣✉r❛ ✶✳✺✿ ❘❡♣r❡s❡♥t❛çã♦ ❞❛s ❢❛s❡s✿ ❛✮ ❊s♠ét✐❝❛ ❆✳ ❜✮ ❊s♠ét✐❝❛ ❈✳
❡♥tr❡ ✉♠ ♣❧❛♥♦ ❡ ♦✉tr♦✱ r❡s✉❧t❛♥❞♦ ❡♠ ✉♠❛ ❡str✉t✉r❛ ❤❡❧✐❝♦✐❞❛❧✳ ❊ss❛ ❡str✉t✉r❛ ❡①❡❝✉t❛ ✉♠❛ t♦rçã♦ ❞❡ ✸✻✵◦ ❡ ♦ ❡s♣❛ç♦ ♥❡❝❡ssár✐♦ ♣❛r❛ t❛♥t♦ é ❝❤❛♠❛❞♦ ❞❡ ♣❛ss♦ ✭p✮✳ P♦❞❡♠♦s
✈❡r ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❡sq✉❡♠át✐❝❛ ❞❛ ❢❛s❡ ♥❛ ✜❣✉r❛ ✶✳✻✳
❊①✐st❡♠ ❜❛s✐❝❛♠❡♥t❡ ❞✉❛s ❢♦r♠❛s ❞❡ ♦❜t❡r♠♦s ✉♠ ❝♦❧❡stér✐❝♦✿ ✉♠❛ é ❡♥❝♦♥tr❛r ❝r✐st❛✐s ❧íq✉✐❞♦s ♥❡♠át✐❝♦s ✐♥tr✐♥s❡❝❛♠❡♥t❡ q✉✐r❛✐s❀ ❛ ♦✉tr❛ ❝♦♥s✐st❡ ❡♠ ❛❞✐❝✐♦♥❛r ✉♠ ❞♦♣❛♥t❡ q✉✐r❛❧ ❛ ✉♠ ♥❡♠át✐❝♦✳ ❊♠ t❡r♠♦s ♣rát✐❝♦s✱ ❛ s❡❣✉♥❞❛ ❢♦r♠❛ é ♠❛✐s út✐❧ q✉❡ ❛ ♣r✐♠❡✐r❛✱ ♣♦✐s ♣♦❞❡♠♦s ❝♦♥tr♦❧❛r ♦ ♣❛ss♦ ♣♦r ♠❡✐♦ ❞❛ ❝♦♥❝❡♥tr❛çã♦ ❞♦ ❞♦♣❛♥t❡ ❬✽❪✳
❋✐❣✉r❛ ✶✳✻✿ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠ ❝♦❧❡stér✐❝♦✳
✶✳✷ Pr♦♣r✐❡❞❛❞❡s ❡❧ást✐❝❛s ❞❡ ❝r✐st❛✐s ❧íq✉✐❞♦s
❖ ♣r♦❜❧❡♠❛ ❡♠ ❞❡t❡r♠✐♥❛r ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❞✐r❡çã♦ ♠é❞✐❛ ❞❛s ♠♦❧é❝✉❧❛s ❡♠ ❝r✐st❛✐s ❧íq✉✐❞♦s é ✉s✉❛❧♠❡♥t❡ ❛❜♦r❞❛❞♦ ❞❡ ❞✉❛s ❢♦r♠❛s✿ ✉♠❛ ❝♦♥s✐st❡ ❡♠ ❛♥❛❧✐s❛r ♠❛❝r♦s❝♦♣✐❝❛✲ ♠❡♥t❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❝♦❧❡t✐✈♦ ❞♦ s✐st❡♠❛❀ ❛ ♦✉tr❛ ❝♦♥s✐st❡ ❡♠ ❞❡s❝r❡✈❡r ♦ s✐st❡♠❛ ♣♦r ♠❡✐♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ♠♦❧❡❝✉❧❛r❡s ✐♥❞✐✈✐❞✉❛✐s✳
❆ ❛❜♦r❞❛❣❡♠ ♠❛❝r♦s❝ó♣✐❝❛ é ❣❡r❛❧♠❡♥t❡ ❢❡✐t❛ ♣♦r ♠♦❞❡❧♦s ❢❡♥♦♠❡♥♦❧ó❣✐❝♦s✳ ◆❡❧❛✱ ♣r♦♣r✐❡❞❛❞❡s ❞♦ s✐st❡♠❛ sã♦ ♦❜t✐❞❛s ♣♦r ♠❡✐♦ ❞❡ ❛♣r♦①✐♠❛çõ❡s ❞❡ ❢✉♥çõ❡s ❝♦♠ ♣r♦♣r✐❡❞❛✲ ❞❡s ❝❛r❛❝t❡ríst✐❝❛s ❞❛ ❢❛s❡✱ ❝♦♠ ❛❧❣✉♠❛s ❝♦♥st❛♥t❡s q✉❡ sã♦ ❛❥✉st❛❞❛s ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡✳ ❯♠ ❡①❡♠♣❧♦ ✐♠♣♦rt❛♥t❡ é ❛ t❡♦r✐❛ ❡❧ást✐❝❛ ❞♦ ❝♦♥tí♥✉♦✱ ❝✉❥♦ ♠ét♦❞♦ t❡♠ ♦❜t✐❞♦ ❜♦♥s r❡✲ s✉❧t❛❞♦s✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ♣❛r❛ ♣r♦❜❧❡♠❛s ❞❡ ✈♦❧✉♠❡ ❬✶✵✕✶✸❪✳ ◆❡ss❛ ❛❜♦r❞❛❣❡♠✱ é ❝♦♠✉♠ ❡♥❝♦♥tr❛r ❡q✉❛çõ❡s q✉❡ ♥ã♦ ♣♦ss✉❡♠ s♦❧✉çõ❡s ❛♥❛❧ít✐❝❛s✱ ♠❛s✱ ♠✉✐t❛s ✈❡③❡s✱ é ♣♦ssí✈❡❧ r❡s♦❧✈ê✲❧❛s ✉t✐❧✐③❛♥❞♦ ♠ét♦❞♦s ♥✉♠ér✐❝♦s ✉s✉❛✐s✱ ❝♦♠♦ ❞✐❢❡r❡♥ç❛s ✜♥✐t❛s ❡ ❡❧❡♠❡♥t♦s ✜♥✐✲ t♦s ❬✶✹✕✶✼❪✳
❆ t❡♦r✐❛ ❡❧ást✐❝❛ ❞♦ ❝♦♥tí♥✉♦ é ❣❡r❛❧♠❡♥t❡ ✉t✐❧✐③❛❞❛ ♣❛r❛ ❞❡s❝r❡✈❡r ❛ ❞✐str✐❜✉✐çã♦ ❞♦ ❞✐r❡t♦r ❡♠ ❝r✐st❛✐s ❧íq✉✐❞♦s✳ P❛r❛ t❛♥t♦✱ é ♣r❡❝✐s♦ ❛❞♠✐t✐r ❞✉❛s ❤✐♣ót❡s❡s✿ ♣r✐♠❡✐r♦✱ é ♣r❡❝✐s♦ ❛❞♠✐t✐r q✉❡ ♦ ❞✐r❡t♦r ❞♦ ❝r✐st❛❧ ❧íq✉✐❞♦ ♣♦ss❛ s❡r ❞❡✜♥✐❞♦ ❡♠ ✉♠ ✈♦❧✉♠❡ ✐♥✜♥✐t❡✲ s✐♠❛❧✱ ❞❡ ❢♦r♠❛ q✉❡ ♦ ❞✐r❡t♦r ✈❛r✐❡ ❝♦♥t✐♥✉❛♠❡♥t❡ ♥♦ ❡s♣❛ç♦✳ ➱ ♣r❡❝✐s♦ q✉❡ ❡ss❡ ✈♦❧✉♠❡ s❡❥❛ ❣r❛♥❞❡ ♦ s✉✜❝✐❡♥t❡ ♣❛r❛ q✉❡ ❡♥❣❧♦❜❡ ✈ár✐❛s ♠♦❧é❝✉❧❛s✱ ♠❛♥t❡♥❞♦ ♦ s❡♥t✐❞♦ ❞❡ ❞✐r❡çã♦ ♠é❞✐❛ ❞♦ ❞✐r❡t♦r ✭❝❤❛♠❛❞❛ ❞❡ ❛♣r♦①✐♠❛çã♦ ♠❡s♦s❝ó♣✐❝❛✮✳
❆ s❡❣✉♥❞❛ ❤✐♣ót❡s❡ ❛ s❡r ❛ss✉♠✐❞❛ é q✉❡ ❛s ✈❛r✐❛çõ❡s q✉❡ ♣♦ss❛♠ ♦❝♦rr❡r ♥♦ ❞✐r❡t♦r ❛♦ ❧♦♥❣♦ ❞♦ ❝r✐st❛❧ ❧íq✉✐❞♦ s❡❥❛♠ ♠✉✐t♦ s✉❛✈❡s✱ ❞❡ ❢♦r♠❛ q✉❡ ✈❛r✐❛çõ❡s s✐❣♥✐✜❝❛t✐✈❛s ♥♦ ❞✐r❡t♦r ❛❝♦♥t❡ç❛♠ ❡♠ ❞✐stâ♥❝✐❛s ❞❡ ♦r❞❡♠ ♠✉✐t♦ ♠❛✐♦r❡s q✉❡ ❛s ❞✐stâ♥❝✐❛s ♠♦❧❡❝✉❧❛r❡s✳
❊ss❛s ❞✉❛s ❤✐♣ót❡s❡s ❡♠ ❝♦♥❥✉♥t♦ ♣❡r♠✐t❡♠ tr❛t❛r ♦ s✐st❡♠❛ ❧íq✉✐❞♦ ❝r✐st❛❧✐♥♦ ❝♦♠♦ ✉♠ ♠❡✐♦ ❡❧ást✐❝♦✱ ❛ss♦❝✐❛♥❞♦ ✉♠❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ♥❛s ✈❛r✐❛çõ❡s ❞♦ ❞✐r❡t♦r ❛♦ ❧♦♥❣♦ ❞♦ ❡s♣❛ç♦✳ ❊ss❛s ✈❛r✐❛çõ❡s ♣♦❞❡♠ s❡r ❡①♣r❡ss❛s ♣♦r ♠❡✐♦ ❞❡ ❞❡r✐✈❛❞❛s ❡s♣❛❝✐❛✐s ❡✱ ❝♦♠♦ ❛s ✈❛r✐❛çõ❡s sã♦ s✉❛✈❡s ✭s❡❣✉♥❞❛ ❤✐♣ót❡s❡✮✱ ♣♦❞❡♠♦s s✉♣♦r✱ ❛✐♥❞❛✱ q✉❡ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ❞❡♣❡♥❞❛ ❛♣❡♥❛s ❞❛ ❞❡r✐✈❛❞❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ ❉❡ss❛ ❢♦r♠❛✱ ♠❡s♠♦ q✉❡ ♥ã♦ s❡❥❛ ❝♦♥❤❡❝✐❞❛ ❛ ❢♦r♠❛ ❞❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛✱ ♣♦❞❡♠♦s ❞❡s❡♥✈♦❧✈ê✲❧❛ ❡♠ sér✐❡ ❞❡ ❚❛②❧♦r ❡ ❞❡t❡r♠✐♥❛r ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ ❛s ❝♦♥st❛♥t❡s ❡♥✈♦❧✈✐❞❛s✳ ❋❛③❡♥❞♦ ✐ss♦✱ ♦❜t❡♠♦s ❛ ❢❛♠♦s❛ ❡♥❡r❣✐❛ ❧✐✈r❡ ❞❡ ❋r❛♥❦✿
ff rank =
1
2K11(∇ ·~n) 2 +1
2K22(~n· ∇ ×~n) 2+ 1
2K33(~n× ∇ ×~n)
2, ✭✶✳✷✮
❡♠ q✉❡ ~né ♦ ❞✐r❡t♦r ❡ K11✱ K22 ❡K33 sã♦ ❝♦♥st❛♥t❡s ❡❧ást✐❝❛s✱ ❞❡t❡r♠✐♥❛❞❛s ❡①♣❡r✐♠❡♥✲
t❛❧♠❡♥t❡✳
❈❛❞❛ ❝♦♥st❛♥t❡ ❡❧ást✐❝❛ ❡stá ❛ss♦❝✐❛❞❛ ❛ ✉♠ t✐♣♦ ❞❡ ❞✐st♦rçã♦ ❞♦ ❞✐r❡t♦r ❞❡ ✉♠ ♥❡♠á✲ t✐❝♦✳ ❆ ❝♦♥st❛♥t❡ K11 ❡stá r❡❧❛❝✐♦♥❛❞❛ ❛ ❞❡❢♦r♠❛çõ❡s ❞❡ ❞✐✈❡r❣ê♥❝✐❛✱ K22 ❛ ❞❡❢♦r♠❛çõ❡s
❞♦ t✐♣♦ t♦rçã♦ ❡ K33 ❛ ❞❡❢♦r♠❛çõ❡s ❞♦ t✐♣♦ ✢❡①ã♦✳ ❉❡✈✐❞♦ à ❢♦r♠❛ ❞❡ss❛s ❞❡❢♦r♠❛çõ❡s✱
❛s ❝♦♥st❛♥t❡s K11✱ K22 ❡ K33 r❡❝❡❜❡♠ ♦ ♥♦♠❡ ❞❡ ❝♦♥st❛♥t❡ ❞❡ ❙♣❧❛②✱ ❚✇✐st ❡ ❇❡♥❞✱ r❡s✲
♣❡❝t✐✈❛♠❡♥t❡✳ ❆s ❢♦r♠❛s ❞❛s ❞✐st♦rçõ❡s ♣♦❞❡♠ s❡r ✈✐s✉❛❧✐③❛❞❛s ♥❛ ✜❣✉r❛ ✶✳✼✳ ❊♠ ✉♠ ❝r✐st❛❧ ❧íq✉✐❞♦ ♥❡♠át✐❝♦✱ ❡ss❛s sã♦ ❛s três ❢♦r♠❛s ❜ás✐❝❛s ❞❡ ❞✐st♦rçõ❡s ♣♦ssí✈❡✐s✳ ❚♦❞❛s ❛s ❞❡♠❛✐s ❢♦r♠❛s ♣♦❞❡♠ s❡r ♦❜t✐❞❛s ❝♦♠♦ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❞❡❧❛s✳
✭❛✮ ✭❜✮
✭❝✮
❋✐❣✉r❛ ✶✳✼✿ ❚✐♣♦s ❞❡ ❞✐st♦rçõ❡s r❡❧❛❝✐♦♥❛❞♦s à ❝♦♥st❛♥t❡✿ ❛✮ K11✭❙♣❧❛②✮✳ ❜✮K22✭❚✇✐st✮✳
❝✮ K33✭❇❡♥❞✮✳
✶✳✸ ❊♥❡r❣✐❛ ❞❡ ❛♥❝♦r❛♠❡♥t♦
❊♠ s✐t✉❛çõ❡s ❞❡ ✐♥t❡r❡ss❡✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ❝r✐st❛✐s ❧íq✉✐❞♦s ❝♦♥✜♥❛❞♦s ♣♦r s✉♣❡r✲ ❢í❝✐❡s ❞❡ ♦✉tr♦s ♠❛t❡r✐❛✐s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❞♦✐s s✉❜str❛t♦s ❞❡ ✈✐❞r♦✳ ❆ ✐♥t❡r❛çã♦ ❡♥tr❡ ♦ ❝r✐st❛❧ ❧íq✉✐❞♦ ❡ ❛ s✉♣❡r❢í❝✐❡ ♣♦❞❡ ❛❧t❡r❛r ❛ ❡♥❡r❣✐❛ ❞❡ s✉♣❡r❢í❝✐❡ ❞♦ s✐st❡♠❛✳ P♦r ❡①❡♠✲ ♣❧♦✱ ♥❛ ❢❛❜r✐❝❛çã♦ ❞❡ ✉♠ ❞✐s♣❧❛②✱ é ✐♥s❡r✐❞❛ ✉♠❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝r✐st❛❧ ❧íq✉✐❞♦ ❡♥tr❡ ❞♦✐s s✉❜str❛t♦s ❞❡ ✈✐❞r♦✳ ❆s ♠♦❧é❝✉❧❛s ♣ró①✐♠❛s à s✉♣❡r❢í❝✐❡ ✐rã♦ ✐♥t❡r❛❣✐r ❝♦♠ ❛ ♣❛r❡❞❡ ❞♦ s✉❜str❛t♦ ❞❡ ❢♦r♠❛ ❞✐❢❡r❡♥t❡ ❞❡ ❝♦♠♦ ❡❧❛s ✐♥t❡r❛❣❡♠ ❝♦♠ ❛s ♠♦❧é❝✉❧❛s ♥♦ ✐♥t❡r✐♦r ❞❛ ❝é❧✉❧❛✱ r❡s✉❧t❛♥❞♦✱ ❛ss✐♠✱ ❡♠ ✉♠❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ s✉♣❡r✜❝✐❛❧✱ ❛ q✉❡ ❝❤❛♠❛♠♦s ❞❡ ❡♥❡r❣✐❛ ❞❡ ❛♥❝♦r❛♠❡♥t♦✳
❊♠❜♦r❛ t❡♥❤❛ s✐❞♦ ❞❡s❝♦❜❡rt❛ ❤á ♠❛✐s ❞❡ tr✐♥t❛ ❛♥♦s✱ ❛ ♥❛t✉r❡③❛ ❡s♣❡❝í✜❝❛ ❞❛ ❡♥❡r❣✐❛ ❞❡ ❛♥❝♦r❛♠❡♥t♦ ❛✐♥❞❛ é ✐♥❝❡rt❛✳ ❙❛❜❡♠♦s q✉❡ ❡❧❛ ❡stá ❧✐❣❛❞❛ ❛ ❢❡♥ô♠❡♥♦s q✉❡ ♦❝♦rr❡♠ ♥❛ ✐♥t❡r❢❛❝❡ ❡♥tr❡ ♦ ❝r✐st❛❧ ❧íq✉✐❞♦ ❡ ✉♠❛ s✉♣❡r❢í❝✐❡✱ ♠❛s ❤á ❡①♣❡r✐♠❡♥t♦s q✉❡ ♠♦str❛r❛♠ q✉❡ ❛ ❡♥❡r❣✐❛ ❞❡ ❛♥❝♦r❛♠❡♥t♦ ♣♦❞❡ ❞❡♣❡♥❞❡r ❞❛ ❡s♣❡ss✉r❛ ❞❛ ❝é❧✉❧❛✳ ❆t✉❛❧♠❡♥t❡✱ ♣♦ss✉í♠♦s ❜♦♥s ♠♦❞❡❧♦s ❢❡♥♦♠❡♥♦❧ó❣✐❝♦s ♣❛r❛ ❞❡s❝r❡✈❡r ❛ ❡♥❡r❣✐❛ ❞❡ ❛♥❝♦r❛♠❡♥t♦✱ ❝♦♥t✉❞♦✱ s✉❛ ♥❛t✉r❡③❛ ❝♦♥t✐♥✉❛ s❡♥❞♦ ♦❜❥❡t♦ ❞❡ ✐♥✈❡st✐❣❛çã♦ ❬✶✽❪✳
❯♠❛ ❝❛r❛❝t❡ríst✐❝❛ ✐♠♣♦rt❛♥t❡ ❞❛ ❡♥❡r❣✐❛ ❞❡ ❛♥❝♦r❛♠❡♥t♦ é s✉❛ ❞❡♣❡♥❞ê♥❝✐❛ ❝♦♠ ❛ ♦r✐❡♥t❛çã♦ ❞♦ ❞✐r❡t♦r ♣ró①✐♠♦ à ✐♥t❡r❢❛❝❡✳ ❊♥q✉❛♥t♦ q✉❡ ✉♠❛ ✐♥t❡r❢❛❝❡ ❡♥tr❡ ✉♠❛ s✉♣❡r❢í❝✐❡ só❧✐❞❛ ❡ ✉♠ ❧íq✉✐❞♦ ✐s♦tró♣✐❝♦ é ❜❛s✐❝❛♠❡♥t❡ ❞❡s❝r✐t❛ ♣❡❧❛ t❡♥sã♦ s✉♣❡r✜❝✐❛❧ ✭❡♠ ✉♠ ♠♦❞❡❧♦ ❢❡♥♦♠❡♥♦❧ó❣✐❝♦✮✱ ♣❛r❛ ✉♠ ❧íq✉✐❞♦ ❛♥✐s♦tró♣✐❝♦✱ ❡ss❛ t❡♥sã♦ ♠✉❞❛ ❝♦♥❢♦r♠❡ ❛ ♦r✐❡♥t❛çã♦ ❞❛s ♠♦❧é❝✉❧❛s✳ ❆ ❞✐r❡çã♦ ♣r❡❢❡r❡♥❝✐❛❧ ✐♥❞✉③✐❞❛ ♣❡❧❛ s✉♣❡r❢í❝✐❡ é ❝❤❛♠❛❞❛ ❞❡ ❡✐①♦ ❢á❝✐❧✳ ◗✉❛♥❞♦ ✉♠ ❝r✐st❛❧ ❧íq✉✐❞♦ é ♦r✐❡♥t❛❞♦ ♥❛ ❞✐r❡çã♦ ❞❡ss❡ ❡✐①♦✱ ❛ ❡♥❡r❣✐❛ ❞❡ s✉♣❡r❢í❝✐❡ é ♠í♥✐♠❛✳
❯♠ ❞♦s ♠♦❞❡❧♦s ❜❛st❛♥t❡ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❞❡s❝r❡✈❡r ❛ ❡♥❡r❣✐❛ ❞❡ ❛♥❝♦r❛♠❡♥t♦ ❢♦✐ ❞❡✲ s❡♥✈♦❧✈✐❞♦ ♣♦r ❘❛♣✐♥✐✲P❛♣♦✉❧❛r ❬✶✾❪✳ ◆❡❧❡✱ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ s✉♣❡r✜❝✐❛❧ t❛♠❜é♠ é ♦❜t✐❞❛ ❢❡♥♦♠❡♥♦❧♦❣✐❝❛♠❡♥t❡✱ ❝✉❥❛ ❡①♣r❡ssã♦ é✿
fs=
1
2Wasen
2(φ−φ s) +
1
2Wpsen
2(θ−θ
s). ✭✶✳✸✮
◆❡ss❛ ❡①♣r❡ssã♦✱ Wa ❡ Wp sã♦ ❛s ❡♥❡r❣✐❛s ❞❡ ❛♥❝♦r❛♠❡♥t♦ ❛③✐♠✉t❛❧ ❡ ♣♦❧❛r❀ θs ❡ φs✱
♦s â♥❣✉❧♦s ♣♦❧❛r ❡ ❛③✐♠✉t❛❧ ❡✐①♦ ❢á❝✐❧❀ ❡θ ❡φ✱ â♥❣✉❧♦s ❞♦ ❞✐r❡t♦r✳ ❆s ❝♦♥st❛♥t❡sWp✱ Wa✱
θs ❡ φs sã♦ ❞❡t❡r♠✐♥❛❞❛s ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡✳
❋✐s✐❝❛♠❡♥t❡✱ ✐♥t❡r♣r❡t❛♠♦s ❛s ❝♦♥st❛♥t❡sWa❡Wp❝♦♠♦ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❛ ✐♥t❡♥s✐❞❛❞❡
❞♦ ❛♥❝♦r❛♠❡♥t♦✳ ◗✉❛♥t♦ ♠❛✐♦r❡s ❢♦r❡♠ ❡ss❛s ❝♦♥st❛♥t❡s✱ ♠❛✐s ❡♥❡r❣✐❛ s❡rá ♥❡❝❡ssár✐♦ ♣❛r❛ ❞❡s✈✐❛r ♦ ❞✐r❡t♦r ❞♦ ❡✐①♦ ❢á❝✐❧✳ ❱❛❧♦r❡s tí♣✐❝♦s ♣❛r❛ ❡ss❛s ❝♦♥st❛♥t❡s ✈❛r✐❛♠ ❡♥tr❡
10−6❏/m2 ❡ 10−3❏/m2✳ ◗✉❛♥❞♦ ❛ ✐♥t❡♥s✐❞❛❞❡ ❞♦ ❛♥❝♦r❛♠❡♥t♦ é ❞❛ ♦r❞❡♠ ❞❡ 10−3J/m2✱
é ♥❡❝❡ssár✐❛ ✉♠❛ ❣r❛♥❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ♣❛r❛ r❡t✐r❛r ♦ ❞✐r❡t♦r ❞♦ ❡✐①♦ ❢á❝✐❧✳ ◆❡ss❛ s✐t✉❛çã♦✱ ❞✐③❡♠♦s q✉❡ ♦ ❛♥❝♦r❛♠❡♥t♦ é ❞♦ t✐♣♦ ❢♦rt❡ ❡✱ ♣❛r❛ ✈❛❧♦r❡s ❞❡ ♦r❞❡♥s ♠❛✐s ❜❛✐①❛s✱ q✉❡ ♦ ❛♥❝♦r❛♠❡♥t♦ é ❞♦ t✐♣♦ ❢r❛❝♦✳
✶✳✹ ❘❡s♣♦st❛ ❡❧étr✐❝❛
❈r✐st❛✐s ❧íq✉✐❞♦s sã♦ ♠❛t❡r✐❛✐s q✉❡ ♥ã♦ ❛♣r❡s❡♥t❛♠ ♣♦❧❛r✐③❛çã♦ ❡s♣♦♥tâ♥❡❛✳ ❊♠❜♦r❛ s✉❛s ♠♦❧é❝✉❧❛s ❣❡r❛❧♠❡♥t❡ ❛♣r❡s❡♥t❡♠ ❞✐♣♦❧♦s ❡❧étr✐❝♦s ✐♥trí♥s❡❝♦s✱ ❛ ❢❛s❡ s❡ ♦r❣❛♥✐③❛ ❞❡ ❢♦r♠❛ ❛ ♥ã♦ ❛♣r❡s❡♥t❛r ♠♦♠❡♥t♦ ❞❡ ❞✐♣♦❧♦ ❬✽❪✳ ❊♥tr❡t❛♥t♦✱ ❛ ♣r❡s❡♥ç❛ ❞❡ ✉♠ ❝❛♠♣♦ ❡❧étr✐❝♦ ♣♦❞❡ ♦r✐❡♥t❛r ♦s ❞✐♣♦❧♦s ♠♦❧❡❝✉❧❛r❡s ❡ ♣r♦✈♦❝❛r ❛ s❡♣❛r❛çã♦ ❞❡ ❝❛r❣❛✱ ♣♦❧❛r✐③❛♥❞♦ ♦ ♠❡✐♦✳
❉❡✈✐❞♦ à ❛♥✐s♦tr♦♣✐❛ ❞✐❡❧étr✐❝❛ ❞♦ s✐st❡♠❛✱ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ♣♦❞❡ ✈❛r✐❛r ♥♦ ✐♥t❡r✐♦r ❞♦ ❝r✐st❛❧ ❧íq✉✐❞♦✳ P❛r❛ q✉❛♥t✐✜❝❛r ❛ r❡s♣♦st❛ ❞♦ ♠❡✐♦ ❛ ❝❛♠♣♦s ❡❧étr✐❝♦s✱ ♣r❡❝✐s❛♠♦s s❛❜❡r q✉❛❧ ♦ t✐♣♦ ❞❡ r❡❣✐♠❡ ❛ q✉❡ ♦ ♣r♦❝❡ss♦ é s✉❜♠❡t✐❞♦✳ ■r❡♠♦s ❛❜♦r❞❛r ❞♦✐s r❡❣✐♠❡s ❞❡ r❡s♣♦st❛ ❡❧étr✐❝❛✿ ♦ ♣r✐♠❡✐r♦ ❛❝♦♥t❡❝❡ q✉❛♥❞♦ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ✐♥❞✉③ ♣❡q✉❡♥❛s ✈❛r✐✲ ❛çõ❡s ♥♦ ❞✐r❡t♦r ❡ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ❝♦♥st❛♥t❡ ❡♠ t♦❞♦ ♦ ♠❡✐♦❀ ♥♦ s❡❣✉♥❞♦✱ r❡❣✐♠❡ ✐r❡♠♦s ❝♦♥s✐❞❡r❛r q✉❡ ❛ ❞✐❢❡r❡♥ç❛ ❞❡ ♣♦t❡♥❝✐❛❧ ♥♦s s✉❜str❛t♦s é ❝♦♥st❛♥t❡✳ ❊ss❡ r❡❣✐♠❡ é ♦❜t✐❞♦ q✉❛♥❞♦ t♦❞♦ ♣r♦❝❡ss♦ ♦❝♦rr❡ ❝♦♥❡❝t❛❞♦ ❛ ✉♠❛ ❜❛t❡r✐❛ q✉❡ ♠❛♥té♠ ❝♦♥st❛♥t❡ ❛ ✈♦❧t❛❣❡♠ ♥♦ s✐st❡♠❛✳
✶✳✹✳✶ ❘❡s♣♦st❛ ❡❧étr✐❝❛ ❛ ❝❛♠♣♦ ❡❧étr✐❝♦ ❝♦♥st❛♥t❡
❊♠ ♠❡✐♦s ✐s♦tró♣✐❝♦s✱ ❛ ♣♦❧❛r✐③❛çã♦ ♣❛r❛ ❝❛♠♣♦s ♣❡q✉❡♥♦s ♣♦❞❡ s❡r ❛♣r♦①✐♠❛❞❛ ♣♦r
~
P =ǫ0χ ~E✱ s❡♥❞♦χ❛ ♣❡r♠✐ss✐✈✐❞❛❞❡ ❞✐❡❧étr✐❝❛ ❞♦ ♠❡✐♦ ❡ P~ ❛ ♣♦❧❛r✐③❛çã♦ ♣♦r ✉♥✐❞❛❞❡ ❞❡
✈♦❧✉♠❡✳
P❛r❛ ♠❡✐♦s ❛♥✐s♦tró♣✐❝♦s✱ ❛ r❡s♣♦st❛ ❡❧étr✐❝❛ ❞❡♣❡♥❞❡ ❞❛ ❞✐r❡çã♦ ❞♦ ❝❛♠♣♦ ❛♣❧✐❝❛❞♦✳ ❊♠ ❝r✐st❛✐s ❧íq✉✐❞♦s✱ ❡❧❛ ❞❡♣❡♥❞❡ ❞❛ ❞✐r❡çã♦ r❡❧❛t✐✈❛ ❡♥tr❡ ♦ ❞✐r❡t♦r ❡ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦✳ P♦rt❛♥t♦✱ ♣❛r❛ ♦❜t❡r♠♦s ❛ ♣♦❧❛r✐③❛çã♦ ❡♠ ❝r✐st❛✐s ❧íq✉✐❞♦s✱ é ♣r❡❝✐s♦ r❡❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ ❞❛ ♣♦❧❛r✐③❛çã♦ ♣❛r❛ ♠❡✐♦s ✐s♦tró♣✐❝♦s ❞❡ ✉♠❛ ❢♦r♠❛ ❝♦♥s✐st❡♥t❡ ❝♦♠ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ♠❡✐♦✳ ■ss♦ ♣♦❞❡ s❡r ❢❡✐t♦ s✉❜st✐t✉✐♥❞♦ ❛ ❝♦♥st❛♥t❡ ❞❡ ♣❡r♠✐ss✐✈✐❞❛❞❡ ❞✐❡❧étr✐❝❛ ✭χ✮ ♣♦r
✉♠ t❡♥s♦r ✭χ✮✱ ❝✉❥❛s ❝♦♠♣♦♥❡♥t❡s ❞❡♣❡♥❞❛♠ ❞♦ ❞✐r❡t♦r ❡ ❞♦ s✐st❡♠❛ ❞❡ r❡❢❡rê♥❝✐❛✳
P❛r❛ ❝r✐st❛✐s ❧íq✉✐❞♦s ✉♥✐❛①✐❛✐s✱ ❛ ♣♦❧❛r✐③❛çã♦ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❞❡ ❢♦r♠❛ ♠❛✐s ❝♦♥✈❡♥✐✲ ❡♥t❡✳ ❆ s✐♠❡tr✐❛ ❛①✐❛❧ ❞♦ ♠❡✐♦ ❢❛③ ❝♦♠ q✉❡ s❡❥❛ ♣♦ssí✈❡❧ s✉❜st✐t✉✐r ♦ t❡♥s♦r χ~ ♣♦r ❞✉❛s
❝♦♥st❛♥t❡s ❞❡ s✉s❝❡t✐❜✐❧✐❞❛❞❡✱ ✉♠❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❡ ♦✉tr❛ ♣❛r❛❧❡❧❛ ❛♦ ❞✐r❡t♦r✳ ❆ss✐♠✱ é ♣♦ssí✈❡❧ ❞❡❝♦♠♣♦r ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ♥❡ss❛s ❞✐r❡çõ❡s✱ t❛❧ ❝♦♠♦ ✐❧✉str❛ ❛ ✜❣✉r❛ ✶✳✽✱ ❡ ❡s❝r❡✈❡r ❛s ❝♦♠♣♦♥❡♥t❡s ❞❛ ♣♦❧❛r✐③❛çã♦ ❝♦♠♦✿
~
Pk =ǫ0χk(~n·E~)~n,
❡
~
P⊥ =χ⊥[E~ −(~n·E~)~n].
✭✶✳✹✮
❖✉ ♥❛ ❢♦r♠❛ ❣❡r❛❧✿
~
P =ǫ0{χk(~n·E~)~n+χ⊥[E~ −(~n·E~)~n]}, ✭✶✳✺✮
❡♠ q✉❡ χk ❡ χ⊥ sã♦ ❛s ❝♦♥st❛♥t❡s ❞❡ ♣❡r♠✐ss✐✈✐❞❛❞❡ ♣❛r❛❧❡❧❛ ❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ❞✐r❡t♦r✱
r❡s♣❡❝t✐✈❛♠❡♥t❡✳
P♦❞❡♠♦s✱ t❛♠❜é♠✱ ❡s❝r❡✈❡r ❛ ♣♦❧❛r✐③❛çã♦ ❡♠ t❡r♠♦s ❞❛s ❝♦♥st❛♥t❡s ❞✐❡❧étr✐❝❛s ❞♦ ♠❡✐♦✳ ❊❧❛s ♣♦❞❡♠ s❡r ♦❜t✐❞❛s ❛tr❛✈és ❞❛s ❝♦♥st❛♥t❡s ❞❡ s✉s❝❡t✐❜✐❧✐❞❛❞❡ ❡❧étr✐❝❛ ♣❡❧❛ r❡❧❛çã♦ ǫ⊥ = χ⊥ + 1 ǫk = χk + 1✱ ❡♠ q✉❡ ǫ⊥ ❡ ǫk sã♦ r❡s♣❡❝t✐✈❛♠❡♥t❡ ❛s ❝♦♥st❛♥t❡s
❞✐❡❧étr✐❝❛s ♣❡r♣❡♥❞✐❝✉❧❛r ❡ ♣❛r❛❧❡❧❛ ❛♦ ❞✐r❡t♦r ❬✾❪✳ ❉❡ss❛s r❡❧❛çõ❡s✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡
∆χ= ∆ǫ, ✭✶✳✻✮
❡♠ q✉❡ ∆ǫ=ǫk−ǫ⊥ é ❝❤❛♠❛❞❛ ❞❡ ❝♦♥st❛♥t❡ ❞❡ ❛♥✐s♦tr♦♣✐❛ ❞✐❡❧étr✐❝❛✳
❙✉❜st✐t✉✐♥❞♦ ❛ ❡q✉❛çã♦ ✭✶✳✻✮ ♥❛ ❡q✉❛çã♦ ✭✶✳✺✮✱ ♦❜t❡♠♦s
~
P =ǫ0[χ⊥E~ + ∆ǫ(E~ ·~n)~n]. ✭✶✳✼✮
◆♦ ♣r♦❝❡ss♦ ❞❡ ♣♦❧❛r✐③❛çã♦✱ ♦❝♦rr❡ tr❛♥s❢❡rê♥❝✐❛ ❞❡ ❡♥❡r❣✐❛ ❡♥tr❡ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ❡ ♦ ♠❡✐♦ q✉❡ ❡stá s❡♥❞♦ ♣♦❧❛r✐③❛❞♦✳ ◗✉❛♥❞♦ ❛ ❛♣❧✐❝❛çã♦ ❞♦ ❝❛♠♣♦ ✐♥❞✉③ ♣❡q✉❡♥❛s ✈❛r✐❛çõ❡s
♥♦ ❞✐r❡t♦r✱ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ E ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ❝♦♥st❛♥t❡ ♥♦ ✐♥t❡r✐♦r ❞♦ ♠❛t❡r✐❛❧✳
❉❡ss❛ ❢♦r♠❛✱ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ❞✐❡❧étr✐❝❛ é ♦❜t✐❞❛ ♣♦r ❬✷✵❪
fe=−
1
2E~ ·P .~ ✭✶✳✽✮
P♦r ✜♠✱ s✉❜st✐t✉✐♥❞♦ ✭✶✳✼✮ ❡♠ ✭✶✳✽✮✱ ♦❜t❡♠♦s✿
fe =−
1
2ǫ0[χ⊥|E|
2+ ∆ǫ(E~
·~n)2]. ✭✶✳✾✮
❖ ♣r✐♠❡✐r♦ t❡r♠♦ ❞❛ ❡q✉❛çã♦ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ♦r✐❡♥t❛çã♦ ❞♦ ❞✐r❡t♦r✱ ❡❧❡ ♣♦❞❡ s❡r ❞❡s✲ ❝❛rt❛❞♦ q✉❛♥❞♦ ❛♣❡♥❛s ❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ❡ ♦ ❝r✐st❛❧ ❧íq✉✐❞♦ ❢♦r ♥❡❝❡ssár✐❛✳ ◆❡ss❛s s✐t✉❛çõ❡s✱ t❡r❡♠♦s
fe =−
1
2ǫ0∆ǫ(E~ ·~n)
2. ✭✶✳✶✵✮
❖❜s❡r✈❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✶✳✶✵✮✱ ♣♦❞❡♠♦s ❝♦♥st❛t❛r q✉❡ ❛ ❝♦♥st❛♥t❡ ❞❡ ❛♥✐s♦tr♦♣✐❛ ❞✐❡❧é✲ tr✐❝❛ ❡①❡r❝❡ ✉♠ ✐♠♣♦rt❛♥t❡ ♣❛♣❡❧ ♥❛ ♦r✐❡♥t❛çã♦ ❞❡ ❝r✐st❛✐s ❧íq✉✐❞♦s✳ ❆ ♦r✐❡♥t❛çã♦ r❡❧❛t✐✈❛ ❞♦s ❞✐r❡t♦r❡s ❡♠ r❡❧❛çã♦ ❛♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ❞❡♣❡♥❞❡ ❞♦ s✐♥❛❧ ❞❡ss❛ ❝♦♥st❛♥t❡✿ q✉❛♥❞♦ ❡❧❛ é ♣♦s✐t✐✈❛✱ ♦ ♠í♥✐♠♦ ❞❡ ❡♥❡r❣✐❛ ❡❧étr✐❝♦ é ❛t✐♥❣✐❞♦ q✉❛♥❞♦ ♦ ❞✐r❡t♦r ❡stá ♦r✐❡♥t❛❞♦ ♣❛r❛✲ ❧❡❧♦ ❛♦ ❝❛♠♣♦ ❡❧étr✐❝♦❀ ❝❛s♦ ❡❧❛ ♣♦ss✉❛ ✉♠ ✈❛❧♦r ♥❡❣❛t✐✈♦✱ ♦ ♠í♥✐♠♦ ❞❡ ❡♥❡r❣✐❛ ❡❧étr✐❝❛ é ♦❜t✐❞♦ q✉❛♥❞♦ ♦ ❞✐r❡t♦r ❡stá ♦r✐❡♥t❛❞♦ ♣❡r♣❡♥❞✐❝✉❧❛r♠❡♥t❡ ❛♦ ❝❛♠♣♦ ❡❧étr✐❝♦✳
✶✳✹✳✷ ❘❡s♣♦st❛ ❡❧étr✐❝❛ à ❞✐❢❡r❡♥ç❛ ❞❡ ♣♦t❡♥❝✐❛❧ ❝♦♥st❛♥t❡
❈❛s♦ ❛ ✈❛r✐❛çã♦ ❞♦ ❞✐r❡t♦r s❡❥❛ ❡①♣r❡ss✐✈❛✱ ♣r❡❝✐s❛♠♦s ❝♦♥s✐❞❡r❛r ❛ ✈❛r✐❛çã♦ ❞♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ♥♦ ✐♥t❡r✐♦r ❞❛ ❛♠♦str❛ ❡ ♦ r❡❣✐♠❡ ❛ q✉❡ é s✉❜♠❡t✐❞♦ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ♥♦ s✐st❡♠❛✳ ❈♦♥s✐❞❡r❛r❡♠♦s ❛❣♦r❛ q✉❡ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ é ❛♣❧✐❝❛❞♦ ❛ ✉♠❛ ❞✐❢❡r❡♥ç❛ ❞❡ ♣♦t❡♥❝✐❛❧ ❝♦♥st❛♥t❡✳ ❉✉r❛♥t❡ ♦ ♣r♦❝❡ss♦✱ ✉♠❛ q✉❛♥t✐❞❛❞❡ ❞❡ tr❛❜❛❧❤♦ We ♣♦❞❡ s❡r r❡❛❧✐③❛❞❛ ♣❡❧❛
❜❛t❡r✐❛ ♣❛r❛ ♠❛♥t❡r ♦ s✐st❡♠❛ ❛ ✉♠❛ ❞✐❢❡r❡♥ç❛ ❞❡ ♣♦t❡♥❝✐❛❧ ❝♦♥st❛♥t❡✳ ❙❡ ♦ ♣r♦❝❡ss♦ ❢♦r r❡❛❧✐③❛❞♦ ❛ t❡♠♣❡r❛t✉r❛ ❡ ♣r❡ssã♦ ❝♦♥st❛♥t❡s✱ ♦ ❡st❛❞♦ ✜♥❛❧ s❡rá ❞❛❞♦ ♣❡❧♦ ♠í♥✐♠♦ ❞❛ ❡♥❡r❣✐❛ ❞❡ ●✐❜❜s ❞♦ s✐st❡♠❛✳ ■♥❝❧✉✐♥❞♦ ♦ tr❛❜❛❧❤♦ r❡❛❧✐③❛❞♦ ♣❡❧❛ ❜❛t❡r✐❛✱ ♦ s✐st❡♠❛ ❡♥❝♦♥tr❛ ♦ ❡q✉✐❧í❜r✐♦ ♥♦ ❡st❛❞♦ q✉❡ ♠✐♥✐♠✐③❛ ♦ ❢✉♥❝✐♦♥❛❧ ❬✷❪
n
E
E
⊥
‖
E
❋✐❣✉r❛ ✶✳✽✿ ❈♦♠♣♦♥❡♥t❡s ❞♦ ✈❡t♦r ❞❡ ♣♦❧❛r✐③❛çã♦✳
