Análise Térmica e Espectroscopia Raman à Temperatura Ambiente de Cristais L-histidina

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❏♦sé ❊♥❡❞✐❧t♦♥ ▼❡❞❡✐r♦s P❡r❡✐r❛

❆♥á❧✐s❡ ❚ér♠✐❝❛ ❡ ❊s♣❡❝tr♦s❝♦♣✐❛ ❘❛♠❛♥ à

❚❡♠♣❡r❛t✉r❛ ❆♠❜✐❡♥t❡ ❞❡ ❈r✐st❛✐s ▲✲❤✐st✐❞✐♥❛

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❏♦sé ❊♥❡❞✐❧t♦♥ ▼❡❞❡✐r♦s P❡r❡✐r❛

❆♥á❧✐s❡ ❚ér♠✐❝❛ ❡ ❊s♣❡❝tr♦s❝♦♣✐❛ ❘❛♠❛♥ à

❚❡♠♣❡r❛t✉r❛ ❆♠❜✐❡♥t❡ ❞❡ ❈r✐st❛✐s ▲✲❤✐st✐❞✐♥❛

▼♦♥♦❣r❛✜❛ ❛♣r❡s❡♥t❛❞❛ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ●r❛✉ ❞❡ ❇❛❝❤❛r❡❧ ❡♠ ❋ís✐❝❛ ♣❡❧❛ ❯♥✐✈❡rs✐✲ ❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá✳

❖r✐❡♥t❛❞♦r✿

Pr♦❢✳ ❉r✳ P❛✉❧♦ ❞❡ ❚❛rs♦ ❈❛✈❛❧❝❛♥t❡ ❋r❡✐r❡

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá

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❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦ ✐♥✈❡st✐❣♦✉✲s❡ ✉♠ ❞♦s ✷✵ ❛♠✐♥♦á❝✐❞♦s q✉❡ ❝♦♠♣õ❡♠ ❛s ♣r♦t❡í♥❛s ❡♥❝♦♥✲ tr❛❞❛s ♥♦s s❡r❡s ✈✐✈♦s✱ ♦ ❛♠✐♥♦á❝✐❞♦ ▲✲❤✐st✐❞✐♥❛ ✭❈6❍9◆3❖2✮✱ ♣♦r ❡s♣❡❝tr♦s❝♦♣✐❛ ❘❛♠❛♥✳

❆ r❡❣✐ã♦ ❡s♣❡❝tr❛❧ ❡st✉❞❛❞❛ ❢♦✐ ❛ ❞❡ ♥ú♠❡r♦s ❞❡ ♦♥❞❛ ❡♥tr❡ ✶✵ cm−1 _{❡ ✸✹✵✵} _cm−1_{✳ ❆s}

❛♠♦str❛s ✉t✐❧✐③❛❞❛s sã♦ ❡♠ ❢♦r♠❛ ❞❡ ♣ó✱ ❝♦♥st✐t✉í❞♦ ♣♦r ♣♦❧✐❝r✐st❛✐s✳ ❋♦r❛♠ r❡❛❧✐③❛❞♦s ❡①♣❡r✐♠❡♥t♦s ❞❡ ❛♥á❧✐s❡ t❡r♠♦❣r❛✈✐♠étr✐❝❛ ✭❚●✮ ❡ ❝❛❧♦r✐♠❡tr✐❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ✈❛rr❡❞✉r❛ ✭❉❙❈✮✱ ♣❛r❛ ♦❜t❡r ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ❛s ♣r♦♣r✐❡❞❛❞❡s tér♠✐❝❛s ❞♦ ♠❛t❡r✐❛❧✳ ❊st❛s ♠❡❞✐❞❛s ✐♥❞✐❝❛r❛♠ q✉❡ ♦ ❛♠✐♥♦á❝✐❞♦ ▲✲❤✐st✐❞✐♥❛ ❛♣r❡s❡♥t❛ ✉♠❛ ❣r❛♥❞❡ ❡st❛❜✐❧✐❞❛❞❡ tér♠✐❝❛ ❛té ❛ t❡♠♣❡r❛t✉r❛ ❞❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✺✺✵ K✱ ♥❛ q✉❛❧ ❡❧❡ s❡ ❞❡❝♦♠♣õ❡✳ ◆❛ s❡❣✉♥❞❛ ♣❛rt❡

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❆❜str❛❝t

■♥ t❤✐s ✇♦r❦ ✇❡ ✐♥✈❡st✐❣❛t❡ ♦♥❡ ♦❢ t❤❡ ✷✵ ❛♠✐♥♦ ❛❝✐❞s t❤❛t ♠❛❦❡ ✉♣ t❤❡ ♣r♦t❡✐♥s ❢♦✉♥❞ ✐♥ ❧✐✈✐♥❣ t❤✐♥❣s✱ t❤❡ ❛♠✐♥♦ ❛❝✐❞ ▲✲❤✐st✐❞✐♥❡ ✭❈6 ❍9 ◆3 ❖2✮✱ ❜② ❘❛♠❛♥ s♣❡❝tr♦s❝♦♣②✳ ❚❤❡

r❡❣✐♦♥ s♣❡❝tr❛❧ st✉❞② ✇❛s t♦ ✇❛✈❡ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✵cm−1_{❛♥❞ ✸✹✵✵}_cm−1_{✳ ❚❤❡ s❛♠♣❧❡s}

❛r❡ ✉s❡❞ ✐♥ ♣♦✇❞❡r ❢♦r♠✱ ❝♦♠♣r✐s✐♥❣ ♣♦❧②❝r②st❛❧s✳ ❊①♣❡r✐♠❡♥ts ✇❡r❡ ❝❛rr✐❡❞ ♦✉t ❛♥❛❧②✲ s✐s ❚❤❡r♠♦❣r❛✈✐♠❡tr✐❝ ✭❚●✮ ❛♥❞ ❞✐✛❡r❡♥t✐❛❧ s❝❛♥♥✐♥❣ ❝❛❧♦r✐♠❡tr② ✭❉❙❈✮ t♦ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ t❤❡r♠❛❧ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♠❛t❡r✐❛❧✳ ❚❤❡s❡ ♠❡❛s✉r❡♠❡♥ts ✐♥❞✐❝❛t❡❞ t❤❛t t❤❡ ❛♠✐♥♦ ❛❝✐❞ ▲✲❤✐st✐❞✐♥❡ ❤❛s ❛ ❤✐❣❤ t❤❡r♠❛❧ st❛❜✐❧✐t② ✉♣ t♦ t❡♠♣❡r❛t✉r❡s ♦❢ ❛♣♣r♦①✐♠❛t❡❧②

550K✱ ❛t ✇❤✐❝❤ ✐t ❞❡❝♦♠♣♦s❡s✳ ■♥ t❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ t❤❡ ♠❡❛s✉r❡♠❡♥t ✇❛s ♣❡r❢♦r♠❡❞ ❛t

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❉❡❞✐❝❛tór✐❛

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❆❣r❛❞❡❝✐♠❡♥t♦s

➚ t♦❞❛ ♠✐♥❤❛ ❢❛♠í❧✐❛ ♣❡❧♦ t♦t❛❧ ❛♣♦✐♦✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ❛♦ ♠❡✉ ♣❛✐✱ ❆♥tô♥✐♦ ▼❛♥♦ P❡r❡✐r❛ ❡ à ♠✐♥❤❛ ♠ã❡✱ ▼❛r✐❛ ❞❡ ❋át✐♠❛ ▼❡❞❡✐r♦s P❡r❡✐r❛ ♣♦r t♦❞♦s ♦s ❡s❢♦rç♦s ♣♦r ❡❧❡s r❡❛❧✐③❛❞♦s ♣❛r❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✱ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ♣♦r t❡r❡♠ ♠❡ ❡♥s✐♥❛❞♦ ♦ q✉❡ é r❡❛❧♠❡♥t❡ ✐♠♣♦rt❛♥t❡ ♣r❛ ✈✐❞❛✳

❆♦ ♣r♦❢❡ss♦r ❉r✳ P❛✉❧♦ ❞❡ ❚❛rs♦ ❈❛✈❛❧❝❛♥t❡ ❋r❡✐r❡✱ ♣❡❧❛ ♦r✐❡♥t❛çã♦ ❞❡❞✐❝❛❞❛ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

❆♦s ♣r♦❢❡ss♦r❡s✿ ❉r✳ ❏♦sé ❆❧✈❡s ❞❡ ▲✐♠❛ ❏ú♥✐♦r ❡ ❉r✳ ▼❛r❝♦s ❆♥tô♥✐♦ ❆r❛ú❥♦ ❙✐❧✈❛ ♣♦r t❡r❡♠ ❛❝❡✐t❛❞♦ ♣❛rt✐❝✐♣❛r ❞❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✳

❆♦ ♣r♦❢❡ss♦r ❉r✳❱á❧❞❡r ◆♦❣✉❡✐r❛ ❋r❡✐r❡✳

➚ ❊✈❡❧✐♥❡ ▼❛❣❛❧❤ã❡s✱ ♣♦r t♦❞♦ ❛♣♦✐♦✱ ❝♦♠♣r❡❡♥sã♦ ❡ ❝✉♠♣❧✐❝✐❞❛❞❡✳ ▼✉✐t♦ ❜♦♠ ❡st❛r ❛♦ s❡✉ ❧❛❞♦✳

❆♦ ❛♠✐❣♦✱ ❏♦sé ●❛❞❡❧❤❛✱ ♣❡❧❛ ❣r❛♥❞✐♦s❛ ❛❥✉❞❛ ♣r❡st❛❞❛ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛✲ ❜❛❧❤♦ ❡ ❡♠ ❞✐✈❡rs♦s ♠♦♠❡♥t♦s ❞✉r❛♥t❡ ❛ ❣r❛❞✉❛çã♦✱ ❝❡rt❛♠❡♥t❡ ✉♠ ♣r✐✈✐❧é❣✐♦ tê✲❧♦ ❝♦♠♦ ❛♠✐❣♦✳

❆♦ ❛♠✐❣♦✱ ❆♥tô♥✐♦ ❏♦❡❧ ❘✳ ❞❡ ❈❛str♦✱ ♣♦r t♦❞❛s ❛s ❡①♣❧✐❝❛çõ❡s✱ ❞✐s❝✉ssõ❡s✱ ♣❡❧♦ tr❡✐✲ ♥❛♠❡♥t♦ ❡♠ ❡q✉✐♣❛♠❡♥t♦s ❞♦ ❧❛❜♦r❛tór✐♦✳ ❈♦♥tr✐❜✉✐çõ❡s ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ❛ ❡①❡❝✉çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

❆♦ ❛♠✐❣♦✱ ❙❛♠✉❡❧ ▼❛rt✐♥s✱ ♣♦r t♦❞♦ ❛♣♦✐♦ ❡ ✐♥❝❡♥t✐✈♦✳

❆♦s ❆♠✐❣♦s✱ ❏♦r❣❡ ▲✉✐③ ❆r❛ú❥♦ ❡ ▼❛r❝♦s ❆✉ré❧✐♦✱ ❝♦♠♣❛♥❤❡✐r♦s ❞❡ ❣r❛❞✉❛çã♦✳

❆♦s ❝♦♠♣❛♥❤❡✐r♦s ❞❡ ❣r❛❞✉❛çã♦ ❡ ❧❛❜♦r❛tór✐♦✿ ❚❤✐❛❣♦ ▼♦✉r❛✱ ❚❤❛✐s ❏♦r❞❛♥❛✱ ❋❡❧✲ ❧✐♣❡ ❘❡❣♦✱ ❉✐❡❣♦ ❘❛❜❡❧♦✱ ❚❤✐❛❣♦ ❉❡ ▼❡❧♦✱ ❏♦r❣❡ ▲✉✐③✲❈❛♣✉❛♥✱ ❏♦ã♦ P❛✉❧♦✱ ▼♦❝✐♥❡✐s ❈❛✈❛❧❝❛♥t❡✱ ❏♦ã♦ ❘✉✜♥♦✱ ■t♦ ▲✐❜❡r❛t♦✱ ▲✉❝❛s ❱✐❡✐r❛✳

❆ t♦❞♦s q✉❡✱ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛✱ ♠❡ ❛❥✉❞❛r❛♠ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

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✏❯♠ ♣♦✉❝♦ ❞❡ ❝✐ê♥❝✐❛ ♥♦s ❛❢❛st❛ ❞❡ ❉❡✉s✳ ▼✉✐t♦✱ ♥♦s ❛♣r♦①✐♠❛✳✑

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❙✉♠ár✐♦

▲✐st❛ ❞❡ ❋✐❣✉r❛s

▲✐st❛ ❞❡ ❚❛❜❡❧❛s

✶ ■♥tr♦❞✉çã♦ ♣✳ ✶✸

✷ ❋✉♥❞❛♠❡♥t♦s ❚❡ór✐❝♦s ♣✳ ✶✺

✷✳✶ ❊❢❡✐t♦ ❘❛♠❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✺

✷✳✶✳✶ ❆❜♦r❞❛❣❡♠ ❝❧áss✐❝❛ ❞♦ ❡❢❡✐t♦ ❘❛♠❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✼

✷✳✷ ❱✐❜r❛çõ❡s ♠♦❧❡❝✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✾

✷✳✷✳✶ ▼♦❞♦s ♥♦r♠❛✐s ❞❡ ✈✐❜r❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✵

✷✳✷✳✷ ❆♥á❧✐s❡ t❡ór✐❝❛ ❞♦s ♠♦❞♦s ♥♦r♠❛✐s ❞❡ ✈✐❜r❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✷

✷✳✷✳✸ ❚✐♣♦s ❞❡ ✈✐❜r❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✹

✷✳✸ ❆♥á❧✐s❡ ✈✐❜r❛❝✐♦♥❛❧ ❡♠ ❝r✐st❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✺

✷✳✸✳✶ ❱✐❜r❛çõ❡s ❡♠ ✉♠❛ r❡❞❡ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✻

✷✳✹ ❆♥á❧✐s❡ tér♠✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✾

✷✳✹✳✶ ❚❡r♠♦❣r❛✈✐♠❡tr✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✵

✷✳✹✳✷ ❈❛❧♦r✐♠❡tr✐❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ✈❛rr❡❞✉r❛ ✭❉❙❈✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✵

✷✳✺ ❆♠✐♥♦á❝✐❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✶

✸ ❖ ❆♠✐♥♦á❝✐❞♦ ▲✲❤✐st✐❞✐♥❛ ♣✳ ✸✷

✹ Pr♦❝❡❞✐♠❡♥t♦ ❊①♣❡r✐♠❡♥t❛❧ ♣✳ ✸✻

(10)

✹✳✷ ❊①♣❡r✐♠❡♥t♦s ❞❡ ❞✐❢r❛çã♦ ❞❡ r❛✐♦s✲❳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✻

✹✳✸ ❊①♣❡r✐♠❡♥t♦s ❞❡ ❛♥á❧✐s❡ tér♠✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✼

✹✳✸✳✶ ❆♥á❧✐s❡ t❡r♠♦❣r❛✈✐♠étr✐❝❛ ✭❚●✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✼

✹✳✸✳✷ ❈❛❧♦r✐♠❡tr✐❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ✈❛rr❡❞✉r❛ ✭❉❙❈✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✼

✹✳✹ ❊①♣❡r✐♠❡♥t♦s ❡s♣❡❝tr♦s❝♦♣✐❛ ❘❛♠❛♥ ❝♦♠ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ✭❋❚✲

❘❛♠❛♥✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✽

✹✳✺ ❊①♣❡r✐♠❡♥t♦s ❞❡ ❡s♣❡❝tr♦s❝♦♣✐❛ ❘❛♠❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✾

✺ Pr♦♣r✐❡❞❛❞❡s tér♠✐❝❛s ❡ ✈✐❜r❛❝✐♦♥❛✐s ❞❛ ▲✲❤✐st✐❞✐♥❛ ♣✳ ✹✷

✺✳✶ Pr♦♣r✐❡❞❛❞❡s tér♠✐❝❛ ❞❛ ▲✲❤✐st✐❞✐♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✷

✺✳✷ Pr♦♣r✐❡❞❛❞❡s ✈✐❜r❛❝✐♦♥❛✐s ❞♦ á❝✐❞♦ ▲✲❤✐st✐❞✐♥❛ à t❡♠♣❡r❛t✉r❛ ❛♠❜✐❡♥t❡ ♣✳ ✹✸

✺✳✷✳✶ ❘❡❣✐ã♦ ❡s♣❡❝tr❛❧ ❡♥tr❡ ✶✵ cm−1 _{❡ ✻✵✵} _cm−1 _{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳} _{♣✳ ✹✹}

✺✳✷✳✷ ❘❡❣✐ã♦ ❡s♣❡❝tr❛❧ ❡♥tr❡ ✻✵✵ cm−1 _{❡ ✶✷✵✵} _cm−1 _{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳} _{♣✳ ✹✺}

✺✳✷✳✸ ❘❡❣✐ã♦ ❡s♣❡❝tr❛❧ ❡♥tr❡ ✶✷✵✵ cm−1 _{❡ ✶✼✵✵} _cm−1 _{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳} _{♣✳ ✹✼}

✺✳✷✳✹ ❘❡❣✐ã♦ ❡s♣❡❝tr❛❧ ❡♥tr❡ ✷✼✵✵ cm−1 _{❡ ✸✹✵✵} _cm−1 _{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳} _{♣✳ ✹✾}

✻ ❈♦♥❝❧✉sõ❡s ♣✳ ✺✷

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▲✐st❛ ❞❡ ❋✐❣✉r❛s

✶ ❊s♣❛❧❤❛♠❡♥t♦s✿✭❛✮ ❘❛♠❛♥ ❙t♦❦❡s❀ ✭❜✮ ❘❛②❧❡✐❣❤❀ ✭❝✮ ❆♥t✐✲❙t♦❦❡s✳❖ ♥í✈❡❧ ♠❛✐s ❛❧t♦ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠ ❡st❛❞♦ ✈✐rt✉❛❧ ❡♥q✉❛♥t♦ q✉❡ ♦s ❞♦✐s ♥í✈❡✐s

❞❡ ♠❡♥♦r ❡♥❡r❣✐❛ ❝♦rr❡s♣♦♥❞❡♠ ❛ ❡st❛❞♦s ✈✐❜r❛❝✐♦♥❛✐s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✻

✷ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ✈❛r✐❛çã♦ ❞❡ ❡♥❡r❣✐❛ ❡♥tr❡ ❞♦✐s ♥í✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✵

✸ ❘❡♣r❡s❡♥t❛çã♦ ❡sq✉❡♠át✐❝❛ ❞❛s ❞✐✈❡rs❛s r❡❣✐õ❡s ❞♦ ❡s♣❡❝tr♦ ❡❧❡tr♦♠❛❣✲

♥ét✐❝♦ ❬✶❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✶

✹ ▼♦❞♦s ❞❡ str❡t❝❤✐♥❣✿✭❛✮ str❡t❝❤✐♥❣ s✐♠étr✐❝♦❀ ✭❜✮ str❡t❝❤✐♥❣ ❛ss✐♠étr✐❝♦✳ ♣✳ ✷✹

✺ ▼♦❞♦s ❞❡ ✈✐❜r❛çõ❡s ❜❡♥❞✐♥❣✿✭❛✮ ❙❝✐ss♦r✐♥❣ ❀ ✭❜✮ ❲❛❣❣✐♥❣ ❀ ✭❝✮ ❚✇✐st✐♥❣ ❀ ✭❞✮ ❘♦❝❦✐♥❣✳ ❖♥❞❡✿ J✐♥❞✐❝❛ ♠♦✈✐♠❡♥t♦ ♣❛r❛ ❢♦r❛ ❞♦ ♣❧❛♥♦ ❞❛ ❢♦❧❤❛ ❡

N ✐♥❞✐❝❛ ♠♦✈✐♠❡♥t♦ ♣❛r❛ ❞❡♥tr♦ ❞♦ ♣❧❛♥♦ ❞❛ ❢♦❧❤❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✺

✻ ❈é❧✉❧❛ ✉♥✐tár✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✺

✼ ❘❡❞❡ ❝r✐st❛❧✐♥❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✺

✽ ❘❡❞❡s ❞❡ ❇r❛✈❛✐s ❬✶❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✻

✾ ❘❡♣r❡s❡♥t❛çã♦ ❡sq✉❡♠át✐❝❛ ❞♦ ❞❡s❧♦❝❛♠❡♥t♦ ❞♦s át♦♠♦s ♥✉♠❛ r❡❞❡ ✉♥✐✲

❞✐♠❡♥s✐♦♥❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✼

✶✵ ❘❡♣r❡s❡♥t❛çã♦ ❡sq✉❡♠át✐❝❛ ❞♦ ❞❡s❧♦❝❛♠❡♥t♦ ❞♦s át♦♠♦s ♣❛r❛ ✉♠❛ r❡❞❡

❜✐❛tô♠✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✽

✶✶ ❈✉r✈❛s ❞❡ ❞✐s♣❡rsã♦ ❞❡ ✉♠❛ r❡❞❡ ❢♦r♠❛❞❛ ♣♦r ❞♦✐s át♦♠♦s ❬✷❪✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✾

✶✷ ❖s ✐sô♠❡r♦s ▲ ✭❡sq✉❡r❞❛✮ ❡ ❉ ✭❞✐r❡✐t❛✮ ❞♦s ❛♠✐♥♦á❝✐❞♦s✳ ❬✸❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✶

✶✸ ❊str✉t✉r❛ ♣❧❛♥❛r ▲✲❤✐st✐❞✐♥❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✷

✶✹ ❊str✉t✉r❛ tr✐❞✐♠❡♥s✐♦♥❛❧ ❞❡ ▲✲❤✐st✐❞✐♥❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✷

✶✺ ❊str✉t✉r❛❧ tr✐❞✐♠❡♥s✐♦♥❛❧ ❞❛ ❝é❧✉❧❛ ✉♥✐tár✐❛ ❞❡ ▲✲❤✐st✐❞✐♥❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✸

✶✻ ❆♠♦str❛ ♣♦❧✐❝r✐st❛❧✐♥❛s ❞❡ ▲✲❤✐st✐❞✐♥❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✻

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✶✽ ❊q✉✐♣❛♠❡♥t♦ ❞❡ ❆♥á❧✐s❡ ❚● ♠♦❞❡❧♦ ❙❚❆ ✹✵✾❈ ❞❛ ◆❡t③s❝❤✱ ▲❛❜♦r❛tór✐♦

❞❡ ❋ís✐❝❛ ❆♣❧✐❝❛❞❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✽

✶✾ ❊q✉✐♣❛♠❡♥t♦ ❞❡ ❉❙❈✱ ♠♦❞❡❧♦ ❉❙❈ ✷✵✹ ❋✶ ❞❛ ◆❡t③s❝❤✱ ▲❛❜♦r❛tór✐♦ ❞❡

❋ís✐❝❛ ❆♣❧✐❝❛❞❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✽

✷✵ ❊q✉✐♣❛♠❡♥t♦s✿ ❱❡rt❡① ✼✵ ❡ ♦ ♠ó❞✉❧♦ ❘❆▼ ■■ ✉t✐❧✐③❛❞♦s ♥♦s ❡①♣❡r✐♠❡♥t♦s

❞❡ ❡s♣❡❝tr♦s❝♦♣✐❛ ❝♦♠ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✾

✷✶ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ ❡s♣❛❧❤❛♠❡♥t♦ ❘❛♠❛♥ ♥❛ ❣❡♦♠❡tr✐❛ ❞❡ r❡tr♦❡s♣❛❧❤❛✲

♠❡♥t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✾

✷✷ ❙✐st❡♠❛ ❞❡ ♠✐❝r♦✲❛♥á❧✐s❡ ❘❛♠❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✵

✷✸ ❉✐❛❣r❛♠❛ ❞♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ó♣t✐❝♦ ❞♦ ❡s♣❡❝trô♠❡tr♦ ❚✻✹✵✵✵ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✶

✷✹ ❋♦♥t❡s ❞❡ ▲❛s❡r ✭à ❡sq✉❡r❞❛✮ ❡ ❡s♣❡❝trô♠❡tr♦ ❚✻✹✵✵✵ ✭à ❞✐r❡✐t❛✮ ❞♦ ▲❛✲ ❜♦r❛tór✐♦ ❞❡ ❊s♣❛❧❤❛♠❡♥t♦ ❞❡ ▲✉③ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛ ❞❛ ❯♥✐✲

✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✶

✷✺ ❆♥á❧✐s❡ ❚❡r♠♦❣r❛✈✐♠étr✐❝❛ ✭❚●✮ ❞❛ ▲✲❤✐st✐❞✐♥❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✸

✷✻ ❆♥á❧✐s❡ ❞❡ ❉❙❈ ❞❛ ▲✲❤✐st✐❞✐♥❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✸

✷✼ ❊s♣❡❝tr♦ ❘❛♠❛♥ ❞♦ ❝r✐st❛❧ ❞❡ ▲✲❤✐st✐❞✐♥❛ à t❡♠♣❡r❛t✉r❛ ❛♠❜✐❡♥t❡ ♥❛

r❡❣✐ã♦ ❡s♣❡❝tr❛❧ ❡♥tr❡ ✶✵ cm−1 _{❡ ✻✵✵} _cm−1 _{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳} _{♣✳ ✹✹}

✷✽ ❊s♣❡❝tr♦ ❘❛♠❛♥ ❞♦ ❝r✐st❛❧ ❞❡ ▲✲❤✐st✐❞✐♥❛ à t❡♠♣❡r❛t✉r❛ ❛♠❜✐❡♥t❡ ♥❛

r❡❣✐ã♦ ❡s♣❡❝tr❛❧ ❡♥tr❡ ✻✵✵ cm−1 _{❡ ✶✷✵✵} _cm−1 _{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳} _{♣✳ ✹✻}

✷✾ ❊s♣❡❝tr♦ ❘❛♠❛♥ ❞♦ ❝r✐st❛❧ ❞❡ ▲✲❤✐st✐❞✐♥❛ à t❡♠♣❡r❛t✉r❛ ❛♠❜✐❡♥t❡ ♥❛

r❡❣✐ã♦ ❡s♣❡❝tr❛❧ ❡♥tr❡ ✶✷✵✵ cm−1 _{❡ ✶✼✵✵}_cm−1 _{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳} _{♣✳ ✹✽}

✸✵ ❊s♣❡❝tr♦ ❘❛♠❛♥ ❞♦ ❝r✐st❛❧ ❞❡ ▲✲❤✐st✐❞✐♥❛ à t❡♠♣❡r❛t✉r❛ ❛♠❜✐❡♥t❡ ♥❛

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▲✐st❛ ❞❡ ❚❛❜❡❧❛s

✶ ❈❧❛ss✐✜❝❛çã♦ ❞❛s ♣r✐♥❝✐♣❛✐s té❝♥✐❝❛s ❞❡ ❛♥á❧✐s❡ tér♠✐❝❛s s❡❣✉♥❞♦ ♦ ♣❛râ✲

♠❡tr♦ ♠❡❞✐❞♦ ❬✹❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✵

✷ ❈♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s ❞♦s át♦♠♦s ❝♦♥st✐t✉✐♥t❡s ❞❡ ✉♠❛ ❝é❧✉❧❛ ✉♥✐tár✐❛

❞❡ ▲✲❤✐st✐❞✐♥❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✸

✸ ❚❛❜❡❧❛ ❞♦s ❝❛r❛❝t❡r❡s ❞♦ ❣r✉♣♦ D2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✹

✹ ▼♦❞♦s ❘❛♠❛♥ ❞♦ ❝r✐st❛❧ ❞❡ ▲✲❤✐st✐❞✐♥❛ ♦❜s❡r✈❛❞♦s ♥♦ ✐♥t❡r✈❛❧♦ ❡s♣❡❝tr❛❧

❞❡ ✶✵ cm−1 _{❛ ✻✵✵} _cm−1 _{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳} _{♣✳ ✹✺}

✺ ▼♦❞♦s ❘❛♠❛♥ ❞♦ ❝r✐st❛❧ ❞❡ ▲✲❤✐st✐❞✐♥❛ ♦❜s❡r✈❛❞♦s ♥♦ ✐♥t❡r✈❛❧♦ ❡s♣❡❝tr❛❧

❞❡ ✻✵✵ cm−1 _{❛ ✶✷✵✵}_cm−1_{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳} _{♣✳ ✹✼}

✻ ▼♦❞♦s ❘❛♠❛♥ ❞♦ ❝r✐st❛❧ ❞❡ ▲✲❤✐st✐❞✐♥❛ ♦❜s❡r✈❛❞♦s ♥♦ ✐♥t❡r✈❛❧♦ ❡s♣❡❝tr❛❧

❞❡ ✶✷✵✵ cm−1 _{❛ ✶✼✵✵} _cm−1_{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳} _{♣✳ ✹✾}

✼ ▼♦❞♦s ❘❛♠❛♥ ❞♦ ❝r✐st❛❧ ❞❡ ▲✲❤✐st✐❞✐♥❛ ♦❜s❡r✈❛❞♦s ♥♦ ✐♥t❡r✈❛❧♦ ❡s♣❡❝tr❛❧

❞❡ ✷✼✵✵ cm−1 _{❛ ✸✹✵✵} _cm−1_{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳} _{♣✳ ✺✵}

✽ ▼♦❞♦s ❘❛♠❛♥ ❞♦ ❝r✐st❛❧ ❞❡ ▲✲❤✐st✐❞✐♥❛ ♦❜s❡r✈❛❞♦s ♥♦ ✐♥t❡r✈❛❧♦ ❡s♣❡❝tr❛❧

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✶✸

✶ ■♥tr♦❞✉çã♦

❖s ❛♠✐♥♦á❝✐❞♦s ♣❛rt✐❝✐♣❛♠ ❞❡ ✈ár✐❛s r❡❛çõ❡s q✉í♠✐❝❛s ❡♠ ♦r❣❛♥✐s♠♦s ✈✐✈♦s✱ ❡ t❛♠✲ ❜é♠ sã♦ ✉♥✐❞❛❞❡s ❜ás✐❝❛s ♥❛ ❢♦r♠❛çã♦ ❞❡ ♣r♦t❡í♥❛s✳ ❆ ♣❛rt✐r ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈✐♥t❡ ❛♠✐♥♦á❝✐❞♦s ♣♦❞❡♠♦s ❝♦♥str✉✐r ❞✐❢❡r❡♥t❡s ♣r♦t❡í♥❛s✱ ❡♥③✐♠❛s ❡ ♦✉tr♦s t❡❝✐❞♦s ❡①✐st❡♥t❡s ♥♦s ♦r❣❛♥✐s♠♦s ✈✐✈♦s✳ ◆❡st❡ tr❛❜❛❧❤♦ ♦ ♦❜❥❡t♦ ❞❡ ❡st✉❞♦ s❡rá ♦ ❛♠✐♥♦á❝✐❞♦ ▲✲❤✐st✐❞✐♥❛ ✭❈6❍9◆3❖2✮✱ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛s s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❢ís✐❝❛s ❡ q✉í♠✐❝❛s é ❞❡ ❣r❛♥❞❡ ✐♥t❡r❡ss❡

♥❛ ❜✉s❝❛ ❞❡ ❛♣❧✐❝❛çõ❡s t❡❝♥♦❧ó❣✐❝❛s ❡ ❢❛r♠❛❝ê✉t✐❝❛s✳

Pr♦♣r✐❡❞❛❞❡s ✈✐❜r❛❝✐♦♥❛✐s✱ ót✐❝❛s ❡ ❡str✉t✉r❛✐s ❞❡ ❝r✐st❛✐s ❞❡ ❛♠✐♥♦á❝✐❞♦s tê♠ ❞❡s✲ ♣❡rt❛❞♦ ❜❛st❛♥t❡ ✐♥t❡r❡ss❡ ♥♦s ú❧t✐♠♦s ❛♥♦s✱ ❞❡✈✐❞♦ s❡✉ ♣♦t❡♥❝✐❛❧ ♣❛r❛ ❛ ✉t✐❧✐③❛çã♦ ♥❛ ✐♥❞ústr✐❛✱ ❡♥tr❡ ♦✉tr❛s r❛③õ❡s✳ ❆ss✐♠ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛s ✐♥t❡r❛çõ❡s ❡♥tr❡ ♠♦❧é❝✉❧❛s ♣♦❧❛r❡s ❡ ♥ã♦ ♣♦❧❛r❡s é ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❛s ❢✉♥çõ❡s ❜✐♦❧ó❣✐✲ ❝❛s❀ ♦ t❡♥s♦r ♣♦❧❛r✐③❛❜✐❧✐❞❛❞❡ ❘❛♠❛♥ ♣♦❞❡ ❢♦r♥❡❝❡r ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ♦ ♠❡❝❛♥✐s♠♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ ❞❡ ❝❛r❣❛ ❡ ✐♥t❡r❛çõ❡s ✐♥t❡r♠♦❧❡❝✉❧❛r❡s❀ ♦s ❡s♣❡❝tr♦s ❘❛♠❛♥ ❡ ✐♥❢r❛✈❡r♠❡✲ ❧❤♦ ♣♦❞❡♠ ❛❥✉❞❛r ❛ ❝♦♠♣r❡❡♥❞❡r ❛ ❝♦♥❢♦r♠❛çã♦ ♠♦❧❡❝✉❧❛r ❡ ❛ ♥❛t✉r❡③❛ ❞❛s ❧✐❣❛çõ❡s ❞❡ ❤✐❞r♦❣ê♥✐♦ ❡♠ s✉❜stâ♥❝✐❛s ❜✐♦❧♦❣✐❝❛♠❡♥t❡ ✐♠♣♦rt❛♥t❡s✱ t❛✐s ❝♦♠♦ ❛s ♣r♦t❡í♥❛s❀ ♣♦❞❡♠♦s ❛❞q✉✐r✐r ❝♦♥❤❡❝✐♠❡♥t♦ s♦❜r❡ ❛ ✐♥t❡r❛çã♦ ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡ ❡♥tr❡ ❛s ♠♦❧é❝✉❧❛s ❡♠ ❢♦r♠❛ ③✇✐tt❡r✐♦♥ ❬✺❪✳ ❆ té❝♥✐❝❛ ❞❡ ❡s♣❡❝tr♦s❝♦♣✐❛ ❘❛♠❛♥ t❡♠ s✐❞♦ ♠✉✐t♦ ✉t✐❧✐③❛❞❛ ♥♦ ❡st✉❞♦ ❞❡ ❝r✐st❛✐s ❞❡ ❛♠✐♥♦á❝✐❞♦s✳ ❈♦♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❡st✉❞♦ ❞❡ss❡s ♠❛t❡r✐❛✐s ♦❜té♠✲s❡ ♠♦❞❡❧♦s ✐♥t❡r❡ss❛♥t❡s ♣❛r❛ ❛ ❛♥á❧✐s❡ ❞❡ s✐st❡♠❛s ❜✐♦❧ó❣✐❝♦s ♠❛✐s ❝♦♠♣❧❡①♦s✱ ❝♦♠♦ ❜✐♦♣♦✲ ❧í♠❡r♦s✳ ❈r✐st❛✐s ❞❡ ❛♠✐♥♦á❝✐❞♦s ❢♦r♠❛♠ ✉♠ ❜♦♠ ♠♦❞❡❧♦ ♣❛r❛ ✐♥✈❡st✐❣❛çõ❡s✱ ❞❡ ❧✐❣❛çõ❡s t✐♣♦ ♣♦♥t❡s ❞❡ ❤✐❞r♦❣ê♥✐♦ ♣♦rq✉❡ ❛❧❣✉♥s ❞❡❧❡s ❛♣r❡s❡♥t❛♠ ♣♦❧✐♠♦r✜s♠♦✱♦✉ s❡❥❛✱ ♣♦❞❡♠ ♦❝♦rr❡r ❝♦♠ ❞✐❢❡r❡♥t❡s ❛rr❛♥❥♦s ❝r✐st❛❧✐♥♦s✳

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✶ ■♥tr♦❞✉çã♦ ✶✹

✉s❛❞♦ t❛♠❜é♠ ♥♦ tr❛t❛♠❡♥t♦ ❞❡ ❞♦❡♥ç❛s ❛❧ér❣✐❝❛s✳ ❆❧é♠ ❞✐ss♦✱ ♦ ♦r❣❛♥✐s♠♦ ♦ ✉t✐❧✐③❛ ♣❛r❛ ❢❛❜r✐❝❛r ❤✐st❛♠✐♥❛✱ q✉❡ é r❡s♣♦♥sá✈❡❧ ♣♦r ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ♣r♦❝❡ss♦s ✜s✐♦❧ó❣✐❝♦s✳ ❙❛❜❡✲s❡ q✉❡ ❤✐st❛♠✐♥❛s ❝❛✉s❛♠ ♦ ✐♥❝❤❛ç♦ ❡ ✈❡r♠❡❧❤✐❞ã♦ ❡♠ ♠✉✐t❛s ✐♥✢❛♠❛çõ❡s ❡ r❡❛çõ❡s ❛❧ér❣✐❝❛s ❬✼❪✳

❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é r❡❛❧✐③❛r ✉♠ ❡st✉❞♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ✈✐❜r❛❝✐♦♥❛✐s ❞♦ ❛♠✐✲ ♥♦á❝✐❞♦ ▲✲❤✐st✐❞✐♥❛ ✭❈6❍9◆3❖2✮✱ à t❡♠♣❡r❛t✉r❛ ❛♠❜✐❡♥t❡✱ ❛tr❛✈és ❞♦ ✉s♦ ❞❡ ❡s♣❡❝tr♦s❝♦✲

♣✐❛ ❘❛♠❛♥✱ ❜❡♠ ❝♦♠♦ ✉♠❛ ✐♥✈❡st✐❣❛çã♦ ♣♦r ❛♥á❧✐s❡ tér♠✐❝❛✳

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✶✺

✷ ❋✉♥❞❛♠❡♥t♦s ❚❡ór✐❝♦s

✷✳✶ ❊❢❡✐t♦ ❘❛♠❛♥

❉❡✜♥✐♠♦s ❡s♣❡❝tr♦s❝♦♣✐❛ ❝♦♠♦ t♦❞❛ té❝♥✐❝❛ q✉❡ ❡st✉❞❛ ❛ ✐♥t❡r❛çã♦ ❡♥tr❡ r❛❞✐❛çã♦ ❡❧❡tr♦♠❛❣♥ét✐❝❛ ❡ ❛ ♠❛tér✐❛✱ ♥♦ q✉❛❧ ✉♠ ❞♦s ♦❜❥❡t✐✈♦s é ❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦s ♥í✈❡✐s ❡♥❡r✲ ❣ét✐❝♦s ❞❡ át♦♠♦s ♦✉ ♠♦❧é❝✉❧❛s✳ ❊♥tr❡ ❛s ❞✐✈❡rs❛s té❝♥✐❝❛s ❡s♣❡❝tr♦s❝ó♣✐❝❛s ❡♥❝♦♥tr❛✲s❡ ❛ ❡s♣❡❝tr♦s❝♦♣✐❛ ❘❛♠❛♥ q✉❡ é ❜❛s❡❛❞❛ ♥♦ ❡❢❡✐t♦ ❘❛♠❛♥✳

❊st❡ ❡❢❡✐t♦ ❢♦✐ ♣r✐♠❡✐r❛♠❡♥t❡ ❞❡s❝r✐t♦ ♣❡❧♦ ❢ís✐❝♦ ✐♥❞✐❛♥♦ ❈❤❛♥❞r❛s❡❦❤❛r ❘❛♠❛♥ ❡♠ ✉♠ ❛rt✐❣♦ ♣✉❜❧✐❝❛❞♦ ♥❛ r❡✈✐st❛ ◆❛t✉r❡✱ ❡♠ ✶✾✷✽✱ ❝♦♠ ❝♦✲❛✉t♦r✐❛ ❞❡ ❑✳ ❙✳ ❑r✐s❤♥❛♥✳ P♦r ❡st❡ tr❛❜❛❧❤♦✱ ❡♠ ✶✾✸✵✱ ❘❛♠❛♥ r❡❝❡❜❡✉ ♦ ♣rê♠✐♦ ◆♦❜❡❧ ❞❡ ❋ís✐❝❛ ❬✽❪✳ ❊st❡ ❡❢❡✐t♦ ❢♦r♥❡❝❡ ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ♦s ❡st❛❞♦s q✉â♥t✐❝♦s ♠♦❧❡❝✉❧❛r❡s ❛tr❛✈és ❞♦ ❡s♣❛❧❤❛♠❡♥t♦ ♣r♦✈♦❝❛❞♦ ♣❡❧♦ ❝❤♦q✉❡ ❞❡ ❢ót♦♥s ❝♦♠ ❛s ♠♦❧é❝✉❧❛s ❞❛ ❛♠♦str❛ ❛♥❛❧✐s❛❞❛✳

❆ ❧✉③ ❛♦ ✐♥❝✐❞✐r s♦❜r❡ ❛ ❛♠♦str❛✱ ♦s ❢ót♦♥s q✉❡ ❛ ❝♦♥st✐t✉✐ ♣♦❞❡♠ ❛tr❛✈❡ss❛r ❛ ❛♠♦s✲ tr❛ s❡♠ s♦❢r❡r ♥❡♥❤✉♠❛ ❛❧t❡r❛çã♦ ♦✉ ♣♦❞❡♠ s❡r ❛❜s♦r✈✐❞♦s ♦✉ ❡s♣❛❧❤❛❞♦s ♣❡❧❛ ♠❛tér✐❛✳ ◗✉❛♥❞♦ ❛ ❡♥❡r❣✐❛ ❞♦ ❢ót♦♥ ❢♦r ✐❣✉❛❧ ❛ ❞✐❢❡r❡♥ç❛ ❞❡ ❡♥❡r❣✐❛ ❡♥tr❡ ♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ ❡ ♦ ❡st❛❞♦ ❡①❝✐t❛❞♦ ❞❛ ♠♦❧é❝✉❧❛✱ ♣♦❞❡rá ♦❝♦rr❡r ❛ ❛❜s♦rçã♦ ❞♦ ❢ót♦♥ ❧❡✈❛♥❞♦ ❛ ♠♦❧é❝✉❧❛ ♣❛r❛ ✉♠ ❡st❛❞♦ ❞❡ ♠❛✐♦r ❡♥❡r❣✐❛✳ ❆❞✐❝✐♦♥❛❧♠❡♥t❡✱ ❛ ❧✉③ ♣♦❞❡rá ❡①❝✐t❛r ❛ ♠♦❧é❝✉❧❛ ♣❛r❛ ✉♠ ❝❤❛♠❛❞♦ ✧❡st❛❞♦ ✈✐rt✉❛❧✧❜❡♠ ♠❛✐s ❡❧❡✈❛❞♦✱ ❞❡❝❛✐♥❞♦ ♣❛r❛ ✉♠ ❡st❛❞♦ ✈✐❜r❛❝✐♦♥❛❧ ❞✐✲ ❢❡r❡♥t❡✳ ❆ ✈❛r✐❛çã♦ ❞❡ ❡♥❡r❣✐❛✱ ❡♥tr❡ ❛ r❛❞✐❛çã♦ ✐♥❝✐❞❡♥t❡ ❡ ❡s♣❛❧❤❛❞❛✱ ❡q✉✐✈❛❧❡ à ❡♥❡r❣✐❛ ❝♦♠ q✉❡ ♦s át♦♠♦s ♥❛ ár❡❛ ❛♥❛❧✐s❛❞❛ ❡stã♦ ✈✐❜r❛♥❞♦✱ ❡ss❛ ❢r❡q✉ê♥❝✐❛ ❞❡ ✈✐❜r❛çã♦ ♣♦s✲ s✐❜✐❧✐t❛ ♦❜t❡r ✐♥❢♦r♠❛çõ❡s s♦❜r❡✿ ❝♦♠♦ ♦s át♦♠♦s ❡stã♦ ❧✐❣❛❞♦s✱ ❛ ❣❡♦♠❡tr✐❛ ♠♦❧❡❝✉❧❛r✱ ❞✐❢❡r❡♥❝✐❛r ♣♦❧✐♠♦r❢♦s ✭♠❡s♠❛ s✉❜stâ♥❝✐❛ ❝♦♠ ❞✐❢❡r❡♥t❡s ❡str✉t✉r❛s ❝r✐st❛❧✐♥❛s✮ ❬✾❪✳

(17)

✷✳✶ ❊❢❡✐t♦ ❘❛♠❛♥ ✶✻

❞♦✐s ❡st❛❞♦s ✈✐❜r❛❝✐♦♥❛✐s✱ ♦♥❞❡ ❛s ❢r❡q✉ê♥❝✐❛s ✈✐❜r❛❝✐♦♥❛✐s sã♦ ♦❜t✐❞❛s ♣❡❧❛ ❞✐❢❡r❡♥ç❛ ❞❛s r❛❞✐❛çõ❡s ❡s♣❛❧❤❛❞❛s ❡ ✐♥❝✐❞❡♥t❡ ❬✶✵❪✳

❙❡ t✐✈❡r♠♦s ❧✉③ ✐♥❝✐❞❡♥t❡ s♦❜r❡ ✉♠❛ ❛♠♦str❛ ❞❡ ❢r❡q✉ê♥❝✐❛ ν0✱ s❡♥❞♦ ❡s♣❛❧❤❛❞♦ ❡♠

t♦❞❛s ❞✐r❡çõ❡s ❝♦♠ ❢r❡q✉ê♥❝✐❛ ν0✱ ♥❡st❡ ❝❛s♦ t❡♠♦s ♦ ❡s♣❛❧❤❛♠❡♥t♦ ❘❛②❧❡✐❣❤✳ ❙❡ ✉♠❛

♣❡q✉❡♥❛ ❢r❛çã♦ ❞❛ ❧✉③ ❡s♣❛❧❤❛❞❛ t✐✈❡r ❢r❡q✉ê♥❝✐❛ ❞✐❢❡r❡♥t❡ ν1✱ t❛❧ q✉❡✱ ∆E ❂ ❤|ν0 −ν1|

s❡❥❛ ❛ ✈❛r✐❛çã♦ ❞❛ ❡♥❡r❣✐❛ ❛❜s♦r✈✐❞❛ ♣❡❧❛ ❛♠♦str❛✱ t❡♠♦s ❛s s❡❣✉✐♥t❡s ♣♦ss✐❜✐❧✐❞❛❞❡s✿

• ν1 ❁ν0 t❡♠♦s ♦ ❡❢❡✐t♦ ❘❛♠❛♥ ❙t♦❦❡s

• ν0 ❁ν1 t❡♠♦s ♦ ❡❢❡✐t♦ ❘❛♠❛♥ ❆♥t✐✲❙t♦❦❡s

❋✐❣✉r❛ ✶✿ ❊s♣❛❧❤❛♠❡♥t♦s✿✭❛✮ ❘❛♠❛♥ ❙t♦❦❡s❀ ✭❜✮ ❘❛②❧❡✐❣❤❀ ✭❝✮ ❆♥t✐✲❙t♦❦❡s✳❖ ♥í✈❡❧ ♠❛✐s ❛❧t♦ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠ ❡st❛❞♦ ✈✐rt✉❛❧ ❡♥q✉❛♥t♦ q✉❡ ♦s ❞♦✐s ♥í✈❡✐s ❞❡ ♠❡♥♦r ❡♥❡r❣✐❛ ❝♦r✲ r❡s♣♦♥❞❡♠ ❛ ❡st❛❞♦s ✈✐❜r❛❝✐♦♥❛✐s✳

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✷✳✶ ❊❢❡✐t♦ ❘❛♠❛♥ ✶✼

✷✳✶✳✶ ❆❜♦r❞❛❣❡♠ ❝❧áss✐❝❛ ❞♦ ❡❢❡✐t♦ ❘❛♠❛♥

❯♠❛ ❡①♣❧✐❝❛çã♦ q✉❛❧✐t❛t✐✈❛ ♣❛r❛ ♦ ❡❢❡✐t♦ ❘❛♠❛♥ ♣♦❞❡ s❡r ♦❜t✐❞❛ ❝♦♠ ❝♦♥❝❡✐t♦s ❞❡ ❡❧❡tr♦❞✐♥â♠✐❝❛ ❝❧áss✐❝❛✳ ❖ ❡❢❡✐t♦ ❘❛♠❛♥ ❡stá r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ♦ ♠♦♠❡♥t♦ ❞❡ ❞✐♣♦❧♦ ✐♥❞✉③✐❞♦ ♥❛ ♠♦❧é❝✉❧❛ ♣❡❧♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ❞❛ r❛❞✐❛çã♦✱ ♦ q✉❡ ❞✐❢❡r❡ ❞♦ ✐♥❢r❛✈❡r♠❡❧❤♦ ♦♥❞❡ s❡ ❝♦♥s✐❞❡r❛ ❛ ✈❛r✐❛çã♦ ❞♦ ♠♦♠❡♥t♦ ❞✐♣♦❧❛r ✐♥trí♥s❡❝♦✳

◗✉❛♥❞♦ t❡♠♦s ✉♠❛ ♠♦❧é❝✉❧❛ ❡♠ ✉♠❛ r❡❣✐ã♦ ❝♦♠ ✉♠ ❝❛♠♣♦ ❡❧étr✐❝♦−→E✱ ✉♠ ♠♦♠❡♥t♦

❞❡ ❞✐♣♦❧♦ ✐♥❞✉③✐❞♦−→P é ♦❜s❡r✈❛❞♦✱ ♣♦✐s ♦ ♣♦❧♦ ♥❡❣❛t✐✈♦ ❞♦ ❝❛♠♣♦ ❛tr❛✐rá ♦ ♥ú❝❧❡♦ ❡ ♦ ♣♦❧♦

♣♦s✐t✐✈♦ ♦s ❡❧étr♦♥s✳ ❖ ❞✐♣♦❧♦ ✐♥❞✉③✐❞♦ é ♣r♦♣♦r❝✐♦♥❛❧ ❛ ✐♥t❡♥s✐❞❛❞❡ ❞♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ❡ ♦s❝✐❧❛ ❝♦♠ s♦❜r❡♣♦s✐çã♦ ❞❡ ❢r❡q✉ê♥❝✐❛s✳ ❖ ♠♦♠❡♥t♦ ❞❡ ❞✐♣♦❧♦ ✐♥❞✉③✐❞♦ −→P✱ ♣♦❞❡ s❡r

❡s❝r✐t♦ ❝♦♠♦✿

−

→_P ₌_α−→_E _{✭✷✳✶✮}

❙❡♥❞♦ α ✉♠ t❡♥s♦r ❞❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡✱ q✉❡ ❞❡♣❡♥❞❡ ❞❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❛ ♠♦❧é❝✉❧❛✱

❝❤❛♠❛❞❛ ❞❡ ♣♦❧❛r✐③❛❜✐❧✐❞❛❞❡ ❡ −→E ♦ ✈❡t♦r ❝❛♠♣♦ ❡❧étr✐❝♦ ❞❛ r❛❞✐❛çã♦ ✐♥❝✐❞❡♥t❡✳ ◗✉❛♥❞♦

❛ ♠♦❧é❝✉❧❛ ♦s❝✐❧❛✱ ❛ ♠❛❣♥✐t✉❞❡ α ✈❛r✐❛✱ ♣♦✐s é ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ❝♦♠♣r✐♠❡♥t♦

❞❛ ❧✐❣❛çã♦ ✭❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s át♦♠♦s✮✳ ❙❡♥❞♦ q ✉♠❛ ❝♦♦r❞❡♥❛❞❛ ✐♥t❡r♥❛ r❡❢❡r❡♥t❡ ❛♦

❞❡s❧♦❝❛♠❡♥t♦ ♥✉❝❧❡❛r✳ ❯t✐❧✐③❛♥❞♦q ❝♦♠♦ ♣❛râ♠❡tr♦ ♣r❛ ❡①♣❛♥❞✐r α ❡♠ sér✐❡ t❡♠♦s✿

α=α0+

dα dq

!

0

+_{· · ·} ✭✷✳✷✮

❖♥❞❡✱ α0 é ❛ ♣♦❧❛r✐③❛❜✐❧✐❞❛❞❡ ♥❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❡

dα dq

0 é ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ❞❡ α

❡♠ r❡❧❛çã♦ ❛q ♠❡❞✐❞❛ ♥❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦✳

❙❡❥❛ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ♦s❝✐❧❛♥t❡ ❞❡s❝r✐t♦ ♣♦r✿

E =E0cos(2πν0t) ✭✷✳✸✮

❚❛❧ q✉❡✱ E0 s❡❥❛ ♦ ✈❛❧♦r ♠á①✐♠♦ ❞♦ ❝❛♠♣♦ ❡ ν0 s✉❛ ❢r❡q✉ê♥❝✐❛✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛

❡q✉❛çã♦ ✭✷✳✶✮ ❝♦♠♦✿

P =αE0cos(2πν0t) ✭✷✳✹✮

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

q=q0cos(2πνt) ✭✷✳✺✮

❯t✐❧✐③❛♥❞♦ ❛s ❡q✉❛çõ❡s ✭✷✳✹✮✱ ✭✷✳✺✮ ❡ ✭✷✳✷✮ ♦♥❞❡ ν ❡ ν0 sã♦ r❡s♣❡❝t✐✈❛♠❡♥t❡ ❛ ❢r❡q✉ê♥❝✐❛

(19)

✷✳✶ ❊❢❡✐t♦ ❘❛♠❛♥ ✶✽

❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

P =α0E0cos(2πν0t) +

dα dq

!

0

q0cos(2πνt)E0cos(2πν0t) ✭✷✳✻✮

❉❡s♣r❡③❛✲s❡ ♦s t❡r♠♦s ❞❡ ♠❛✐s ❛❧t❛ ♦r❞❡♠ ♣❛r❛ ✉♠❛ ♣❡q✉❡♥❛ ✈❛r✐❛çã♦ ❞❡ q✱ ❡ ✉t✐❧✐③❛♥❞♦ ❛ ✐❞❡♥t✐❞❛❞❡ tr✐❣♦♥♦♠étr✐❝❛✿ cos(a)cos(b) = 1₂[cos(a+b) +cos(a₋b)] ❛ ❡q✉❛çã♦ ✭✷✳✻✮

t♦r♥❛✲s❡✿

P =α0E0cos(2πν0t) +

1 2 dα dq ! 0

q0Eocos[(2π(ν0 +ν)t] +cos[(2π(ν0−ν)t] ✭✷✳✼✮

❖ ♣r✐♠❡✐r♦ t❡r♠♦ ❞❛ ❡q✉❛çã♦ ✭✷✳✼✮ ❝♦♥té♠ s♦♠❡♥t❡ r❛❞✐❛çã♦ ✐♥❝✐❞❡♥t❡ ν0✱ ❡st❡ t❡r♠♦

❝♦rr❡s♣♦♥❞❡ ❛♦ ❡s♣❛❧❤❛♠❡♥t♦ ❘❛②❧❡✐❣❤ ✭❡s♣❛❧❤❛♠❡♥t♦ ❡❧ást✐❝♦✮✳ ◆♦ s❡❣✉♥❞♦ t❡r♠♦ t❡♠♦s r❛❞✐❛çã♦ ❡s♣❛❧❤❛❞❛ ❝♦♠ ❢r❡q✉ê♥❝✐❛ ✭ν0 − ν✮✱ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛♦ ❡s♣❛❧❤❛♠❡♥t♦ ❘❛♠❛♥

❙t♦❦❡s ❡ ❝♦♠ ❢r❡q✉ê♥✐❝❛ ✭ν0 +ν✮ ❡s♣❛❧❤❛♠❡♥t♦ ❘❛♠❛♥ ❆♥t✐✲❙t♦❦❡s✳ P♦❞❡♠♦s ♦❜s❡r✈❛r

♣❡❧❛ ❡q✉❛çã♦ ✭✷✳✼✮ q✉❡ s❡

dα dq

0 = 0✱ ♥ã♦ t❡r❡♠♦s ♦ ❡❢❡✐t♦ ❞❛ ♣♦❧❛r✐③❛❜✐❧✐❞❛❞❡✱ ❧♦❣♦ ♣❛r❛

t❡r♠♦s ♦ ❡❢❡✐t♦ ❘❛♠❛♥ é ♥❡❝❡ssár✐♦ q✉❡ ♦❝♦rr❛ ✉♠ ♣❡q✉❡♥♦ ❞❡s❧♦❝❛♠❡♥t♦ ♥❛ ❝♦♦r❞❡♥❛❞❛ q ❡♠ t♦r♥♦ ❞♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❬✶✵❪✳

❖ ♠♦♠❡♥t♦ ❞❡ ❞✐♣♦❧♦ é ✉♠ ✈❡t♦r ❝✉❥❛ ❞✐r❡çã♦ é ❛ ♠❡s♠❛ ❞❛ ❧✐♥❤❛ q✉❡ ✉♥❡ ♦ ❝❡♥tr♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❞❛s ❝❛r❣❛s ♣♦s✐t✐✈❛s ❡ ♥❡❣❛t✐✈❛s ♥❛ ♠♦❧é❝✉❧❛✳ P❛r❛ ✉♠❛ ♠♦❧é❝✉❧❛ ❞❡ ♠❛✐s ❞❡ ❞♦✐s át♦♠♦s ✐s♦tró♣✐❝❛✱ ♦r✐❡♥t❛❞❛ ♣♦r ✉♠ ❝❛♠♣♦ ❡❧étr✐❝♦✱ ♣❛r❛ ❝❛❞❛ ❝♦♦r❞❡♥❛❞❛ ❝❛rt❡s✐❛♥❛ ✉♠❛ ❝♦♠♣♦♥❡♥t❡ ❞❡ ❞✐♣♦❧♦ ❡❧étr✐❝♦ é ❛ss♦❝✐❛❞❛✳ P♦r ❡①❡♠♣❧♦✱ ♣❛r❛ ✉♠ ❝❛♠♣♦ ♥❛ ❞✐r❡çã♦ ①✱ ❛ ❝♦♠♣♦♥❡♥t❡ ❞♦ ♠♦♠❡♥t♦ ♥❡st❛ ❞✐r❡çã♦ s❡r✐❛✿

Px =αEx ✱ Py =αEy ❡ Pz =αEz

P♦rt❛♥t♦ t❡♠♦s três ❝♦♠♣♦♥❡♥t❡s ❝❛rt❡s✐❛♥❛s ❞♦ ♠♦♠❡♥t♦ ❞❡ ❞✐♣♦❧♦ ♣❛r❛ ❝❛❞❛ ♠♦✲ ❧é❝✉❧❛✳ ❆ ♣♦❧❛r✐③❛❜✐❧✐❞❛❞❡ é ❛ ♠❡s♠❛ ❡♠ t♦❞❛s ❛s ❞✐r❡çõ❡s✳ P❛r❛ ❛❧❣✉♠❛s ♠♦❧é❝✉❧❛s ❛ ♣♦❧❛r✐③❛❜✐❧✐❞❛❞❡α♣♦❞❡ s❡r ❞✐❢❡r❡♥t❡ ♣❛r❛ ❛s ❞✐r❡çõ❡s ①✱ ② ❡ ③✱ ❛ss✐♠ ♦ ♠♦♠❡♥t♦ ❞❡ ❞✐♣♦❧♦

✐♥❞✉③✐❞♦ ♣♦❞❡ ♥ã♦ s❡r ♣❛r❛❧❡❧♦ ❛♦ ❝❛♠♣♦ ❡❧étr✐❝♦✱ ❛ ♠♦❧é❝✉❧❛ é ❞✐t❛ ❛♥✐s♦tró♣✐❝❛✳ ◆❡st❡ ❝❛s♦ ❛ ❝♦♠♣♦♥❡♥t❡ ❞♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ♥❛ ❞✐r❡çã♦ ① ✭Ex✮ ♥ã♦ ✐♥❞✉③ ❛♣❡♥❛s ✉♠ ❞✐♣♦❧♦ ♥❛

❞✐r❡çã♦ ①✱ ♠❛s t❛♠❜é♠ ♥❛s ❞✐r❡çõ❡s ② ❡ ③✿

Px=αxxEx+αxyEy+αxzEz ✭✷✳✽✮

Py =αyxEx+αyyEy+αyzEz ✭✷✳✾✮

(20)

✷✳✷ ❱✐❜r❛çõ❡s ♠♦❧❡❝✉❧❛r❡s ✶✾

◆❛ ❢♦r♠❛ ♠❛tr✐❝✐❛❧✱ t❡♠♦s✿

     Px Py Pz      =     

αxx αxy αxz

αyx αyy αyz

αzx αzy αzz

          Ex Ey Ey      ✭✷✳✶✶✮

❖ t❡♥s♦r ♣♦❧❛r✐③❛❜✐❧✐❞❛❞❡ ❞❡✈❡ s❡r s✐♠étr✐❝♦ ❬✶✶❪✱ ✐st♦ é✿

αxy =αyx αxz =αzx αyz =αzy✳

P❛r❛ ♠♦❧é❝✉❧❛s ♣♦❧✐❛tô♠✐❝❛s ❛ ❝♦♥❞✐çã♦ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ q✉❡ ♦❝♦rr❛ tr❛♥s✐çã♦ ❡♥tr❡ ♦s ❡st❛❞♦s ✈✐❜r❛❝✐♦♥❛✐sm ❡ n✱ é q✉❡ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❞❛s ❝♦♠♣♦♥❡♥t❡s(αij)mn✱ ❞❛ ♣♦❧❛✲

r✐③❛❜✐❧✐❞❛❞❡ ✈❛r✐❡ ❝♦♠ ♦ ♠♦✈✐♠❡♥t♦ ❞❡ ✈✐❜r❛çã♦ ❞♦ s✐st❡♠❛✱ ✐st♦ é✱ s❡❥❛ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✳ ❖♥❞❡i ❡j sã♦ ❛s ❝♦♠♣♦♥❡♥t❡s ①✱ ② ♦✉ ③✳

✷✳✷ ❱✐❜r❛çõ❡s ♠♦❧❡❝✉❧❛r❡s

P♦❞❡♠♦s ❝♦♥s✐❞❡r❛r q✉❡ ❛ ❡♥❡r❣✐❛ ❞❡ ✉♠❛ ♠♦❧é❝✉❧❛ é ❢♦r♠❛❞❛ ♣♦r ❝♦♠♣♦♥❡♥t❡s ❞✐st✐♥✲ t❛s ❛ss♦❝✐❛❞❛s ❝♦♠✿ ♦ ♠♦✈✐♠❡♥t♦ ❞♦s ❡❧étr♦♥s ♥❛ ♠♦❧é❝✉❧❛✱ ❝♦♠ ❛s ✈✐❜r❛çõ❡s ❞♦s át♦♠♦s ❝♦♥st✐t✉✐♥t❡s ❡ ❝♦♠ ❛s r♦t❛çõ❡s ❞❛ ♠♦❧é❝✉❧❛ ❝♦♠♦ ✉♠ t♦❞♦✱ ❡ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ ❡q✉❛çã♦✿

Etotal =Eel+Evib +Erot, ✭✷✳✶✷✮

❖♥❞❡ ❛ ❡♥❡r❣✐❛ ❞❛ tr❛♥s✐çã♦ ❞♦s ❡❧étr♦♥s ♥♦r♠❛❧♠❡♥t❡ ❞á ♦r✐❣❡♠ ❛ ❛❜s♦rçã♦ ♦✉ ❡♠✐s✲ sã♦ ♥♦ ✐♥❢r❛✈❡r♠❡❧❤♦ ❡ ♥❛ r❡❣✐ã♦ ❞♦ ✉❧tr❛✈✐♦❧❡t❛ ❡ ✈✐sí✈❡❧ ❞♦ ❡s♣❡❝tr♦ ❡❧❡tr♦♠❛❣♥ét✐❝♦✳ ❆ r♦t❛çã♦ ♣✉r❛ ❞á ♦r✐❣❡♠ ❛ ❛❜s♦rçã♦ ♥❛ r❡❣✐ã♦ ❞❡ ♠✐❝r♦♦♥❞❛s ♦✉ ♦ ✐♥❢r❛✈❡r♠❡❧❤♦✳ ❱✐❜r❛çã♦ ♠♦❧❡❝✉❧❛r ❞á ♦r✐❣❡♠ ❛ ❜❛♥❞❛s ❞❡ ❛❜s♦rçã♦ ❞✉r❛♥t❡ ❛ ♠❛✐♦r ♣❛rt❡ ❞❛ r❡❣✐ã♦ ✐♥❢r❛✈❡r♠❡❧❤❛ ❞♦ ❡s♣❡❝tr♦✳ P❛r❛ ✉♠❛ ♠♦❧é❝✉❧❛ ❡♠ ✉♠❛ r❡❣✐ã♦ ❝♦♠ ❝❛♠♣♦ ❡❧❡tr♦♠❛❣♥ét✐❝♦✱ ❛ tr❛♥s✲ ❢❡rê♥❝✐❛ ❞❡ ❡♥❡r❣✐❛ ❞♦ ❝❛♠♣♦ ♣❛r❛ ❛ ♠♦❧é❝✉❧❛ ♦❝♦rr❡rá q✉❛♥❞♦ ❛ ❡q✉❛çã♦ ❞❡ ❇♦❤r ❢♦r s❛t✐s❢❡✐t❛✿

∆E =hν, ✭✷✳✶✸✮

♦♥❞❡ ∆E é ❛ ❞✐❢❡r❡♥ç❛ ❞❡ ❡♥❡r❣✐❛s ❡♥tr❡ ❞♦✐s ❡st❛❞♦s q✉❛♥t✐③❛❞♦s✱ h é ❛ ❝♦♥st❛♥t❡ ❞❡

P❧❛♥❝❦ ❡ν é ❛ ❢r❡q✉ê♥❝✐❛ ❞❛ ❧✉③ ✭♥ú♠❡r♦ ❞❡ ✈✐❜r❛çõ❡s ♣♦r ✉♥✐❞❛❞❡ ❞❡ t❡♠♣♦✮✳

❆ r❛❞✐❛çã♦ ❡❧❡tr♦♠❛❣♥ét✐❝❛ é ❝❛r❛❝t❡r✐③❛❞❛ ♣♦r ✉♠ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ λ ✭❝♦♠♣r✐✲

♠❡♥t♦ ❞❡ ✉♠❛ ♦♥❞❛ ❡❧❡tr♦♠❛❣♥ét✐❝❛✮ q✉❡ s❡ r❡❧❛❝✐♦♥❛ ❝♦♠ ❛ ❢r❡q✉ê♥❝✐❛ ν✱ ♣❡❧❛ ❢ór♠✉❧❛✿

ν = c

(21)

✷✳✷ ❱✐❜r❛çõ❡s ♠♦❧❡❝✉❧❛r❡s ✷✵

❙❡♥❞♦c❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❧✉③✳ P♦ré♠ ❡♠ ❡s♣❡❝tr♦s❝♦♣✐❛ ✈✐❜r❛❝✐♦♥❛❧ ✉♠ ♣❛râ♠❡tr♦ ♠❛✐s

✉t✐❧✐③❛❞♦ é ♦ ♥ú♠❡r♦ ❞❡ ♦♥❞❛ν˜ ❝♦♠ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ cm−1_{✱ ❞❡✜♥✐❞♦ ♣♦r✿}

˜ ν = 1

λ ✭✷✳✶✺✮

❞❛ q✉❛❧ ♣♦❞❡♠♦s ♦❜t❡r ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦✿

ν =c˜ν ✭✷✳✶✻✮

❯s❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✷✳✶✻✮ ♥❛ ❝♦♥❞✐çã♦ ❞❡ ❇♦❤r✱ ❡q✉❛çã♦ ✭✷✳✶✸✮ t❡♠♦s✿

∆E =hc˜ν ✭✷✳✶✼✮

❉❛❞♦s ❞♦✐s ❡st❛❞♦s ❡①❝✐t❛❞♦s ❝♦♠ r❡s♣❡❝t✐✈❛s ❡♥❡r❣✐❛sE1 ❡ E2 ❞❡ ❢♦r♠❛ q✉❡✿

∆E =E2−E1 =hc˜ν ✭✷✳✶✽✮

❛ ♠♦❧é❝✉❧❛ ✧❛❜s♦r✈❡✧∆E q✉❛♥❞♦ ❡①❝✐t❛❞❛ ❞❡ E1 ♣❛r❛ E2 ❡ ✧❡♠✐t❡✧∆E q✉❛♥❞♦ ♣❛ss❛ ❞❡

E2 ♣❛r❛E1✳ ❆ ✜❣✉r❛ ✷ ♠♦str❛ ✉♠ ❡sq✉❡♠❛ ❞❡ tr❛♥s✐çã♦ ❡♥tr❡ ♦s ♥í✈❡✐s E1 ❡E2✳

❋✐❣✉r❛ ✷✿ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ✈❛r✐❛çã♦ ❞❡ ❡♥❡r❣✐❛ ❡♥tr❡ ❞♦✐s ♥í✈❡✐s ✳

◆❡st❡ tr❛❜❛❧❤♦ ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦s ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥❛s tr❛♥s✐çõ❡s ✈✐❜r❛❝✐♦♥❛✐s ♦❜s❡r✲ ✈❛❞❛s ❝♦♠ ♦ ❡s♣❡❝tr♦ ❘❛♠❛♥ ❡ ✐♥❢r❛✈❡r♠❡❧❤♦✳ ❊st❛s tr❛♥s✐çõ❡s sã♦ ♦❜s❡r✈❛❞❛s ♥❛ r❡❣✐ã♦ ❞❡102 _∼₁₀4_cm−1 _{♦r✐❣✐♥❛❞❛s ♣♦r ✈✐❜r❛çõ❡s ❞♦ ♥ú❝❧❡♦ q✉❡ ❝♦♥st✐t✉✐ ❛ ♠♦❧é❝✉❧❛ ❬✶❪✳}

✷✳✷✳✶ ▼♦❞♦s ♥♦r♠❛✐s ❞❡ ✈✐❜r❛çã♦

(22)

✷✳✷ ❱✐❜r❛çõ❡s ♠♦❧❡❝✉❧❛r❡s ✷✶

❋✐❣✉r❛ ✸✿ ❘❡♣r❡s❡♥t❛çã♦ ❡sq✉❡♠át✐❝❛ ❞❛s ❞✐✈❡rs❛s r❡❣✐õ❡s ❞♦ ❡s♣❡❝tr♦ ❡❧❡tr♦♠❛❣♥ét✐❝♦ ❬✶❪✳

❞❡ ❡q✉✐❧í❜r✐♦ t❛♠❜é♠ ❛♦ ♠❡s♠♦ t❡♠♣♦✱ ♣♦ré♠ ❝♦♠ ❛♠♣❧✐t✉❞❡s ❞❡ ♦s❝✐❧❛çã♦ ❡♠ ❣❡r❛❧ ❞✐❢❡r❡♥t❡s✳

▼♦❧é❝✉❧❛s ❝♦♠ n át♦♠♦s ❡♠ ❣❡r❛❧ ♣♦ss✉❡♠ ✸n ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡✱ ❝❛❞❛ át♦♠♦ ♣♦❞❡

♠♦✈❡r✲s❡ ❡♠ três ❞✐r❡çõ❡s ✭①✱②✱③✮✳ ❚rês ❞❡st❡s ❞❡s❝r❡✈❡♠ ♦ ♠♦✈✐♠❡♥t♦ tr❛♥s❧❛❝✐♦♥❛❧ ❡ ❣❡r❛❧♠❡♥t❡✱ três ❞❡s❝r❡✈❡♠ ❛s ♣♦ssí✈❡✐s r♦t❛çõ❡s✳ ❆ss✐♠✱ ♦ ♥ú♠❡r♦ ❞❡ ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡✱ ♦✉ ❞❡ ♠♦❞♦s ♥♦r♠❛✐s ❞❡ ✈✐❜r❛çã♦✱ é ✸n ✲ ✻✳ P❛r❛ ♠♦❧é❝✉❧❛s ❝♦♠ át♦♠♦s ❧✐♥❡❛r❡s ❡st❡

♥ú♠❡r♦ s❡ r❡❞✉③ ❛ ✸n ✲ ✺✱ ❥á q✉❡ ♥❡st❡ ❝❛s♦ ♥ã♦ ❤á r♦t❛çã♦ ❡♠ t♦r♥♦ ❞♦ ❡✐①♦ ♠♦❧❡❝✉❧❛r✳

P❛r❛ ✈✐❜r❛çõ❡s ❡♠ ✉♠ ♣❧❛♥♦✱ ❝♦♠ tr❛♥s❢♦r♠❛çõ❡s ❛❞❡q✉❛❞❛s ❡ ❡s❝♦❧❤❡♥❞♦ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❛♣r♦♣r✐❛❞♦ ♣♦❞❡♠♦s ❡❧✐♠✐♥❛r ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❞❡ r♦t❛çã♦ ❡ tr❛♥s❧❛çã♦✳ ◆♦ ♣❧❛♥♦ t❡♠♦s ✷n ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡✱ ❝♦♠♦ ❞♦✐s sã♦ ❞❡ tr❛♥s❧❛çã♦ ❡ ✉♠ ❞❡ r♦t❛çã♦✳

●❡r❛❧♠❡♥t❡ ♥♦ ♣❧❛♥♦ ♦❜t❡♠♦s ✷n ✲ ✸ ✈✐❜r❛çõ❡s ♥♦r♠❛✐s✱ ♣♦rt❛♥t♦ ♦s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡

❞♦ át♦♠♦ ❢♦r❛ ❞♦ ♣❧❛♥♦ sã♦✿ ✭✸n ✲ ✻✮ ✲ ✭✷n ✲ ✸✮ ❂ ✭n ✲ ✸✮

❊♠ ♠♦❧é❝✉❧❛s ❧✐♥❡❛r❡s ♣♦❞❡♠♦s t❡r ✈✐❜r❛çõ❡s tr❛♥s✈❡rs❛✐s ❡ ❧♦♥❣✐t✉❞✐♥❛✐s✱ ❡st❛s ♦❝♦r✲ r❡♠ ❛♦ ❧♦♥❣♦ ❞❛ ❧✐♥❤❛ q✉❡ ❧✐❣❛ ♦s át♦♠♦s✳ P❛r❛n át♦♠♦s✱ t❡♠♦s n ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❛♦

❧♦♥❣♦ ❞❛ ❧✐♥❤❛ ♣♦ré♠ ✉♠ ❞❡❧❡s é ❞❡ tr❛♥s❧❛çã♦✱ ❛ss✐♠ t❡♠♦s ✭n ✲ ✶✮ ♣♦ssí✈❡✐s ♠♦❞♦s ❞❡

✈✐❜r❛çõ❡s ♥❛ ❞✐r❡çã♦ ❧♦♥❣✐t✉❞✐♥❛❧ ♣❛r❛n át♦♠♦s✳ ❊①✐st✐♥❞♦ ✭✸n ✲ ✺✮ ♠♦❞♦s ✈✐❜r❛❝✐♦♥❛✐s

♣❛r❛ ✉♠❛ ♠♦❧é❝✉❧❛ ❧✐♥❡❛r✱ ❡♥tã♦ ❞❡✈❡♠♦s t❡r✿ ✭✸n ✲ ✺✮ ✲ ✭n ✲ ✶✮ ❂ ✭2n ✲ ✹✮ ✈✐❜r❛çõ❡s

(23)

✷✳✷ ❱✐❜r❛çõ❡s ♠♦❧❡❝✉❧❛r❡s ✷✷

✷✳✷✳✷ ❆♥á❧✐s❡ t❡ór✐❝❛ ❞♦s ♠♦❞♦s ♥♦r♠❛✐s ❞❡ ✈✐❜r❛çã♦

P❛r❛ ❝❛❞❛ ✉♠ ❞♦s N át♦♠♦s✱ ♣♦❞❡♠♦s r❡♣r❡s❡♥t❛r s✉❛s r❡s♣❡❝t✐✈❛s ♣♦s✐çõ❡s ♣❡❧♦

❝♦♥❥✉♥t♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ✭x1✱y1✱z1✮ ♣❛r❛ ♦ át♦♠♦ ✶✱ ♣❛r❛ ♦ át♦♠♦ ✷ ✭x2✱y2✱z2✮✱ ♣❛r❛ ♦

n✲és✐♠♦ át♦♠♦ ✭xN✱yN✱zN✮✱ ♠❛✐s ❣❡r❛❧♠❡♥t❡ q1✱q2✱q3✱q4✱✳✳✳✱q3N✳ ❆s ❝♦♦r❞❡♥❛❞❛s qi✱ ♦♥❞❡

i = 1,2,3, ...,3N ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞❛s ♣r❛ r❡♣r❡s❡♥t❛r ❛ ❡♥❡r❣✐❛ ❝✐♥ét✐❝❛ T ❡ ❛ ❡♥❡r❣✐❛

♣♦t❡♥❝✐❛❧V✳ P❛r❛ ♣❡q✉❡♥♦s ❞❡s❧♦❝❛♠❡♥t♦s ♦❜t❡♠♦s ❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s✿

2V = 3N X i=1 3N X j=1

fijqiqj e 2T =

3N X i=1 3N X j=1

mijq˙iq˙j ✭✷✳✶✾✮

♦♥❞❡q˙i❂ ✭dq_dt✮ ❡qi✱qj sã♦ ❝♦♦r❞❡♥❛❞❛s q✉❡ ♣♦❞❡♠ s❡r ❞✐❢❡r❡♥t❡s ♦✉ ♥ã♦✱ ✐st♦ é✱i=j ♦✉i6=

j✳ ❖s ✈❛❧♦r❡s ❞❡fij sã♦ ❢♦rç❛s ❝♦♥st❛♥t❡s ❡ ♦s ✈❛❧♦r❡smij sã♦ ❢✉♥çõ❡s ❞❛s ♠❛ss❛s ❛tô♠✐❝❛s✱

❛ ❡①♣r❡ssã♦ q✉❡ ❞❡s❝r❡✈❡ ❛ ❡♥❡r❣✐❛ ❝✐♥ét✐❝❛ é ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❛ ❡♥❡r❣✐❛ ❞❡ ✉♠❛ ú♥✐❝❛ ♣❛rtí❝✉❧❛✱ ✭1

2mv

2_{✮✱ ❡ ❛ ❡①♣r❡ssã♦ q✉❡ ❢♦r♥❡❝❡ ❛ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❡}

✭1 2X

2_{✮ ♣❛r❛ ✉♠❛ ♣❛rtí❝✉❧❛ r❡❛❧✐③❛♥❞♦ ♦s❝✐❧❛çõ❡s ❤❛r♠ô♥✐❝❛s ♦♥❞❡ ❛ ❢♦rç❛ ❞❡ r❡st❛✉r❛çã♦}

✭−f X✮ é ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ❞❡s❝♦❧❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❬✶✸❪✳

❆s ❡q✉❛çõ❡s ❞❡ ♠♦✈✐♠❡♥t♦ ♣❛r❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ át♦♠♦s✱ ♦♥❞❡T ❞❡♣❡♥❞❡ s♦♠❡♥t❡ ❞❡ ˙

qi ❡ V s♦♠❡♥t❡ ❞❡ qi✱ ♣♦❞❡ s❡r ❡s❝r✐t❛ ♥❛ s❡❣✉✐♥t❡ ❢♦r♠❛ ▲❛❣r❛♥❣❡❛♥❛✿

d dt(

∂T ∂q˙i

) + ∂T ∂qi

= 0 ✭✷✳✷✵✮

❉❛ ❡q✉❛çã♦ ✭✷✳✶✾✮ ♦❜t❡♠♦s ❝❛❞❛ ✈❛❧♦r ❞❡ q˙i✳

∂V ∂qi = 3N X j=1

fijqj

∂T ∂q˙i

=

3N

X

j=1

mijq˙j

d dt(

∂T ∂q˙i

) =

3N

X

j=1

mijq¨j ✭✷✳✷✶✮

❆♣❧✐❝❛♥❞♦ ♦s ✈❛❧♦r❡s ❞❛ ❡q✉❛çã♦ ✭✷✳✷✶✮ ♥❛ ❡q✉❛çã♦ ✭✷✳✷✵✮ ♦❜t❡♠♦s ♣❛r❛ ❝❛❞❛ 3N ✉♠

✈❛❧♦r ♣❛r❛i✳

3N

X

j=1

mijq¨j +

3N

X

j=1

fijqj = 0 ✭✷✳✷✷✮

❊st❛s 3N ❡q✉❛çõ❡s t❡♠ s♦❧✉çã♦ ❣❡r❛❧ ❞♦ t✐♣♦✿

qj =Ajsen(

√

λt+α) ✭✷✳✷✸✮

♦♥❞❡ √λ ❂ 2πν✳ ❛ ❡q✉❛çã♦ ✭✷✳✷✸✮ é ❛ ❡q✉❛çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ✉♠ ♦s❝✐❧❛❞♦r ❤❛r♠ô♥✐❝♦

s✐♠♣❧❡s✱ ❝♦♠ ❢r❡q✉ê♥❝✐❛ ν ❂ √λ

2π✱ ❛♠♣❧✐t✉❞❡ ♠á①✐♠❛ Aj ❡ ❢r❡q✉ê♥❝✐❛ ❞❡ ❢❛s❡ ✐❣✉❛❧ ❛ α✳

❈❛❧❝✉❧❛♥❞♦ ❛ ❞❡r✐✈❛❞❛ s❡❣✉♥❞❛ ❞❛ ❡q✉❛çã♦ ✭✷✳✷✸✮ ♦❜t❡♠♦s✿

˙ qj =

√

λAjcos(

√

(24)

✷✳✷ ❱✐❜r❛çõ❡s ♠♦❧❡❝✉❧❛r❡s ✷✸

¨

qj =−λAjsen(

√

λt+α) ✭✷✳✷✺✮

❙✉❜st✐t✉✐♥❞♦qj ❡ q¨j ♥❛ ❡q✉❛çã♦ ✭✷✳✷✷✮ ♦❜t❡♠♦s✿

3N

X

j=1

fijAjsen(

√

λt+α)₋

3N

X

j=1

mijλAjsen(

√

λt+α) = 0 ✭✷✳✷✻✮

3N

X

j=1

(fij −mijλ)Aj = 0 ✭✷✳✷✼✮

P♦rt❛♥t♦✿

(f11−m11λ)A1+ (f12−m12λ)A2+· · ·+ (f1,3N −m1,3Nλ)A3N = 0

(f21−m21λ)A1+ (f22−m22λ)A2+· · ·+ (f2,3N −m2,3Nλ)A3N = 0

✳✳✳

(f3N,1−m3N,1λ)A1 + (f3N,2−m3N,2λ)A2+· · ·+ (f3N,3N −m3N,3Nλ)A3N = 0

. ✭✷✳✷✽✮

❆ ❡q✉❛çã♦ ✭✷✳✷✽✮ é ❝❤❛♠❛❞❛ ❞❡ ❡q✉❛çã♦ s❡❝✉❧❛r✱ ❝♦♠♣♦st❛ ♣♦r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s ❤♦♠♦❣ê♥❡❛s✳ ❖❜t❡♠♦s ❛ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧ ♣❛r❛ ♦s ✈❛❧♦r❡s ❞❡A s♦♠❡♥t❡ q✉❛♥❞♦

t❡♠♦s ✈❛❧♦r❡s ❞❡ λ ❡s♣❡❝í✜❝♦s✳ P❛r❛ ❡ss❡s ✈❛❧♦r❡s ❞❡ λ✱ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③ ❞♦s

❝♦❡✜❝✐❡♥t❡sA ❞❡✈❡ s❡r ♥✉❧♦✱ ♦✉ s❡❥❛✿

f11−m11λ f12−m12λ · · · f1,3N −m1,3Nλ

f21−m21λ f22−m22λ · · · f2,3N −m2,3Nλ

· · · ·

f3N,1−m3N,1λ f3N,2−m3N,2λ · · · f3N,3N −m3N,3Nλ

= 0. ✭✷✳✷✾✮

❉❛ ❡q✉❛çã♦ ✭✷✳✷✾✮✱ q✉❛♥❞♦ ❡①♣❛♥❞✐❞❛ ♣❛r❛ ❡q✉❛çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ♦r❞❡♠3N✱ ♦❜t❡r❡✲

♠♦s 3N ✈❛❧♦r❡s ❞❡ λ ❡♠ t❡r♠♦s ❞❡ f ❡ m✳ ❈❛❞❛ ✈❛❧♦r ❞❡ λ ♦❜t✐❞♦ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞♦ ♥❛

❡q✉❛çã♦ ✭✷✳✷✽✮ ❡ ❛ss✐♠ ❞❡t❡r♠✐♥❛r ♦s r❡s♣❡❝t✐✈♦s ✈❛❧♦r❡s ❞❡Aj✳ ❉❡ ❢❛t♦✱ s♦♠❡♥t❡ ❛s r❡❧❛✲

çõ❡s ❡♥tr❡ ♦s ✈❛❧♦r❡s ❞❡Aj✱ ✐st♦ é✱ r❡❧❛çõ❡s ❡♥tr❡ ❛s ❛♠♣❧✐t✉❞❡s ♣❛r❛ ❝❛❞❛ λ❞❡t❡r♠✐♥❛❞♦✱

♣♦❞❡♠ s❡r ❝❛❧❝✉❧❛❞❛s✳ P♦ré♠ ❡st❛ r❡❧❛çã♦ é s✉✜❝✐❡♥t❡ ♣r❛ ❞❡s❝r❡✈❡r ❛ ✈✐❜r❛çã♦✳ ❆ s♦❧✉çã♦ ❣❡r❛❧ ✭✷✳✷✸✮ ✐♥❞✐❝❛ q✉❡ ❝❛❞❛ át♦♠♦ ❡stá r❡❛❧✐③❛♥❞♦ ✉♠ ♠♦✈✐♠❡♥t♦ ♦s❝✐❧❛tór✐♦ ❡♠ r❡❧❛çã♦ ❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❝♦♠ ❛♠♣❧✐t✉❞❡ Aj✱ ❣❡r❛❧♠❡♥t❡ ❞✐❢❡r❡♥t❡ ♣❛r❛ ❝❛❞❛ ❝♦♦r❞❡♥❛❞❛

♣♦ré♠ ❝♦♠ ❛ ♠❡s♠❛ ❢r❡q✉ê♥❝✐❛ ν ❂ √λ

2π ❡ ❝♦♥st❛♥t❡ ❞❡ ❢❛s❡ α✱ ✐st♦ ✐♠♣❧✐❝❛ q✉❡ t♦❞♦s ♦s

(25)

✷✳✷ ❱✐❜r❛çõ❡s ♠♦❧❡❝✉❧❛r❡s ✷✹

✷✳✷✳✸ ❚✐♣♦s ❞❡ ✈✐❜r❛çõ❡s

❖s ♠♦❞♦s ✈✐❜r❛❝✐♦♥❛✐s ❞❡ ✉♠❛ ♠♦❧é❝✉❧❛ sã♦ ❝❧❛ss✐✜❝❛❞♦s ❛tr❛✈és ❞♦ ♠♦❞♦ ❝♦♠♦ s❡✉s át♦♠♦s ❝♦♥st✐t✉✐♥t❡s s❡ ♠♦✈✐♠❡♥t❛♠✳ ❚❡♠♦s ❞♦✐s t✐♣♦s ❜ás✐❝♦s ❞❡ ♠♦✈✐♠❡♥t♦s✱ ❞❡♥♦♠✐✲ ♥❛❞♦s✿ str❡t❝❤✐♥❣ ❡ ❜❡♥❞✐♥❣✳

◆♦ ♠♦❞♦ str❡t❝❤✐♥❣ ✭❡st✐r❛♠❡♥t♦✮ ♦❜s❡r✈❛♠♦s ❡♥tr❡ ♦s át♦♠♦s ♠♦✈✐♠❡♥t♦s ✈✐❜r❛❝✐♦✲ ♥❛✐s ♣❡r✐ó❞✐❝♦s ❞❡ ❡st✐r❛♠❡♥t♦ ❡ r❡❧❛①❛♠❡♥t♦✱ ❞♦s q✉❛✐s ♣♦❞❡♠♦s ❞✐st✐♥❣✉✐r ❞♦✐s ♠♦❞♦s ❞✐st✐♥t♦s✿

•❙tr❡t❝❤✐♥❣ s✐♠étr✐❝♦ ✭νs✮✿ ❞♦✐s át♦♠♦s ❡①tr❡♠♦s s❡ ♠♦✈❡♠✱ ❡♠ ❢❛s❡✱ ♣❛r❛ ❞❡♥tr♦ ❡ ♣❛r❛

❢♦r❛ ❞♦ ♣❧❛♥♦ ♠♦❧❡❝✉❧❛r❀

•❙tr❡t❝❤✐♥❣ ❛ss✐♠étr✐❝♦ ✭νa✮✿ ❞♦✐s át♦♠♦s ❡①tr❡♠♦s s❡ ♠♦✈❡♠✱ ❛❧t❡r♥❛❞❛♠❡♥t❡✱ ♣❛r❛ ❞❡♥✲

tr♦ ❡ ♣❛r❛ ❢♦r❛ ❞♦ ♣❧❛♥♦ ♠♦❧❡❝✉❧❛r❀ ❆ ✜❣✉r❛ ✹ ✐❧✉str❛ ❡ss❡s ♠♦❞♦s✳

❋✐❣✉r❛ ✹✿ ▼♦❞♦s ❞❡ str❡t❝❤✐♥❣✿✭❛✮ str❡t❝❤✐♥❣ s✐♠étr✐❝♦❀ ✭❜✮ str❡t❝❤✐♥❣ ❛ss✐♠étr✐❝♦✳

◆♦ ♠♦❞♦ ❜❡♥❞✐♥❣ ✭❞♦❜r❛♠❡♥t♦✮✱ ♦❜s❡r✈❛♠♦s ♠♦✈✐♠❡♥t♦s ✈✐❜r❛❝✐♦♥❛✐s ♥♦s q✉❛✐s à ❞✐✲ r❡çã♦ é ♣❡r♣❡♥❞✐❝✉❧❛r à ❧✐❣❛çã♦ ❡♥tr❡ ♦s át♦♠♦s q✉❡ ❢♦r♠❛♠ ❛ ♠♦❧é❝✉❧❛✳ ❊st❡ ❣r✉♣♦ é s✉❜❞✐✈✐❞✐❞♦ ❡♠ q✉❛tr♦ ❡ ❡stã♦ r❡♣r❡s❡♥t❛❞♦s ♥❛ ✜❣✉r❛ ✺✳

•❙❝✐ss♦r✐♥❣ ✭δ✮✿ ♦s át♦♠♦s s❡ ♠♦✈✐♠❡♥t❛♠ ♥❛ ❞✐r❡çã♦ ❞❛ ♠✉❞❛♥ç❛ ❞❡ â♥❣✉❧♦✱ ♣♦ré♠ ♥♦

♠❡s♠♦ ♣❧❛♥♦❀

• ❲❛❣❣✐♥❣ ✭ω✮✿ ♦s át♦♠♦s ♠♦✈❡♠✲s❡ ♣❛r❛ ❝✐♠❛ ❡ ♣❛r❛ ❜❛✐①♦ ❞♦ ♣❧❛♥♦✱ s❡♠ q✉❡ ❤❛❥❛

♠✉❞❛♥ç❛ ❞♦ â♥❣✉❧♦ ❡♥tr❡ ❛s ❧✐❣❛çõ❡s❀

(26)

✷✳✸ ❆♥á❧✐s❡ ✈✐❜r❛❝✐♦♥❛❧ ❡♠ ❝r✐st❛✐s ✷✺

♠❡♥t♦s ❝♦♠ ❢❛s❡s ❞✐❢❡r❡♥t❡s❀

•❘♦❝❦✐♥❣ ✭ρ✮✿ ♦s át♦♠♦s ♦s❝✐❧❛♠ ❡♠ ❢❛s❡ ♥♦ ♠❡s♠♦ ♣❧❛♥♦ ❬✶✺❪✳

❋✐❣✉r❛ ✺✿ ▼♦❞♦s ❞❡ ✈✐❜r❛çõ❡s ❜❡♥❞✐♥❣✿✭❛✮ ❙❝✐ss♦r✐♥❣ ❀ ✭❜✮ ❲❛❣❣✐♥❣ ❀ ✭❝✮ ❚✇✐st✐♥❣ ❀ ✭❞✮ ❘♦❝❦✐♥❣✳ ❖♥❞❡✿ J ✐♥❞✐❝❛ ♠♦✈✐♠❡♥t♦ ♣❛r❛ ❢♦r❛ ❞♦ ♣❧❛♥♦ ❞❛ ❢♦❧❤❛ ❡N ✐♥❞✐❝❛ ♠♦✈✐♠❡♥t♦

♣❛r❛ ❞❡♥tr♦ ❞♦ ♣❧❛♥♦ ❞❛ ❢♦❧❤❛✳

✷✳✸ ❆♥á❧✐s❡ ✈✐❜r❛❝✐♦♥❛❧ ❡♠ ❝r✐st❛✐s

❈r✐st❛❧ é ✉♠ só❧✐❞♦ ❝❛r❛❝t❡r✐③❛❞♦ ♣♦r ✉♠❛ ♦r❞❡♥❛❞❛ ❞✐s♣♦s✐çã♦ ❞♦s át♦♠♦s q✉❡ r❡♣❡t❡✲ s❡ ♣❡r✐♦❞✐❝❛♠❡♥t❡✱ ❢♦r♠❛♥❞♦ ✉♠❛ ❡str✉t✉r❛ tr✐❞✐♠❡♥s✐♦♥❛❧✳ ❆ ✉♥✐❞❛❞❡ ❡❧❡♠❡♥t❛r ❞❡ ✉♠ ❝r✐st❛❧ é ❞❡♥♦♠✐♥❛❞❛ ❝é❧✉❧❛ ✉♥✐tár✐❛ ❞❡✜♥✐❞❛ ♣♦r três ✈❡t♦r❡s ♥ã♦ ❝♦♣❧❛♥❛r❡s −→a✱−→b ✱−→c✱

♦♥❞❡ ♦ ♠ó❞✉❧♦ ❞❡ −→a✱−→b ✱−→c ❡ ♦s ✈❛❧♦r❡s ❞♦s â♥❣✉❧♦s α, β, γ sã♦ ♦s ♣❛râ♠❡tr♦s ❞❛ r❡❞❡

❝r✐st❛❧✐♥❛ ❬✶✻❪✱ ❝♦♠♦ ✐♥❞✐❝❛❞♦ ♥❛ ✜❣✉r❛ ✻✳

❋✐❣✉r❛ ✻✿ ❈é❧✉❧❛ ✉♥✐tár✐❛ ❋✐❣✉r❛ ✼✿ ❘❡❞❡ ❝r✐st❛❧✐♥❛✳

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✷✳✸ ❆♥á❧✐s❡ ✈✐❜r❛❝✐♦♥❛❧ ❡♠ ❝r✐st❛✐s ✷✻

❝♦♠ r❡♣❡t✐çã♦ ❞❡ ✉♥✐❞❛❞❡s ✐❞ê♥t✐❝❛s ❇r❛✈❛✐s ♠♦str♦✉ q✉❡ ♣♦❞❡✲s❡ ❢♦r♠❛r ✶✹ ❞✐❢❡r❡♥t❡s t✐♣♦s ❞❡ r❡❞❡s ❝r✐st❛❧✐♥❛s✿

❋✐❣✉r❛ ✽✿ ❘❡❞❡s ❞❡ ❇r❛✈❛✐s ❬✶❪✳

✷✳✸✳✶ ❱✐❜r❛çõ❡s ❡♠ ✉♠❛ r❡❞❡ ✉♥✐❞✐♠❡♥s✐♦♥❛❧

(28)

✷✳✸ ❆♥á❧✐s❡ ✈✐❜r❛❝✐♦♥❛❧ ❡♠ ❝r✐st❛✐s ✷✼

❈♦♥s✐❞❡r❛♥❞♦ ✉♠❛ ❝❛❞❡✐❛ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ❞❡ át♦♠♦s ❞❡ ♠❛ss❛ m✱ ❝♦♠ ❞✐stâ♥❝✐❛ a

❡♥tr❡ ❡❧❡s✳ ❙❡♥❞♦ un ♦ ❞❡s❧♦❝❛♠❡♥t♦ ❞♦ át♦♠♦ ❞❡ ♣♦s✐çã♦ n ❛ ♣❛rt✐r ❞♦ s❡✉ ❡st❛❞♦ ❞❡

❡q✉✐❧í❜r✐♦ ❡ f ✉♠❛ ❝♦♥st❛♥t❡ ❞❡ ❢♦rç❛ ♥❛ r❡❞❡✱ ❛♣❧✐❝❛♥❞♦ ❛s ❧❡✐s ❞❡ ◆❡✇t♦♥ t❡♠♦s✿

Fn=f(un+1−un)−f(un−un−1). ✭✷✳✸✵✮

❋✐❣✉r❛ ✾✿ ❘❡♣r❡s❡♥t❛çã♦ ❡sq✉❡♠át✐❝❛ ❞♦ ❞❡s❧♦❝❛♠❡♥t♦ ❞♦s át♦♠♦s ♥✉♠❛ r❡❞❡ ✉♥✐❞✐♠❡♥✲ s✐♦♥❛❧✳

❆ ❡q✉❛çã♦ ❞♦ ♠♦✈✐♠❡♥t♦ ♣♦❞❡ s❡r ❡s❝r✐t❛ ♥❛ ❢♦r♠❛✿

mu¨n=f(un+1+un−1−2un). ✭✷✳✸✶✮

❆ s♦❧✉çã♦ ❞❡st❛ ❡q✉❛çã♦ é✿

un=u0ei(ωt+kna). ✭✷✳✸✷✮

❙✉❜st✐t✉✐♥❞♦ ✭✷✳✸✷✮ ❡♠ ✭✷✳✸✶✮ ♦❜t❡♠♦s✿

ω=_±2

s

f m|sen

ka 2

!

|. ✭✷✳✸✸✮

❆ ❡q✉❛çã♦ ✭✷✳✸✸✮ q✉❡ ❝♦♥❡❝t❛ ❢r❡q✉ê♥❝✐❛ ❡ ♥ú♠❡r♦ ❞❡ ♦♥❞❛ ♣♦❞❡ s❡r ❞❡s✐❣♥❛❞❛ ♣♦r r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❞❛ r❡❞❡✳ ❚❡r❡♠♦s ❛ ❢r❡q✉ê♥❝✐❛ ♠á①✐♠❛ ❞❡ ✈✐❜r❛çã♦ ωm✱ q✉❛♥❞♦ ♦

♥ú♠❡r♦ ❞❡ ♦♥❞❛ km ❂ ±(π_a)✳ ❖s ✈❛❧♦r❡s ❞❡ k ❡♥tr❡ −(π_a) ❡ (π_a) ♣❡rt❡♥❝❡♠ ❛ ③♦♥❛ ❞❡

❇r✐❧❧♦✉✐♥ ❞❛ r❡❞❡✳ ❊♠ ❝r✐st❛✐s|km| ❂ π_a ≈ 108 cm−1✱ ωm ≈ 1013sec−1✳ ❊st❛ é ❛ ♠á①✐♠❛

❢r❡q✉ê♥❝✐❛ s✐t✉❛❞❛ ♥❛ r❡❣✐ã♦ ❞♦ ✐♥❢r❛✈❡r♠❡❧❤♦✳

P❛r❛ ✉♠❛ r❡❞❡ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ❝♦♠♣♦st❛ ♣♦rN át♦♠♦s✱ t❡♠♦sN s♦❧✉çõ❡s ♣❛r❛ ✭✷✳✸✷✮

♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ t❡♠♦sN ✈❛❧♦r❡s ❞❡k s✐t✉❛❞♦s ♥❛ ♣r✐♠❡✐r❛ ③♦♥❛ ❞❡ ❇r✐❧❧♦✉✐♥✳ ▲✐♠✐t❛♥❞♦

❛ r❡❞❡ ❛ ✉♠ ❝♦♠♣r✐♠❡♥t♦ L✱ t❛❧ q✉❡✱ L ❂(N ₋1)a✳ ❆❞✐❝✐♦♥❛♥❞♦ ✉♠ ♣❛râ♠❡tr♦ ♣❛r❛ ❛s

s♦❧✉çõ❡s ♣❡r✐ó❞✐❝❛s ♥♦ ❝♦♠♣r✐♠❡♥t♦ L✱ ✐st♦ é✱ ✉♠❛ ❝♦♥❞✐çã♦ ❞❡ ♣❡r✐♦❞✐❝✐❞❛❞❡

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✷✳✸ ❆♥á❧✐s❡ ✈✐❜r❛❝✐♦♥❛❧ ❡♠ ❝r✐st❛✐s ✷✽

♦♥❞❡L ❂ N a✱ ♣❛r❛ ✭✷✳✸✷✮ ♦s ✈❛❧♦r❡s ❞❡k ♣❡r♠✐t✐❞♦s sã♦✿

k =_±2π L ,±

4π

L ,· · ·,±

(N/2)π

L ✭✷✳✸✺✮

❡ss❡s ✈❛❧♦r❡s ❞❡ k ❞❡✜♥❡♠ ♦s N ♠♦❞♦s ✈✐❜r❛❝✐♦♥❛✐s ❞❛ r❡❞❡ ❬✷❪✳ ❆❣♦r❛✱ ❝♦♥s✐❞❡r❛♥❞♦ ❛s

♠❡s♠❛s ❝♦♥❞✐çõ❡s ❞❡s❝r✐t❛s ❛❝✐♠❛ ♣♦ré♠ ❛ r❡❞❡ é ❢♦r♠❛❞❛ ♣♦r ❞♦✐s t✐♣♦s ❞❡ át♦♠♦s ❝♦♠ ♠❛ss❛sm1 ❡m2✱ t❛❧ q✉❡ m1 ❃m2✳ ◆❡st❡ ❝❛s♦ ❛ ❝é❧✉❧❛ ✉♥✐tár✐❛ ♣♦ss✉✐ ❞♦✐s át♦♠♦s✳

❋✐❣✉r❛ ✶✵✿ ❘❡♣r❡s❡♥t❛çã♦ ❡sq✉❡♠át✐❝❛ ❞♦ ❞❡s❧♦❝❛♠❡♥t♦ ❞♦s át♦♠♦s ♣❛r❛ ✉♠❛ r❡❞❡ ❜✐❛tô✲ ♠✐❝❛✳

❆ ❡q✉❛çã♦ ❞♦ ♠♦✈✐♠❡♥t♦ ♣❛r❛ ❝❛❞❛ át♦♠♦ é✿

m1u¨2n=f(u2n+1+u2n−1−2u2n)

m2u¨2n+1=f(u2n+2+u2n−2u2n+1)

. ✭✷✳✸✻✮

❆s s♦❧✉çõ❡s sã♦✿

u2n =A1ei(ωt+2nka)

u2n+1 =A2ei(ωt+(2n+1)ka)

. ✭✷✳✸✼✮

❙✉❜st✐t✉✐♥❞♦ ✭✸✳✸✼✮ ❡♠ ✭✸✳✸✻✮ ♦❜t❡♠♦s✿

−ω2_m

1A1 =f A2(eika+e−ika)−2f A1

−ω2_m

2A2 =f A1(eika+e−ika)−2f A2

. ✭✷✳✸✽✮

❖ s✐st❡♠❛ ✭✸✳✸✽✮ t❡♠ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧ s❡✿

2f₋m1ω2 −2f cos(ka)

2f cos(ka) 2f ₋m2ω2

= 0. ✭✷✳✸✾✮

❘❡s♦❧✈❡♥❞♦ ❡st❛ ❡q✉❛çã♦ ♦❜t❡♠♦s✿

ω_±2 =f

1 m1 + 1 m2 ±f " 1 m1 + 1 m2 2 − 4sen 2_(ka)

m1m2

#1

2

. ✭✷✳✹✵✮

P❛r❛ ω2

+ ♦❜t❡♠♦s ♦ r❛♠♦ ót✐❝♦✱ ♥♦ q✉❛❧ ♦s ❞♦✐s át♦♠♦s ✈✐③✐♥❤♦s ♠♦✈❡♠✲s❡ ❡♠ ❞✐r❡çõ❡s

♦♣♦st❛s✱ ♠❛s ♦ ❝❡♥tr♦ ❞❡ ❣r❛✈✐❞❛❞❡ ❞❛ ❝é❧✉❧❛ ✉♥✐tár✐❛ ♣❡r♠❛♥❡❝❡ ✐♥❛❧t❡r❛❞♦✱ ♣❛r❛ ω2

− ♦❜t❡♠♦s ♦ r❛♠♦ ❛❝úst✐❝♦✱ t♦❞♦s ♦s át♦♠♦s ♠♦✈❡♠✲s❡ ♥❛ ♠❡s♠❛ ❞✐r❡çã♦✱✐st♦ é✱ ♠♦✈✐♠❡♥t♦ tr❛♥s❧❛❝✐♦♥❛❧ ❡♠ t♦❞❛ ❛ r❡❞❡ ❝♦♠ ❢r❡q✉ê♥❝✐❛ ③❡r♦✳ k✐♥❞✐❝❛ ❛ ❞✐❢❡r❡♥ç❛ ❞❡ ❢❛s❡ ❡♥tr❡ át♦♠♦s

❡q✉✐✈❛❧❡♥t❡s ❡♠ ❝❛❞❛ ❝é❧✉❧❛ ✉♥✐tár✐❛ ❞❛ r❡❞❡ ❬✶✻❪✳ P❛r❛ ♦s ✈❛❧♦r❡s ❞❡k ❡♥tr❡₋(π

2a)❡ ( π

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✷✳✹ ❆♥á❧✐s❡ tér♠✐❝❛ ✷✾

❡♥❝♦♥tr❛♠✲s❡ ♥❛ ♣r✐♠❡✐r❛ ③♦♥❛ ❞❡ ❇r✐❧❧♦✉✐♥✳

P❛r❛k ❂ ✵ t❡♠♦s ❛s s❡❣✉✐♥t❡s r❛í③❡s ♣❛r❛ ❛ ❡q✉❛çã♦ ✭✸✳✹✵✮✿

ω ❂ ✵✱ ♣❛r❛ ♦ r❛♠♦ ❛❝úst✐❝♦✳

ω ❂h2f_m1

1 +

1

m2

i1

2✱ ♣❛r❛ ♦ r❛♠♦ ót✐❝♦✳

P❛r❛k ❂(π

2a) ♦❜t❡♠♦s✿

ω ❂q_m2f

1✱ ♣❛r❛ ♦ r❛♠♦ ❛❝úst✐❝♦✳

ω ❂q_m2f

2✱ ♣❛r❛ ♦ r❛♠♦ ót✐❝♦✳

❋✐❣✉r❛ ✶✶✿ ❈✉r✈❛s ❞❡ ❞✐s♣❡rsã♦ ❞❡ ✉♠❛ r❡❞❡ ❢♦r♠❛❞❛ ♣♦r ❞♦✐s át♦♠♦s ❬✷❪✳

❈♦♠ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s✱ ✈❡r✐✜❝❛♠♦s q✉❡ ♣❛r❛ ♦ ♠♦❞♦ ❛❝úst✐❝♦ ❛ ❢r❡q✉ê♥❝✐❛ ♠á①✐♠❛

k❂(π

2a)❞❡♣❡♥❞❡ ❞❛ ♠❛ss❛ ❞♦ át♦♠♦ ♠❛✐s ♣❡s❛❞♦✱ ❥á ♥♦ ót✐❝♦ ❞❡♣❡♥❞❡ ❞❛ ♠❛ss❛ ❞♦ át♦♠♦

♠❛✐s ❧❡✈❡✳ P❛r❛ k ❂ ✵ ❛ ❢r❡q✉ê♥❝✐❛ ♠á①✐♠❛ ♦❝♦rr❡ ♥♦ r❛♠♦ ót✐❝♦✱ q✉❡ ❞❡♣❡♥❞❡ ❞❛s ❞✉❛s

♠❛ss❛s ❬✶✵❪✳

❊♠ ✉♠❛ r❡❞❡ tr✐❞✐♠❡♥s✐♦♥❛❧ ❝♦♠ N ❝é❧✉❧❛s ✉♥✐tár✐❛s t❡rí❛♠♦s 3Nn ♠♦❞♦s ❞❡ ✈✐❜r❛✲

çõ❡s✱ ♦♥❞❡ n é ♦ ♥ú♠❡r♦ ❞❡ ♣❛rtí❝✉❧❛s ♣♦r ❝é❧✉❧❛ ✉♥✐tár✐❛✳ ❖ ♠ét♦❞♦ ❛♣r❡s❡♥t❛❞♦ ♣❛r❛

❝❛❧❝✉❧❛r ♦s ♠♦❞♦s ✈✐❜r❛❝✐♦♥❛✐s ❡♠ ✉♠❛ r❡❞❡ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ♣♦❞❡ s❡r ❣❡♥❡r❛❧✐③❛❞♦ ♣❛r❛ r❡❞❡s ❜✐ ❡ tr✐❞✐♠❡♥s✐♦♥❛✐s ❬✷❪✳