❋✐♥✐t❡ ❞❡♥s✐t② ❡q✉❛t✐♦♥ ♦❢ st❛t❡ ♦❢ ◗❈❉ ❜② ♠❡❛♥s ♦❢ r❡s✉♠♠❡❞ ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② ❙②❧✈❛✐♥ ▼♦❣❧✐❛❝❝✐ ❇✐❡❧❡❢❡❧❞ ❯♥✐✈❡rs✐t② ✭s②❧✈❛✐♥❅♣❤②s✐❦✳✉♥✐✲❜✐❡❧❡❢❡❧❞✳❞❡✮ ❋❛✐r♥❡ss ■■ ❇❡r❧✐♥✱ ❙❡♣t❡♠❜❡r ✶✻✱ ✷✵✶✸ ▼❛✐♥ r❡❢❡r❡♥❝❡s✿ ❏❡♥s ❖✳ ❆♥❞❡rs❡♥✱ ❙✳▼✳✱ ◆❛♥ ❙✉ ❛♥❞ ❆❧❡❦s✐ ❱✉♦r✐♥❡♥❀ P❘❉ ✽✼✱ ✵✼✹✵✵✸ ❬❛r❳✐✈✿✶✷✶✵✳✵✾✶✷❪ ❙✳▼✳✱ ❏❡♥s ❖✳ ❆♥❞❡rs❡♥✱ ▼✐❝❤❛❡❧ ❙tr✐❝❦❧❛♥❞✱ ◆❛♥ ❙✉ ❛♥❞ ❆❧❡❦s✐ ❱✉♦r✐♥❡♥❀ s✉❜♠✐tt❡❞ t♦ ❏❍❊P ❬❛r❳✐✈✿✶✸✵✼✳✽✵✾✽❪
■♥tr♦❞✉❝t✐♦♥ ✶ ▼♦t✐✈❛t✐♦♥s ❍❛r❞✲❚❤❡r♠❛❧✲▲♦♦♣ ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② ✷ ❘❡s✉♠♠❛t✐♦♥ ❢♦r♠❛❧✐s♠ ❋✉❧❧ ✶✲❧♦♦♣ ♣r❡ss✉r❡ ❉✐♠❡♥s✐♦♥❛❧ r❡❞✉❝t✐♦♥ ✐♥s♣✐r❡❞ r❡s✉♠♠❛t✐♦♥ ✸ ◗❈❉ ❛♥❞ ✐ts ❉❘ ❢r❛♠❡✇♦r❦ ❘❡s✉♠♠❡❞ ✹✲❧♦♦♣ ✜♥✐t❡ µ f ♣r❡ss✉r❡ ❘❡s✉❧ts ❛♥❞ ❝♦♠♣❛r✐s♦♥ ✇✐t❤ ▲❛tt✐❝❡ ✹ ❈❤♦✐❝❡ ♦❢ ♣❛r❛♠❡t❡rs ❋r♦♠ ✷♥❞ t♦ ✻t❤ ♦r❞❡r ◗◆❙ µ f ❞❡♣❡♥❞❡♥t ♣❛rt ♦❢ t❤❡ ♣r❡ss✉r❡ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❍❚▲♣t tr✉♥❝❛t❡❞ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥ ✺ ❙✉♠♠❛r② ❛♥❞ ♣❡rs♣❡❝t✐✈❡s ❙②❧✈❛✐♥ ▼♦❣❧✐❛❝❝✐✱ ❇✐❡❧❡❢❡❧❞ ❯♥✐✈❡rs✐t② ❋✐♥✐t❡ µ f ❊♦❙ ✐♥ ◗❈❉ ✈✐❛ r❡s✉♠♠❡❞ P❚ ❙❡♣t❡♠❜❡r ✶✻✱ ✷✵✶✸ ✵ ✴ ✶✺
■♥tr♦❞✉❝t✐♦♥ ✶ ■♥tr♦❞✉❝t✐♦♥ ▼♦t✐✈❛t✐♦♥s ✷ ❍❛r❞✲❚❤❡r♠❛❧✲▲♦♦♣ ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② ❘❡s✉♠♠❛t✐♦♥ ❢♦r♠❛❧✐s♠ ❋✉❧❧ ✶✲❧♦♦♣ ♣r❡ss✉r❡ ✸ ❉✐♠❡♥s✐♦♥❛❧ r❡❞✉❝t✐♦♥ ✐♥s♣✐r❡❞ r❡s✉♠♠❛t✐♦♥ ◗❈❉ ❛♥❞ ✐ts ❉❘ ❢r❛♠❡✇♦r❦ ❘❡s✉♠♠❡❞ ✹✲❧♦♦♣ ✜♥✐t❡ µ f ♣r❡ss✉r❡ ✹ ❘❡s✉❧ts ❛♥❞ ❝♦♠♣❛r✐s♦♥ ✇✐t❤ ▲❛tt✐❝❡ ❈❤♦✐❝❡ ♦❢ ♣❛r❛♠❡t❡rs ❋r♦♠ ✷♥❞ t♦ ✻t❤ ♦r❞❡r ◗◆❙ µ f ❞❡♣❡♥❞❡♥t ♣❛rt ♦❢ t❤❡ ♣r❡ss✉r❡ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❍❚▲♣t tr✉♥❝❛t❡❞ r❡s✉❧ts ✺ ❈♦♥❝❧✉s✐♦♥ ❙✉♠♠❛r② ❛♥❞ ♣❡rs♣❡❝t✐✈❡s ❙②❧✈❛✐♥ ▼♦❣❧✐❛❝❝✐✱ ❇✐❡❧❡❢❡❧❞ ❯♥✐✈❡rs✐t② ❋✐♥✐t❡ µ f ❊♦❙ ✐♥ ◗❈❉ ✈✐❛ r❡s✉♠♠❡❞ P❚ ❙❡♣t❡♠❜❡r ✶✻✱ ✷✵✶✸ ✵ ✴ ✶✺
■♥tr♦❞✉❝t✐♦♥ ▼♦t✐✈❛t✐♦♥s ❲❤② ❤♦t ❛♥❞ ❞❡♥s❡ ◗❈❉ ✈✐❛ r❡s✉♠♠❡❞ ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r②❄ ❈♦♥♥❡❝t ❧❛tt✐❝❡ r❡❣✐♦♥ t♦ ❛s②♠♣t♦t✐❝❛❧❧② ❤✐❣❤ ❡♥❡r❣✐❡s ◗✉❛❧✐t❛t✐✈❡ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ ♣❧❛s♠❛ ♣r♦♣❡rt✐❡s✱ ❡✈❡♥ ❛t ❧❛r❣❡ µ f / T ❙❤♦✇ ✉s❡❢✉❧♥❡ss ♦❢ ✇❡❛❦ ❝♦✉♣❧✐♥❣ ♠❡t❤♦❞s ❢♦r ✜♥✐t❡ µ f ♣r♦❜❧❡♠s ❲❤② ❞✐❛❣♦♥❛❧ q✉❛r❦ ♥✉♠❜❡r s✉s❝❡♣t✐❜✐❧✐t✐❡s❄ ◆❛t✉r❛❧ q✉❛♥t✐t✐❡s ❢♦r ❞✐r❡❝t ❝♦♠♣❛r✐s♦♥ ✇✐t❤ ❧❛tt✐❝❡ ❈❛rr✐❡s ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ r❡s♣♦♥s❡ ♦❢ ❛ s②st❡♠ t♦ ♥♦♥③❡r♦ µ f ❆ ❢❡✇ r❡♠✐♥❞❡rs✿ ■♠♣❧❡♠❡♥t✐♥❣ µ f � = ✵ ✐♥ ✭✐♠❛❣✐♥❛r② t✐♠❡ ❢♦r♠❛❧✐s♠✮ t❤❡r♠❛❧ ♣◗❈❉✿ k f ✵ = ( ✷ n + ✶ ) π T → ( ✷ n + ✶ ) π T − i µ f ◗✉❛r❦ ♥✉♠❜❡r s✉s❝❡♣t✐❜✐❧✐t✐❡s ✭◗◆❙✮ ❛s ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ♣r❡ss✉r❡✿ � ∂ i + j + k + ... p ( T , { µ f } ) � � χ u i d j s k ... ( T ) ≡ � j ∂µ i u ∂µ d ∂µ k s ... { µ f } = ✵ ❙②❧✈❛✐♥ ▼♦❣❧✐❛❝❝✐✱ ❇✐❡❧❡❢❡❧❞ ❯♥✐✈❡rs✐t② ❋✐♥✐t❡ µ f ❊♦❙ ✐♥ ◗❈❉ ✈✐❛ r❡s✉♠♠❡❞ P❚ ❙❡♣t❡♠❜❡r ✶✻✱ ✷✵✶✸ ✶ ✴ ✶✺
❍❚▲♣t ✶ ■♥tr♦❞✉❝t✐♦♥ ▼♦t✐✈❛t✐♦♥s ✷ ❍❛r❞✲❚❤❡r♠❛❧✲▲♦♦♣ ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② ❘❡s✉♠♠❛t✐♦♥ ❢♦r♠❛❧✐s♠ ❋✉❧❧ ✶✲❧♦♦♣ ♣r❡ss✉r❡ ✸ ❉✐♠❡♥s✐♦♥❛❧ r❡❞✉❝t✐♦♥ ✐♥s♣✐r❡❞ r❡s✉♠♠❛t✐♦♥ ◗❈❉ ❛♥❞ ✐ts ❉❘ ❢r❛♠❡✇♦r❦ ❘❡s✉♠♠❡❞ ✹✲❧♦♦♣ ✜♥✐t❡ µ f ♣r❡ss✉r❡ ✹ ❘❡s✉❧ts ❛♥❞ ❝♦♠♣❛r✐s♦♥ ✇✐t❤ ▲❛tt✐❝❡ ❈❤♦✐❝❡ ♦❢ ♣❛r❛♠❡t❡rs ❋r♦♠ ✷♥❞ t♦ ✻t❤ ♦r❞❡r ◗◆❙ µ f ❞❡♣❡♥❞❡♥t ♣❛rt ♦❢ t❤❡ ♣r❡ss✉r❡ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❍❚▲♣t tr✉♥❝❛t❡❞ r❡s✉❧ts ✺ ❈♦♥❝❧✉s✐♦♥ ❙✉♠♠❛r② ❛♥❞ ♣❡rs♣❡❝t✐✈❡s ❙②❧✈❛✐♥ ▼♦❣❧✐❛❝❝✐✱ ❇✐❡❧❡❢❡❧❞ ❯♥✐✈❡rs✐t② ❋✐♥✐t❡ µ f ❊♦❙ ✐♥ ◗❈❉ ✈✐❛ r❡s✉♠♠❡❞ P❚ ❙❡♣t❡♠❜❡r ✶✻✱ ✷✵✶✸ ✶ ✴ ✶✺
❍❚▲♣t ❘❡s✉♠♠❛t✐♦♥ ❢♦r♠❛❧✐s♠ ❍❚▲ ❣❛✉❣❡ ✐♥✈❛r✐❛♥t ❡✛❡❝t✐✈❡ ❛❝t✐♦♥✿ ❋r❡♥❦❡❧✱ ❚❛②❧♦r ❬◆✉❝❧ P❤②s ❇ ✸✸✹ ✱ ✶✾✾❪ ❛♥❞ ❇r❛❛t❡♥✱ P✐s❛rs❦✐ ❬◆✉❝❧ P❤②s ❇ ✸✸✼ ✱ ✺✻✾❪ ❘❡♦r❣❛♥✐③❛t✐♦♥ ♦❢ t❤❡ ♣❡rt✉r❜❛t✐✈❡ s❡r✐❡s ♦❢ t❤❡r♠❛❧ ♣◗❈❉✿ � �� � L ❍❚▲♣t = L ◗❈❉ + L ❍❚▲ δ g + ∆ L ❍❚▲ � √ g → ❲✐t❤ t❤❡ ❣❛✉❣❡ ✐♥✈❛r✐❛♥t ❍❚▲ ✐♠♣r♦✈❡♠❡♥t t❡r♠✿ � � � y α y β � − ✶ ✷ ( ✶ − δ ) m ✷ F µ L ❍❚▲ = D ❚r F µα ( y · D ) ✷ β y � y µ � N f � q f ¯ m ✷ ψ f γ µ +( ✶ − δ ) i ψ f y · D y f δ ✿ ❋♦r♠❛❧ ❡①♣❛♥s✐♦♥ ♣❛r❛♠❡t❡r✱ δ = ✶ ✐♥ t❤❡ ❡♥❞ → ◗❈❉✦ ❈♦✉♥t t❤❡ ★ ♦❢ ❍❚▲ ❞r❡ss❡❞ ❧♦♦♣s ∆ L ❍❚▲ ✿ ❍❚▲ ❝♦✉♥t❡rt❡r♠✭s✮❀ y µ ✿ ▲✐❣❤t❧✐❦❡ ✹✲✈❡❝t♦r m D / m q f ✿ ❉❡❜②❡✴q✉❛r❦ t❤❡r♠❛❧ ♠❛ss ♣❛r❛♠❡t❡rs ❆❞❞✐♥❣ L ❍❚▲ ✿ ✏❙❍■❋❚❙✑ t❤❡ ❣r♦✉♥❞ st❛t❡ t♦ ❛♥ ✐❞❡❛❧ ❣❛s ♦❢ t❤❡r♠❛❧ ✭♠❛ss✐✈❡✦✮ q✉❛s✐♣❛rt✐❝❧❡s ❙②❧✈❛✐♥ ▼♦❣❧✐❛❝❝✐✱ ❇✐❡❧❡❢❡❧❞ ❯♥✐✈❡rs✐t② ❋✐♥✐t❡ µ f ❊♦❙ ✐♥ ◗❈❉ ✈✐❛ r❡s✉♠♠❡❞ P❚ ❙❡♣t❡♠❜❡r ✶✻✱ ✷✵✶✸ ✷ ✴ ✶✺
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