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❆♣♣❧✐❝❛t✐♦♥ ♦❢ ●❛s ❉②♥❛♠✐❝❛❧ ❋r✐❝t✐♦♥ t♦ P❧❛♥❡t❡s✐♠❛❧s ❊✈❣❡♥✐ ●r✐s❤✐♥ ✫ ❍❛❣❛✐ ❇✳ P❡r❡ts ✶ ▲✉♥❞ ❖❜s❡r✈❛t♦r②✱ ▲✉♥❞✱ ❙✇❡❞❡♥ ❙✉♣♣♦rt❡❞ ❜② ❊✉r♦♣❡❛♥ ❋P✼❈❆● ❣r❛♥❞ ❊①♦♣❧❛♥❡ts ✐♥ ▲✉♥❞✱ ✵✻✳✵✺✳✷✵✶✺ ✶ ❚❡❝❤♥✐♦♥✱ ■sr❛❡❧ ■♥st✐t✉t❡ ♦❢ ❚❡❝❤♥♦❧♦❣②✱ ❍❛✐❢❛✱ ■sr❛❡❧

●❛s ♣❧❛♥❡t❡s✐♠❛❧ ✐♥t❡r❛❝t✐♦♥ ♣❧❛② ❛♥ ✐♠♣♦rt❛♥t r♦❧❡ ✐♥ ♣❧❛♥❡t ❢♦r♠❛t✐♦♥ ❢❡✇ × ✶✵ ✻ ②r ❞✐s❦ ❧✐❢❡t✐♠❡s ✭P❢❛❧③♥❡r✳✱ ✷✵✶✹✮ ●❛s ❞r❛❣ ✐s ✐♠♣♦rt❛♥t ❢♦r s♠❛❧❧ ♣❧❛♥❡t❡s✐♠❛❧s ❚②♣❡ ■ ♠✐❣r❛t✐♦♥ ✐s ✐♠♣♦rt❛♥t ❢♦r ❧❛r❣❡ ♣❧❛♥❡t❛r② ❡♠❜r②♦s ❍♦✇ ❞♦❡s ❣❛s ❛✛❡❝t ✐♥t❡r♠❡❞✐❛t❡ ♠❛ss ♣❧❛♥❡t❡s✐♠❛❧s❄

❆❡r♦❞②♥❛♠✐❝ ❣❛s ❞r❛❣ ✐s ❡✛❡❝t✐✈❡ ❢♦r s♠❛❧❧ ♣❧❛♥❡s✐♠❛❧s ●❛s ❉r❛❣ ❋♦r♠✉❧❛ ❚✐❣❤t❧② ❝♦✉♣❧❡s s♠❛❧❧ ❣r❛✐♥s ❋ ❞ = − ✶ ✷ ❈ ❉ ( ❘❡ ) ❆ ρ ❣ ✈ ✷ r❡❧ ■♥❡✣❝t✐✈❡ ❢♦r ❧❛r❣❡ ♣❧❛♥❡t❡s✐♠❛❧s ❈ ❉ ✲ ❉r❛❣ ❝♦❡✣❝✐❡♥t ❑❡❡♣s r❡❧❛t✐✈❡ ✈❡❧♦❝✐t✐❡s ❧♦✇ ❘❡ ✲ ❘❡②♥♦❧❞s ♥✉♠❜❡r ■♥❝r❡❛s❡s ❣r♦✇t❤ ❆ ✲ ❈r♦ss s❡❝t✐♦♥ ✈ r❡❧ ✲ r❡❧❛t✐✈❡ ✈❡❧♦❝✐t②

P❧❛♥❡t❛r② ♠✐❣r❛t✐♦♥ ✐s ❞♦♠✐♥❛♥t ❢♦r ❧❛r❣❡ ♣r♦t♦♣❧❛♥❡ts ✭▼❛ss❡t✳✱ ✶✾✾✾✮ P❧❛♥❡t❛r② ▼✐❣r❛t✐♦♥ ❊①❝❤❛♥❣❡ ♦❢ ❛♥❣✉❧❛r ♠♦♠❡♥t✉♠ ✇✐t❤ t❤❡ ❣❛s ✭▲✐♥ ✫ P❛♣❛❧♦✐③♦✉✱ ✶✾✼✾✮ ❙♣✐r❛❧ ❞❡♥s✐t② ✇❛✈❡ ✭●♦❧❞r❡✐❝❤ ✫ ❚r❡♠❛✐♥❡✱ ✶✾✽✵✮ ❘❡s♦♥❛♥t ▲✐♥❞❜❧❛❞ ❛♥❞ ❝♦r♦t❛t✐♦♥ t♦rq✉❡s ♠ | Ω( r ) − Ω ♣ ( r ) | = ± κ ( r ) , ♠ ∈ Z ❊✛❡❝t✐✈❡ ❢♦r ♠❛ss❡s ♦❢ ♠ � ✶✵ ✷✺ ❣ ✭❍♦✉r✐❣❛♥ ✫ ❲❛r❞✱ ✶✾✽✹❀ ❚❛❦❛♥❦❛ ✫ ■❞❛✱ ✶✾✾✾✮

❙❝❛❧✐♥❣ ✇✐t❤ P❧❛♥❡t❡s✐♠❛❧ ▼❛ss

❉②♥❛♠✐❝❛❧ ❋r✐❝t✐♦♥ ✭❉❋✮ ✐s ❛♥ ❡✛❡❝t✐✈❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❞r❛❣ ♠❡❝❤❛♥✐s♠ ❉❋ ✐s ❛ ▲♦ss ♦❢ ♠♦♠❡♥t✉♠ ♦❢ ❛ ♠❛ss✐✈❡ ♦❜❥❡❝t ✐♥ ❛ ❜❛❝❦❣r♦✉♥❞ ♠❡❞✐✉♠✱ ❜② ❝r❡❛t✐♥❣ ❛♥ ♦✈❡r✲❞❡♥s✐t② ❣r❛✈✐t❛t✐♦♥❛❧ ✇❛❦❡ ❈♦❧❧✐s✐♦♥❧❡ss s②st❡♠s ✭❈❤❛♥❞r❛s❡❦❤❛r✱ ✶✾✹✸✮ ●❛s❡♦✉s ♠❡❞✐✉♠ ✭❖str✐❦❡r✱ ✶✾✾✾✮ ●r❛✈✐t❛t✐♦♥❛❧ ♣❡rt✉r❜❛t✐t✐♦♥ ♦♥ ✉♥✐❢♦r♠ ❣❛s❡♦✉s ♠❡❞✐✉♠✿ ❈❛❧❝✉❧❛t❡ t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ✇❛❦❡ α ( ① , t ) = ∆ ρ ( ① , t ) / ρ ✵ � ρ ∇ Φ ❡①t ❞ ✸ r ❈❛❧❝✉❧❛t❡ t❤❡ ❡✛❡❝t✐✈❡ ❢♦r❝❡ ❋ ●❉❋ =

❉②♥❛♠✐❝❛❧ ❋r✐❝t✐♦♥ ✭❉❋✮ ✐s ❛♥ ❡✛❡❝t✐✈❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❞r❛❣ ♠❡❝❤❛♥✐s♠ ❉❋ ✐s ❛ ▲♦ss ♦❢ ♠♦♠❡♥t✉♠ ♦❢ ❛ ♠❛ss✐✈❡ ♦❜❥❡❝t ✐♥ ❛ ❜❛❝❦❣r♦✉♥❞ ♠❡❞✐✉♠✱ ❜② ❝r❡❛t✐♥❣ ❛♥ ♦✈❡r✲❞❡♥s✐t② ❣r❛✈✐t❛t✐♦♥❛❧ ✇❛❦❡ ❈♦❧❧✐s✐♦♥❧❡ss s②st❡♠s ✭❈❤❛♥❞r❛s❡❦❤❛r✱ ✶✾✹✸✮ ●❛s❡♦✉s ♠❡❞✐✉♠ ✭❖str✐❦❡r✱ ✶✾✾✾✮ ●r❛✈✐t❛t✐♦♥❛❧ ♣❡rt✉r❜❛t✐t✐♦♥ ♦♥ ✉♥✐❢♦r♠ ❣❛s❡♦✉s ♠❡❞✐✉♠✿ ❈❛❧❝✉❧❛t❡ t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ✇❛❦❡ α ( ① , t ) = ∆ ρ ( ① , t ) / ρ ✵ � ρ ∇ Φ ❡①t ❞ ✸ r ❈❛❧❝✉❧❛t❡ t❤❡ ❡✛❡❝t✐✈❡ ❢♦r❝❡ ❋ ●❉❋ =

❉②♥❛♠✐❝❛❧ ❋r✐❝t✐♦♥ ✐♥ ●❛s❡♦✉s ▼❡❞✐✉♠ ✭●❉❋✮ ▲✐♥❡❛r ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② ②✐❡❧❞s ❛♥ ♦✉t❣♦✐♥❣ ♣r❡ss✉r❡ ✇❛✈❡ ❙♦❧✈✐♥❣ ■♥❤♦♠♦❣❡♥♦✉s ✇❛✈❡ ❡q✉❛t✐♦♥ ✇✐t❤ r❡t❛r❞❡❞ ♣♦t❡♥t✐❛❧ P♦✐♥t ♠❛ss ♣❡rt✉r❜❡r

●❉❋ ♣❡❛❦s ♥❡❛r ✈ ∼ ❝ s ❋ = ❋ ✵ × I ( M ) ✇❤❡r❡ ❋ ✵ = ✹ π ● ✷ ▼ ✷ ρ ✵ ❝ ✷ s ▼ ✲ ♦❜❥❡❝t ♠❛ss ❝ s ✲ s♣❡❡❞ ♦❢ s♦✉♥❞ ✶ I ( M ) = M ✷ × � ✶ � ✶ + M � ✷ ❧♥ − M M < ✶ ✶ − M ✶ ✷ ❧♥ ( ✶ − M − ✷ )+ ❧♥ Λ M > ✶ ❆♣♣r♦①✐♠❛t❡ ❢♦r♠✉❧❛✿ � M / ✸ M ≪ ✶ I ( M ) = ❧♥ Λ / M ✷ M ≫ ✶ Λ = r ♠❛① / r ♠✐♥ ✐s ❝❛❧❧❡❞ ❈♦✉❧♦♠❜ ❧♦❣❛r✐t❤♠

Pr❡✈✐♦✉s ✇♦r❦s ❝♦♥s✐❞❡r❡❞ ♦♥❧② ♠❛ss❡s ♦❢ � ▼ ⊕ ●❉❋ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ♣❧❛♥❡t ❢♦r♠❛t✐♦♥✿ ❱❡rt✐❝❛❧❧② ❛✈❡r❛❣❡❞✱ st❡❛❞② st❛t❡ ●❉❋ ✭▼✉t♦ ❡❧✳ ❛❧✳✱ ✷✵✶✶✮ ●❉❋ ❞♦♠✐♥❛♥t ❢♦r ❤✐❣❤❧② ❡❝❝❡♥tr✐❝ ♦r❜✐t ❉✐s❦ ♣❧❛♥❡t ✐♥t❡r❛❝t✐♦♥s ❢♦r ❤✐❣❤❧② ✐♥❝❧✐♥❡❞ ♦r❜✐ts ✭❘❡✐♥✳✱ ✷✵✶✷✮ ■♥t❡r❛❝t✐♦♥ ♦❢ ❛❝❝r❡t✐♥❣ ♣❧❛♥❡t ✭▲❡❡ ✫ ❙t❛❤❧❡r✳✱ ✷✵✶✷❀ ❈❛♥t♦ ❡t✳ ❛❧✳✱ ✷✵✶✷✮ ❙❡❝✉❧❛r ✐♥t❡r❛❝t✐♦♥ ♦❢ s❡❧❢ ❣r❛✈✐t❛t✐♥❣ ❞✐s❦ ✭❚❡②ss❛♥❞✐❡r ❡t✳ ❛❧✳✱ ✷✵✶✸✮ ❆❧❧ ❝♦♥s✐❞❡r ♠❛ss❡s ♦❢ ❢✉❧❧② ❡✈♦❧✈❡❞ ♣❧❛♥❡ts✱ ❛t ❧❡❛st � ▼ ⊕

P♦✇❡r ❧❛✇ ❉✐s❦ ❙tr✉❝t✉r❡ Pr♦t♦♣❧❛♥❡t❛r② ❉✐s❦ ❙tr✉❝t✉r❡ ✭●♦❧❞r❡✐❝❤ ✫ ❈❤✐❛♥❣✳✱ ✶✾✾✼✮ ❘❛❞✐❛❧ ❙tr✉❝t✉r❡✿ ❚❡♠♣❡r❛t✉r❡ ♣r♦✜❧❡✿ ❚ ❞✐s❦ ≈ ✶✷✵ ( ❛ / ❆❯ ) − ✸ / ✼ ❑ ❙♦✉♥❞ s♣❡❡❞ ❝ s ≈ ✹ . ✼ × ✶✵ ✹ ( ❛ / ❆❯ ) − ✸ / ✶✹ ❝♠ / s ❆s♣❡❝t r❛t✐♦ ❍ ✵ = ❤ ( ❛ ) / ❛ = ✵ . ✵✷✷ ( ❛ / ❆❯ ) ✷ / ✼ ❘❛❞✐❛❧ ❣❛s ❞❡♥s✐t②✿ ρ ❣ ( ❛ ) = ✸ × ✶✵ − ✾ ( ❛ / ❆❯ ) − ✶✻ / ✼ ❣ / ❝♠ ✸ ❱❡rt✐❝❛❧ str✉❝t✉r❡✿ ❱❡rt✐❝❛❧ ●❛s ❞❡♥s✐t②✿ ρ ❣ ( ❛ ✵ , ③ ) ∼ ρ ❣ ( ❛ ✵ , ✵ ) × ❡①♣ ( − ③ ✷ / ✷ ❤ ✷ ) ❣ / ❝♠ ✸ ■s♦t❤❡r♠❛❧ ❞✐s❦ ❘❡❧❛t✐✈❡ ✈❡❧♦❝✐t② ❞✉❡ t♦ ♣r❡ss✉r❡ ❣r❛❞✐❡♥ts✿ Pr❡ss✉r❡ ❣r❛❞✐❡♥t✿ P ∼ ( ❛ / ❆❯ ) − β ✇❤❡r❡ β = ✶✾ / ✼ ✈ r❡❧ = | ✈ ❑ − ✈ ϕ , ❣❛s | ∼ β ❍ ✷ ✵ ✈ ❑ ≪ ❝ s

P♦✇❡r ❧❛✇ ❉✐s❦ ❙tr✉❝t✉r❡ Pr♦t♦♣❧❛♥❡t❛r② ❉✐s❦ ❙tr✉❝t✉r❡ ✭●♦❧❞r❡✐❝❤ ✫ ❈❤✐❛♥❣✳✱ ✶✾✾✼✮ ❘❛❞✐❛❧ ❙tr✉❝t✉r❡✿ ❚❡♠♣❡r❛t✉r❡ ♣r♦✜❧❡✿ ❚ ❞✐s❦ ≈ ✶✷✵ ( ❛ / ❆❯ ) − ✸ / ✼ ❑ ❙♦✉♥❞ s♣❡❡❞ ❝ s ≈ ✹ . ✼ × ✶✵ ✹ ( ❛ / ❆❯ ) − ✸ / ✶✹ ❝♠ / s ❆s♣❡❝t r❛t✐♦ ❍ ✵ = ❤ ( ❛ ) / ❛ = ✵ . ✵✷✷ ( ❛ / ❆❯ ) ✷ / ✼ ❘❛❞✐❛❧ ❣❛s ❞❡♥s✐t②✿ ρ ❣ ( ❛ ) = ✸ × ✶✵ − ✾ ( ❛ / ❆❯ ) − ✶✻ / ✼ ❣ / ❝♠ ✸ ❱❡rt✐❝❛❧ str✉❝t✉r❡✿ ❱❡rt✐❝❛❧ ●❛s ❞❡♥s✐t②✿ ρ ❣ ( ❛ ✵ , ③ ) ∼ ρ ❣ ( ❛ ✵ , ✵ ) × ❡①♣ ( − ③ ✷ / ✷ ❤ ✷ ) ❣ / ❝♠ ✸ ■s♦t❤❡r♠❛❧ ❞✐s❦ ❘❡❧❛t✐✈❡ ✈❡❧♦❝✐t② ❞✉❡ t♦ ♣r❡ss✉r❡ ❣r❛❞✐❡♥ts✿ Pr❡ss✉r❡ ❣r❛❞✐❡♥t✿ P ∼ ( ❛ / ❆❯ ) − β ✇❤❡r❡ β = ✶✾ / ✼ ✈ r❡❧ = | ✈ ❑ − ✈ ϕ , ❣❛s | ∼ β ❍ ✷ ✵ ✈ ❑ ≪ ❝ s