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UBC Theses and Dissertations

Solid state NMR investigations of protein based biomaterials: spider silk, recombinant spider silk proteins,… Katz, David Samuel 2010

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❙♦❧✐❞ ❙t❛t❡ ◆▼❘ ■♥✈❡st✐❣❛t✐♦♥s ♦❢ Pr♦t❡✐♥ ❇❛s❡❞ ❇✐♦♠❛t❡r✐❛❧s ❙♣✐❞❡r ❙✐❧❦✱ ❘❡❝♦♠❜✐♥❛♥t ❙♣✐❞❡r ❙✐❧❦ Pr♦t❡✐♥s✱ ❛♥❞ ❊❧❡❝tr♦s♣✉♥ ❘❡❝♦♠❜✐♥❛♥t ❙♣✐❞❡r ❙✐❧❦ Pr♦t❡✐♥s  ❜② ❉❛✈✐❞ ❙❛♠✉❡❧ ❑❛t③ ❇✳❆✳ ❘❡❡❞ ❈♦❧❧❡❣❡✱ ✷✵✵✻ ❆ ❚❍❊❙■❙ ❙❯❇▼■❚❚❊❉ ■◆ P❆❘❚■❆▲ ❋❯▲❋■▲▲▼❊◆❚ ❖❋ ❚❍❊ ❘❊◗❯■❘❊▼❊◆❚❙ ❋❖❘ ❚❍❊ ❉❊●❘❊❊ ❖❋ ▼❛st❡r ♦❢ ❙❝✐❡♥❝❡ ✐♥ ❚❤❡ ❋❛❝✉❧t② ♦❢ ●r❛❞✉❛t❡ ❙t✉❞✐❡s ✭P❤②s✐❝s✮ ❚❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❇r✐t✐s❤ ❈♦❧✉♠❜✐❛ ✭❱❛♥❝♦✉✈❡r✮ ❖❝t♦❜❡r ✷✵✶✵  ➞ ❉❛✈✐❞ ❙✳ ❑❛t③ ✷✵✶✵  ❆❜str❛❝t  13  ❈ ◆✉❝❧❡❛r ▼❛❣♥❡t✐❝ ❘❡s♦♥❛♥❝❡ ✇❛s ❡♠♣❧♦②❡❞ t♦ ✐♥✈❡st✐❣❛t❡ t❤❡ str✉❝t✉r❡ ♦❢ s♣✐❞❡r ❞r❛❣❧✐♥❡  s✐❧❦✱ ♣♦✇❞❡r❡❞ r❡❝♦♠❜✐♥❛♥t ♠❛❥♦r ❛♠♣✉❧❛t❡ s♣✐❞r♦✐♥ ✶ ✭▼❛❙♣✶✮ ❛♥❞ ✷ ✭▼❛❙♣✷✮ t❤❛t ✇❡r❡ ♣r♦❞✉❝❡❞ ✐♥ t❤❡ ♠✐❧❦ ♦❢ ❣❡♥❡t✐❝❛❧❧② ❡♥❣✐♥❡❡r❡❞ ❣♦❛ts✱ ❛♥❞ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶✳ ❈r♦ss ♣♦❧❛r✐③❛✲ t✐♦♥ s♣❡❝tr❛ ✇❡r❡ ✉s❡❞ t♦ ❛ss✐❣♥ s❡❝♦♥❞❛r② str✉❝t✉r❡s t♦ t❤❡ ♣r♦t❡✐♥ r❡s✐❞✉❡s✱ ❛♥❞ ❧♦♥❣✐t✉❞✐♥❛❧ r❡❧❛①❛t✐♦♥ ♠❡❛s✉r❡♠❡♥ts ✇❡r❡ ✉s❡❞ t♦ ✐♥✈❡st✐❣❛t❡ t❤❡ ♠♦❧❡❝✉❧❛r t❤❡r♠❛❧ ♠♦t✐♦♥✳ ❚❤❡ ❝r②st❛❧❧✐♥❡ r❡❣✐♦♥s ♦❢ s♣✐❞❡r s✐❧❦ ✇❡r❡ ❢♦✉♥❞ t♦ ❡①❤✐❜✐t ♥❛♥♦s❡❝♦♥❞ s❝❛❧❡ t❤❡r♠❛❧ ♠♦✲ t✐♦♥✱ s✉❜❥❡❝t t♦ ✈❡r② r✐❣✐❞ ♠♦t✐♦♥❛❧ ❧✐♠✐ts✳ ❚❤❡ r❡❝♦♠❜✐♥❛♥t ▼❛❙♣✶ ❛♥❞ ▼❛❙♣✷ ✇❡r❡ ❢♦✉♥❞ t♦ ❤❛✈❡ ✈❡r② s✐♠✐❧❛r str✉❝t✉r❡s t❤❛t ❡①❤✐❜✐t❡❞ ❛❜✉♥❞❛♥t β s❤❡❡t ❝r②st❛❧❧✐♥❡ r❡❣✐♦♥s✳ ❊❧❡❝✲ tr♦s♣✉♥ ▼❛❙♣✶ ❤♦✇❡✈❡r ❛♣♣❡❛rs t♦ ❜❡ ❤✐❣❤❧② ❞✐s♦r❞❡r❡❞ ❛♥❞ ✐s ♣❡r❤❛♣s ❜❡st ❝❤❛r❛❝t❡r✐③❡❞ ❛s ❞❡♥❛t✉r❡❞✳ ❚❤❡s❡ r❡s✉❧ts ❛r❡ ✐♥ ❝♦♥tr❛st t♦ ♣r❡✈✐♦✉s ✜♥❞✐♥❣s ♦❢ s♣✐❞❡r s✐❧❦ ♣r♦t❡✐♥s ✐♥ ♥♦♥✲✜❜❡r st❛t❡s✱ ✇❤❡r❡ ♥♦ ❛♣♣r❡❝✐❛❜❧❡ ❝r②st❛❧❧✐♥❡ ❝♦♠♣♦♥❡♥t ✇❛s ♦❜s❡r✈❡❞✱ ❛♥❞ ❛♣♣❡❛rs t♦ ❜❡ ✐♥❝♦♥s✐st❡♥t ✇✐t❤ ♣r❡✈✐♦✉s ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✐♥❢r❛r❡❞ s♣❡❝tr♦s❝♦♣② ♦❢ s✐♠✐❧❛r❧② ♣r❡♣❛r❡❞ s❛♠♣❧❡s✳ ❘❡❝♦♥s✐❞❡r❛t✐♦♥ ♦❢ t❤❡ ❋❚■❘ ❞❛t❛ ❤♦✇❡✈❡r r❛✐s❡s ❝♦♥❝❡r♥s ❛❜♦✉t t❤❡ ✐♥t❡r♣r❡t❛✲ t✐♦♥ ♦❢ t❤♦s❡ ❞❛t❛✱ ♣♦ss✐❜❧② ❡①♣❧❛✐♥✐♥❣ t❤❡ ❞✐s❛❣r❡❡♠❡♥t✳ ❚❤✐s ✇♦r❦ s✉❣❣❡sts t❤❛t t❤❡ ❧❛❝❦ ♦❢ r❡❣✉❧❛r str✉❝t✉r❡ ❢♦✉♥❞ ✐♥ t❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ✐s t❤❡ ❝❛✉s❡ ♦❢ t❤❡ ✈❡r② ♣♦♦r ♠❡❝❤❛♥✐❝❛❧ ♣r♦♣❡rt✐❡s ♣r❡✈✐♦✉s❧② ♠❡❛s✉r❡❞ ❢♦r t❤✐s ♠❛t❡r✐❛❧✳  ✐✐  ❈♦♥t❡♥ts  ❆❜str❛❝t  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✐✐  ❈♦♥t❡♥ts  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✐✐✐  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✈✐  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✈✐✐  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✐①  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✶  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✸  ✷✳✶  ❩❡❡♠❛♥ ■♥t❡r❛❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✸  ✷✳✷  ❚❤❡ ◆▼❘ ❊①♣❡r✐♠❡♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✻  ✷✳✸  ❈❧❛ss✐❝❛❧ ❘❡❧❛①❛t✐♦♥  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✽  ✷✳✹  ❉❡♥s✐t② ▼❛tr✐① ❘❡♣r❡s❡♥t❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✽  ✷✳✺  ❘♦t❛t✐♥❣ ❋r❛♠❡ ❚r❛♥s❢♦r♠❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✶✵  ✷✳✻  ❆❞✈❛♥❝❡❞ ◆▼❘ ■♥t❡r❛❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✶✶  ✷✳✻✳✶  ❈❤❡♠✐❝❛❧ ❙❤✐❢t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✶✷  ✷✳✻✳✷  ❉✐♣♦❧❛r ❈♦✉♣❧✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✶✹  ▼✐❝r♦s❝♦♣✐❝ ❉❡s❝r✐♣t✐♦♥ ♦❢ ❘❡❧❛①❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✶✻  ✷✳✼✳✶  ❉✐♣♦❧❛r ❘❡❧❛①❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✶✾  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✷✸  ✸✳✶  ❍❡t❡r♦♥✉❝❧❡❛r ❉❡❝♦✉♣❧✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✷✸  ✸✳✷  ▼❛❣✐❝ ❆♥❣❧❡ ❙♣✐♥♥✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✷✹  ✸✳✸  ❈r♦ss P♦❧❛r✐③❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✷✺  ✸✳✹  ❊❧❡❝tr♦s♣✐♥♥✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✷✽  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✸✵  ▲✐st ♦❢ ❚❛❜❧❡s  ▲✐st ♦❢ ❋✐❣✉r❡s  ❆❝❦♥♦✇❧❡❞❣❡♠❡♥ts  ✶  ■♥tr♦❞✉❝t✐♦♥  ✷  ◆✉❝❧❡❛r ▼❛❣♥❡t✐❝ ❘❡s♦♥❛♥❝❡  ✷✳✼  ✸  ✹  ❊①♣❡r✐♠❡♥t❛❧ ❚❡❝❤♥✐q✉❡s  Pr♦t❡✐♥s  ✐✐✐  ✹✳✶  ■♥tr♦❞✉❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✸✵  ✹✳✷  ❙❡❝♦♥❞❛r② ❙tr✉❝t✉r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✸✶  ✹✳✷✳✶  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✸✷  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✸✷  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✸✸  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✸✹  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✸✽  ♦❢ t❤❡ ❚r❛♥s❣❡♥✐❝ ▼❛❙♣✶ ❛♥❞ ▼❛❙♣✷ ✳ ✳ ✳ ✳ ✳  ✸✾  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✹✷  ✺✳✶  ❙❛♠♣❧❡s Pr❡♣❛r❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✹✷  ✺✳✷  ❈r♦ss P♦❧❛r✐③❛t✐♦♥ ❊①♣❡r✐♠❡♥ts ❛♥❞ ❈❤❡♠✐❝❛❧ ❙❤✐❢t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✹✸  ✺✳✸  ❘❡❧❛①❛t✐♦♥ ▼❡❛s✉r❡♠❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✹✸  ✺✳✹  ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❙❡❝♦♥❞❛r② ❙tr✉❝t✉r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✹✻  ✺✳✹✳✶  ❆❧❛♥✐♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✹✼  ✺✳✹✳✷  ●❧②❝✐♥❡ α ❈❛r❜♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✹✾  ✺✳✹✳✸  ●❧✉t❛♠✐♥❡ α/β/γ ❈❛r❜♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✺✵  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✺✷  ✻✳✶  ❆❧❛♥✐♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✺✷  ✻✳✷  ●❧②❝✐♥❡ α ❈❛r❜♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✺✺  ✻✳✸  ●❧✉t❛♠✐♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✺✼  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✻✶  ▼❛❙♣✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✻✶  ✼✳✶✳✶  ❆❧❛♥✐♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✻✶  ✼✳✶✳✷  ●❧②❝✐♥❡ α ❈❛r❜♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✻✸  ✼✳✶✳✸  ●❧✉t❛♠✐♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✻✸  ✼✳✶✳✹  ❉✐s❝✉ss✐♦♥ ♦❢ ▼❛❙♣✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✻✺  ▼❛❙♣✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✻✼  ✼✳✷✳✶  ❆❧❛♥✐♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✻✼  ✼✳✷✳✷  ●❧②❝✐♥❡ α ❈❛r❜♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✻✽  ✼✳✷✳✸  ●❧✉t❛♠✐♥❡ ❛♥❞ ❙❡r✐♥❡ α ❈❛r❜♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✻✾  ✼✳✷✳✹  Pr♦❧✐♥❡ α ❛♥❞ ❙❡r✐♥❡ β ❈❛r❜♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✼✵  ✼✳✷✳✺  ●❧✉t❛♠✐♥❡ β ✴γ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✼✶  ✼✳✷✳✻  ▼❛❙♣✷ ❘❡s✉❧ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✼✶  ❊❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✼✷  ✹✳✷✳✷ ✹✳✸  ❙✐❧❦ ✳ ✹✳✸✳✶ ✹✳✸✳✷  ✺  ✻  ✼  α✲❍❡❧✐① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ β ✲❙❤❡❡ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❙♣✐❞❡r ❙✐❧❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚r❛♥s❣❡♥✐❝ ❙♣✐❞❡r ❙✐❧❦ ✹✳✸✳✷✳✶ ❋❚■❘ ❘❡s✉❧ts  ❊①♣❡r✐♠❡♥t❛❧ ▼❛t❡r✐❛❧s✱ ▼❡t❤♦❞s✱ ❛♥❞ ❘❡s✉❧ts  ❙♣✐❞❡r ❙✐❧❦ ◆▼❘ ❘❡s✉❧ts ❛♥❞ ❆♥❛❧②s✐s  ❘❡❝♦♠❜✐♥❛♥t ▼❛♠♠❛❧✐❛♥ ❙✐❧❦  ✼✳✶  ✼✳✷  ✼✳✸  ✐✈  ✽  ✼✳✸✳✶  ❆❧❛♥✐♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✼✸  ✼✳✸✳✷  ●❧②❝✐♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✼✺  ✼✳✸✳✸  ●❧✉t❛♠✐♥❡ β/γ ❈❛r❜♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✼✺  ✼✳✸✳✹  ❊❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ❈♦♥❝❧✉s✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✼✻  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✼✽  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✽✵  ❈♦♥❝❧✉❞✐♥❣ ❘❡♠❛r❦s  ❇✐❜❧✐♦❣r❛♣❤②  ✈  ▲✐st ♦❢ ❚❛❜❧❡s  ✹✳✶  ❚♦rs✐♦♥ ❛♥❣❧❡s ❢♦r ✈❛r✐♦✉s s❡❝♦♥❞❛r② str✉❝t✉r❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✸✷  ✹✳✷  ▼❡❝❤❛♥✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ ✈❛r✐♦✉s ♠❛♥ ♠❛❞❡ ♠❛t❡r✐❛❧s ❛♥❞ ❞r❛❣❧✐♥❡ s✐❧❦✳ ✳ ✳ ✳  ✸✺  ✹✳✸  ❇r❡❛❦❞♦✇♥ ❜② ❛♠✐♥♦ ❛❝✐❞ ♦❢  ✸✽  ✹✳✹  ▼❡❝❤❛♥✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ ♥❛t✉r❛❧ s♣✐❞❡r s✐❧❦ ❛♥❞ ✈❛r✐♦✉s ❡❧❡❝tr♦s♣✉♥ s✐❧❦s ♠❛❞❡  ◆✳ ❝❧❛✈✐♣❡s  ❞r❛❣❧✐♥❡ s✐❧❦✱ ▼❛❙♣✶✱ ❛♥❞ ▼❛❙♣✷✳ ✳  ❢r♦♠ ▼❛❙♣✶ ❛♥❞ ▼❛❙♣✷✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✸✾  ✹✳✺  ❙❡❝♦♥❞❛r② str✉❝t✉r❡ ❛ss✐❣♥♠❡♥ts ❢r♦♠ ❋❚■❘ ✇❛✈❡♥✉♠❜❡rs✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✸✾  ✹✳✻  ❘❡♣♦rt❡❞ ✇❛✈❡♥✉♠❜❡rs ❢r♦♠ ❋✐❣✉r❡ ✹✳✻ ❛♥❞ ❡st✐♠❛t❡s ♦❢ ❛❝t✉❛❧ ✇❛✈❡♥✉♠❜❡rs ✇❤✐❝❤ ❢♦✉♥❞ ✉s✐♥❣ ✐♠❛❣❡ ❛♥❛❧②s✐s s♦❢t✇❛r❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✹✵  ✺✳✶  ❈❤❡♠✐❝❛❧ s❤✐❢ts ❢♦r s♣✐❞❡r s✐❧❦✱ ▼❛❙♣✶✱ ▼❛❙♣✷✱ ❛♥❞ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶✳ ✳ ✳ ✳  ✹✼  ✺✳✷  ❑♥♦✇♥ ❝❤❡♠✐❝❛❧ s❤✐❢ts ❢♦r ❛♠✐♥♦ ❛❝✐❞s ✐♠♣♦rt❛♥t ✐♥ s♣✐❞❡r s✐❧❦ ❛♥❞ ✈❛r✐♦✉s s❡❝♦♥❞❛r② str✉❝t✉r❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✹✽  ✺✳✸  ❘❡❧❛①❛t✐♦♥ ♣❛r❛♠❡t❡rs ❢♦r ❛❧❛♥✐♥❡ α ❝❛r❜♦♥s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✹✽  ✺✳✹  ❘❡❧❛①❛t✐♦♥ ♣❛r❛♠❡t❡rs ❢♦r ❛❧❛♥✐♥❡ β ❝❛r❜♦♥s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✹✾  ✺✳✺  ❘❡❧❛①❛t✐♦♥ ♣❛r❛♠❡t❡rs ❢♦r ❣❧②❝✐♥❡ α ❝❛r❜♦♥s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✺✵  ✺✳✻  ❘❡❧❛①❛t✐♦♥ ♣❛r❛♠❡t❡rs ❢♦r ❣❧✉t❛♠✐♥❡ β ✴γ ❝❛r❜♦♥s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✺✶  ✈✐  ▲✐st ♦❢ ❋✐❣✉r❡s  ✷✳✶  ❚❤❡ ♣r❡❝❡ss✐♦♥ ♦❢ ❛ ♠❛❣♥❡t✐❝ ♠♦♠❡♥t ❛❜♦✉t ❛ st❛t✐❝ ♠❛❣♥❡t✐❝ ✜❡❧❞✳ ✳ ✳ ✳ ✳ ✳  ✹  ✷✳✷  ❚❤❡ ❡♥❡r❣② ❧❡✈❡❧s ♦❢ ❛  s♣✐♥ s②st❡♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✺  ✷✳✸  ❆ s❛♠♣❧❡ ❢r❡❡ ✐♥❞✉❝t✐♦♥ ❞❡❝❛②✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✼  ✷✳✹  ❚❤❡ r♦t❛t✐♥❣ ❢r❛♠❡ tr❛♥s❢♦r♠❛t✐♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✶✶  ✷✳✺  ❚❤❡ ❞✐♣♦❧❛r ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ t✇♦ s♣✐♥s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✶✺  ✸✳✶  ❆ s❝❤❡♠❛t✐❝ ♦❢ ♠❛❣✐❝ ❛♥❣❧❡ s♣✐♥♥✐♥❣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✷✹  ✸✳✷  ❆ ❝r♦ss ♣♦❧❛r✐③❛t✐♦♥ ♣✉❧s❡ s❡q✉❡♥❝❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✷✺  ✸✳✸  ❈♦♠♣♦♥❡♥ts ♦❢ ❛♥ ❡❧❡❝tr♦s♣✐♥♥✐♥❣ ❛♣♣❛r❛t✉s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✷✾  ✹✳✶  ❆ ❝♦♥♥❡❝t✐♦♥ ♦❢ t✇♦ ✉♥s♣❡❝✐✜❡❞ ❛♠✐♥♦ ❛❝✐❞s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✸✶  ✹✳✷  ❚❤❡ str✉❝t✉r❡ ♦❢ t❤❡ β str❛♥❞ ❛♥❞ β s❤❡❡ts✳  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✸✸  ✹✳✸  ❚❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ ▼❛❙♣✶ ♣r♦t❡✐♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✸✺  ✹✳✹  ❚❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ ▼❛❙♣✷ ♣r♦t❡✐♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✸✻  ✹✳✺  ❆ ♣♦ss✐❜❧❡ str✉❝t✉r❡ ♦❢ s♣✐❞❡r ❞r❛❣❧✐♥❡ s✐❧❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✸✼  ✹✳✻  ❆♥❛❧②s✐s ♦❢ ▼❛❙♣✶ ❋❚■❘ r❡s✉❧ts✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✹✶  ✺✳✶  T1 ❡①♣❡r✐♠❡♥t ♣✉❧s❡ s❡q✉❡♥❝❡ ❡♠♣❧♦②✐♥❣ ❝r♦ss✲♣♦❧❛r✐③❛t✐♦♥ t♦ st✉❞② t❤❡ ❣✐t✉❞✐♥❛❧ r❡❧❛①❛t✐♦♥ ♦❢ t❤❡ 13 C ♥✉❝❧❡✐✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚❤❡ ❝r♦ss ♣♦❧❛r✐③❛t✐♦♥ s♣❡❝tr❛ ❢♦r ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶✱ ▼❛❙♣✶✱ ▼❛❙♣✷✱ s♣✐❞❡r s✐❧❦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✺✳✷  1 2  ❧♦♥✲ ✳ ✳ ✳  ✹✹  ❛♥❞ ✳ ✳ ✳  ✹✻  ✻✳✶  ❚❤❡ r❡❧❛①❛t✐♦♥ ❝✉r✈❡ ♦❢ t❤❡ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥s ✐♥ s♣✐❞❡r s✐❧❦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✺✸  ✻✳✷  P❧♦t ♦❢ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ✈❡rs✉s  ❢♦r t❤❡ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✺✹  ✻✳✸  ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ s♣✐❞❡r s✐❧❦ ❛❧❛♥✐♥❡ β  1 T1  ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✹  ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ s♣✐❞❡r s✐❧❦ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳  ✻✳✺  ✺✺  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  P❧♦t ♦❢ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ✈❡rs✉s  1 T1  ❢♦r t❤❡ ❣❧②❝✐♥❡ α ❝❛r❜♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✈✐✐  ✺✻ ✺✼  ✻✳✻  ▼✉❧t✐♣❧❡ s♣❡❝tr❛ ❢r♦♠ t❤❡ s♣✐❞❡r s✐❧❦ r♦♦♠ t❡♠♣❡r❛t✉r❡ T1 r❡❧❛①❛t✐♦♥ ❡①♣❡r✐✲ ♠❡♥t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✻✳✼  ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ s♣✐❞❡r s✐❧❦ ❣❧✉t❛♠✐♥❡ β ✴γ ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✼✳✶  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✻✷  ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ▼❛❙♣✶ ❛❧❛♥✐♥❡ β ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳  ✼✳✸  ✺✾  ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ▼❛❙♣✶ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳  ✼✳✷  ✺✽  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✻✸  ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ▼❛❙♣✶ ❣❧②❝✐♥❡ α✲❈ r❡✲ ❧❛①❛t✐♦♥ ❞❛t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✻✹  ✼✳✹  ▼✉❧t✐♣❧❡ s♣❡❝tr❛ ❢r♦♠ t❤❡ ▼❛❙♣✶ T1 r❡❧❛①❛t✐♦♥ ❡①♣❡r✐♠❡♥t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✻✹  ✼✳✺  ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ▼❛❙♣✶ ❣❧②❝✐♥❡ β ✴γ ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳  ✼✳✻  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✻✺  ❙♣❡❝tr❛ ♦❢ ❧②♦♣❤✐❧✐③❡❞ ❣❧❛♥❞ s✐❧❦ ✭❇✮ ❛♥❞ ❧②♦♣❤✐❧✐③❡❞ ❞❡♥❛t✉r❡❞ s✐❧❦ ✭❉✮✱ ❝♦♠✲ ♣❛r❡❞ t♦ ▼❛❙♣✶ ✭❆✮ ❛♥❞ ▼❛❙♣✷ ✭❈✮ ❛♥❞ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶✳ ▲②♦♣❤✐❧✐③❡❞ s♣❡❝tr❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✼✳✼  ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ▼❛❙♣✷ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✼✳✽  ✻✼  ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ▼❛❙♣✷ ❛❧❛♥✐♥❡ β ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳  ✼✳✾  ✻✻  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✻✽  ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ▼❛❙♣✷ ❣❧②❝✐♥❡ α ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✻✾  ✼✳✶✵ ❚❤❡ r❡❧❛①❛t✐♦♥ ❝✉r✈❡ ♦❢ t❤❡ ❣❧✉t❛♠✐♥❡ α ❝❛r❜♦♥s ✭❛♥❞ ♣♦ss✐❜❧② s❡r✐♥❡ α ❝❛r✲ ❜♦♥s✮ ✐♥ ▼❛❙♣✷✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✼✵  ✼✳✶✶ ❚❤❡ r❡❧❛①❛t✐♦♥ ❝✉r✈❡ ♦❢ t❤❡ ♣r♦❧✐♥❡ α ❝❛r❜♦♥s ✐♥ ▼❛❙♣✷✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✼✶  ✼✳✶✷ ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ▼❛❙♣✷ ❣❧✉t❛♠✐♥❡ β ✴γ ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✼✷  ✼✳✶✸ ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ❛❧❛✲ ♥✐♥❡ α ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✼✸  ✼✳✶✹ ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ❛❧❛✲ ♥✐♥❡ β ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✼✹  ✼✳✶✺ ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ❣❧②❝✐♥❡  α ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✳✶✻ ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ❣❧✉✲ t❛♠✐♥❡ β ✴γ ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✈✐✐✐  ✼✺ ✼✻  ❆❝❦♥♦✇❧❡❞❣♠❡♥ts  ❲❤✐❧❡ ❛ t❤❡s✐s ♠❛② ♦♥❧② ❤❛✈❡ ♦♥❡ ❛✉t❤♦r✱ ✐t ✐s ♥♦t ❛♥ ✐♥❞✐✈✐❞✉❛❧ ❡✛♦rt✳ ❲✐t❤♦✉t t❤❡ ♣❡♦♣❧❡ ❜❡❤✐♥❞ ♠❡ t❤✐s ✇♦r❦ ✇♦✉❧❞ ♥❡✈❡r ❤❛✈❡ ❜❡❡♥ ✇r✐tt❡♥✳ ▼♦r❡ t❤❛♥ ❛♥②♦♥❡✱ ■ ♦✇❡ t❤❛♥❦s t♦ ♠② ❛❞✈✐s❡r✱ ❈❛r❧ ▼✐❝❤❛❧✱ ✇❤♦✬s ♣❛t✐❡♥❝❡ ✇✐t❤ ♠❡ ♠❛② ❣♦ ❞♦✇♥ ❛s ❛♥ ♦✣❝✐❛❧ ♠✐r❛❝❧❡ ✐❢ ❤❡ ✐s ❡✈❡r ✭❛♥❞ ❞❡s❡r✈❡❞❧②✮ ❝❛♥♦♥✐③❡❞✳ ❍✐s ✐♥s✐❣❤t ✐♥t♦ ❡✈❡r② ❛s♣❡❝t ♦❢ t❤✐s t❤❡s✐s ✇❛s ✐♥str✉♠❡♥t❛❧ ❛♥❞ ■ ❝♦✉❧❞ ♥♦t ❤❛✈❡ ❤♦♣❡❞ t♦ ❤❛✈❡ ❛ ❜❡tt❡r ❢r✐❡♥❞ ❛♥❞ ♠❡♥t♦r t❤r♦✉❣❤♦✉t t❤✐s ♣r♦❝❡ss✳ ▼② ♣❡❡rs ❤❡❧♣❡❞ ♠❡ ❛❧♦♥❣ t❤✐s ♣r♦❝❡ss ❛s ✇❡❧❧✳ ❏❡♥♥② ❈❤✐❡♥✲❍s✐♥✱ ✇❤♦ ❤❡❧♣❡❞ ♠❡ ✇♦r❦ t❤r♦✉❣❤ t❤❡ ✐♥✐t✐❛❧ ❧❡❛r♥✐♥❣ ❝✉r✈❡ ♦❢ ◆▼❘ t❤❡♦r②❀ ❚♦♠ ❉❡♣❡✇✱ ✇❤♦ t❛✉❣❤t ♠❡ ♣r❡tt② ♠✉❝❤ ❡✈❡r②t❤✐♥❣ r❡❣❛r❞✐♥❣ ❤♦✇ t♦ ❛❝t✉❛❧❧② r✉♥ ❛ s♣❡❝tr♦♠❡t❡r❀ ❆♥❞② ❘❡❞❞✐♥✱ ✇❤♦ ✇❛s ❛❧✇❛②s ✇✐❧❧✐♥❣ t♦ ❧❡♥❞ ♠❡ ❛ ♣❡♥❝✐❧✶ ✱ ❛♥❞ ❛ ✈❡r② s♣❡❝✐❛❧ t❤❛♥❦s t♦ ❈❧❛r❦ ▲❡♠❦❡✱ ✇❤♦ ❣❛✈❡ ♠❡ t❤❡ ♠♦t✐✈❛t✐♦♥ t♦ ❣r❛❞✉❛t❡✳ ■ ✇♦✉❧❞ ❛❧s♦ ❧✐❦❡ t♦ t❤❛♥❦ ❘♦♥❛❧❞ ❉♦♥❣❀ ❤❛✈✐♥❣ ❛ ❜r✐❧❧✐❛♥t s❝✐❡♥t✐st s✐tt✐♥❣ ♥❡①t t♦ ♠❡ ❞❡✜✲ ♥✐t❡❧② ❤❡❧♣❡❞ ♠❡ ♦✈❡r❝♦♠❡ ♠♦r❡ t❤❛♥ ♦♥❡ ❝♦♥❝❡♣t✉❛❧ ❤✉r❞❧❡✳ ❚❤❛♥❦ ②♦✉ ❙t❡♣❤❡♥ ❘❡✐♥s❜❡r❣ ❢♦r ❤❡❧♣✐♥❣ ♠❡ ❝♦♥✈❡rt ♦♥ 4th ❛♥❞ ❧♦♥❣✳ ■ ✇♦✉❧❞ ❛❧s♦ ❧✐❦❡ t♦ t❤❛♥❦ ♠② ❝♦❧❧❛❜♦r❛t♦rs ❢r♦♠ t❤❡ ❆❞✈❛♥❝❡❞ ❋✐❜r♦✉s ▼❛t❡r✐❛❧ ▲❛❜♦r❛t♦r②✳ ❋r❛♥❦ ❑♦ ♠❛❞❡ s✉r❡ t❤❛t t❤✐s t❤❡s✐s ❣♦t ♦✛ t❤❡ ❣r♦✉♥❞✳ ■t ✐s ❛ r❛r❡ ♠❛♥ t❤❛t ✇♦✉❧❞ tr✉st ❛ ✜rst ②❡❛r ♠❛st❡rs st✉❞❡♥t ❢r♦♠ ❛ ❞✐✛❡r❡♥t ❞❡♣❛rt♠❡♥t ❛❢t❡r ❛ s✐♥❣❧❡ ♠❡❡t✐♥❣ ✇✐t❤ ♠❛t❡r✐❛❧ ❛s ✐♥❝r❡❞✐❜❧② ❤❛r❞ t♦ ❛❝q✉✐r❡ ❛s ◆❡①✐❛✬s ▼❛❙♣✶ ❛♥❞ ▼❛❙♣✷ ♣♦✇❞❡r✳ ❆❞r✐❡♥♥❡ ❈❤❛♥❣ ♣r♦✈✐❞❡❞ t❤❡ ♠♦st ✉s❡❢✉❧ ❤♦✉r t❤❛t ❛ ❤✐❣❤ s❝❤♦♦❧ s✉♠♠❡r ✐♥t❡r♥ ❝♦✉❧❞ ♣♦ss✐❜❧② ❣✐✈❡ ✇❤❡♥ ❤❡ t❛✉❣❤t ♠❡ ❡❧❡❝tr♦s♣✐♥♥✐♥❣ ❛♥❞ s♣✉♥ ✇❤❛t ✇♦✉❧❞ ❜❡ ♠② ♣r✐♠❛r② s❛♠♣❧❡✳ ■ ✇♦✉❧❞ ❛❧s♦ ❧✐❦❡ t♦ t❤❛♥❦ ❍❡❡❥❛② ❨❛♥❣✱ ❱✐❝t♦r ▲❡✉♥❣✱ ❛♥❞ ❙t❡✈❡ ❨❡♦❤ ♦❢ t❤❡ ❆❋▼▲✳ ■ ✇♦✉❧❞ ❧✐❦❡ t♦ t❤❛♥❦ ❛❧❧ ♦❢ ♠② ❢r✐❡♥❞s❀ ❏❛♠❡s ❛♥❞ ❛ss♦❝✐❛t❡s✱ ❩❛❝✱ ▼✐❦❡✱ ♠② ❢r✐❡♥❞s ❢r♦♠ ❍✐❧❧❡❧✱ t❤❡ ❝r❡✇ ❛t ❈❡✐❧✐✬s✱ ❛♥❞ t❤❡ ❤❡❛t❤❡♥s ✭❘❡❡❞✐❡s✮✳ ❨♦✉ ❛❧❧ ♠❛❞❡ ♠② ❧✐❢❡ ❡♥❥♦②❛❜❧❡ ♦✈❡r t❤❡ ♣❛st ❢❡✇ ②❡❛rs✳ ❑❛②❧✐❡✱ ②♦✉r ❧♦✈❡ ❛♥❞ s✉♣♣♦rt ❤❛s ❤❡❧♣❡❞ t♦ ❦❡❡♣ ♠❡ s❛♥❡ t❤r♦✉❣❤ t❤❡s❡ ❤❛r❞ ②❡❛rs✳ ❚❤❛♥❦ ②♦✉✳ ❋✐♥❛❧❧② ❛ ❤✉❣❡ t❤❛♥❦s t♦ ♠② ❢❛♠✐❧② ❢♦r r❡❛❧❧② ❤❡❧♣✐♥❣ ♠❡ t❤r♦✉❣❤ ❛❧❧ ♦❢ t❤✐s✱ ♠② ♠♦♠ ❛♥❞ ❞❛❞ ❢♦r ♦✛❡r✐♥❣ t♦ ❞r✐✈❡ ♠❡ ❤♦♠❡ ❢r♦♠ ❝❛♠♣✉s ♦♥ ❝♦❧❞ ❛♥❞ ✇❡t ♥✐❣❤ts✱ ♠② ❜r♦t❤❡r ❛♥❞ s✐st❡r ✐♥✲❧❛✇ ❢♦r ♠♦t✐✈❛t✐♦♥✱ ♠② ♥✐❡❝❡ ❘✉❜✐ ❢♦r s♦ ♠❛♥② ❤✉❣s✱ ❛♥❞ ♠② ♥❡♣❤❡✇ ❆✈✐ ❢♦r ♥♦t ❦✐❝❦✐♥❣ ♠❡ ✐♥ t❤❡ ❤❡❛❞ ❛s ❤❛r❞ ❛s ❤❡ ❝❛♥✳✳✳ t♦♦ ♦❢t❡♥✳ ✶ ❲✐❧❧✐♥❣  ❛♥❞ ❜♦rr♦✇ ♠❛② ❜❡ s❧✐❣❤t❧② ✐♥❛❝❝✉r❛t❡✱ ❆ ♠♦r❡ ❛❝❝✉r❛t❡ ❛❝❦♥♦✇❧❡❞❣♠❡♥t ✇♦✉❧❞ ❜❡ t❤❛♥❦ ②♦✉  ❢♦r ♥♦t ❧♦❝❦✐♥❣ ②♦✉r ❞❡s❦ ❛t ♥✐❣❤t✳  ✐①  ❈❤❛♣t❡r ✶  ■♥tr♦❞✉❝t✐♦♥  ❚❤❡ ♦r❜✲✇❡❛✈✐♥❣ s♣✐❞❡r✱  ◆❡♣❤✐❧❛ ❝❧❛✈✐♣❡s  ♣r♦❞✉❝❡s ❛ ❞r❛❣❧✐♥❡ s✐❧❦ t❤❛t ❤❛s ❛ ❤✐❣❤❡r t❡♥s✐❧❡  str❡♥❣t❤ t❤❛♥ st❡❡❧✱ ❛♥❞ ❤❛s ❛ ❤✐❣❤❡r ❜r❡❛❦ ❡♥❡r❣② t❤❛♥ s②♥t❤❡t✐❝ ❤✐❣❤ ♣❡r❢♦r♠❛♥❝❡ ✜❜❡rs✳ ❯♥t✐❧ r❡❝❡♥t❧② t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ♠❛ss ♣r♦❞✉❝t✐♦♥ ♦❢ s♣✐❞❡r s✐❧❦ ❤❛s ❜❡❡♥ ✐♠♣♦ss✐❜❧❡ ❞✉❡ t♦ t❤❡ ❝❛♥♥✐❜❛❧✐st✐❝ ♥❛t✉r❡ ♦❢ s♣✐❞❡rs✳ ❆❞✈❛♥❝❡s ✐♥ ❣❡♥❡t✐❝ ❡♥❣✐♥❡❡r✐♥❣ ❤❛✈❡ ❛❧❧♦✇❡❞ ❢♦r t❤❡ ♣r♦❞✉❝t✐♦♥ ♦❢ t❤❡ s♣✐❞❡r s✐❧❦ ♣r♦t❡✐♥s ✇✐t❤♦✉t t❤❡ ♥❡❡❞ ❢♦r s♣✐❞❡rs✳ Pr♦❞✉❝✐♥❣ ❤✐❣❤ ♣❡r❢♦r♠❛♥❝❡ ✜❜❡rs ❢r♦♠ t❤❡s❡ ♣r♦t❡✐♥s ✐s ❛ ❝❤❛❧❧❡♥❣❡ ✐♥ ✐ts ♦✇♥ r✐❣❤t✳ ❚❤✐s ❝❤❛❧❧❡♥❣❡ ✐s t✇♦ ❢♦❧❞✳ ❚❤❡ ✜rst ✐s t❤❡ ❣❡♥❡r❛t✐♦♥ ♦❢ ✜❜❡rs t❤❛t ❤❛✈❡ ❛ s✐♠✐❧❛r ♣r♦t❡✐♥ s❡❝♦♥❞❛r② str✉❝t✉r❡ t♦ t❤♦s❡ ♣r♦❞✉❝❡❞ ❜② s♣✐❞❡rs✳ ❚❤❡ s❡❝♦♥❞ ❝❤❛❧❧❡♥❣❡ ✐s ✜♥❞✐♥❣ ❛ ♠❡t❤♦❞ ♦❢ ❝r❡❛t✐♥❣ ✜❜❡rs ✇✐t❤ ❛♥ ❛s♣❡❝t r❛t✐♦ ❝♦♠♣❛r❛❜❧❡ t♦ t❤❡ ✜❜❡rs ♣r♦❞✉❝❡❞ ❜② s♣✐❞❡rs✳ ❖♥❡ ♠❡t❤♦❞ ❢♦r ❝r❡❛t✐♥❣ ✜❜❡rs ✉s❡s ❛ ♣r♦❝❡ss ❝❛❧❧❡❞ ❡❧❡❝tr♦s♣✐♥♥✐♥❣ ✇❤✐❝❤ ✈✐♦❧❡♥t❧② ✇❤✐♣s ❛ ❝❤❛r❣❡❞ ❥❡t ♦❢ ♣♦❧②♠❡r s♦❧✉t✐♦♥ ✐♥t♦ ♥❛♥♦✜❜❡rs✳ ❚❤✐s ♠❡t❤♦❞ ❤❛❞ ❜❡❡♥ ❦♥♦✇♥ t♦ ♣r♦❞✉❝❡ ✜❜❡rs ✇✐t❤ ❡①tr❡♠❡❧② ❧❛r❣❡ ❛s♣❡❝t r❛t✐♦s✱ ✉♥❢♦rt✉♥❛t❡❧② t❤❡ ♠❡❝❤❛♥✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✐❧❦ ✜❜❡rs ❝r❡❛t❡❞ ❜② t❤✐s ♠❡t❤♦❞ ❞♦ ♥♦t ❝♦♠♣❛r❡ ✇✐t❤ t❤♦s❡ ♣r♦❞✉❝❡❞ ✐♥ ♥❛t✉r❡✳ ❙♦❧✐❞ st❛t❡ ◆▼❘ ✐s ❛ ✉s❡❢✉❧ t❡❝❤♥✐q✉❡ ❢♦r t❤❡ st✉❞② ♦❢ ❜♦t❤ t❤❡ s♣✐❞❡r s✐❧❦ ♣r♦t❡✐♥s ❛♥❞ t❤❡ ❡❧❡❝tr♦s♣✉♥ ✜❜❡rs t♦ ✜♥❞ ♦✉t ❤♦✇ t❤❡s❡ ✜❜❡rs ❛r❡ ❞✐✛❡r❡♥t ❢r♦♠ t❤♦s❡ ♣r♦❞✉❝❡❞ ❜② ❛ s♣✐❞❡r✳ ❇❡❝❛✉s❡ ◆▼❘ ❛❧❧♦✇s ✉s t♦ ♣r♦❜❡ t❤❡ ❧♦❝❛❧ ♥✉❝❧❡❛r ❡♥✈✐r♦♥♠❡♥t✱ ✇❡ ❛r❡ ❛❜❧❡ t♦ ✉s❡ ✐t t♦ ❧♦♦❦ ❛t ❝❤❛r❛❝t❡r✐st✐❝s s✉❝❤ ❛s ♠♦❧❡❝✉❧❛r ♦r✐❡♥t❛t✐♦♥✱ ❞②♥❛♠✐❝s✱ ❛♥❞ ❝♦♥❢♦r♠❛t✐♦♥✳ ❖♥❡ ✐♠♣♦rt❛♥t ♠❡❛s✉r❡♠❡♥t t❤❛t ❝❛♥ ❜❡ t❛❦❡♥ ✐♥ ◆▼❘ ✐s ✜♥❞✐♥❣ t❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ r❡❧❛①❛t✐♦♥ t✐♠❡✳ ❚❤✐s r❡❧❛①❛t✐♦♥ t✐♠❡ ✐s ❝❛✉s❡❞ ❜② ✢✉❝t✉❛t✐♦♥s ✐♥ t❤❡ ♠❛❣♥❡t✐❝ ✐♥t❡r❛❝t✐♦♥s ❜❡t✇❡❡♥ ♥✉❝❧❡✐ t❤❛t ❛r❡ ❛ r❡s✉❧t ♦❢ t❤❡r♠❛❧ ♠♦t✐♦♥✳ ❑♥♦✇✐♥❣ t❤❡ ❞♦♠✐♥❛♥t ✐♥t❡r❛❝t✐♦♥ ❝❛♥ ②✐❡❧❞ t❤❡ ❢r❡q✉❡♥❝② ♦❢ t❤❡ t❤❡r♠❛❧ ✢✉❝t✉❛t✐♦♥s✱ ♦r t❤❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡✳ ❚❤❡s❡ ♠❡❛s✉r❡♠❡♥ts t❤✉s ✶  ❝❛♥ ❜❡ ✉s❡❞ t♦ ❝❤❛r❛❝t❡r✐③❡ ♠♦❧❡❝✉❧❛r ♠♦t✐♦♥✱ t♦ ❞✐✛❡r❡♥t✐❛t❡ str✉❝t✉r❛❧ r❡❣✐♦♥s✱ ❛♥❞ t♦ ✜♥❞ ♦r❞❡r ♣❛r❛♠❡t❡rs t❤❛t ❞❡s❝r✐❜❡ ♠♦t✐♦♥❛❧ r❡str✐❝t✐♦♥s✳ ■♥ s✐♠✐❧❛r ♠❛t❡r✐❛❧s✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ✉s❡ ❝❤❛♥❣❡s ✐♥ r❡❧❛①❛t✐♦♥ t✐♠❡ t♦ s❡❡ ✐❢ t❤❡r❡ ❛r❡ ❝❤❛♥❣❡s ✐♥ t❤❡ ♦r❞❡r ♣❛r❛♠❡t❡r✱ ✇❤✐❝❤ ❝❛♥ ❤❡❧♣ t❡❧❧ ✉s ✇❤✐❝❤ s②st❡♠ ❤❛s ♠♦r❡ str✉❝t✉r❡✳ ❚❤✐s t❤❡s✐s ❜❡❣✐♥s ✇✐t❤ ❛♥ ✐♥tr♦❞✉❝t✐♦♥ t♦ ◆▼❘ ✐♥ ❈❤❛♣t❡r ✷✳ ❈❤❛♣t❡r ✸ ✇✐❧❧ ♣r❡s❡♥t s♦♠❡ ♦❢ t❤❡ ❡①♣❡r✐♠❡♥t❛❧ t♦♦❧s ✉s❡❞ ✐♥ ◆▼❘✱ ✐♥❝❧✉❞✐♥❣ ❛ ❞❡r✐✈❛t✐♦♥ r❡❧❛t✐♥❣ t❤❡r♠❛❧ ♠♦t✐♦♥ t♦ t❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ r❡❧❛①❛t✐♦♥ t✐♠❡ ❛♥❞ ❛ ❜r✐❡❢ ✐♥tr♦❞✉❝t✐♦♥ t♦ t❤❡ ♣r♦❝❡ss ♦❢ ❡❧❡❝tr♦s♣✐♥♥✐♥❣✳ ❈❤❛♣t❡r ✹ ✇✐❧❧ ❜❡ ❞❡✈♦t❡❞ t♦ ✐♥tr♦❞✉❝✐♥❣ ♣r♦t❡✐♥ str✉❝t✉r❡✱ t❤❡ ❝✉rr❡♥t ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ s♣✐❞❡r s✐❧❦✱ ❛♥❞ s♦♠❡ ♦❢ t❤❡ ♣❛st ✇♦r❦ ♦♥ r❡❝♦♠❜✐♥❛♥t s♣✐❞❡r s✐❧❦ ♣r♦t❡✐♥s✳ ❈❤❛♣t❡r ✺ ♣r♦✈✐❞❡s ❛♥ ♦✈❡r✈✐❡✇ ♦❢ t❤❡ s♣❡❝✐✜❝ ❡①♣❡r✐♠❡♥t❛❧ t❡❝❤♥✐q✉❡s ✉s❡❞✱ t❤❡ ♠❛t❡r✐❛❧s ❜❡✐♥❣ st✉❞✐❡❞✱ ❤♦✇ t❤❡② ✇✐❧❧ ❜❡ ❛♥❛❧②③❡❞✱ ❛♥❞ t❤❡ ♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❡①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts✳ ❈❤❛♣t❡r ✻ ❝♦✈❡rs t❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ◆▼❘ ❡①♣❡r✐♠❡♥ts ♦♥ s♣✐❞❡r s✐❧❦ ❛♥❞ ❈❤❛♣t❡r ✼ ❝♦✈❡rs t❤❡ ❛♥❛❧②s✐s ♦❢ r❡s✉❧ts ❢r♦♠ t❤❡ ◆▼❘ ❡①♣❡r✐♠❡♥ts ♦♥ t❤❡ r❡❝♦♠❜✐♥❛♥t ♣r♦t❡✐♥s ✐♥ ❜♦t❤ ♣♦✇❞❡r ❛♥❞ ❡❧❡❝tr♦s♣✉♥ ✜❜❡r ❢♦r♠✳ ❈❤❛♣t❡r ✽ ❝♦♥❝❧✉❞❡s t❤❡ t❤❡s✐s ❛♥❞ s✉❣❣❡sts s❡✈❡r❛❧ ♣♦ss✐❜❧❡ ❛✈❡♥✉❡s ♦❢ ❢✉t✉r❡ r❡s❡❛r❝❤✳  ✷  ❈❤❛♣t❡r ✷  ◆✉❝❧❡❛r ▼❛❣♥❡t✐❝ ❘❡s♦♥❛♥❝❡  ◆✉❝❧❡❛r ♠❛❣♥❡t✐❝ r❡s♦♥❛♥❝❡ ✭◆▼❘✮ ✐s ❛ ♣❤❡♥♦♠❡♥♦♥ t❤❛t ✐s ❡①❤✐❜✐t❡❞ ❜② ♥✉❝❧❡❛r s♣❡❝✐❡s ✇✐t❤ ❛♥ ✐♥tr✐♥s✐❝ ♥✉❝❧❡❛r ♠❛❣♥❡t✐❝ ♠♦♠❡♥t✳ ❚❤✐s ♦❝❝✉rs ❛s ❛ r❡s✉❧t ♦❢ t❤❡ ♥✉❝❧❡✉s ❝♦♥t❛✐♥✐♥❣ ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ ❜❛r②♦♥s✳ ■t ✐s ♣♦ss✐❜❧❡ t♦ st✉❞② t❤❡s❡ ♥✉❝❧❡✐ ❛♥❞ t❤❡✐r ❝❤❡♠✐❝❛❧ ❡♥✈✐r♦♥♠❡♥t ❜② ❛❧✐❣♥✐♥❣ t❤❡s❡ ♠♦♠❡♥ts ✐♥ ❛ ♠❛❣♥❡t✐❝ ✜❡❧❞ ❛♥❞ ♦❜s❡r✈✐♥❣ t❤❡✐r r❡s♣♦♥s❡ t♦ r❛❞✐♦ ❢r❡q✉❡♥❝② ♣✉❧s❡s✳ ▲❡✈✐tt ❬✶❪ ♣r♦✈✐❞❡s ❛ ❜r♦❛❞✱ ❧❡ss ♠❛t❤❡♠❛t✐❝❛❧❧② ♦r✐❡♥t❡❞ ♦✈❡r✈✐❡✇ ♦❢ ◆▼❘ ❛♥❞ ❝❛♥ ❛❝t ❛s ❛♥ ❡①❝❡❧❧❡♥t ✐♥tr♦❞✉❝t✐♦♥ t♦ t❤❡ ✜❡❧❞ ❢♦r ❛♥ ✉♥❞❡r❣r❛❞✉❛t❡ ✐♥t❡r❡st❡❞ ✐♥ ◆▼❘ ♦r ❢♦r r❡s❡❛r❝❤❡rs ❢r♦♠ ❧❡ss ❛♥❛❧②t✐❝❛❧ ✜❡❧❞s✳ ❆ ♠♦r❡ ♠❛t❤❡♠❛t✐❝❛❧❧② r✐❣♦r♦✉s tr❡❛t♠❡♥t ♦❢ t❤❡ ♠❛t❡r✐❛❧ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❆❜r❛❣❛♠ ❬✷❪ ♦r ❙❧✐❝❤t❡r ❬✸❪✳ ❚❤❡ ❜♦♦❦ ❜② ❉✉❡r ❬✹❪ ♣r♦✈✐❞❡s ❛♥ ❡①❝❡❧❧❡♥t ♠✐❞❞❧❡ ❣r♦✉♥❞ t❤❛t ♠❛♥❛❣❡s t♦ ❝♦✈❡r t❤❡ s✉❜❥❡❝t ✇✐t❤ ♠❛t❤❡♠❛t✐❝❛❧ r✐❣♦r ✇✐t❤♦✉t ❡✈❡r ❧♦s✐♥❣ t❤❡ ♣❡rs♣❡❝t✐✈❡ ♦❢ ❡①♣❡r✐♠❡♥t❛❧ ❛♣♣❧✐❝❛t✐♦♥ ❛♥❞ s❤♦✉❧❞ ❜❡ t❤❡ ✜rst s♦✉r❝❡ ❢♦r ❛♥② ❡①♣❡r✐♠❡♥t❛❧✐st ✇✐t❤ ❛ s♦❧✐❞ ♠❛t❤❡♠❛t✐❝❛❧ ❜❛❝❦❣r♦✉♥❞ ✐♥t❡r❡st❡❞ ✐♥ ◆▼❘✳ ▼✉❝❤ ♦❢ t❤✐s ❝❤❛♣t❡r ✐s ❞r❛✇♥ ❢r♦♠ ✈❛r✐♦✉s tr❡❛t♠❡♥ts ♦❢ t❤❡ s✉❜❥❡❝t ❜② t❤❡s❡ ❛✉t❤♦rs✳  ✷✳✶  ❩❡❡♠❛♥ ■♥t❡r❛❝t✐♦♥  ❚❤❡ st❛rt✐♥❣ ♣♦✐♥t ✐♥ ✉♥❞❡rst❛♥❞✐♥❣ ◆▼❘ ✐s ✉♥❞❡rst❛♥❞✐♥❣ t❤❡ ❡✛❡❝t t❤❛t ❛ st❛t✐❝ ♠❛❣♥❡t✐❝ ✜❡❧❞✱ Bo ❤❛s ♦♥ ❛ ♥✉❝❧❡✉s✳ ❚❤❡ ♥✉❝❧❡✉s ✐♥t❡r❛❝ts ✇✐t❤ ♠❛❣♥❡t✐❝ ✜❡❧❞s ✈✐❛ ✐ts ♠❛❣♥❡t✐❝ ♠♦♠❡♥t✱ ✇❤✐❝❤ ✐s ❣✐✈❡♥ ❛s  µ = γI,  ✸  ✭✷✳✶✮  ❋✐❣✉r❡ ✷✳✶✿ ❚❤❡ ♣r❡❝❡ss✐♦♥ ♦❢ ❛ ♠❛❣♥❡t✐❝ ♠♦♠❡♥t ❛❜♦✉t ❛ st❛t✐❝ ♠❛❣♥❡t✐❝ ✜❡❧❞✳ ✇❤❡r❡ µ ✐s t❤❡ ♠❛❣♥❡t✐❝ ♠♦♠❡♥t ❛♥❞ I ✐s t❤❡ ♥✉❝❧❡❛r s♣✐♥ ❛♥❣✉❧❛r ♠♦♠❡♥t✉♠✳ γ ✐s t❤❡ ❣②r♦♠❛❣♥❡t✐❝ r❛t✐♦ ♦❢ t❤❡ ♥✉❝❧❡✉s ❛♥❞ ✐s ✉♥✐q✉❡ ❢♦r ❡❛❝❤ ♥✉❝❧❡❛r s♣❡❝✐❡s✳ ❆♥ ❡①t❡r♥❛❧ ✜❡❧❞ ❤❛s t❤❡ ❡✛❡❝t ♦❢ ❡①❡rt✐♥❣ ❛ t♦rq✉❡ ♦♥ t❤❡ ♥✉❝❧❡✉s✳ ❚❤❡ t♦rq✉❡ ✐s ❣✐✈❡♥ ❛s t❤❡ t✐♠❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ♥✉❝❧❡❛r s♣✐♥  dI . ✭✷✳✷✮ dt ❲❡ ❝❛♥ ❢✉rt❤❡r ♥♦t❡ ❜② ❡❧❡♠❡♥t❛r② ♣❤②s✐❝s t❤❛t t❤❡ t♦rq✉❡ ✇✐❧❧ ❜❡ t❤❡ ❝r♦ss ♣r♦❞✉❝t ♦❢ t❤❡ ❡①t❡r♥❛❧ ✜❡❧❞ ❛♥❞ t❤❡ ♠❛❣♥❡t✐❝ ♠♦♠❡♥t✱ ✇❤✐❝❤ ❣✐✈❡s T=  T = µ × Bo .  ✭✷✳✸✮  ❆❢t❡r s♦♠❡ r❡❛rr❛♥❣✐♥❣✱ t❤❡ ❛❜♦✈❡ t❤r❡❡ ❡q✉❛t✐♦♥s ❝❛♥ ❜❡ ❝♦♠❜✐♥❡❞ t♦ ❣✐✈❡ t❤❡ ❝❧❛ss✐❝❛❧ ❡q✉❛t✐♦♥ ♦❢ ♠♦t✐♦♥ ❢♦r ❛ ♥✉❝❧❡❛r s♣✐♥❀  dµ = γµ × Bo . dt  ✭✷✳✹✮  ❚❤❡ s♦❧✉t✐♦♥ t♦ t❤✐s ❡q✉❛t✐♦♥ s❤♦✇s t❤❛t t❤❡ ♠❛❣♥❡t✐❝ ♠♦♠❡♥t ♣r❡❝❡ss❡s ❛❜♦✉t t❤❡ ♠❛❣♥❡t✐❝ ✜❡❧❞✱ ❛s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✷✳✶✳ ❚❤❡ r❛t❡ ♦❢ ♣r❡❝❡ss✐♦♥ ✐s ❝❛❧❧❡❞ t❤❡ ▲❛r♠♦r ❢r❡q✉❡♥❝②✱ ❛♥❞ ✐s ❡q✉❛❧ t♦  ✹  ❋✐❣✉r❡ ✷✳✷✿ ❚❤❡ ❡♥❡r❣② ❧❡✈❡❧s ♦❢ ❛  1 2  s♣✐♥ s②st❡♠✳  ✭✷✳✺✮  ω = γB0 .  ❚❤❡ q✉❛♥t✉♠ ♠❡❝❤❛♥✐❝❛❧ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤✐s s②st❡♠ ✐s ❢♦✉♥❞ ❜② ✇r✐t✐♥❣ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ♦❢ t❤❡ s②st❡♠ ✐♥ t❡r♠s ♦❢ q✉❛♥t✉♠ ♠❡❝❤❛♥✐❝❛❧ s♣✐♥ ♦♣❡r❛t♦rs✳ ❋♦r ❡①❛♠♣❧❡✱ I ❜❡❝♦♠❡s Iˆ✳  ˆ Z ✱ ✐s t❤❡r❡❢♦r❡ ❚❤❡ ❍❛♠✐❧t♦♥✐❛♥ ❢♦r t❤❡ ❩❡❡♠❛♥ ✜❡❧❞✱ H ˆ Z = −Bo · µ. ˆ H  ✭✷✳✻✮  ❇② ❝♦♥✈❡♥t✐♦♥✱ t❤❡ st❛t✐❝ ♠❛❣♥❡t✐❝ ✜❡❧❞ ✐s ❛ss✐❣♥❡❞ t♦ t❤❡ ③✲❛①✐s✳ ❚❤✐s ❝♦♥✈❡♥t✐♦♥ ✐s ✐♥t✉✐t✐✈❡ ✇✐t❤ t❤❡ ❢❛❝t t❤❛t ♠♦st ◆▼❘ ♠❛❣♥❡ts ✉s❡ ❛ ✈❡rt✐❝❛❧ ✜❡❧❞✳ ❙✉❜st✐t✉t✐♥❣ ✐♥ t❤❡ r❡❧❛t✐♦♥ ❢♦r t❤❡ ♠❛❣♥❡t✐❝ ♠♦♠❡♥t ✇❡ ❣❡t  ˆ Z = γ Bo Iˆz H  ✭✷✳✼✮  ❚❤❡ ❝❛s❡ ♦❢ t❤❡ ± 12 s♣✐♥ s②st❡♠ ✐s t❤❡ ♠♦st tr✐✈✐❛❧✱ ❛s ✐t ♦♥❧② ❤❛s t✇♦ st❛t❡s✱ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✷✳✷✱ ❛♥❞ ♦♥❡ ♦❢ t❤❡ ♠♦st ❝♦♠♠♦♥ ✐♥ ◆▼❘ ❡①♣❡r✐♠❡♥ts ♦❢ ❜✐♦❧♦❣✐❝❛❧ s②st❡♠s✱ ❛s ✐t ❛♣♣❧✐❡s t♦ ❜♦t❤ 13 ❈ ❛♥❞ ♣r♦t♦♥s✳ ❯s✐♥❣ t❤❡ ♣♦ss✐❜❧❡ s♣✐♥ ✈❛❧✉❡s ♦❢ ± 1 ❢♦r Iˆz ✐♥ ❊q✉❛t✐♦♥ ✭✷✳✼✮ ✇❡ ❣❡t 2  t❤❡ ♣♦ss✐❜❧❡ ❡♥❡r❣✐❡s ♦❢ ❛ ♥✉❝❧❡✉s ✐♥ t❤✐s s②st❡♠✱ ✇❤✐❝❤ ❛r❡❀  1 E±1/2 = ± γ Bo . 2  ✭✷✳✽✮  ■♥ t❤✐s ❝❛s❡✱ t❤❡ ❡♥❡r❣② s❡♣❛r❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ st❛t❡s ❝❛♥ ❜❡ s❤♦✇♥ t♦ ❜❡✿  ∆E = γB0 = − ωo .  ✭✷✳✾✮  ❆s ◆▼❘ ❡①♣❡r✐♠❡♥ts t②♣✐❝❛❧❧② ❞❡❛❧ ✇✐t❤ ❡①tr❡♠❡❧② ❧❛r❣❡ ♥✉♠❜❡rs ♦❢ ♣❛rt✐❝❧❡s✱ ✇❡ ❝❛♥ ❛♣♣❧② st❛t✐st✐❝❛❧ ♠❡❝❤❛♥✐❝s ✐♥ ❛❧❧ ❝❛s❡s r❡❧❡✈❛♥t t♦ t❤✐s ✇♦r❦✳ ❆ ❝♦❧❧❡❝t✐♦♥ ♦❢ ✐♥t❡r❛❝t✐♥❣ ♥✉❝❧❡✐✱ ✺  ❦♥♦✇♥ ❛s ❛ s♣✐♥ ❡♥s❡♠❜❧❡✱ ✇✐t❤ t❤❡ ❩❡❡♠❛♥ ❡♥❡r❣② st❛t❡s ❣✐✈❡♥ ❛❜♦✈❡ ✇✐❧❧ ❤❛✈❡ ❛ t❤❡r♠❛❧ ❡q✉✐❧✐❜r✐✉♠ ♣♦♣✉❧❛t✐♦♥ ❞✐str✐❜✉t✐♦♥ ❣✐✈❡♥ ❜② t❤❡ ❇♦❧t③♠❛♥♥ ❞✐str✐❜✉t✐♦♥❀  p±1/2 =  exp−E±1/2 /kB T , Z  ✭✷✳✶✵✮  ✇❤❡r❡ p± ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛ ♥✉❝❧❡✉s ❜❡✐♥❣ ✐♥ st❛t❡ ✉♣ ✭+✮ ♦r ❞♦✇♥ ✭−✮✱ kB ✐s t❤❡ ❇♦❧t③♠❛♥♥ ❝♦♥st❛♥t✱ ❚ ✐s t❤❡ t❡♠♣❡r❛t✉r❡ ❛♥❞ ❩ ✐s t❤❡ ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜② t❤❡ ❡q✉❛t✐♦♥❀  exp−Em /kB T .  Z=  ✭✷✳✶✶✮  m  ✷✳✷  ❚❤❡ ◆▼❘ ❊①♣❡r✐♠❡♥t  ❚❤❡ t②♣✐❝❛❧ ◆▼❘ ❡①♣❡r✐♠❡♥t ♦❜s❡r✈❡s t❤❡ r❡s♣♦♥s❡ ♦❢ ❛ s②st❡♠ ♦❢ ♥✉❝❧❡✐ ❛❢t❡r ✐t ✐s s✉❜❥❡❝t❡❞ t♦ ❛ r❛❞✐♦ ❢r❡q✉❡♥❝② ✭r❢✮ ♠❛❣♥❡t✐❝ ♣✉❧s❡✳ ❚❤❡ ♣✉❧s❡ ✐s ❝r❡❛t❡❞ ✐♥ ❛ ❝♦✐❧ s✉rr♦✉♥❞✐♥❣ t❤❡ s❛♠♣❧❡ ❜② ❤❛✈✐♥❣ ❛♥ ❆❈ ❝✉rr❡♥t ❛♣♣❧✐❡❞ t♦ ✐t✳ ❋♦r t❤❡r❡ t♦ ❜❡ ❛ ♥♦t✐❝❡❛❜❧❡ ❡✛❡❝t ❢r♦♠ t❤❡ r❢ ♣✉❧s❡✱ t❤❡ ❢r❡q✉❡♥❝② ♦❢ t❤❡ ♣✉❧s❡ ♠✉st ❜❡ ♦♥ ♦r ❝❧♦s❡ t♦ r❡s♦♥❛♥❝❡❀ t❤❡ ▲❛r♠♦r ❢r❡q✉❡♥❝②✳ ❚❤✐s r❡q✉✐r❡♠❡♥t ✐s ❡①tr❡♠❡❧② ❢♦rt✉♥❛t❡✱ ❛s ✐t ❛❧❧♦✇s ❢♦r ❡❛s② ✐s♦❧❛t✐♦♥ ♦❢ ❞✐✛❡r❡♥t s♣❡❝✐❡s ♦❢ ♥✉❝❧❡✐✳ ❚❤❡ ❝❧❛ss✐❝❛❧ ❢♦r♠✉❧❛ ❢♦r s♣✐♥s ✐♥ ♠❛❣♥❡t✐❝ ✜❡❧❞s ❝❛♥ ❜❡ ❛❧t❡r❡❞ ✐♥t♦ ❛ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ t♦t❛❧ ♠❛❣♥❡t✐③❛t✐♦♥ ♦❢ ❛ s❛♠♣❧❡✿  dM = γM × B. ✭✷✳✶✷✮ dt ❚❤✐s ✐s ❦♥♦✇♥ ❛s t❤❡ ❇❧♦❝❤ ❡q✉❛t✐♦♥✱ ❛♥❞ ✐s t❤❡ ❝❧❛ss✐❝❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ♥✉❝❧❡❛r s♣✐♥ s②st❡♠✳ ❚❤✐s ❡q✉❛t✐♦♥✱ ✇❤✐❝❤ ✐s ✐♥ ❢❛❝t ❛ s❡t ♦❢ t❤r❡❡ ❡q✉❛t✐♦♥s✱ ❞❡s❝r✐❜✐♥❣ t❤❡ t✐♠❡ ❞❡♣❡♥❞❡♥t ❝❤❛♥❣❡ ♦❢ t❤❡ ♠❛❣♥❡t✐③❛t✐♦♥ ❛❧♦♥❣ t❤❡ ❳✱ ❨✱ ❛♥❞ ❩ ❛①❡s✱ ♣r♦✈✐❞❡s ❛ s✐♠♣❧❡✱ ✐♥t✉✐t✐✈❡✱ ❛♥❞ ❝❧❛ss✐❝❛❧ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ♠❛❣♥❡t✐③❛t✐♦♥ s✐❞❡st❡♣♣✐♥❣ ❝♦♠♣❧❡t❡❧② ❛♥② ✐♥✈♦❧✈❡♠❡♥t ❢r♦♠ q✉❛♥t✉♠ ♠❡❝❤❛♥✐❝s✳ ❚❤✐s ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❇❧♦❝❤ ❡q✉❛t✐♦♥ ♥❡❣❧❡❝ts r❡❧❛①❛t✐♦♥✳ ❆❢t❡r t❤❡ ✐♥✐t✐❛❧ ❘❋ ♣✉❧s❡✱ t❤❡ ♦s❝✐❧❧❛t✐♥❣ ♠❛❣♥❡t✐③❛t✐♦♥ ✐♥❞✉❝❡s ❛ ✈♦❧t❛❣❡ ✐♥ t❤❡ ❘❋ ❝♦✐❧✳ ❚❤✐s ✈♦❧t❛❣❡ ♣r♦❞✉❝❡s ❛ s✐❣♥❛❧ ❦♥♦✇♥ ❛s t❤❡ ❢r❡❡ ✐♥❞✉❝t✐♦♥ ❞❡❝❛② ✭❋■❉✮ ♦❢ t❤❡ ♠❛❣♥❡t✐③❛t✐♦♥ ♦❢ t❤❡ s②st❡♠✳ ❆♥ ❡①❛♠♣❧❡ ♦❢ ❛♥ ❋■❉ ❝❛♥ ❜❡ s❡❡♥ ✐♥ ❋✐❣✉r❡ ✷✳✸ ❚❤❡s❡ s✐❣♥❛❧s ❛r❡ ♠✐①❡❞ ❞♦✇♥ ❜② ❛ q✉❛❞r❛t✉r❡ r❡❝❡✐✈❡r ✇✐t❤ t❤❡ r❢✲♣✉❧s❡ ❢r❡q✉❡♥❝② ωrf ✳ ❚❤✐s ♠✐①✐♥❣ ❞♦✇♥ ♦❢ t❤❡ ❢r❡q✉❡♥❝② r❡❞✉❝❡s t❤❡ ❞❡♠❛♥❞ ♦♥ t❤❡ ❡❧❡❝tr♦♥✐❝s ❛♥❞ t❤❡ ❛♠♦✉♥t ♦❢  ✻  ❋✐❣✉r❡ ✷✳✸✿ ❆ ❢r❡❡ ✐♥❞✉❝t✐♦♥ ❞❡❝❛② ♦❢ ❛ s❛♠♣❧❡ ✐♥ ❛ ♣♦♦r❧② s❤✐♠♠❡❞ ♠❛❣♥❡t ❝♦✉rt❡s② ♦❢ ●②r♦▼❛❣✐❝✐❛♥ ❢r♦♠ t❤❡ ❲✐❦✐♣❡❞✐❛ ❝♦♠♠♦♥s✳ ❞❛t❛ ♥❡❡❞❡❞✳ ❚❤❡ r❛♥❣❡ ♦❢ ❢r❡q✉❡♥❝✐❡s ♦❜s❡r✈❡❞ ✐♥ t❤❡ s❛♠♣❧❡ ❛r❡ ❞❡♣❡♥❞❡♥t ♦♥ ✈❛r✐♦✉s ❡♥✈✐r♦♥♠❡♥t❛❧ ❢❛❝t♦rs ♦❢ t❤❡ ♥✉❝❧❡❛r s♣✐♥s✱ s✉❝❤ ❛s ❡❧❡❝tr♦♥✐❝ str✉❝t✉r❡ ❛♥❞ s♣✐♥ ❝♦✉♣❧✐♥❣s ✭t❤❡s❡ ✇✐❧❧ ❜❡ ❞❡t❛✐❧❡❞ ✐♥ ❛ ❧❛t❡r s❡❝t✐♦♥✮✳ ❚❤❡ ❢r❡q✉❡♥❝② r❛♥❣❡ ♦❢ Ω0 = ω0 − ωrf ✐s t②♣✐❝❛❧❧② ❛ ❢❡✇ ❤✉♥❞r❡❞ ❦❍③ ❛t ♠♦st✳ ❆❢t❡r q✉❛❞r❛t✉r❡ ❞❡t❡❝t✐♦♥✱ t❤❡ ❋■❉ ❝♦♥s✐sts ♦❢ t✇♦ s✐❣♥❛❧s ♦❢ t❤❡ ❢♦r♠ j  ✭✷✳✶✸✮  cos(Ωj t) exp−t/T2  sR (T ) ∝ j  j  ✭✷✳✶✹✮  sin(Ωj t) exp−t/T2  sI (T ) ∝ j  ❚❤❡s❡ s✐❣♥❛❧s ❛r❡ s✉♣❡r♣♦s✐t✐♦♥s ♦❢ s✐❣♥❛❧s ✇✐t❤ ✈❛r✐♦✉s ❢r❡q✉❡♥❝✐❡s r❡♣r❡s❡♥t❡❞ ❜② Ωj ✳ ❚❤❡s❡ s✐❣♥❛❧s ❝❛♥ ❜❡ ❝♦♠❜✐♥❡❞ ✐♥t♦ ❛ s✐♥❣❧❡ ❝♦♠♣❧❡① t✐♠❡ ❞❡♣❡♥❞❡♥t ❢✉♥❝t✐♦♥ s(t)✱ ✇✐t❤ sR r❡♣r❡✲ s❡♥t✐♥❣ t❤❡ r❡❛❧ ❝♦♠♣♦♥❡♥t✱ ❛♥❞ sI r❡♣r❡s❡♥t✐♥❣ t❤❡ ✐♠❛❣✐♥❛r② ❝♦♠♣♦♥❡♥t✶ ✳ ❚❤✐s ②✐❡❧❞s t❤❡ ❢✉♥❝t✐♦♥ s(t) = sR (t)+isI (t)✳ ❉♦✐♥❣ t❤✐s ❞✐st✐♥❣✉✐s❤❡s t❤❡ ❢r❡q✉❡♥❝✐❡s t❤❛t ❛r❡ ❢❛st❡r t❤❛♥ t❤❡ tr❛♥s♠✐tt❡r ❢r❡q✉❡♥❝② ❢r♦♠ t❤♦s❡ s❧♦✇❡r t❤❛♥ t❤❡ tr❛♥s♠✐tt❡r ❢r❡q✉❡♥❝②✳ ❚❤❡ t✐♠❡ ❞♦♠❛✐♥ s✐❣♥❛❧ ❝❛♥ ❜❡ tr❛♥s❢❡rr❡❞ t♦ ❛ ❢r❡q✉❡♥❝② s♣❛❝❡ ❜② ❝♦♥❞✉❝t✐♥❣ ❛ ❋♦✉r✐❡r tr❛♥s❢♦r♠❛t✐♦♥  ˆ  ∞  ✭✷✳✶✺✮  s(t) exp−iΩt dt.  S(Ω) = 0  ❚❤✐s ②✐❡❧❞s  1 T2j  S(Ω) = j ✶ ❆❧t❤♦✉❣❤  (1/T2j )2  + (Ω − Ωj  )2  −i  t❤✐s ✐s ♠♦r❡ ♦❢ ❝♦♥✈❡♥t✐♦♥✱ t❤❡r❡ ✐s ♥♦ r❡❛s♦♥ t❤❛t  ♦t❤❡r t❤❛♥ t❤❡ ♥♦t❛t✐♦♥ ❝♦✉❧❞ ❣❡t ❝♦♥❢✉s✐♥❣  ✼  Ω − Ωj + (Ω − Ωj )2  (1/T2j )2 sR  ✭✷✳✶✻✮  ❝♦✉❧❞ ♥♦t ❜❡ t❤❡ ✐♠❛❣✐♥❛r② ❝♦♠♣♦♥❡♥t  ❚❤❡ r❡❛❧ ❝♦♠♣♦♥❡♥t ✐s t❤❡ ❛❜s♦r♣t✐♦♥ ❛♥❞ t❤❡ ✐♠❛❣✐♥❛r② ✐s t❤❡ ❞✐s♣❡rs✐♦♥ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ▲♦r❡♥t③✐❛♥ ♣❡❛❦✳ ❚❤❡ r❛t❡ ♦❢ t❤❡ ❞❡❝❛② ❛✛❡❝ts t❤❡ ♣❡❛❦ ✇✐❞t❤✱ ✇❤❡r❡ s❧♦✇❡r ❞❡❝❛② s❤❛r♣❡♥s t❤❡ ♣❡❛❦✳  ✷✳✸  ❈❧❛ss✐❝❛❧ ❘❡❧❛①❛t✐♦♥  ❘❡❧❛①❛t✐♦♥ ✐s t❤❡ ♣r♦❝❡ss ✐♥ ✇❤✐❝❤ ❛♥ ❡①❝✐t❡❞ s②st❡♠ ♠♦✈❡s ❜❛❝❦ t♦ t❤❡r♠❛❧ ❡q✉✐❧✐❜r✐✉♠ ♦r ❛ st❛❜❧❡ ❧♦❝❛❧ ♠✐♥✐♠❛ ❡♥❡r❣② ❝♦♥✜❣✉r❛t✐♦♥✳ ■♥ ◆▼❘✱ t❤❡ s②st❡♠ r❡❢❡rr❡❞ t♦ ✐♥ r❡❧❛①❛t✐♦♥ ✐s t❤❡ ♥❡t ♠❛❣♥❡t✐③❛t✐♦♥✳ Pr❛❝t✐❝❛❧❧② s♣❡❛❦✐♥❣✱ r❡❧❛①❛t✐♦♥ ✐s ♠❡❛s✉r❡❞ ❜② t❤❡ t✐♠❡ t❛❦❡♥ ❢♦r t❤❡ ♠❛❣♥❡t✐③❛t✐♦♥ t♦ r❡✈❡rt t♦ t❤❡r♠❛❧ ❡q✉✐❧✐❜r✐✉♠ ❛❢t❡r ❛ r❡s♦♥❛♥t ❘❋ ♣✉❧s❡✳ ❚❤❡r❡ ❛r❡ t✇♦ ♣❛r❛♠❡t❡rs t❤❛t ❝❤❛r❛❝t❡r✐③❡ t❤❡ t✐♠❡s❝❛❧❡s ❢♦r ♥✉❝❧❡❛r ❡①❝✐t❛t✐♦♥ t♦ r❡❧❛① ❜❛❝❦ ✐♥t♦ t❤❡r♠❛❧ ❡q✉✐❧✐❜r✐✉♠✳ ❚❤❡ ✜rst ✐s t❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ r❡❧❛①❛t✐♦♥✱ T1 ✳ ❚❤❡ s❡❝♦♥❞ ✐s t❤❡ tr❛♥s✈❡rs❡ r❡❧❛①❛t✐♦♥✱ T2 ✳ ❚❤❡ r❡❧❛①❛t✐♦♥ ♦❢ ♠❛❣♥❡t✐③❛t✐♦♥ ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❝❧❛ss✐❝❛❧❧② ❜② ❛❞❞✐♥❣ r❡❧❛①❛t✐♦♥ t❡r♠s t♦ t❤❡ ❇❧♦❝❤ ❊q✉❛t✐♦♥s✱ ✇❤✐❝❤ ❝❛♥ t❤❡♥ ❜❡ ❡①♣r❡ss❡❞ ❛s❀ −t  Mx (t) = Mx (0) cos(ωo t)e T2  ✭✷✳✶✼✮  −t T2  ✭✷✳✶✽✮  My (t) = Mx (0) sin(ωo t)e  −t T1  Mz (t) = M0 + (Mz (0) − Mo )e ,  ✭✷✳✶✾✮  ✇❤❡r❡ M0 ✐s t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♠❛❣♥❡t✐③❛t✐♦♥✳ ❯s✐♥❣ t❤✐s✱ t❤❡ ❇❧♦❝❤ ❊q✉❛t✐♦♥s ❜❡❝♦♠❡❀  dM = γM × B − R(M − M0 ), dt ✇❤❡r❡  R=  1 T2  0  0  0  1 T2  0  0  0  1 T1  .  ✭✷✳✷✵✮  ✭✷✳✷✶✮  ❚❤❡s❡ r❡❧❛①❛t✐♦♥ t✐♠❡s T1 ❛♥❞ T2 ❛r❡ ❧✐♥❦❡❞ t♦ ❞❡❡♣❡r q✉❛♥t✉♠ ♠❡❝❤❛♥✐❝❛❧ ♣r♦❝❡ss❡s t❤❛t ❛r❡ ❞✐s❝✉ss❡❞ ✐♥ ❛ ❧❛t❡r s❡❝t✐♦♥✳  ✷✳✹  ❉❡♥s✐t② ▼❛tr✐① ❘❡♣r❡s❡♥t❛t✐♦♥  ❚❤❡ ❞❡♥s✐t② ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ✐s ❛ s✐♠♣❧❡ ✇❛② t♦ r❡♣r❡s❡♥t t❤❡ st❛t❡ ♦❢ ❛ s❛♠♣❧❡ t❤❛t ✐s ♦❢t❡♥ ♠♦r❡ ❝♦♥✈❡♥✐❡♥t t❤❛♥ ✉s✐♥❣ ❛♥ ❡♥s❡♠❜❧❡ ♦❢ s✐♥❣❧❡ q✉❛♥t✉♠ st❛t❡s✳ ❘❡♣r❡s❡♥t✐♥❣ ✽  ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ s♣✐♥s✱ ✇❤❡r❡ ❡❛❝❤ ❤❛s ❛ ♣r♦❜❛❜✐❧✐t② ♦❢ ❜❡✐♥❣ ✐♥ st❛t❡ ψ ❜② ❛ s✉♣❡r♣♦s✐t✐♦♥ st❛t❡ Ψ r❡q✉✐r❡s ♦♥❡ t♦ ❛❝❝♦✉♥t ❢♦r ❡❛❝❤ s♣✐♥ ✐♥❞✐✈✐❞✉❛❧❧②✳ ❚❤✐s ❝❛♥ q✉✐❝❦❧② ❞❡❣❡♥❡r❛t❡ ✐♥t♦ ❡①tr❡♠❡❧② ♠❡ss② s✉♠♠❛t✐♦♥ ♠❛♥✐♣✉❧❛t✐♦♥s✳ ❚❤❡ ❡①♣❡❝t❛t✐♦♥ ✈❛❧✉❡ ❢♦r s♦♠❡ ♦♣❡r❛t♦r Aˆ ✐♥ t❤❡ s♣✐♥ ❡♥s❡♠❜❧❡ ♠✉st ❜❡ s✉♠♠❡❞ ♦✈❡r ❛❧❧ ♣♦ss✐❜❧❡ st❛t❡s✿  ˆ >= < Aˆ >=< Ψ|A|Ψ  ˆ > pψ < ψ|A|ψ  ✭✷✳✷✷✮  ψ  ❘❡♣r❡s❡♥t✐♥❣ t❤❡ s♣✐♥ st❛t❡s ✐♥ t❡r♠s ♦❢ ❜❛s✐s st❛t❡s s✐♠♣❧✐✜❡s t❤❡ ❛❜♦✈❡ s♣✐♥ ❡♥s❡♠❜❧❡ ✇✐t❤ ♣♦ss✐❜❧❡ st❛t❡s  ψ=  ✭✷✳✷✸✮  cψi φi i  ✇❤❡r❡ φi ❛r❡ t❤❡ ❜❛s✐s st❛t❡s✳ ❚❤✐s ❡①♣❛♥❞s t❤❡ ❡①♣❡❝t❛t✐♦♥ ✈❛❧✉❡ t♦✿  < Aˆ >=  ˆ j> c∗ψi cψj < φi |A|φ  pψ  ✭✷✳✷✹✮  i,j  ψ  ❚❤✐s ❛♣♣r♦❛❝❤ ✐s ❛❞✈❛♥t❛❣❡♦✉s ❜❡❝❛✉s❡ t❤❡ ♠❛tr✐① ❡❧❡♠❡♥ts ♦❢ Aˆ ✐♥ t❤✐s ❜❛s✐s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t  pψ c∗ψi cψj ❛s t❤❡ ❝♦♠♣♦♥❡♥ts ρji ♦❢ ❛♥♦t❤❡r ♠❛tr✐①✱ ✇❡ ❝❛♥ s❡❡ t❤❛t t❤❡ ❡①♣❡❝t❛t✐♦♥ ✈❛❧✉❡ ♦❢ Aˆ ❜❡❝♦♠❡s  ♦❢ t❤❡ s♣❡❝✐✜❝ st❛t❡ ♦❢ Ψ✳ ■❢ ✇❡ ❞❡✜♥❡ t❤❡ st❛t❡ ❞❡♣❡♥❞❡♥t ❝♦❡✣❝✐❡♥ts  < Aˆ >=  ρji Aij  ψ  ✭✷✳✷✺✮  i,j  ✇❤❡r❡ Aij ❛r❡ t❤❡ ♠❛tr✐① ❡❧❡♠❡♥ts ♦❢ t❤❡ ♦♣❡r❛t♦r Aˆ✳ ρji ❛r❡ t❤❡ ♠❛tr✐① ❡❧❡♠❡♥ts ♦❢ ✇❤❛t ✐s ❝❛❧❧❡❞ t❤❡ ❞❡♥s✐t② ♠❛tr✐①✱ ρ✳ ❚❤❡ ❞❡♥s✐t② ♠❛tr✐① ❝♦rr❡s♣♦♥❞s t♦ ❛♥ ♦♣❡r❛t♦r ✇❤♦✬s ❡❧❡♠❡♥ts ❝❛♥ ❜❡ s❤♦✇♥ t♦ ❜❡  pi |φi >< φi |.  ρˆ =  ✭✷✳✷✻✮  i  ❆t t❤❡r♠❛❧ ❡q✉✐❧✐❜r✐✉♠✱ t❤❡ ❞❡♥s✐t② ♠❛tr✐① t❛❦❡s t❤❡ ❢♦r♠  ρˆ =  1 ˆ exp−H/kT Z  ✭✷✳✷✼✮  ✇❤❡r❡ t❤❡ ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥ ✐s ˆ  Z = T r(exp−H/kT )  ✭✷✳✷✽✮  ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ ✉s❡ t❤✐s t♦ ✜♥❞ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ t❤❡r♠❛❧ ❡q✉✐❧✐❜r✐✉♠ ❞❡♥s✐t② ♠❛tr✐① ❢♦r t❤❡ ❩❡❡♠❛♥ ✐♥t❡r❛❝t✐♦♥✳ ❘❡❝❛❧❧✐♥❣ t❤❛t t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ❢♦r t❤✐s ✐♥t❡r❛❝t✐♦♥ ✐s  ˆ = −γ Iˆz B0 = ω0 Iˆz H ✾  ✭✷✳✷✾✮  ❲❡ ❝❛♥ t❤❡♥ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❞❡♥s✐t② ♠❛tr✐① t♦ ✜rst ♦r❞❡r ❛s❀  ρˆeq =  1 Z  1+  ω0 ˆ Iz , kT  ✭✷✳✸✵✮  ✇❤✐❝❤ ❝❛♥ ❜❡ tr❡❛t❡❞ ❛s❀  1 ω0 ˆ Iz , Z kT ❜❡❝❛✉s❡ t❤❡ ✶ ❞♦❡s ♥♦t ❧❡❛❞ t♦ ❛♥② ♦❜s❡r✈❛❜❧❡s✳ ❚❤✐s ✐s ❛ ✈❛❧✐❞ ❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r t❤❡ ✜❡❧❞ str❡♥❣t❤s ❛♥❞ t❡♠♣❡r❛t✉r❡s ✉s❡❞ ✐♥ t②♣✐❝❛❧ ◆▼❘ ❡①♣❡r✐♠❡♥ts✳ ❚❤❡ ❡q✉❛t✐♦♥ ❞❡s❝r✐❜✐♥❣ t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ s②st❡♠ ✐s t❤❡ ▲✐♦✉✈✐❧❧❡✲✈♦♥ ◆❡✉♠❛♥♥ ❡q✉❛t✐♦♥✱ ✇❤✐❝❤ ✐s ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ ❙❝❤r♦❞✐♥❣❡r ❡q✉❛t✐♦♥ ❛♥❞ ✐s✿ dˆ ρ ˆ ρˆ]. = −i[H, ✭✷✳✸✶✮ dt ❚❤❡ s♦❧✉t✐♦♥ t♦ t❤✐s ✐s❀ ρˆ(t) = Uˆ (t, 0)ˆ ρ(0)Uˆ † (t, 0), ✭✷✳✸✷✮ ρˆeq =  ˆ (t, 0) ✐s t❤❡ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♦♣❡r❛t♦r✳ ❚❤❡ t✐♠❡ ✇❤❡r❡ ρˆ(0) ✐s t❤❡ ✐♥✐t✐❛❧ ❞❡♥s✐t② ♦♣❡r❛t♦r ❛♥❞ U ❡✈♦❧✉t✐♦♥ ♦♣❡r❛t♦r ✐s ❛❧s♦ ❝❛❧❧❡❞ t❤❡ ♣r♦♣❛❣❛t♦r ❛♥❞ ✐s ❞❡✜♥❡❞ ❢♦r t❤❡ ❣❡♥❡r❛❧ t✐♠❡✲❞❡♣❡♥❞❡♥t ❍❛♠✐❧t♦♥✐❛♥s ❛s❀  ˆ exp− i Uˆ (t, 0) = µT  ´t 0  ˆ )dt H(t  ,  ✭✷✳✸✸✮  ✇❤❡r❡ Tˆ ✐s t❤❡ ❉②s♦♥ t✐♠❡ ♦r❞❡r✐♥❣ ♦♣❡r❛t♦r✳  ✷✳✺  ❘♦t❛t✐♥❣ ❋r❛♠❡ ❚r❛♥s❢♦r♠❛t✐♦♥  ❚❤❡ ✐♥✈♦❧✈❡♠❡♥t ♦❢ ♦s❝✐❧❧❛t✐♥❣ ❝♦♠♣♦♥❡♥ts ♦❢ s♣✐♥ ♦♣❡r❛t♦rs ✐♥ st❛t✐❝ ♠❛❣♥❡t✐❝ ✜❡❧❞s ♠❡❛♥ t❤❛t ◆▼❘ ❡①♣❡r✐♠❡♥ts ❛r❡ ❛❧♠♦st ❛❧✇❛②s s✐♠♣❧✐✜❡❞ ❜② ✇♦r❦✐♥❣ ✇✐t❤✐♥ t❤❡ r♦t❛t✐♥❣ ❢r❛♠❡✳ ❚❤❡ r♦t❛t✐♥❣ ❢r❛♠❡ r♦t❛t❡s ❛t t❤❡ ❛♣♣❧✐❡❞ r❢✲❢r❡q✉❡♥❝② ❛❜♦✉t t❤❡ ③✲❛①✐s ♦❢ t❤❡ ❧❛❜ ❢r❛♠❡ ❛s ❞❡✜♥❡❞ t♦ ❜❡ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ♠❛❣♥❡t✐❝ ✜❡❧❞✳ ❚❤❡ ❢r❛♠❡ tr❛♥s❢♦r♠❛t✐♦♥ ✐s ❛ ❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ❧❛❜ ❢r❛♠❡ ✉s✐♥❣ t❤❡ r♦t❛t✐♦♥ ♦♣❡r❛t♦r  ˆ = exp−iφIˆz R  ✶✵  ✭✷✳✸✹✮  ❋✐❣✉r❡ ✷✳✹✿ ❚❤❡ r♦t❛t✐♥❣ ❢r❛♠❡ (x , y , z )✿ ❆ ❢r❛♠❡ t❛❦❡♥ t♦ ♣r❡❝❡ss ❛r♦✉♥❞ t❤❡ st❛t✐❝ ✜❡❧❞✱ B0 ❛t ✇✐t❤ ❛ ❢r❡q✉❡♥❝② ♦❢ ωrf ✳ ■♥ t❤❡ r♦t❛t✐♥❣ ❢r❛♠❡ t❤❡ s♣✐♥s ❡✈♦❧✈❡ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❡✛❡❝t✐✈❡ ✜❡❧❞ ✇❤✐❝❤ ✐♥❝❧✉❞❡s ❛ −ωrf zˆ t❡r♠ ❛r✐s✐♥❣ ❢r♦♠ t❤❡ ❢r❛♠❡ tr❛♥s❢♦r♠❛t✐♦♥✳ ✇❤✐❝❤ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❣❡♥❡r❛t❡ ❛ r♦t❛t✐♥❣ ❢r❛♠❡ ❍❛♠✐❧t♦♥✐❛♥ ✇❤✐❝❤ t❛❦❡s t❤❡ ❢♦r♠❀  ˆR = R ˆ −1 H ˆR ˆ − ωrf Iˆz . H  ✭✷✳✸✺✮  ■♥ ❛♥ ◆▼❘ ❡①♣❡r✐♠❡♥t✱ t❤❡ ♣✉❧s❡s ❛r❡ ❛♣♣❧✐❡❞ ♦♥ ♦r ♥❡❛r r❡s♦♥❛♥❝❡✱ ♠❡❛♥✐♥❣ t❤❛t ωrf ≈ ωo ✳ ❚❤❡ ❢r❛♠❡ r♦t❛t❡s ❛t t❤❡ ▲❛r♠♦r ❢r❡q✉❡♥❝②✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t ❢r♦♠ t❤✐s ♣❡rs♣❡❝t✐✈❡ t❤❡ s♣✐♥ ✈❡❝t♦r ❞♦❡s ♥♦t r♦t❛t❡ ❛r♦✉♥❞ t❤❡ Z ❛①✐s✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ t❡r♠s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❩❡❡♠❛♥ t❡r♠s ❛r❡ ❡✛❡❝t✐✈❡❧② ❣r❡❛t❧② r❡❞✉❝❡❞✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ❡✛❡❝t✐✈❡ ✜❡❧❞ Bef f ✐s t❤❛t ♦❢ t❤❡ ♣✉❧s❡ B1 ✐♥ t❤❡ r♦t❛t✐♥❣ ❢r❛♠❡✱ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✷✳✹✳  ✷✳✻  ❆❞✈❛♥❝❡❞ ◆▼❘ ■♥t❡r❛❝t✐♦♥s  ❚❤❡r❡ ❛r❡ ❛ ✇✐❞❡ ✈❛r✐❡t② ♦❢ ✐♥t❡r❛❝t✐♦♥s t❤❛t ❛✛❡❝t ❛♥ ◆▼❘ ❛❝t✐✈❡ s❛♠♣❧❡✳ ❚❤❡s❡ ✐♥t❡r❛❝t✐♦♥s ❛r❡ ❞r✐✈❡♥ ❜② ✈❛r✐♦✉s ❡❧❡❝tr♦♠❛❣♥❡t✐❝ ✜❡❧❞s t❤❛t t❤❡ ✈❛r✐♦✉s s❛♠♣❧❡ ♥✉❝❧❡✐ ❛r❡ ❡①♣♦s❡❞ t♦✳ ❚❤❡ s✐♠♣❧❡st ♦♥❡ ✐s t❤❡ ✐♥t❡r❛❝t✐♦♥ ✇✐t❤ t❤❡ ❩❡❡♠❛♥ ✜❡❧❞✳ ▼♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ♦♥❡s ❛r❡ ❛ r❡s✉❧t ♦❢ s❤✐❡❧❞✐♥❣ ❢r♦♠ t❤❡ ❡❧❡❝tr♦♥ ❝❧♦✉❞s ❛♥❞ ✈❛r✐♦✉s ♥✉❝❧❡❛r ♠❛❣♥❡t✐❝ ❝♦✉♣❧✐♥❣s✳ ❚✇♦ ♦❢ t❤❡s❡ ✐♥t❡r❛❝t✐♦♥s ❛r❡ r❡❧❡✈❛♥t ❢♦r ❝♦♥t❡①t ♦❢ t❤✐s ✇♦r❦✳  ✶✶  ✷✳✻✳✶  ❈❤❡♠✐❝❛❧ ❙❤✐❢t  ❲❤❡♥ ❛ ♠❛❣♥❡t✐❝ ✜❡❧❞ ✐s ❛♣♣❧✐❡❞ t♦ ❛♥ ❛t♦♠✱ t❤❡ ❡❧❡❝tr♦♥✐❝ str✉❝t✉r❡ ❞♦❡s ♥♦t s✐t ♦♥ t❤❡ s✐❞❡❧✐♥❡s✳ ❚❤✐s ✜❡❧❞ ❤❛s ❛ ♣✉rt❛❜✐t♦r② ❡✛❡❝t t❤❛t ❛❧t❡rs ❡✈❡r s♦ s❧✐❣❤t❧② t❤❡ ❡✛❡❝t✐✈❡ ♠❛❣♥❡t✐❝ ✜❡❧❞ ❛t t❤❡ ♥✉❝❧❡✉s✳ ❚❤❡ str❡♥❣t❤ ♦❢ t❤❡ ❝❤❛♥❣❡ ✐♥ t❤❡ ✜❡❧❞ ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❝❤❡♠✐❝❛❧ ❡♥✈✐r♦♥♠❡♥t ♦❢ t❤❡ ❛t♦♠✱ s♣❡❝✐✜❝❛❧❧② t❤❡ t❤❡ str❡♥❣t❤✱ ❧♦❝❛t✐♦♥✱ ❞✐st❛♥❝❡✱ ❛♥❞ t②♣❡ ♦❢ ❜♦♥❞s ✐t s❤❛r❡s ✇✐t❤ ✐ts ♥❡✐❣❤❜♦rs✳ ❚❤✐s ♣❤❡♥♦♠❡♥♦♥ ✐s ❦♥♦✇♥ ❛s ❝❤❡♠✐❝❛❧ s❤✐❡❧❞✐♥❣✳ ❚❤❡ ❝❤❡♠✐❝❛❧ s❤✐❡❧❞✐♥❣ ♦❢ ❛ ♥✉❝❧❡✉s ❝❤❛♥❣❡s t❤❡ ❧♦❝❛❧ ♠❛❣♥❡t✐❝ ✜❡❧❞✳ ❚❤✐s ✐♥ t✉r♥ ❝❤❛♥❣❡s t❤❡ ❢r❡q✉❡♥❝② ♦❢ t❤❡ ♥✉❝❧❡❛r ♣r❡❝❡ss✐♦♥✳ ❚❤✐s ♣❤❡♥♦♠❡♥♦♥ ✐s ❦♥♦✇♥ ❛s ❝❤❡♠✐❝❛❧ s❤✐❢t ❛♥❞ t❤❡ st✉❞② ♦❢ ✐t ✐s ♦♥❡ ♦❢ t❤❡ ♣r✐♠❛r② ♠❡t❤♦❞s ♦❢ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ✐♥ t❤✐s t❤❡s✐s✳ ❚❤❡ ❝❤❡♠✐❝❛❧ s❤✐❡❧❞✐♥❣ ✐s ❞❡s❝r✐❜❡❞ ❜② t❤❡ ❝❤❡♠✐❝❛❧ s❤✐❡❧❞✐♥❣ t❡♥s♦r✱ σ cs ✭t♦ ❜❡ r❡❢❡rr❡❞ t♦ ❛s σ ❢♦r t❤✐s s❡❝t✐♦♥ ♦♥❧②✮✳ ❚❤❡ ♥♦♥ ❝♦♦r❞✐♥❛t❡ s♣❡❝✐✜❝ ❡❧❡♠❡♥ts ♦❢ t❤✐s s❡❝♦♥❞ r❛♥❦ t❡♥s♦r ❞❡s❝r✐❜❡ t❤❡ t❤r❡❡ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ♠❛❣♥❡t✐❝ ✜❡❧❞ ❣❡♥❡r❛t❡❞ ✇❤❡♥ ❛ ♠❛❣♥❡t✐❝ ✜❡❧❞ ✐s ❛♣♣❧✐❡❞✳ ❚❤❡ ❍❛♠✐❧t♦♥✐❛♥ ✇✐❧❧ ❜❡✿  ˆ =H ˆo + H ˆ cs = γ(ˆI · Bo − ˆI · σ · Bo ). H  ✭✷✳✸✻✮  ■♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ✜❡❧❞ ❛❧✐❣♥❡❞ ❛❧♦♥❣ ③✱ t❤❡ r❡s✉❧t✐♥❣ ✜❡❧❞ ✇✐❧❧ ❜❡ ❣✐✈❡♥ ❜② ❊q✉❛t✐♦♥ ✭✷✳✸✼✮✳ ❆s ✇✐❧❧ ❜❡ s❤♦✇♥✱ t❤❡ ① ❛♥❞ ② ❝♦♠♣♦♥❡♥ts ❝❛♥ ❜❡ ✐❣♥♦r❡❞ ✐❢ σ  1✳ ❚❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❝❛❧❧❡❞ t❤❡ s❡❝✉❧❛r ❛♣♣r♦①✐♠❛t✐♦♥✱ ❛♥❞ ❝❛♥ ❜❡ s❤♦✇♥ ✇✐t❤ t❤❡ ✉s❡ ♦❢ ❛✈❡r❛❣❡ ❍❛♠✐❧t♦♥✐❛♥ t❤❡♦r②✳ ❚❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ ❝❛♥ ❛❧s♦ ❜❡ s❡❡♥ t♦ ❜❡ tr✉❡ ❜② ❧♦♦❦✐♥❣ ❛t t❤❡ ✜rst ♦r❞❡r ❝❤❛♥❣❡ ✐♥ ♠❛❣♥✐t✉❞❡ t♦ t❤❡ ♠❛❣♥❡t✐❝ ✜❡❧❞✳ ❚❤❡ ♠❛❣♥❡t✐❝ ✜❡❧❞ ❛s ❛ r❡s✉❧t ♦❢ t❤❡ s❤✐❡❧❞✐♥❣ ✇✐❧❧ ❜❡✿ B = (−Bz σzx , −Bz σzy , Bz (1 − σzz )).  ✭✷✳✸✼✮  ❚❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ♠❛❣♥❡t✐❝ ✜❡❧❞ ✇✐❧❧ t❤❡r❡❢♦r❡ ❜❡✿ 2 2 2 2 2 B 2 = Bz2 (σzx + σzy + (1 + σzz )2 ) = Bz2 (σzx + σzy + σzz − 2σzz + 1).  ✭✷✳✸✽✮  ❆ss✉♠✐♥❣ t❤❛t ❡❛❝❤ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❝❤❡♠✐❝❛❧ s❤✐❡❧❞✐♥❣ t❡♥s♦r ✐s ♠✉❝❤ s♠❛❧❧❡r t❤❛♥ ♦♥❡ ✭t❤❡ ❝❤❛♥❣❡ ✐♥ ❢r❡q✉❡♥❝② ❜② ❝❤❡♠✐❝❛❧ s❤✐❢t ✐s ♠❡❛s✉r❡❞ ✐♥ ♣❛rts ♣❡r ♠✐❧❧✐♦♥ ♦❢ t❤❡ ▲❛r♠♦r ❢r❡q✉❡♥❝②✮✱ t❤❡♥ t♦ t❤❡ ✜rst ♦r❞❡r✱ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ♠❛❣♥❡t✐❝ ✜❡❧❞ ✐s✿  B ≈ Bz (1 − σzz ).  ✭✷✳✸✾✮  ❋♦r t❤✐s r❡❛s♦♥✱ ✇❡ ❝❛♥ ✐❣♥♦r❡ t❤❡ tr❛♥s✈❡rs❡ ❝♦♠♣♦♥❡♥ts✳ ❋r♦♠ ❤❡r❡ ✇❡ ❝❛♥ r❡✇r✐t❡ t❤❡  ✶✷  ❍❛♠✐❧t♦♥✐❛♥ ❛s✿  ˆo + H ˆ cs = γ(1 − σzz )Bo Iˆz = ωo (1 − σzz )Iˆz . H  ✭✷✳✹✵✮  ❇② ✐♥s♣❡❝t✐♦♥ ✇❡ ❝❛♥ s❡❡ t❤❛t t❤❡ ❝❤❡♠✐❝❛❧ s❤✐❢t ❢r❡q✉❡♥❝② ✐s✿ ❬✺❪  ωcs = −ωo σzz .  ✭✷✳✹✶✮  ❲❡ ❝❛♥ t❛❦❡ t❤✐s ❢✉rt❤❡r ❜② ♥♦t✐♥❣ t❤❛t  σzz = bTo σbo ,  ✭✷✳✹✷✮  ✇❤❡r❡ bo ✐s t❤❡ ✉♥✐t ✈❡❝t♦r ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ♠❛❣♥❡t✐❝ ✜❡❧❞✳ ❊q✉❛t✐♦♥ ✭✷✳✹✷✮ ❤♦❧❞s tr✉❡ r❡❣❛r❞❧❡ss ♦❢ t❤❡ ❝❤♦✐❝❡ ♦❢ ❝♦♦r❞✐♥❛t❡ s②st❡♠✳ ❋✉rt❤❡r♠♦r❡✱ ❜❡❝❛✉s❡ t❤❡ t❡r♠ σzz ✐s ♦♥ t❤❡ ❞✐❛❣♦♥❛❧✱ t❤❡ ♦♥❧② r❡❧❡✈❛♥t ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ s❤✐❡❧❞✐♥❣ t❡♥s♦r ❛❢t❡r tr❛♥s❢♦r♠❛t✐♦♥ ✇♦✉❧❞ ❤❛✈❡ t♦ ❜❡ s②♠♠❡tr✐❝✳ ❚❤✐s ♠❡❛♥s t❤❛t ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❝❤♦♦s❡ ❛ ❝♦♦r❞✐♥❛t❡ s②st❡♠ t❤❛t ✇♦✉❧❞ ❞✐❛❣♦♥❛❧✐③❡ t❤❡ s❤✐❡❧❞✐♥❣ t❡♥s♦r✳ ■♥ t❤✐s ❝♦♦r❞✐♥❛t❡ s②st❡♠ t❤❡ ♠❛❣♥❡t✐❝ ✜❡❧❞ ❞✐r❡❝t✐♦♥ ✐s s♣❡❝✐✜❡❞ ❜② ♣♦❧❛r ❛♥❞ ❛③✐♠✉t❤❛❧ ❛♥❣❧❡s θ ❛♥❞ φ✳  bPo AS = (sin θ cos φ, sin θ sin φ, cos θ),  ✭✷✳✹✸✮  ❚❤❡ ❝❤❡♠✐❝❛❧ s❤✐❢t ❢r❡q✉❡♥❝② ❝♦♥tr✐❜✉t✐♦♥ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ t❡r♠s ♦❢ t❤❡ P❆❋ ❝♦♠♣♦♥❡♥ts✿ P AS P AS P AS ωcs = −ωo (σxx (sin θ cos φ)2 + σyy (sin θ sin φ)2 + σzz cos2 θ).  ✭✷✳✹✹✮  ❇② ♠❛❦✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉❜st✐t✉t✐♦♥s✿ P AS σxx = σxP AS + σiso  ✭✷✳✹✺✮  P AS σyy = σyP AS + σiso  ✭✷✳✹✻✮  P AS σzz = σzP AS + σiso ,  ✭✷✳✹✼✮  ✇❤❡r❡ t❤❡ ❛♥✐s♦tr♦♣✐❝ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ❞✐❛❣♦♥❛❧ ❛r❡ σαP AS ❛♥❞ t❤❡ ✐s♦tr♦♣✐❝ ❝♦♠♣♦♥❡♥t ✐s✿  1 P AS P AS P AS + σyy + σzz ), σiso = (σxx 3  ✭✷✳✹✽✮  t❤❡ ❝❤❡♠✐❝❛❧ s❤✐❢t ❢r❡q✉❡♥❝② ❝❛♥ ❜❡ r❡❞✉❝❡❞ t♦✿  1 ωcs = ωiso + ωaniso = −ωo σiso − δ(3 cos2 θ − 1 − η sin2 θ cos(2φ)). 2  ✶✸  ✭✷✳✹✾✮  ❚❤❡ t✇♦ ♥❡✇ ♣❛r❛♠❡t❡rs ❛r❡ t❤❡ ❛s②♠♠❡tr② ♣❛r❛♠❡t❡r✿  η=  ωy − ωx σy − σx = σz ωz  ✭✷✳✺✵✮  ❛♥❞ t❤❡ ❛♥✐s♦tr♦♣② ♣❛r❛♠❡t❡r✿ ✭✷✳✺✶✮  δ = ωz = −ωo σz .  ❚❤❡ t♦t❛❧ ❢r❡q✉❡♥❝② ♦❢ t❤❡ s❛♠♣❧❡ ✇✐❧❧ ❜❡ ❡q✉❛❧ t♦ t❤❡ ❝❤❡♠✐❝❛❧ s❤✐❢t ❢r❡q✉❡♥❝② ♣❧✉s t❤❡ ▲❛r♠♦r ❢r❡q✉❡♥❝②✳ ❚❤❡ ❝❤❡♠✐❝❛❧ s❤✐❢t ✐s ♥♦t ❤♦✇❡✈❡r✱ ♠❡❛s✉r❡❞ ✐♥ ❛❜s♦❧✉t❡ t❡r♠s ❜❡❝❛✉s❡ ♦❢ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦♥ t❤❡ str❡♥❣t❤ ♦❢ t❤❡ ❡①t❡r♥❛❧ ♠❛❣♥❡t✐❝ ✜❡❧❞✱ ❜✉t ✐♥ t❤❡ r❡❧❛t✐✈❡ t❡r♠s ♦❢ ♣❛rts ♣❡r ♠✐❧❧✐♦♥✳ ❚❤❡ ♦r✐❡♥t❛t✐♦♥ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ❝❤❡♠✐❝❛❧ s❤✐❢t ❞♦❡s ♥♦t ♣r❡s❡♥t ❛ ♣r♦❜❧❡♠ ✐♥ ❧✐q✉✐❞s ◆▼❘ ❜❡❝❛✉s❡ t❤❡ r❛♥❞♦♠ ♠♦t✐♦♥ ❛✈❡r❛❣❡s ♦✉t t❤❡ ❛♥✐s♦tr♦♣②✳ ❍♦✇❡✈❡r✱ ✐♥ s♦❧✐❞s ♠♦❧❡❝✉❧❛r ♠♦t✐♦♥ ✐s ♥♦t ❢❛st ❡♥♦✉❣❤ t♦ ❞♦ t❤✐s ❛♥❞ ❛s ❛ r❡s✉❧t ✐♥ t❤❡ t②♣✐❝❛❧ s♦❧✐❞✱ ❛ s♣❡❝tr✉♠ ♦❢ ❛❧❧ ♦r✐❡♥t❛t✐♦♥s ✇✐❧❧ ❛♣♣❡❛r✱ ❛♥❞ ❛s ❛ r❡s✉❧t t❤❡ s♣❡❝tr✉♠ ✇✐❧❧ ❡①♣❡r✐❡♥❝❡ ❧✐♥❡ ❜r♦❛❞❡♥✐♥❣✳ ❚❤❡ s♦❧✉t✐♦♥ t♦ t❤✐s ✇✐❧❧ ❜❡ ❞❡❛❧t ✇✐t❤ ❧❛t❡r ✇❤❡♥ ✇❡ ✇♦r❦ ✇✐t❤ ♠❛❣✐❝ ❛♥❣❧❡ s♣✐♥♥✐♥❣✳  ✷✳✻✳✷  ❉✐♣♦❧❛r ❈♦✉♣❧✐♥❣  ■♥ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♥✉❝❧❡✐✱ t❤❡ ♥✉❝❧❡❛r ♠❛❣♥❡t✐❝ ♠♦♠❡♥ts ♦❢ t❤❡ s♣✐♥s ✐♥t❡r❛❝t ✇✐t❤ ❡❛❝❤ ♦t❤❡r t❤r♦✉❣❤ t❤❡ ❞✐♣♦❧❛r ✐♥t❡r❛❝t✐♦♥✳ ■♥ ❡✛❡❝t t❤❡ ❧♦❝❛❧ ✜❡❧❞ ♦❢ ❡❛❝❤ s♣✐♥ ❝r❡❛t❡s ❛ ❧♦❝❛❧ ♠❛❣♥❡t✐❝ ✜❡❧❞ ✇❤✐❝❤ ✐♥t❡r❛❝ts ✇✐t❤ t❤❡ s✉rr♦✉♥❞✐♥❣ s♣✐♥s✳ ❚❤✐s ✐♥t❡r❛❝t✐♦♥ ✐s ❝❧❛ss✐❝❛❧❧② ❛♥❛❧♦❣♦✉s t♦ t❤❡ ✐♥t❡r❛❝t✐♦♥ ♦❢ ❜❛r ♠❛❣♥❡ts ❝❧♦s❡ t♦ ❡❛❝❤ ♦t❤❡r✳ ■♥ ❧✐q✉✐❞s✱ t❤✐s ♣❤❡♥♦♠❡♥♦♥ ✐s ❛✈❡r❛❣❡❞ t♦ ✐ts ✐s♦tr♦♣✐❝ ✈❛❧✉❡ ♦❢ ✵ t❤r♦✉❣❤ ♠♦❧❡❝✉❧❛r t✉♠❜❧✐♥❣✳ ❍♦✇❡✈❡r✱ ✐♥ ❛ s♦❧✐❞ t❤✐s ✐s ♥♦t t❤❡ ❝❛s❡ ❛♥❞ ❞✐♣♦❧❛r ❝♦✉♣❧✐♥❣ ❝❛♥ ❜❡ ♦♥❡ ♦❢ t❤❡ ♠❛❥♦r ❝❛✉s❡s ♦❢ s♦❧✐❞ st❛t❡ ❧✐♥❡ ❜r♦❛❞❡♥✐♥❣✳ ❚❤❡ r❛♥❣❡ ♦❢ t❤✐s ✐♥t❡r❛❝t✐♦♥ ✐s ❡✛❡❝t✐✈❡❧② ✈❡r② s❤♦rt✱ ✇✐t❤ ✐ts r❛♥❣❡ ❞r♦♣♣✐♥❣ ♦✛ ❛s t❤❛t ♦❢ ❛ ♠❛❣♥❡t✐❝ ❞✐♣♦❧❡❀  1 ✳ r3  ■♥ ❛❞❞✐t✐♦♥✱ t❤❡ ♦r✐❡♥t❛t✐♦♥ ♦❢ t❤❡ ♠♦❧❡❝✉❧❡ ✐s ♦❢ ❦❡② ✐♠♣♦rt❛♥❝❡ ❛s s❡❡♥  ✐♥ ❋✐❣✉r❡ ✷✳✺✳ ❚❤❡ ❝❧❛ss✐❝❛❧ ❡♥❡r❣② ❜❡t✇❡❡♥ t✇♦ ♠❛❣♥❡t✐❝ ❞✐♣♦❧❡s µ1 ❛♥❞ µ2 ✐s ❣✐✈❡♥ ❜②✿  U=  µ1 · µ2 (µ1 · r)(µ2 · r) −3 3 r r5  µ0 . 4π  ✭✷✳✺✷✮  ❲❡ ❝❛♥ ✉s❡ t❤✐s t♦ ✜♥❞ t❤❡ ❛♥❛❧♦❣♦✉s q✉❛♥t✉♠ ♠❡❝❤❛♥✐❝❛❧ ❍❛♠✐❧t♦♥✐❛♥ ❜② ✉s✐♥❣ t❤❡ q✉❛♥t✉♠  ✶✹  z  B0  S θ  r  I y x  φ  ❋✐❣✉r❡ ✷✳✺✿ ❚❤❡ ❞✐♣♦❧❛r ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ t✇♦ s♣✐♥s ■ ❛♥❞ ❙❀ ❛s ❛ r❡s✉❧t ♦❢ t❤❡ ❝♦✉♣❧✐♥❣ ♦❢ t❤❡ ♠❛❣♥❡t✐❝ ✜❡❧❞s t❤❛t ❛r❡ ♣r♦❞✉❝❡❞ ❜② t❤❡✐r ❞✐♣♦❧❡ ♠♦♠❡♥ts✳ θ ❛♥❞ φ ❛r❡ t❤❡ ♣♦❧❛r ❛♥❞ ❛③✐♠✉t❤❛❧ ❛♥❣❧❡s ♦❢ t❤❡ ✐♥t❡r❛❝t✐♦♥ ✈❡❝t♦r r ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ st❛t✐❝ ✜❡❧❞✱ B0 ✳  ˆ ✐♥ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥✳ ❚❤✐s ②✐❡❧❞s✿ ♠♦♠❡♥t ♦❢ µˆ1 = γ1 ˆI ❛♥❞ µˆ1 = γ2 S ˆ ˆ ˆ ˆ ˆ dd = − µ0 γI γS ( I · S − 3 (I · r)(S · r) ). H 4π r3 r3  ✭✷✳✺✸✮  ❚❤✐s ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ✐♥ s♣❤❡r✐❝❛❧ ❝♦♦r❞✐♥❛t❡s ✇✐t❤ ❡①♣❛♥❞❡❞ s❝❛❧❛r ♣r♦❞✉❝ts ❛s✿  ˆ dd = d(A + B + C + D + E + F ), H  ✭✷✳✺✹✮  A = Iˆz Sˆz (3 cos2 θ − 1) 1 B = − [Iˆ+ Sˆ− + Iˆ− Sˆ+ ](3 cos2 θ − 1) 4 3 ˆ ˆ C = [IZ S+ + Iˆ+ Sˆz ] sin θ cos θ exp−iφ 2 3 ˆ ˆ D = [IZ S− + Iˆ− Sˆz ] sin θ cos θ expiφ 2 3 ˆ ˆ [I+ S+ ] sin2 θ exp−2iφ E = 4 3 ˆ ˆ F = [I− S− ] sin2 θ exp2iφ , 4  ✭✷✳✺✺✮  ✇✐t❤  ✭✷✳✺✻✮ ✭✷✳✺✼✮ ✭✷✳✺✽✮ ✭✷✳✺✾✮ ✭✷✳✻✵✮  ✇❤❡r❡ t❤❡ I± ❛♥❞ S± r❡♣r❡s❡♥t t❤❡ r❛✐s✐♥❣ ❛♥❞ ❧♦✇❡r✐♥❣ ♦♣❡r❛t♦rs ❛❝t✐♥❣ ♦♥ t❤❡ s♣✐♥s I ❛♥❞ ✶✺  S ❛♥❞ ❞ ✐s t❤❡ ❞✐♣♦❧❛r ❝♦✉♣❧✐♥❣ ❝♦♥st❛♥t ❞❡✜♥❡❞ ❛s d=−  µo γI γS . 4π r3  ✭✷✳✻✶✮  ❉✐♣♦❧❛r ❝♦✉♣❧✐♥❣s ❝❛♥ ❜❡ s♣❧✐t ✐♥t♦ t✇♦ ❧♦❣✐❝❛❧ s✉❜❣r♦✉♣s❀ ❞✐♣♦❧❛r ❝♦✉♣❧✐♥❣s ❜❡t✇❡❡♥ ❧✐❦❡ s♣✐♥s✱ ❦♥♦✇♥ ❛s ❤♦♠♦♥✉❝❧❡❛r ❝♦✉♣❧✐♥❣s✱ ❛♥❞ ❜❡t✇❡❡♥ ❞✐✛❡r❡♥t s♣✐♥s✱ ❤❡t❡r♦♥✉❝❧❡❛r ❞✐♣♦❧❛r ❝♦✉♣❧✐♥❣s✳ ❚❤❡ ❍❛♠✐❧t♦♥✐❛♥ ❢♦r ❜♦t❤ ❦✐♥❞s ♦❢ ❞✐♣♦❧❛r ❝♦✉♣❧✐♥❣s ❛r❡ s✐♠♣❧✐✜❡❞ ✐♥ t❤❡ r♦t❛t✐♥❣ ❢r❛♠❡✳ ❚❤✐s ❤❛♣♣❡♥s ❜❡❝❛✉s❡ t❤❡ ❩❡❡♠❛♥ ✐♥t❡r❛❝t✐♦♥ ✐s s♦ ♠✉❝❤ ❧❛r❣❡r t❤❛♥ t❤❡ ❞✐♣♦❧❛r ˆ z ♠❛tt❡r✳ ❚❤❡ ✐♥t❡r❛❝t✐♦♥s t❤❛t t♦ ✜rst ♦r❞❡r✱ ♦♥❧② t❤❡ ❝♦♠♣♦♥❡♥ts t❤❛t ❝♦♠♠✉t❡ ✇✐t❤ H s✐♠♣❧✐✜❡❞ ❍❛♠✐❧t♦♥✐❛♥s ❜❡❝♦♠❡  1 homo ˆ dd H = −d · (3 cos2 θ − 1)[3Iˆz Sˆz − I · S] 2 hetero ˆ Hdd = −d(3 cos2 θ − 1)Iˆz Sˆz .  ✷✳✼  ✭✷✳✻✷✮ ✭✷✳✻✸✮  ▼✐❝r♦s❝♦♣✐❝ ❉❡s❝r✐♣t✐♦♥ ♦❢ ❘❡❧❛①❛t✐♦♥  ❲❤✐❧❡ t❤❡ ❇❧♦❝❤ ❊q✉❛t✐♦♥s ❛r❡ ✉s❡❢✉❧ ❢♦r ♣r♦✈✐❞✐♥❣ ✐♥s✐❣❤t ✐♥t♦ t❤❡ ❝♦♥s❡q✉❡♥❝❡s ♦❢ r❡❧❛①✲ ❛t✐♦♥✱ t❤❡② ♣r♦✈✐❞❡ ✈❡r② ❧✐tt❧❡ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ♣❤②s✐❝❛❧ ❝❛✉s❡s ♦❢ r❡❧❛①❛t✐♦♥✳ ❚♦ ❛❝q✉✐r❡ t❤✐s✱ ❛ ❞❡❡♣❡r ✉♥❞❡rst❛♥❞✐♥❣ ✐♥t♦ t❤❡ ♠❡❝❤❛♥✐s♠s ♦❢ r❡❧❛①❛t✐♦♥ ♠✉st ❜❡ ✉♥❞❡rst♦♦❞✳ ❚❤❡ r❡❧❛①❛t✐♦♥ ♦❢ ❛ s♣✐♥  1 2  ♥✉❝❧❡✉s ✐s ❝❛✉s❡❞ ❜② ✢✉❝t✉❛t✐♦♥s ✐♥ t❤❡ ✈❛r✐♦✉s ◆▼❘ ✐♥t❡r❛❝t✐♦♥s  ❡①♣❡r✐❡♥❝❡❞ ❜② t❤❡ ♥✉❝❧❡✐ s✉❝❤ ❛s ❝❤❡♠✐❝❛❧ s❤✐❢t ❛♥✐s♦tr♦♣② ❛♥❞ ❞✐♣♦❧❛r ❝♦✉♣❧✐♥❣s✳ ❚❤❡s❡ ✢✉❝t✉❛t✐♦♥s ❛r❡ ❝❛✉s❡❞ ❜② t❤❡r♠❛❧ ♠♦t✐♦♥✱ ✇❤♦s❡ ❢r❡q✉❡♥❝②✱ r❛♥❣❡ ♦❢ ♠♦t✐♦♥ ❛♥❞ t②♣❡ ♦❢ ✢✉❝t✉❛t✐♦♥ ❞❡t❡r♠✐♥❡ t❤❡ r❛t❡ ♦❢ r❡❧❛①❛t✐♦♥✳ ❚❤❡ t✇♦ ❛❜♦✈❡ ✐♥t❡r❛❝t✐♦♥s ❛r❡ ❜♦t❤ ♦r✐❡♥t❛t✐♦♥ ❞❡♣❡♥❞❡♥t ❛♥❞ s♦ ❛♥② ♠♦t✐♦♥ ♦❢ t❤❡ ♥✉❝❧❡✐ r❡s✉❧ts ✐♥ ❛ ❝♦♥tr✐❜✉t✐♦♥ ♦❢ t✐♠❡ ❞❡♣❡♥❞❡♥❝❡ t♦ t❤❡ ✐♥t❡r❛❝t✐♦♥✳ ❚❤❡ t✐♠❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ❧♦❝❛❧ ✜❡❧❞s ✐s ❞❡s❝r✐❜❡❞ ❜② ❛ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥✳ ❚❤✐s ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥ ❞❡s❝r✐❜❡s ❤♦✇ ♠✉❝❤ t❤❡ ❧♦❝❛❧ ♠❛❣♥❡t✐❝ ✜❡❧❞ ✇✐❧❧ ❡✈♦❧✈❡ ❢r♦♠ t✐♠❡  t t♦ t✐♠❡ t + τ ❛♥❞ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❡♥s❡♠❜❧❡ ❛✈❡r❛❣❡✿ G(τ ) = B(t)B(t + τ ),  ✭✷✳✻✹✮  ✇❤❡r❡ t❤❡ ♦✈❡r❜❛r r❡♣r❡s❡♥ts t❤❡ ❛✈❡r❛❣❡ ♦✈❡r t✐♠❡ t ✇✐t❤✐♥ t❤❡ s②st❡♠✳ G(τ ) r❡♣r❡s❡♥ts t❤❡ t❤❡ ❧✐❦❡❧✐❤♦♦❞ t❤❛t t❤❡ ♠❛❣♥❡t✐❝ ✜❡❧❞ ✇✐❧❧ ❜❡ t❤❡ s❛♠❡ ❛t s♦♠❡ ❧❛t❡r ♣♦✐♥t ✐♥ t✐♠❡✳ ❋♦r r❛♥❞♦♠ ♠♦t✐♦♥s✱ ✐t ✐s ❡①♣❡❝t❡❞ t❤❛t G(τ ) ✇✐❧❧ ♠♦♥♦t♦♥✐❝❛❧❧② ❞❡❝❛② t♦ ✵✳ ❚❤❡ q✉❛♥t✉♠ ♠❡❝❤❛♥✐❝❛❧ ❍❛♠✐❧t♦♥✐❛♥ ❞❡s❝r✐❜✐♥❣ t❤❡ ❩❡❡♠❛♥ ✜❡❧❞ ❛♥❞ ❛ s❡❝♦♥❞ ❛r❜✐tr❛r② ✶✻  ✐♥t❡r❛❝t✐♦♥✱ α ✐s ❣✐✈❡♥ ❜②✿  ˆ = Hˆz + Hˆα . H  ✭✷✳✻✺✮  ❚❤❡ s❡❝♦♥❞ ❍❛♠✐❧t♦♥✐❛♥ ✐s r❡s♣♦♥s✐❜❧❡ ❢♦r t❤❡ r❛♥❞♦♠ ✢✉❝t✉❛t✐♦♥ t❤❛t ❧❡❛❞ t♦ r❡❧❛①❛t✐♦♥✳ ❚❤❡ ❞❡♥s✐t② ♦♣❡r❛t♦r ✐s t❤❡♥ ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ r♦t❛t✐♥❣ ❢r❛♠❡ ❜② t❤❡ ▲✐♦✉✈✐❧❧❡ ✈♦♥ ◆❡✉♠❛♥♥ ❡q✉❛t✐♦♥✿  dˆ ρ = −i[Hˆα , ρˆ]. ✭✷✳✻✻✮ dt ❋r♦♠ t❤✐s✱ ✇❡ ❝❛♥ s♦❧✈❡ ❢♦r t❤❡ t✐♠❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ ρˆ(t) t❤r♦✉❣❤ ✐♥t❡❣r❛t✐♦♥ ❛♥❞ ✉s❡ ♦❢ t❤❡ ▼❛❣♥✉s ❡①♣❛♥s✐♦♥ ˆ  ˆ  t  ρˆ(t) = ρˆ(0) − i 0  ˆ  t  [Hˆα (t ), ρˆ(0)]dt −  t  [Hˆα (t ), [Hˆα (t ), ρˆ(0)]]dt + ...  dt  ✭✷✳✻✼✮  0  0  ❆ s♠❛❧❧ t ✐s s❤♦✇♥ s♦ t❤❛t t❤❡ s❡❝♦♥❞ ♦r❞❡r ❞❡s❝r✐♣t✐♦♥ ✐s ❡♥♦✉❣❤ t♦ ❞❡s❝r✐❜❡ t❤❡ ❞❡♥s✐t② ♦♣❡r❛t♦r✳ ❚❤❡ ✜rst ♦r❞❡r t❡r♠ ❞r♦♣s ♦✉t ❜❡❝❛✉s❡ t❤❡ ❛✈❡r❛❣❡ ✐s ✵ ❜❡❝❛✉s❡ r❛♥❞♦♠ ✢✉❝t✉❛t✐♦♥s ˆ α ❛s ❛ s✉♠♠❛t✐♦♥ ♠❛❦❡ ❛❧❧ ♦r✐❡♥t❛t✐♦♥s ❡q✉❛❧❧② ❧✐❦❡❧②✳ ❲❡ ❝❛♥ ❞❡s❝r✐❜❡ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ H ♦❢ t❤❡ ❢♦r♠✿  ˆα = H  Fm Tˆm eiωt ,  ✭✷✳✻✽✮  m  ✇❤❡r❡ t❤❡ Tˆm ❛r❡ t❤❡ s♣✐♥ ♦♣❡r❛t♦rs ❛♥❞ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢❛❝t♦r ✐s ❛ r❡s✉❧t ♦❢ t❤❡ r♦t❛t✐♥❣ ❢r❛♠❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦♥ t❤❡ s♣✐♥ ♦♣❡r❛t♦rs✳ ❚❤❡ Fm ❛r❡ t❤❡ ♦r✐❡♥t❛t✐♦♥ ❞❡♣❡♥❞❡♥t ♣❛rt ♦❢  Hα ✱ ✇❤✐❝❤ ✢✉❝t✉❛t❡ ❞✉❡ t♦ t❤❡r♠❛❧ ♠♦t✐♦♥✳ ❲❡ ❞❡✜♥❡ t❤❡ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥ t♦ ❜❡✿ G(τ )mm = Fm (t)Fm† (t + τ ).  ✭✷✳✻✾✮  ❊①♣❡r✐♠❡♥t❛❧❧②✱ t❤❡ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥ ✐s ✇❡❧❧ r❡♣r❡s❡♥t❡❞ ❜② ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✱ ✇❤✐❧❡ ✐t ✐s ♥♦t ❧✐♠✐t❡❞ t♦ t❤✐s ❢♦r♠ ✐♥ t❤❡♦r②✱ ✐♥ ❣❡♥❡r❛❧ ♣r❛❝t✐❝❡ ✐t ✐s ❛s✉♠❡❞ t♦ ❜❡✿ τ  G(τ ) = Ce− τc .  ✭✷✳✼✵✮  ❚❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❝♦♥st❛♥t ❈ ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ t②♣❡ ♦❢ ✐♥t❡r❛❝t✐♦♥✭s✮ ❞♦♠✐♥❛t✐♥❣ t❤❡ r❡❧❛①❛t✐♦♥ ♣r♦❝❡ss✱ ❛♥❞ t❤❡ ✈❛❧✉❡ τc ✐s t❤❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡✱ ✇❤✐❝❤ r❡♣r❡s❡♥ts t❤❡ t✐♠❡s❝❛❧❡ ♦❢ t❤❡ ✐♥t❡r♠♦❧❡❝✉❧❛r ♠♦t✐♦♥✳ ◆♦t✐♥❣ t❤❛t ✐♥ ♣r❛❝t✐❝❡✿  Fm (t) = Fm† (−t),  ✶✼  ✭✷✳✼✶✮  ˆ α ❢r♦♠ ❊q✉❛t✐♦♥ ✭✷✳✻✽✮ ✐♥t♦ ❊q✉❛t✐♦♥ ✭✷✳✻✼✮ ✇❡ ❣❡t ❛♥❞ s✉❜st✐t✉t✐♥❣ ✐♥ t❤❡ ❢✉♥❝t✐♦♥ ❢♦r H ˆ  ˆ  t  ρˆ(t) − ρˆ(0) = [Tˆm , [Tˆm† , ρˆ(0)]]  t  Gmm (t − t )ei(ωm t −ωm t ) dt ,  dt  ✭✷✳✼✷✮  0  0  ˆ dρ dt  ❖❢t❡♥ t❤❡ ♦♥❧② s✐❣♥✐✜❝❛♥t ❝♦♥tr✐❜✉t✐♦♥s t♦  ❛r❡ ✇❤❡♥ m = m ✳ ▼❛❦✐♥❣ ❛ ✈❛r✐❛❜❧❡ ❝❤❛♥❣❡  ♦❢✿ ✭✷✳✼✸✮  τ =t −t . ✇❡ ❣❡t✿  ˆ  ˆ  t  ˆ  t i(ωm t −ωm t )  Gmm (t − t )e  dt  ✭✷✳✼✹✮  0  0  0  t  (t − τ )Gmm eiωm τ dτ.  dt =  ❆t t❤✐s ♣♦✐♥t✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ s♣❡❝tr❛❧ ❞❡♥s✐t② J(ω) ✇❤✐❝❤ ✐s t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ t❤❡ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥✳ ■t r❡♣r❡s❡♥ts t❤❡ ❛❧❧♦✇❛❜❧❡ ❢r❡q✉❡♥❝✐❡s ❢♦r ❡♥❡r❣② tr❛♥s♠✐ss✐♦♥ ❛♥❞ ❝❤❛r❛❝t❡r✐③❡s t❤❡ ❡♥❡r❣② s♣❡❝tr✉♠ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡r♠❛❧ ♠♦t✐♦♥✳ ❆ss✉♠✐♥❣ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ ❢✉♥❝t✐♦♥✱ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ r❡s✉❧ts ✐♥ ❛ ▲♦r❡♥t③✐❛♥✿  ˆ  ∞  τ  e− τc e−iωτ dτ = C  J(ω) = C −∞  2τc . 1 + (τc ω)2  ✭✷✳✼✺✮  ❲❡ t❤❡♥ ♥♦t❡ t❤❛t ✇✐t❤ r❡❣❛r❞s t♦ ❊q✉❛t✐♦♥ ✭✷✳✼✹✮ t❤❛t t❤❡ t✐♠❡s❝❛❧❡ t ✐s ♠✉❝❤ ❣r❡❛t❡r t❤❛♥ τ ✳ ❚❤✐s ❛❧s♦ ✐♠♣❧✐❡s t❤❛t t❤❡ ❝♦♥tr✐❜✉t✐♦♥s ♦✉ts✐❞❡ ♦❢ t❤❡ ❜♦✉♥❞s ♦❢ t❤❡ ✐♥t❡❣r❛❧ ❛r❡ ♥❡❣❧✐❣✐❜❧❡✱ ❛♥❞ t❤❡ ❜♦✉♥❞s ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❣❡t✿  ˆ  ˆ  t  (t − τ )Gmm e  iωm τ  tGmm eiωm τ dτ = tJmm (ωm ).  ✭✷✳✼✻✮  0  0 ˆ  = dρ(t) ❜❡❝❛✉s❡ t ✐s s♠❛❧❧ ❡♥♦✉❣❤ t❤❛t t❤❡ ❞❡♥s✐t② ♦♣❡r❛t♦r ✉♥❞❡r❣♦❡s dt s♠❛❧❧ ❝❤❛♥❣❡s✱ ✇❡ ❣❡t t❤❡ ♠❛st❡r ❡q✉❛t✐♦♥✿ ◆♦t✐♥❣ t❤❛t  ρˆ(t)−ˆ ρ(0) t  ∞  dτ =  dˆ ρ = [Tˆm , [Tˆm† , ρˆ]]Jmm (ωm ) dt  ✭✷✳✼✼✮  ❲❡ ❝❛♥ ❞❡t❡r♠✐♥❡ t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ ❛♥② ♣❤②s✐❝❛❧ ♦❜s❡r✈❛❜❧❡ ✉s✐♥❣✿  d ρ ˆ >= T r(Q dˆ <Q )= dt dt  ˆ Tˆm , [Tˆm† , ρˆ])Jmm (ωm ). T r(Q[ m  ✶✽  ✭✷✳✼✽✮  ❇② s✉❜st✐t✉t✐♥❣ ✐♥ ρˆ = ρˆ − ρˆeq t♦ ❛❝❝♦✉♥t ❢♦r t❤❡ ✐♥✐t✐❛❧ t❤❡r♠❛❧ ❡q✉✐❧✐❜r✐✉♠ ✈❛❧✉❡ ♦❢ t❤❡ ❞❡♥s✐t② ♦♣❡r❛t♦r ✇❡ ❣❡t✿  d ˆ >= <Q dt  ˆ Tˆm ], Tˆ† ] > − < [[Q, ˆ Tˆm ], Tˆ† ] >eq ). Jmm (ωm )(< [[Q, m m  ✭✷✳✼✾✮  m  ❲❡ ❝❛♥ ✉s❡ t❤❡ ❛❜♦✈❡ t♦ ❞❡r✐✈❡ t❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ r❡❧❛①❛t✐♦♥ t✐♠❡✱ T1 ✱ ❜② ✐♥s❡rt✐♥❣ t❤❡ ♦❜s❡r✈✲ ❛❜❧❡ ♦❢ t❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ ♠❛❣♥❡t✐③❛t✐♦♥ ♦❢ Iˆz ✳ ❲❡ ❝❛♥ t❤❡♥ ✉s❡ t❤❡ ♠❛st❡r ❡q✉❛t✐♦♥ t♦ r❡❧❛t❡ t❤❡ T1 t♦ t❤❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡✳  ✷✳✼✳✶  ❉✐♣♦❧❛r ❘❡❧❛①❛t✐♦♥  ❚❤❡ ♣r✐♠❛r② ♠❡❝❤❛♥✐s♠ ♦❢ r❡❧❛①❛t✐♦♥ r❡❧❡✈❛♥t t♦ t❤✐s ✇♦r❦ ✐s ❞✐♣♦❧❛r ❝♦✉♣❧✐♥❣✳ ❇② r❡✇♦r❦✐♥❣ t❤❡ ❞✐♣♦❧❛r ❍❛♠✐❧t♦♥✐❛♥ ❢r♦♠ ❊q✉❛t✐♦♥ ✭✷✳✺✸✮ ✐♥t♦ ❛ ❢♦r♠ s✉✐t❛❜❧❡ ❢♦r r❡❧❛①❛t✐♦♥ ✐♥ t❡r♠s ♦❢ Fm ❛♥❞ Tˆm ✇❡ ❝❛♥ ❛♣♣❧② t❤❡ r❡s✉❧ts ♦❢ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✳ Fm ❛r❡ t❤❡ s♣❛t✐❛❧ ❢✉♥❝t✐♦♥s t❤❛t ✢✉❝t✉❛t❡ r❛♥❞♦♠❧② ✇❤✐❝❤ r❡s✉❧ts ✐♥ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥✳ ❚❤❡ ❢✉♥❝t✐♦♥s ❢♦r Fm ❛r❡✿  3 (cos2 θ − 1) 2 = ∓3 sin θ cos θ exp±iφ 3 2 = sin θ exp±2iφ . 2  F0 = F±1 F±2  ✭✷✳✽✵✮ ✭✷✳✽✶✮ ✭✷✳✽✷✮  ❚❤❡ s♣✐♥ ♦♣❡r❛t♦rs ❢♦r Tˆm ❛r❡ t❤❡ s♣❤❡r✐❝❛❧ t❡♥s♦rs✿  1 ˆ Tˆ0 = √ (3Iˆz Sˆz − ˆI · S) 6 1 Tˆ±1 = ∓ (Iˆ± Sˆz + Iˆz Sˆ± ) 2 1 Tˆ±2 = Iˆ± Sˆ± . 2  ✭✷✳✽✸✮ ✭✷✳✽✹✮ ✭✷✳✽✺✮  ❋r♦♠ t❤✐s ✇❡ ❣❡t t❤❡ ❞✐♣♦❧❛r ❍❛♠✐❧t♦♥✐❛♥ ✐♥ t❤❡ ❢♦r♠ ♦❢✿  ˆ d = − µ0 γI γs H 3 4π rIS  +2  ˆ S), ˆ Fm (θ, φ)Tˆm (I,  ✭✷✳✽✻✮  −2  ✇❤❡r❡ ■ ❛♥❞ ❙ r❡♣r❡s❡♥t ❞✐✛❡r❡♥t s♣✐♥s✳ ❚❤❡ ❞❡r✐✈❛t✐♦♥ t❤❛t ❢♦❧❧♦✇s ♦♥❧② ❛♣♣❧✐❡s t♦ s②st❡♠s ✇❤❡r❡ ❜♦t❤ s♣✐♥s ❛r❡  1 2  ✇❤✐❝❤ ✇✐t❤✐♥ t❤❡ ❝♦♥t❡①t ♦❢ t❤✐s t❤❡s✐s ♦❢ ♦♥❧② ✉s✐♥❣ ❤②❞r♦❣❡♥ ❛♥❞ ✶✾  ❝❛r❜♦♥✱ ✐s tr✉❡✳ ■t ✐s ✇♦rt❤ ♥♦t✐♥❣ t❤❛t ✐♥ r❡❧❛①❛t✐♦♥ st✉❞✐❡s✱ t❤❡ ♥♦♥ s❡❝✉❧❛r t❡r♠s ❢r♦♠ t❤❡ ❞✐♣♦❧❛r ❍❛♠✐❧t♦♥✐❛♥ ❊q✉❛t✐♦♥s ✭✷✳✺✻✲✷✳✻✵✮ ❛r❡ ✐♠♣♦rt❛♥t✳ ❯s✐♥❣ t❤✐s ❍❛♠✐❧t♦♥✐❛♥ ✇✐t❤ t❤❡ ♦❜s❡r✈❛❜❧❡s Iˆz ❛♥❞ Sˆz t❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ ♠❛❣♥❡t✐③❛t✐♦♥ ✇✐t❤ ❊q✉❛t✐♦♥ ✷✳✽✻ ♦♥❡ ♦❜t❛✐♥s t✇♦ ❝♦✉♣❧❡❞ ❡①♣r❡ss✐♦♥s ♦❢ t❤❡ ❢♦r♠✿  d ˆ Iz = −RI ( Iˆz − Iˆz dt d ˆ Sz = −RIS ( Iˆz − Iˆz dt  eq )  − RIS ( Sˆz − Sˆz  eq )  ✭✷✳✽✼✮  eq )  − RS ( Sˆz − Sˆz  eq ).  ✭✷✳✽✽✮  ❚❤❡ Rα r❡♣r❡s❡♥t r❛t❡s✳ ❚❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ r❡❧❛①❛t✐♦♥ t✐♠❡ ❢♦r I ✐s T1 =  1 ✳ RI  ❚❤❡ t❡r♠ RIS  r❡♣r❡s❡♥ts ❝r♦ss r❡❧❛①❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ I ❛♥❞ S s♣✐♥s✳ ❚❤❡ r❡❧❛t✐♦♥s❤✐♣s ❜❡t✇❡❡♥ Rα ❛♥❞ t❤❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ τc ❛r❡✿  3 1 6 1 µ0 γI2 γS2 2 τc + + ✭✷✳✽✾✮ RI = 6 2 2 10 4π rIS 1 + (ωI τc ) 1 + ((ωI − ωS )τc ) 1 + ((ωI + ωS )τc )2 6 1 µ0 γI2 γS2 2 1 τc + RIS = . ✭✷✳✾✵✮ 6 2 10 4π rIS 1 + ((ωI − ωS )τc ) 1 + ((ωI + ωS )τc )2 ❲❤✐❧❡ t❤✐s r❡s✉❧t ♦♥❧② ❞❡s❝r✐❜❡s ✉♥✐❢♦r♠ ✐s♦tr♦♣✐❝ ♠♦t✐♦♥✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❣❛✐♥ ❛ ❣❡♥❡r❛❧ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ♠♦t✐♦♥ ❜② ❧♦♦❦✐♥❣ ❛t t❤❡ ♠❛❣♥❡t✐❝ ✜❡❧❞ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ T1 ✳ ❆s ❝❛♥ ❜❡ s❡❡♥ ❢r♦♠ ❊q✉❛t✐♦♥ ✭✷✳✽✾✮ ✇❤❡♥ τc ωI  1✱ r❡♣r❡s❡♥t✐♥❣ t❤❡ ❢❛st ♠♦t✐♦♥ r❡❣✐♠❡ ✇❤❡r❡ ❛ s❤♦rt τc ✐♠♣❧✐❡s ❢❛st ♠♦t✐♦♥✱ ❊q✉❛t✐♦♥ ✭✷✳✽✾✮ ❜❡❝♦♠❡s✿ µ0 γI2 γS2 2 τc . 6 4π rIS  RI ≈  ▲✐❦❡✇✐s❡✱ ✐♥ t❤❡ s❧♦✇ ♠♦t✐♦♥ ❧✐♠✐t ✇❤❡r❡ τc ωI  RI = ❇❡❝❛✉s❡ RI ✐s  1 ✱ T1  1 µ0 γI2 γS2 6 10 4π rIS  2  1 τc  ✭✷✳✾✶✮  1✱ ✇❡ s❡❡ ❊q✉❛t✐♦♥ ✭✷✳✽✾✮ ❜❡❝♦♠❡✿  3 1 6 + + 2 2 ωI (ωI − ωS ) (ωI + ωS )2  .  ✭✷✳✾✷✮  ❛♥❞ ωx = −γx B0 ✱ ❛♥❞ t❤❡ ❢❛❝t t❤❛t ✐♥ ♠♦st ❝❛s❡s t❤❡ ❝♦rr❡❝t ♣r❡❢❛❝t♦r ✇✐❧❧  ❜❡ ✉♥❦♥♦✇♥✱ ✇❡ ❤❛✈❡  1 τc ∝ τc B02 .  T1f ast ∝ T1slow  ✭✷✳✾✸✮ ✭✷✳✾✹✮  ❙♦ ❢♦r ♠♦t✐♦♥ ✐♥ t❤❡ ❢❛st r❡❣✐♠❡ t❤❡ r❡❧❛①❛t✐♦♥ t✐♠❡ s❝❛❧❡s ✐♥❞❡♣❡♥❞❡♥t❧② ♦❢ t❤❡ ♠❛❣♥❡t✐❝ ✷✵  ✜❡❧❞✱ ❛♥❞ ✐♥✈❡rs❡ t♦ t❤❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡✱ ✇❤✐❧❡ ❢♦r t❤❡ s❧♦✇ ❧✐♠✐t t❤❡ r❡❧❛①❛t✐♦♥ t✐♠❡ s❝❛❧❡s ♣r♦♣♦rt✐♦♥❛❧❧② t♦ t❤❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ❛♥❞ t♦ t❤❡ ♠❛❣♥❡t✐❝ ✜❡❧❞ sq✉❛r❡❞✳ ❊✈❡♥ ✐❢ ♠♦t✐♦♥ ✐s ♥♦t ✐♥ ❡✐t❤❡r ❧✐♠✐t✱ ✇❡ ❝❛♥ ❢✉rt❤❡r r❡✜♥❡ t❤✐s ❜② ♥♦t✐♥❣ t❤❛t ❛ str♦♥❣❡r r❡❧❛t✐♦♥s❤✐♣ t♦ t❤❡ ♠❛❣♥❡t✐❝ ✜❡❧❞ ♠❡❛♥s t❤❛t t❤❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ✐s ❧♦♥❣❡r✳ ❚❤❡ s♣❡❝tr❛❧ ❞❡♥s✐t② ✉s❡❞ ❛❜♦✈❡ r❡♣r❡s❡♥ts ❛♥ ✐❞❡❛❧ ❝❛s❡ ✇❤❡r❡ t❤❡r❡ ✐s ❛ s✐♥❣❧❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ✉♥❞❡r❣♦✐♥❣ ✐s♦tr♦♣✐❝ ♠♦t✐♦♥✳ ■♥ r❡❛❧✐t② t❤✐s ✐s q✉✐t❡ r❛r❡ ✇✐t❤ ♠❛❝r♦♠♦❧❡❝✉❧❡s✳ ■♥ ❧❛r❣❡ ♠♦❧❡❝✉❧❡s s✉❝❤ ❛s t❤❡ ♦♥❡s t❤❛t ❛r❡ t❤❡ s✉❜❥❡❝t ♦❢ t❤✐s t❤❡s✐s✱ t❤❡r❡ ✐s ♠♦r❡ ❧✐❦❡❧② t♦ ❡①✐st ❛ ❞✐str✐❜✉t✐♦♥ ♦❢ ❝♦rr❡❧❛t✐♦♥ t✐♠❡s✳ ❍♦✇ t❤❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡s ❛r❡ ❞✐str✐❜✉t❡❞ ❤❛s ❛ s✐❣♥✐✜❝❛♥t ❡✛❡❝t ♦♥ t❤❡ s♣❡❝tr❛❧ ❞❡♥s✐t②✳ ❖♥❡ ♦❢ t❤❡ ♠♦st s✉❝❝❡ss❢✉❧ s♣❡❝tr❛❧ ❞❡♥s✐t② ❞✐str✐❜✉t✐♦♥s ✉s❡❞ t♦ ✐♥t❡r♣r❡t r❡❧❛①❛t✐♦♥ ❡①♣❡r✐♠❡♥ts ✐♥ s♦❧✐❞s ✐s t❤❡ ❉❛✈✐❞s♦♥✲❈♦❧❡ ❞✐str✐❜✉t✐♦♥✳ ❚❤✐s ❞✐str✐❜✉t✐♦♥ ❞❡♥s✐t② ✐s ❣✐✈❡♥ ❜② ❬✻❪ ✿  sin(σπ) ρ(τ, η, σ) = π  1 τ /η − 1  σ  ,  ✭✷✳✾✺✮  ✇❤❡r❡ ρ ✐s t❤❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ❞✐str✐❜✉t✐♦♥✱ τ ✐s t❤❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡✱ η ❞❡t❡r♠✐♥❡s t❤❡ ❞✐str✐❜✉t✐♦♥ ❝❡♥t❡r✱ ❛♥❞ σ ❞❡t❡r♠✐♥❡s t❤❡ ❞✐str✐❜✉t✐♦♥ ✇✐❞t❤ ❛♥❞ r❛♥❣❡s ❢r♦♠ ✵ t♦ ✶✱ ✇✐t❤ ❛ ✈❛❧✉❡ ♦❢ ✶ r❡s✉❧t✐♥❣ ✐♥ ❛ ✇✐❞t❤ ♦❢ ✵✳ ❚❤✐s ❞✐str✐❜✉t✐♦♥ ❤❛s ❛ s♣❡❝tr❛❧ ❞❡♥s✐t② ♦❢✿  JDC (ω, τ, σ) =  2 sin(σ arctan(ωτ )) ω (1 + ω 2 τ 2 )σ/2  ✭✷✳✾✻✮  ❆ s❡❝♦♥❞ ❛♥❞ s✐♠♣❧❡r ♦♣t✐♦♥ t♦ ✉s❡ ❛ ♠❡❛♥ ❝♦rr❡❧❛t✐♦♥ t✐♠❡✳ ❚❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ t❤✐s ✐s t❤❛t ✐t ❝❛♥ ❜❡ tr❡❛t❡❞ ❛s ❛ s✐♥❣❧❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ✇❤✐❝❤ r❡✈❡rts ❜❛❝❦ t♦ ❊q✉❛t✐♦♥ ✭✷✳✽✾✮ r❡❞✉❝✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ✉♥❦♥♦✇♥s ❜② ✶✳ ❚❤✐s ♠❛❦❡s ✐t ❡❛s✐❡r t♦ s♦❧✈❡ ❢♦r t❤❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡✳ ❍♦✇❡✈❡r✱ ❛s ❛ r❡s✉❧t ♦❢ t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥✱ t❤❡r❡ ✐s ❛ s✐❣♥✐✜❝❛♥t ❧♦ss ♦❢ ✐♥❢♦r♠❛t✐♦♥✳ ❚❤❡ ✜rst ♣✐❡❝❡ ♦❢ ✐♥❢♦r♠❛t✐♦♥ ❧♦st ✐s t❤❡ ✇✐❞t❤ ♦❢ t❤❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ❞✐str✐❜✉t✐♦♥✳ ■t ✇✐❧❧ ♥♦t ❜❡ ❦♥♦✇♥ ✐❢ t❤❡ ♠♦t✐♦♥ ✐s ❝♦♥✜♥❡❞ t♦ ❛ s✐♥❣❧❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ❢♦r t❤❡ ❡♥t✐r❡ s②st❡♠✱ ♦r ❛ ✈❡r② ✇✐❞❡ r❛♥❣❡ t❤❛t ❝♦✉❧❞ ❡①t❡♥❞ ✇❡❧❧ ♦✈❡r ❛♥ ♦r❞❡r ♠❛❣♥✐t✉❞❡✳ ❚❤❡ s❡❝♦♥❞ ♣✐❡❝❡ ♦❢ ✐♥❢♦r♠❛t✐♦♥ ❧♦st ✐s ❤♦✇ ❢❛r ❢r♦♠ t❤❡ ❛❝t✉❛❧ ❝❡♥t❡r ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ t❤❡ ❛✈❡r❛❣❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ✐s✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ t❤❡ ❉❛✈✐❞s♦♥ ❈♦❧❡ ❞✐str✐❜✉t✐♦♥ ✐s ❛s②♠♠❡tr✐❝✱ s♦ t❤❡ ♠❡❛♥ ❞♦❡s ♥♦t ❛❝t✉❛❧❧② ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ❞✐str✐❜✉t✐♦♥ ✬❝❡♥t❡r✬ ♣❛r❛♠❡t❡r η ❞✐s❝✉ss❡❞ ✐♥ ❊q✉❛t✐♦♥ ✷✳✾✺✳ ❍♦✇❡✈❡r ❡✈❡♥ ✇✐t❤ t❤✐s ❛✈❡r❛❣❡❞ ❛ss✉♠♣t✐♦♥✱ t❤✐s st✐❧❧ ❤❛s ♥♦t t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ♠♦t✐♦♥❛❧ r❡str✐❝t✐♦♥s✳ ❋♦r ❛ s✐♥❣❧❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ✉♥❞❡r❣♦✐♥❣ ❛♥✐s♦tr♦♣✐❝ ♠♦t✐♦♥✱ t❤❡ s♣❡❝tr❛❧ ❞❡♥s✐t②  ✷✶  ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ t❤r♦✉❣❤ ❛ ♠♦❞❡❧ ❢r❡❡ ❛♣♣r♦❛❝❤ ❜② t❤❡ ❢✉♥❝t✐♦♥ ❬✼❪✿  2 J(ω) = 5  S 2 τM (1 − S 2 )τ + 1 + (τM ω)2 1 + (τ ω)2  ✭✷✳✾✼✮  ,  ✇✐t❤ ✭✷✳✾✽✮  −1 τ −1 = τM + τc−1 ,  ✇❤❡r❡ τM r❡♣r❡s❡♥ts t❤❡ ♦✈❡r❛❧❧ ♠♦❧❡❝✉❧❛r t✉♠❜❧✐♥❣✱ τc r❡♣r❡s❡♥ts t❤❡ ✐♥t❡r♥❛❧ ♠♦❧❡❝✉❧❛r ✈✐❜r❛t✐♦♥s✱ ❛♥❞ S ✐s ❛ ❣❡♥❡r❛❧✐③❡❞ ♦r❞❡r ♣❛r❛♠❡t❡r ✇❤✐❝❤ ♠❡❛s✉r❡s t❤❡ ❞❡❣r❡❡ ♦❢ s♣❛t✐❛❧ r❡str✐❝t✐♦♥ ♦❢ t❤❡ ♠♦t✐♦♥ ❛♥❞ r❛♥❣❡s ❢r♦♠ ✵ ✭♥♦ r❡str✐❝t✐♦♥s✮ t♦ ✶ ✭❝♦♠♣❧❡t❡❧② r❡str✐❝t❡❞✮✳ ❋♦r ❝❛s❡s ✇❤❡r❡ t❤❡ ♦✈❡r❛❧❧ ♠♦t✐♦♥ ✐s r❡❧❛t✐✈❡❧② s❧♦✇ ❝♦♠♣❛r❡❞ t♦ t❤❡ ✐♥t❡r♥❛❧ ♠♦t✐♦♥ s✉❝❤ t❤❛t τM  τc ❊q✉❛t✐♦♥ ✭✷✳✾✼✮ r❡❞✉❝❡s t♦ J(ω) =  2 5  (1 − S 2 )τc S 2 τM + 1 + (τM ω)2 1 + (τc ω)2  ✭✷✳✾✾✮  .  ❋✉rt❤❡r♠♦r❡✱ ✐❢ t❤❡ ♦✈❡r❛❧❧ ♠♦❧❡❝✉❧❛r ♠♦t✐♦♥ ✐s r❡❧❛t✐✈❡❧② st❛t✐❝ ❛♥❞ t❤❡ ✐♥t❡r♠♦❧❡❝✉❧❛r ♠♦t✐♦♥ ✐s ❢❛st s✉❝❤ t❤❛t ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s✱ τM ω  1✱ τc ω t❤❡♥ ❊q✉❛t✐♦♥ ✭✷✳✾✾✮ ❝❛♥ ❜❡ ❢✉rt❤❡r r❡❞✉❝❡❞ ✐♥t♦ J(ω) =  2 (1 − S 2 )τc . 5 1 + (τc ω)2  1✱ ❛♥❞  S2 τM ω 2  (1−S 2 )τc ✱ 1+(τc ω)2  ❤♦❧❞ tr✉❡  ✭✷✳✶✵✵✮  ❚❤✐s r❡s✉❧t✱ ❛❧♦♥❣ ✇✐t❤ t❤❛t ❢♦r t❤❡ ♠❡❛♥ r❡❧❛①❛t✐♦♥ t✐♠❡✱ ♠❡❛♥s t❤❛t ✇❡ ❝❛♥ r❡♣❧❛❝❡ t❤❡ s♣❡❝✲ tr❛❧ ❞❡♥s✐t② ❛♥❞ T1 ♦❢ ❊q✉❛t✐♦♥ ✭✷✳✽✾✮ t♦ ❞❡✈❡❧♦♣ ❛ r♦✉❣❤ ✐❞❡❛ ♦❢ t❤❡ s❝❛❧❡ ♦❢ t❤❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ❛♥❞ ♦r❞❡r ♣❛r❛♠❡t❡r ✇✐t❤✐♥ ❛ ♣r♦t❡✐♥✳  ✷✷  ❈❤❛♣t❡r ✸  ❊①♣❡r✐♠❡♥t❛❧ ❚❡❝❤♥✐q✉❡s  ❖♥❡ ❛❞✈❛♥t❛❣❡ ♦❢ ◆▼❘ ✐s t❤❛t t❤❡r❡ ❛r❡ ❧✐t❡r❛❧❧② ❤✉♥❞r❡❞s ♦❢ ✈❛r✐♦✉s t♦♦❧s ❛♥❞ t❡❝❤♥✐q✉❡s t♦ ♣r♦❜❡ t❤❡ ♠♦❧❡❝✉❧❛r str✉❝t✉r❡s ♦❢ ♠❛tt❡r ❛♥❞ ✐♠♣r♦✈❡ s♣❡❝tr✉♠ r❡s♦❧✉t✐♦♥✳ ❚❤❡s❡ r❛♥❣❡ ❢r♦♠ ✈❛r✐♦✉s ♣✉❧s❡ s❡q✉❡♥❝❡s✱ ♠❛❣♥❡t✐❝ ✜❡❧❞ ❝♦rr❡❝t✐♦♥s ✭❛ ♣r♦❝❡ss ❝❛❧❧❡❞ s❤✐♠♠✐♥❣✮✱ t♦ s♣✐♥♥✐♥❣ t❤❡ s❛♠♣❧❡ ❡①tr❡♠❡❧② ❢❛st✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥t❛✐♥s t❤❡ ◆▼❘ t❡❝❤♥✐q✉❡s r❡❧❡✈❛♥t t♦ t❤✐s t❤❡s✐s✳ ■♥ ❛❞❞✐t✐♦♥✱ ❛ ♣r♦❝❡ss ✉s❡❞ t♦ ♠❛❦❡ ♥❛♥♦✜❜❡rs ❦♥♦✇♥ ❛s ❡❧❡❝tr♦s♣✐♥♥✐♥❣ ✇✐❧❧ ❜❡ ❞❡s❝r✐❜❡❞ ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ s❡❝t✐♦♥✳  ✸✳✶  ❍❡t❡r♦♥✉❝❧❡❛r ❉❡❝♦✉♣❧✐♥❣  ❉✐❧✉t❡ s♣✐♥s ♥❡❛r ❛❜✉♥❞❛♥t s♣✐♥s s✉❝❤ ❛s t❤❡ ♣r♦①✐♠✐t② ♦❢ t❤❡ ❞✐❧✉t❡  C t♦ t❤❡ ❤✐❣❤❧② ❛❜✉♥❞❛♥t H ✐♥ ♦r❣❛♥✐❝ ♠❛t❡r✐❛❧s ❝❛✉s❡s ❛ ❜r♦❛❞❡♥✐♥❣ ♦❢ s♣❡❝tr❛❧ ❧✐♥❡s✳ ❚❤✐s ❝❛♥ r❡❞✉❝❡❞ t❤r♦✉❣❤ ❤❡t❡r♦♥✉❝❧❡❛r ❞❡❝♦✉♣❧✐♥❣✱ ❛ t❡❝❤♥✐q✉❡ t❤❛t ❛♣♣❧✐❡s ❤✐❣❤ ♣♦✇❡r r❢ ✐rr❛❞✐❛t✐♦♥ t♦ t❤❡ ❛❜✉♥❞❛♥t s♣✐♥ ❝❤❛♥♥❡❧ ❞✉r✐♥❣ ❛❝q✉✐s✐t✐♦♥ ♦❢ t❤❡ t❛r❣❡t ♦❜s❡r✈❛t✐♦♥ ❝❤❛♥♥❡❧✳ ❚❤❡ ❤✐❣❤ ♣♦✇❡r ✐rr❛❞✐❛t✐♦♥ ❝❛✉s❡s t❤❡ ❛❜✉♥❞❛♥t s♣✐♥s t♦ ✉♥❞❡r❣♦ tr❛♥s✐t✐♦♥s ❛t ❛ r❛t❡ ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ r❢ ❛♠♣❧✐t✉❞❡✳ Pr♦t♦♥s t❤❡♥ ❤❛✈❡ ❛ t✐♠❡ ❛✈❡r❛❣❡❞ ❞✐♣♦❧❛r ❝♦✉♣❧✐♥❣ ❝♦♥tr✐❜✉t✐♦♥ ♦❢ ③❡r♦ ❜❡❝❛✉s❡ t❤❡ s♣✐♥ q✉❛♥t✉♠ ♥✉♠❜❡r ♦s❝✐❧❧❛t❡s ❜❡t✇❡❡♥ ± 21 ✳ ❚❤✐s t✐♠❡ ❛✈❡r❛❣✐♥❣ ♦❢ t❤❡ ♣r♦t♦♥ s♣✐♥ st❛t❡s ❡❧✐♠✐♥❛t❡s t❤❡ ❞✐♣♦❧❛r ❝♦♥tr✐❜✉t✐♦♥ t♦ t❤❡ 13 ❈ ❬✽❪✳ 1  ✷✸  13  ❋✐❣✉r❡ ✸✳✶✿ ❆ s❝❤❡♠❛t✐❝ ♦❢ ♠❛❣✐❝ ❛♥❣❧❡ s♣✐♥♥✐♥❣✳ ✸✳✷  ▼❛❣✐❝ ❆♥❣❧❡ ❙♣✐♥♥✐♥❣  ▼❛❣✐❝ ❛♥❣❧❡ s♣✐♥♥✐♥❣ ✭▼❆❙✮ ✐s ❛ r♦✉t✐♥❡✱ s✐♠♣❧❡✱ ❛♥❞ ❡①tr❡♠❡❧② ❡✛❡❝t✐✈❡ t❡❝❤♥✐q✉❡ ✉s❡❞ ✐♥ s♦❧✐❞✲st❛t❡ ◆▼❘ ❡①♣❡r✐♠❡♥ts t❤❛t ❡✛❡❝t✐✈❡❧② r❡♠♦✈❡s ❜r♦❛❞❡♥✐♥❣ ❢r♦♠ ❝❤❡♠✐❝❛❧ s❤✐❢t ❛♥✐s♦tr♦♣② ❛s ✇❡❧❧ ❛s ❤❡❧♣✐♥❣ ✐♥ t❤❡ r❡♠♦✈❛❧ ♦❢ ❤❡t❡r♦♥✉❝❧❡❛r ❞✐♣♦❧❛r ❝♦✉♣❧✐♥❣s✳ ■t ✐s ❛❧s♦ ❡✛❡❝t✐✈❡ ❢♦r r❡❞✉❝✐♥❣ t❤❡ ❡✛❡❝ts ♦❢ q✉❛❞r✉♣♦❧❛r ❝♦✉♣❧✐♥❣ ❛♥❞ ❛t ❤✐❣❤ ❡♥♦✉❣❤ s♣✐♥♥✐♥❣ ❢r❡✲ q✉❡♥❝✐❡s✱ ✐t ✐s ❡✛❡❝t✐✈❡ ❢♦r r❡♠♦✈✐♥❣ ❤♦♠♦♥✉❝❧❡❛r ❞✐♣♦❧❛r ❝♦✉♣❧✐♥❣s✳ ❆♥ ♦❜s❡r✈❛♥t r❡❛❞❡r ✇✐❧❧ ♥♦t✐❝❡ t❤❛t t❤❡ t❤❡ ♠❛❥♦r✐t② ♦❢ t❤❡ ✐♥t❡r❛❝t✐♦♥s t❤❛t ▼❆❙ ❛✛❡❝ts ❛r❡ ♦♥❡s t❤❛t ❞♦ ♥♦t ❛✛❡❝t ❧✐q✉✐❞ ◆▼❘✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ ▼❆❙ ✐♥ s♦♠❡ ✇❛②s ♠✐♠✐❝s t❤❡ r❛♣✐❞ ✐s♦tr♦♣✐❝ t✉♠❜❧✐♥❣ ♦❢ t❤❡ ♠♦❧❡❝✉❧❡s ✐♥ ❛ s♦❧✉t✐♦♥✳ ❚❤✐s t✉♠❜❧✐♥❣ ❛✈❡r❛❣❡s t❤❡ ♠♦❧❡❝✉❧❛r ♦r✐❡♥t❛t✐♦♥ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ tr❛♥s✐t✐♦♥ ❢r❡q✉❡♥❝✐❡s t♦ ③❡r♦✳ ▼❛❣✐❝ ❛♥❣❧❡ s♣✐♥♥✐♥❣ r❡♣❧✐❝❛t❡s t❤✐s ♣❤❡♥♦♠❡♥♦♥ ❜② r♦t❛t✐♥❣ t❤❡ s❛♠♣❧❡ ❛❧♦♥❣ ❛♥ ❛①✐s ❛t ❛ ♠❛❣✐❝ ❛♥❣❧❡ t❤❛t ❛✈❡r❛❣❡s ♦✉t ✐s♦tr♦♣✐❝ ❡✛❡❝ts✳ ❆s ♥♦t❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡r✱ t❤❡r❡ ❛r❡ ♠❛♥② ♠♦❧❡❝✉❧❛r ♦r✐❡♥t❛t✐♦♥ ❞❡♣❡♥❞❡♥t ✐♥t❡r❛❝t✐♦♥s ♦❢ t❤❡ ❢♦r♠ 3 cos2 θ − 1✱ ✇❤❡r❡ θ ✐s t❤❡ ❛♥❣❧❡ ♦❢ t❤❡ ♦r✐❡♥t❛t✐♦♥ ♦❢ t❤❡ ✐♥t❡r❛❝t✐♦♥ t❡♥s♦r✳ ■♥ ❛ ♣♦✇❞❡r s❛♠♣❧❡✱ θ t❛❦❡s ♦♥ ❛❧❧ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ❜❡❝❛✉s❡ ❛❧❧ ♠♦❧❡❝✉❧❛r ♦r✐❡♥t❛t✐♦♥s ❛r❡ r❡♣r❡s❡♥t❡❞ ✐♥ ❛ r❛♥❞♦♠ ❞✐str✐❜✉t✐♦♥✳ ■❢ t❤❡ s❛♠♣❧❡ ✐s s♣✉♥ ❛r♦✉♥❞ ❛♥ ❛♥❣✉❧❛r ❛①✐s✱ θR ♦✛ ♦❢ t❤❡ ❛♣♣❧✐❡❞ ✜❡❧❞✱ t❤❡♥ θ ✇✐❧❧ ✈❛r② ✇✐t❤ t✐♠❡ ❛s t❤❡ ♠♦❧❡❝✉❧❡s r♦t❛t❡ ❛r♦✉♥❞ t❤❡ ❛①✐s✳ ❚❤❡ ❛✈❡r❛❣❡ ♦r✐❡♥t❛t✐♦♥ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ✐♥t❡r❛❝t✐♦♥ ✇✐❧❧ t❤❡♥ ❜❡  1 < 3 cos2 θ − 1 >= (3 cos2 θR − 1)(3 cos2 β − 1) 2  ✷✹  ✭✸✳✶✮  π 2 1  H  CP  13  C  CP  ❋✐❣✉r❡ ✸✳✷✿ ❆ ❝r♦ss ♣♦❧❛r✐③❛t✐♦♥ ♣✉❧s❡ s❡q✉❡♥❝❡✳ ❚❤❡ ✜rst ❜❧♦❝❦ r❡♣r❡s❡♥ts ❛ ❜❧♦❝❦s ❧❛❜❡❧❡❞ ❈P t❤❡ ❝r♦ss ♣♦❧❛r✐③❛t✐♦♥ ♣✉❧s❡s✳  π 2  ♣✉❧s❡✳ ❚❤❡  ✇❤❡r❡ β = θR − θ✳ ❇❡❝❛✉s❡ θ ✐s t✐♠❡ ❞❡♣❡♥❞❡♥t✱ s♦ ✐s β ✳ ❍♦✇❡✈❡r✱ t❤❡ ❛♥❣❧❡ ♦❢ r♦t❛t✐♦♥ ❝❛♥ ❜❡ ❝❤♦s❡♥ ❛s ❛♥ ❡①♣❡r✐♠❡♥t❛❧ ♣❛r❛♠❡t❡r✳ ❲❤❡♥ θR ✐s s❡t t♦ ❜❡ 54.74o ✱ t❤❡♥ t❤❡ str❡♥❣t❤ ♦❢ t❤❡ ✐♥t❡r❛❝t✐♦♥ ✇✐❧❧ ❜❡ ❛✈❡r❛❣❡❞ t♦ ③❡r♦✳ ❆ s❝❤❡♠❛t✐❝ ♦❢ ❛ ▼❆❙ s②st❡♠ ❝❛♥ ❜❡ s❡❡♥ ✐♥ ❋✐❣✉r❡ ✸✳✶ ❬✹❪✳  ✸✳✸  ❈r♦ss P♦❧❛r✐③❛t✐♦♥  ❈r♦ss ♣♦❧❛r✐③❛t✐♦♥ ✭❈P✮ s❡q✉❡♥❝❡s ❛r❡ ❛ ❝r✉❝✐❛❧ ♣❛rt ♦❢ t❡❝❤♥✐q✉❡s ✉s❡❞ t♦ st✉❞② ❜✐♦❧♦❣✐❝❛❧ ♠❛t❡r✐❛❧s✳ ❆♥ ✐♠♣♦rt❛♥t ❡❧❡♠❡♥t ✐♥ ♣r♦❜✐♥❣ t❤❡s❡ ♠❛t❡r✐❛❧s ✈✐❛ ◆▼❘ ♦❢t❡♥ ❧✐❡s ✐♥ ♣❡r❢♦r♠✐♥❣ ◆▼❘ ❡①♣❡r✐♠❡♥ts ♦♥ t❤❡ ❝❛r❜♦♥ ❛t♦♠s✳ ❚❤❡ ◆▼❘ ❛❝t✐✈❡ ❝♦♠♣♦♥❡♥t✱  13  ❈✱ ❤❛s ❛ ♥❛t✉r❛❧  ❛❜✉♥❞❛♥❝❡ ♦❢ s❧✐❣❤t❧② ❛❜♦✈❡ ✶✳✶ ♣❡r❝❡♥t ♠❛❦✐♥❣ t❤❡ s✐❣♥❛❧ t♦ ♥♦✐s❡ r❛t✐♦ ✈❡r② ❧♦✇✳ ❖♥❡ ✇❛② t♦ ✐♥❝r❡❛s❡ t❤❡ s✐❣♥❛❧ t♦ ♥♦✐s❡ ✐s t♦ ✐♥❝r❡❛s❡ t❤❡ ❛♠♦✉♥t ♦❢ ✐t ✇✐t❤  13  13  ❈ ✇✐t❤✐♥ t❤❡ s❛♠♣❧❡ ❜② ❞♦♣✐♥❣  ❈ ❡♥r✐❝❤❡❞ ♠❛t❡r✐❛❧✳ ❍♦✇❡✈❡r✱ t❤✐s ✐s ♦❢t❡♥ ♥♦t ♣♦ss✐❜❧❡ ♦r ♣r❛❝t✐❝❛❧✳ ❆♥ ❡❛s✐❡r ✇❛②  t♦ ✐♥❝r❡❛s❡ t❤❡ s✐❣♥❛❧ t♦ ♥♦✐s❡ ✐s t♦ ✉s❡ ❛ ❝r♦ss ♣♦❧❛r✐③✐♥❣ ♣✉❧s❡ s❡q✉❡♥❝❡✳ ■♥ ❛ ❈P s❡q✉❡♥❝❡✱ t❤❡ ❡♥❡r❣② ❣❛♣ ❢♦r  13  ❈ ✇✐t❤✐♥ t❤❡ s❛♠♣❧❡ ✐s ♠❛❞❡ t♦ ❜❡ t❤❡ s❛♠❡ ❛s t❤❛t ♦❢ 1 ❍✳ ❚❤✐s ❛❧❧♦✇s  ♠❛❣♥❡t✐③❛t✐♦♥ t♦ ❜❡ tr❛♥s❢❡rr❡❞ ❢r♦♠ t❤❡ ❛❜✉♥❞❛♥t ♥✉❝❧❡✐ t♦ t❤❡ ❞✐❧✉t❡ ♥✉❝❧❡✐ t❤❛t ❛r❡ ❜❡✐♥❣ ♦❜s❡r✈❡❞✳ ❈r♦ss ♣♦❧❛r✐③❛t✐♦♥ ✐s ♥♦t ❧✐♠✐t❡❞ t♦ 1 ❍✴13 ❈ ♥✉❝❧❡✐✱ ❛♥② ❛❜✉♥❞❛♥t ♥✉❝❧❡✉s ✇✐t❤ ❛ ❤✐❣❤ γ ❛♥❞ s❤♦rt T1 ✐♥ ❝❧♦s❡ ♣r♦①✐♠✐t② t♦ t❤❡ ❞✐❧✉t❡ t❛r❣❡t ❝♦✉❧❞ ❜❡ ✉s❡❞✳  ✷✺  ❚❤❡ ♣✉❧s❡ s❡q✉❡♥❝❡ ✉s❡❞ ✐s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✸✳✷✱ ❛♥❞ ❜❡❣✐♥s ✇✐t❤ ❛  π 2  ♣✉❧s❡ t♦ t❤❡ 1 ❍ t♦  ❝r❡❛t❡ ✲② ❛①✐s ♠❛❣♥❡t✐③❛t✐♦♥ ✐♥ t❤❡ 1 ❍ r♦t❛t✐♥❣ ❢r❛♠❡✳ ❆❢t❡r t❤✐s✱ ❛♥ ❘❋ ❝r♦ss ♣♦❧❛r✐③❛t✐♦♥ ✭❈P✮ ♣✉❧s❡ ✐s ❛♣♣❧✐❡❞✳ ❚❤✐s ♣✉❧s❡ ✐s r❡s♦♥❛♥t ❛♥❞ ❞✐r❡❝t❡❞ ❛❧♦♥❣ t❤❡ ✲② ❛①✐s ✇❤✐❝❤ ❧♦❝❦s t❤❡ s♣✐♥s ♦♥t♦ t❤❡ ❛①✐s✳ ❚❤❡ ❡✛❡❝t ♦❢ t❤✐s ✐s t❤❡ ❝r❡❛t✐♦♥ ♦❢ ❛ q✉❛♥t✉♠ ❡♥✈✐r♦♥♠❡♥t s✐♠✐❧❛r t♦ t❤❛t ♦❢ t❤❡ ❩❡❡♠❛♥ st❛t✐❝ ✜❡❧❞✳ ❚❤❡ ❡♥❡r❣② ❣❛♣ ❢♦r t❤✐s ♥❡✇ st❛t❡ ✐s ∆EH = γH BH ✇❤❡r❡  BH ✐s t❤❡ str❡♥❣t❤ ♦❢ t❤❡ ❘❋ ♠❛❣♥❡t✐❝ ✜❡❧❞ ❛♣♣❧✐❡❞ ✐♥ t❤❡ ♣✉❧s❡✳ ❙✐♠✉❧t❛♥❡♦✉s❧② t♦ t❤✐s ♣✉❧s❡✱ ❛ ❈P ♣✉❧s❡ ✐s ❛♣♣❧✐❡❞ t♦ t❤❡ ❝❛r❜♦♥ ♥✉❝❧❡✐✳ ❚❤❡ r❡s✉❧t✐♥❣ ❡♥❡r❣② ❣❛♣ ❢♦r t❤❡ 13 ❈ ✐s ∆EC = γC BH ✳ ❯♥❞❡r t❤❡ ❝♦♥❞✐t✐♦♥ BH =  γC BC γH  ✭✸✳✷✮  ✇❡ ❣❡t ✭✸✳✸✮  ∆EC = ∆EH .  ❚❤✐s ♠❛t❝❤✐♥❣ ♦❢ t❤❡ ❡♥❡r❣② ❧❡✈❡❧ s❡♣❛r❛t✐♦♥ ♦❢ t❤❡ r♦t❛t✐♥❣ ❢r❛♠❡ s♣✐♥ st❛t❡s ✐s ❝❛❧❧❡❞ ❍❛rt♠❛♥✲❍❛♥♥ ♠❛t❝❤✐♥❣✳ ❚❤✐s ❝♦♥❞✐t✐♦♥ ❛❧❧♦✇s ❛ tr❛♥s❢❡r ♦❢ ♠❛❣♥❡t✐③❛t✐♦♥ ❢r♦♠ t❤❡ ♣r♦t♦♥s t♦ t❤❡  13  ❈ ♥✉❝❧❡✐✳ ❚❤✐s tr❛♥s❢❡r ✐s ♠❡❞✐❛t❡❞ t❤r♦✉❣❤ ♠✉t✉❛❧ s♣✐♥ ✢✐♣s ✈✐❛ t❤❡ ❤❡t❡r♦♥✉❝❧❡❛r  ❞✐♣♦❧❛r ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ♣r♦t♦♥s ❛♥❞ ❝❛r❜♦♥ ♥✉❝❧❡✐✳ ❚❤❡ ❍❛♠✐❧t♦♥✐❛♥ ❢♦r t❤✐s s②st❡♠ ✐s x ˆ =H ˆZ + H ˆ HH + H ˆ HC + H ˆ CC + H ˆ pulse H .  ❍♦✇❡✈❡r✱ ❜❡❝❛✉s❡ t❤❡  ✭✸✳✹✮  C ✐s s♦ ❞✐❧✉t❡✱ ✇❡ ❝❛♥ ✐❣♥♦r❡ ✐ts ❤♦♠♦♥✉❝❧❡❛r ❞✐♣♦❧❛r ✐♥t❡r❛❝t✐♦♥✳ ▼♦✈✐♥❣ ✐♥t♦ t❤❡ r♦t❛t✐♥❣ ❢r❛♠❡✱ t❤❡ ❩❡❡♠❛♥ ✐♥t❡r❛❝t✐♦♥ t❡r♠ ✈❛♥✐s❤❡s ❛s ✇❡❧❧✳ ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t ✐♥ t❤❡ r♦t❛t✐♥❣ ❢r❛♠❡✱ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ❜❡❝♦♠❡s 13  x ˆ rot = H ˆ HH + H ˆ HC + H ˆ pulse,rot H  ✭✸✳✺✮  x ˆ pulse,rot H = ω1H  ✭✸✳✻✮  ✇❤❡r❡  IˆizH + ω1C SˆxC i H/C  ✇❤❡r❡ t❤❡ s✉❜s❝r✐♣t i r❡♣r❡s❡♥ts t❤❡ ith ♣r♦t♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ 13 ❈ ❛♥❞ ω1  = γH/C BH/C ✳  ❚❤❡ ♥❡①t st❡♣ ❢r♦♠ ❤❡r❡ ✐s t♦ ❞♦ ❛ ❢✉rt❤❡r ❢r❛♠❡ tr❛♥s❢♦r♠❛t✐♦♥ ✐♥t♦ ✇❤❛t ✐s t❡r♠❡❞ t❤❡ t♦❣❣❧✐♥❣ ❢r❛♠❡ ♦r ✐♥t❡r❛❝t✐♦♥ r❡♣r❡s❡♥t❛t✐♦♥✳ ❚❤❡ t♦❣❣❧✐♥❣ ❢r❛♠❡ ✐s ❛♥ ❡①t❡♥s✐♦♥ ❜② ❛♥❛❧♦❣② ♦❢ t❤❡ r♦t❛t✐♥❣ ❢r❛♠❡ ✇❤❡r❡ t❤❡ ❛♣♣❧✐❡❞ ♣✉❧s❡ ✐♥ t❤❡ r♦t❛t✐♥❣ ❢r❛♠❡ ✐s r❡♠♦✈❡❞ t❤r♦✉❣❤ ❛ s❡❝♦♥❞ r♦t❛t✐♦♥ tr❛♥s❢♦r♠❛t✐♦♥✳ ❚❤✐s ✇♦r❦s ✐♥ t❤❡ s❛♠❡ ✇❛② t❤❛t ❛❞♦♣t✐♥❣ ❛ ❢r❛♠❡ r♦t❛t✐♥❣ ❛r♦✉♥❞ t❤❡ ❩❡❡♠❛♥ ❛①✐s ❛t t❤❡ ▲❛r♠♦r ❢r❡q✉❡♥❝②✱ t❤❡ t♦❣❣❧✐♥❣ ❢r❛♠❡ r♦t❛t❡s ❛❜♦✉t t❤❡ ❛♣♣❧✐❡❞ ✜❡❧❞ ✐♥ t❤❡ r♦t❛t✐♥❣ ❢r❛♠❡✳ ❚❤✐s r❡q✉✐r❡s t❤❡ ❝r❡❛t✐♦♥ ♦❢ ❛ ❞♦✉❜❧② r♦t❛t✐♥❣ ❢r❛♠❡ ❢♦r ✷✻  ❜♦t❤ t❤❡ ♣r♦t♦♥s ❛♥❞ ❝❛r❜♦♥ ♥✉❝❧❡✐✱ ❛♥❞ ✐s ❝r❡❛t❡❞ ❜② ❛♣♣❧②✐♥❣ t❤❡ r♦t❛t✐♦♥  ˆ = exp−iωH R  i Iix  ˆ  expiωC Sx  ✭✸✳✼✮  t♦ t❤❡ r♦t❛t✐♥❣ ❢r❛♠❡ ❍❛♠✐❧t♦♥✐❛♥✱ ✇❤✐❝❤ ❣✐✈❡s t❤❡ t♦❣❣❧✐♥❣ ❢r❛♠❡ r❡♣r❡s❡♥t❛t✐♦♥  ˆ tog = R ˆ −1 H ˆ rot R ˆ−ω H  Iˆix − ωC Sˆx. .  ✭✸✳✽✮  i  ❚❤❡ ❡①tr❛ Iˆix ❛♥❞ Sˆx t❡r♠s ❝❛♥❝❡❧ ♣❡r❢❡❝t❧② ✇✐t❤ t❤❡ ♣✉❧s❡ t❡r♠s ❢r♦♠ t❤❡ r♦t❛t✐♥❣ ❢r❛♠❡ ❍❛♠✐❧t♦♥✐❛♥✳ ❚❤✐s ❛❧❧♦✇s t❤❡ t♦❣❣❧✐♥❣ ❢r❛♠❡ ❍❛♠✐❧t♦♥✐❛♥ t♦ ❜❡ r❡✇r✐tt❡♥ ❛s  ˆ tog = H ˆ tog + H ˆ tog H HH HC ✇❤❡r❡  ˆ tog = − 1 H HH 2  CijHH (ˆIi · ˆIj − 3Iˆix Iˆjx )  ✭✸✳✾✮  ✭✸✳✶✵✮  i<j  ❛♥❞  ˆ tog = − H HC  Iˆiz Sˆz +  CiHC [( i  i  Iˆiy Sˆz  + ( i  Iˆiy Sˆy ) cos(ωH − ωC )t i  Iˆiz Sˆy ) sin(ωH − ωC )t]  ✭✸✳✶✶✮  i  ✇✐t❤ ❝♦❡✣❝✐❡♥ts C ❜❡✐♥❣ 2 1 µo γH (cos2 θij − 1) 3 2 4π rij 1 µo γH γC (cos2 θi − 1). = 3 2 4π rij  CijHH =  ✭✸✳✶✷✮  CiHC  ✭✸✳✶✸✮  ■♥ t❤❡ ❛❜♦✈❡ ❝♦❡✣❝✐❡♥ts✱ ri r❡♣r❡s❡♥ts t❤❡ ✐♥t❡r♥✉❝❧❡❛r ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ i❚❤ ♣r♦t♦♥ ❛♥❞ t❤❡ ❝❛r❜♦♥ ♥✉❝❧❡✉s✳ θi r❡♣r❡s❡♥ts t❤❡ ♣♦❧❛r ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ ✈❡❝t♦r ri ❛♥❞ t❤❡ ❛♣♣❧✐❡❞ ✜❡❧❞✳ ▲✐❦❡✇✐s❡✱ rij r❡♣r❡s❡♥ts ✐♥t❡r♥✉❝❧❡❛r ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ ♣r♦t♦♥s i ❛♥❞ j ❛♥❞ θij r❡♣r❡s❡♥ts t❤❡ ♣♦❧❛r ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ ✈❡❝t♦r rij ❛♥❞ t❤❡ ❛♣♣❧✐❡❞ ✜❡❧❞✳ ❲❤✐❧❡ t❤❡ ❤♦♠♦♥✉❝❧❡❛r ❝♦✉♣❧✐♥❣ ❜❡t✇❡❡♥ ♣r♦t♦♥s ❛✛❡❝ts t❤❡ r❡❞✐str✐❜✉t✐♦♥ ♦❢ ♠❛❣♥❡t✐③❛t✐♦♥✱ ✐t ❞♦❡s ♥♦t r❡♣r❡s❡♥t ❛ s✐❣♥✐✜❝❛♥t ✐♥t❡r❛❝t✐♦♥ ❢♦r ❝r♦ss✲ ♣♦❧❛r✐③❛t✐♦♥✳ ❚❤❡ ❤❡t❡r♦♥✉❝❧❡❛r ❝♦✉♣❧✐♥❣ ♣❧❛②s t❤❡ ❞♦♠✐♥❛♥t r♦❧❡✱ ❜✉t ❜❡❝❛✉s❡ ♦❢ t❤❡ ❍❛rt♠❛♥✲❍❛❤♥ ❝♦♥❞✐t✐♦♥ ♦❢ ωH ≈ ωC ✱ t❤❡ t✐♠❡ ❞❡♣❡♥❞❡♥❝❡ ✐s r❡♠♦✈❡❞ ❢r♦♠ t❤❡ ❤❡t❡r♦♥✉❝❧❡❛r ❍❛♠✐❧t♦♥✐❛♥✳ ❚❤❡ t❡r♠s ♦❢ t❤❡ ✷✼  r❡s✉❧t✐♥❣ ❍❛♠✐❧t♦♥✐❛♥ ❛r❡ ♦❢ t❤❡ ❢♦r♠ Iˆz Sˆz ❛♥❞ Iˆy Sˆy ✳ ❚❤❡ ❧❛tt❡r t❡r♠ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s Iˆ+ Sˆ− + Iˆ− Sˆ+ ✇❤✐❝❤ ✐s t❤❡ ❇ t❡r♠ ❢r♦♠ t❤❡ ❞✐♣♦❧❛r ❝♦✉♣❧✐♥❣ ✐♥t❡r❛❝t✐♦♥ t❤❛t ♣r♦♠♦t❡s ♠✉t✉❛❧ s♣✐♥ ✢✐♣s ❜❡t✇❡❡♥ t❤❡ I ❛♥❞ S s♣✐♥s✳ ❇❡❝❛✉s❡ t❤❡ ❡♥❡r❣② ❧❡✈❡❧ s❡♣❛r❛t✐♦♥ ✐s t❤❡ s❛♠❡ ❢♦r ❜♦t❤ s♣❡❝✐❡s ✉♥❞❡r t❤❡ ❍❛rt♠❛♥✲❍❛❤♥ ❝♦♥❞✐t✐♦♥✱ t❤❡r❡ ✐s ♥♦ ♥❡t ❝❤❛♥❣❡ ♦❢ ♠❛❣♥❡t✐③❛t✐♦♥ ♦r ❧♦ss ♦❢ ❡♥❡r❣② ✇❤❡♥ t❤❡r❡ ✐s ❛♥ ❡①❝❤❛♥❣❡ ♦❢ ♠❛❣♥❡t✐③❛t✐♦♥ ❢r♦♠ t❤❡ ♣r♦t♦♥s t♦ t❤❡ ❝❛r❜♦♥ ♥✉❝❧❡✐✳ ❚❤❡ r❡s✉❧t ♦❢ t❤✐s ✐s t❤❛t t❤❡ ❝❛r❜♦♥ ❛♥❞ ♣r♦t♦♥s ❡q✉✐❧✐❜r✐❛t❡ t♦ ❡q✉❛❧ ♠❛❣♥❡t✐③❛t✐♦♥s✱ ❛♥❞ ❜❡❝❛✉s❡ t❤❡ ❝❛r❜♦♥ ♣♦♣✉❧❛t✐♦♥ ✐s ♥❡❣❧✐❣✐❜❧❡ ❝♦♠♣❛r❡❞ t♦ t❤❛t ♦❢ t❤❡ ♣r♦t♦♥s✱ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♠❛❣♥❡t✐③❛t✐♦♥ ✐s ❛♣♣r♦①✐♠❛t❡❧② t❤❛t ♦❢ t❤❡ ♣r♦t♦♥✳ ■♥✐t✐❛❧ ✐♥t✉✐t✐♦♥ s❤♦✉❧❞ ❧❡❛✈❡ ♦♥❡ s❦❡♣t✐❝❛❧ ♦❢ t❤✐s r❡s✉❧t✱ ❛❢t❡r ❛❧❧ t❤❡ ♣♦♣✉❧❛t✐♦♥ ♦❢ t❤❡ ❞✐❧✉t❡  C ✐s st✐❧❧ q✉✐t❡ s♠❛❧❧✱ ❛♥❞ ❛❧❧ t❤❛t ❤❛s ❤❛♣♣❡♥❡❞ ✐s t❤❛t t❤❡ ♣❡r❝❡♥t❛❣❡ ♦❢ ◆▼❘ ❛❝t✐✈❡ ❝❛r❜♦♥s t❤❛t ❛r❡ ✐♥ t❤❡ ❤✐❣❤ ❡♥❡r❣② st❛t❡ ✐s ❡q✉❛❧ t♦ t❤❛t ♦❢ t❤❡ ❡①❝✐t❡❞ ♣r♦t♦♥s✳ ❖r ♠♦r❡ s✉❝❝✐♥❝t❧②✱ ✇♦✉❧❞♥✬t ❛ ❞✐r❡❝t ♣♦❧❛r✐③❛t✐♦♥ ❝❛r❜♦♥ s♣❡❝tr✉♠ ❛❝❝♦♠♣❧✐s❤ t❤❡ s❛♠❡ r❡s✉❧t❄ ❚❤❡ r❡❛s♦♥ ❢♦r t❤✐s t❡❝❤♥✐q✉❡ ❝♦♠❡s ❞♦✇♥ t♦ t❤❡ ❝♦♥st❛♥ts✳ ❚❤❡ γ ❢❛❝t♦r ❢♦r ♣r♦t♦♥s ✐s ❛♣♣r♦①✐♠❛t❡❧② ❛ ❢❛❝t♦r ♦❢ ✹ ❣r❡❛t❡r t❤❛♥ t❤❛t ♦❢ ❝❛r❜♦♥ ♠❡❛♥✐♥❣ t❤❛t t❤❡② ❛r❡ ♠✉❝❤ ♠♦r❡ ❧✐❦❡❧② t♦ ❜❡ ♣♦❧❛r✐③❡❞ ✐♥ t❤❡r♠❛❧ ❡q✉✐❧✐❜r✐✉♠✳ ❚❤❡r❡ ✐s ❛❧s♦ ❛ s❡❝♦♥❞❛r② ❛❞✈❛♥t❛❣❡ t❤❛t t❤❡ T1 ❢♦r ♣r♦t♦♥s ✐s ❣❡♥❡r❛❧❧② ♠✉❝❤ s❤♦rt❡r t❤❛♥ t❤❛t ♦❢ ❝❛r❜♦♥✳ ❚❤✐s ❛❧❧♦✇s t❤❡ ❡①♣❡r✐♠❡♥t t♦ ❜❡ r❡♣❡❛t❡❞ ♠♦r❡ ❢r❡q✉❡♥t❧②✱ r❡s✉❧t✐♥❣ ✐♥ ❛♥ ✐♥❝r❡❛s❡ ✐♥ t❤❡ s✐❣♥❛❧ t♦ ♥♦✐s❡ r❛t✐♦ ❬✾❪✳  ✸✳✹  13  ❊❧❡❝tr♦s♣✐♥♥✐♥❣  ❊❧❡❝tr♦s♣✐♥♥✐♥❣ ✐s ❛ ♥♦✈❡❧ t❡❝❤♥✐q✉❡ ❝❛♣❛❜❧❡ ♦❢ ♣r♦❞✉❝✐♥❣ ♥❛♥♦s❝❛❧❡ ✜❜❡rs ❢r♦♠ ♣♦❧②♠❡rs t❤❛t ❤❛s r❡❝❡♥t❧② ❝♦♠❡ ✐♥t♦ ✈♦❣✉❡ ✐♥ ❡♥❣✐♥❡❡r✐♥❣ r❡s❡❛r❝❤✳ ❯♥❧✐❦❡ ♠♦st t❡❝❤♥✐q✉❡s ❢♦r ♥❛♥♦✲ ♠❛t❡r✐❛❧s✱ ❡❧❡❝tr♦s♣✐♥♥✐♥❣ ❤❛s ❜❡❡♥ ✉s❡❞ s✐♥❝❡ t❤❡ ❡❛r❧② t✇❡♥t✐❡t❤ ❝❡♥t✉r②✳ ❚❤❡ ♦r✐❣✐♥s ♦❢ t❤❡ t❡❝❤♥✐q✉❡ ❞❛t❡ ❜❛❝❦ ❤✉♥❞r❡❞s ♦❢ ②❡❛rs t♦ ❛♥ ❛♥❝✐❧❧❛r② ♣❤❡♥♦♠❡♥♦♥❀ ❡❧❡❝tr♦s♣r❛②✐♥❣✳ ❆ ❜♦♦❦ ♣✉❜❧✐s❤❡❞ ✐♥ ✶✻✷✽ ❜② ❲✐❧❧✐❛♠ ●✐❧❜❡rt ❞❡t❛✐❧❡❞ ❛ ♣❤❡♥♦♠❡♥♦♥ ✇❤❡r❡ ✇❛t❡r ✇✐t❤ ❛ ✈♦❧t❛❣❡ s♦✉r❝❡ ✐♥ ❛ ❝❛♣✐❧❧❛r② ✐♥ ♣r♦①✐♠✐t② ♦❢ ❛ ❣r♦✉♥❞❡❞ s♦✉r❝❡ ✇♦✉❧❞ ❤❛✈❡ ❛ ❝♦♥❡ ♦❢ ✇❛t❡r ❢♦r♠ ❛t t❤❡ t✐♣ t❤❛t ✇♦✉❧❞ ❡❥❡❝t s♠❛❧❧ ❞r♦♣❧❡ts t♦✇❛r❞s t❤❡ ✈♦❧t❛❣❡ s♦✉r❝❡✳ ❚❤❡ t❡❝❤♥✐q✉❡ ♦❢ ❡❧❡❝tr♦✲ s♣✐♥♥✐♥❣ ✐ts❡❧❢ ❞✐❞ ♥♦t ❝♦♠❡ ❛r♦✉♥❞ ✉♥t✐❧ ✶✾✵✷ ✇❤❡♥ ✐t ✇❛s ❢♦✉♥❞ t❤❛t ✉s✐♥❣ t❤❡ t❡❝❤♥✐q✉❡s ♦❢ ❡❧❡❝tr♦s♣r❛②✐♥❣ ✉s✐♥❣ ❛ ✈✐s❝♦✉s ♣♦❧②♠❡r s♦❧✉t✐♦♥ ✐♥st❡❛❞ ♦❢ ✇❛t❡r r❡s✉❧ts ✐♥ t❤❡ ❡❥❡❝t✐♦♥ ♦❢ ❛ s✐♥❣❧❡ ❥❡t ✐♥st❡❛❞ ♦❢ ♦❢ ❞r♦♣❧❡ts✳ ❚❤❡ ❡❧❡❝tr♦s♣✐♥♥✐♥❣ t❡❝❤♥✐q✉❡✱ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✸✳✸✱ ✐♥✈♦❧✈❡s ❣❡♥❡r❛t✐♥❣ ❛ ❤✐❣❤ ❡❧❡❝tr✐❝ ♣♦✲ t❡♥t✐❛❧ ❜❡t✇❡❡♥ ❛ ♣♦❧②♠❡r s♦❧✉t✐♦♥ ✐♥ ❛ r❡s❡r✈♦✐r✱ t②♣✐❝❛❧❧② ❛ ❣❧❛ss s②r✐♥❣❡ ♦r ♣✐♣❡tt❡ ✇✐t❤ ❛ ❝❛♣✐❧❧❛r② ♦r ♥❡❡❞❧❡ ❛t t❤❡ t✐♣ ❛♥❞ ❛ ❝♦♥❞✉❝t✐♥❣ ❝♦❧❧❡❝t✐♦♥ ♣❧❛t❡✳ ❆s t❤❡ ✈♦❧t❛❣❡ ✐♥❝r❡❛s❡s  ✷✽  ❋✐❣✉r❡ ✸✳✸✿ ❈♦♠♣♦♥❡♥ts ♦❢ ❛♥ ❡❧❡❝tr♦s♣✐♥♥✐♥❣ ❛♣♣❛r❛t✉s✳ ■♠❛❣❡ ❢r♦♠ ●❛♥❞❤✐ ❬✶✵❪✳ ❢r♦♠ ✵✱ t❤❡ ❞r♦♣❧❡t ❛t t❤❡ t✐♣ ♦❢ t❤❡ ♥❡❡❞❧❡ ♦r ❝❛♣✐❧❧❛r② ❜❡❣✐♥s t♦ ❞❡❢♦r♠ ✐♥t♦ ❛ ❝♦♥❡ ♣♦✐♥t✐♥❣ t♦✇❛r❞s t❤❡ ♣❧❛t❡✳ ❆t ❛ ❝r✐t✐❝❛❧ ✈♦❧t❛❣❡ t❤❡ ❡❧❡❝tr♦♥✐❝ ❢♦r❝❡ ♦✈❡r❝♦♠❡s t❤❡ s✉r❢❛❝❡ t❡♥s✐♦♥ ♦❢ t❤❡ ❝♦♥❡ ❛♥❞ ❛ ❥❡t ✐s ♣r♦❞✉❝❡❞✳ ❚❤❡ ❞✐❛♠❡t❡r ♦❢ t❤✐s ❥❡t ❞❡❝r❡❛s❡s ✉♥❞❡r ❡❧❡❝tr♦❤②❞r♦❞②♥❛♠✐❝ ❢♦r❝❡s✱ ❛♥❞ ✉♥❞❡r❣♦❡s ❛ ❞❡st❛❜✐❧✐③❛t✐♦♥ t❤❛t r❡s✉❧ts ✐♥ ❛ r❛♣✐❞ ✇❤✐♣♣✐♥❣ ♠♦t✐♦♥ ✇❤✐❝❤ r❡s✉❧ts ✐♥ ❡①t❡♥s✐✈❡ str❡t❝❤✐♥❣ ♦♥ ✐t✬s ♣❛t❤ t♦ t❤❡ ♣❧❛t❡✳ ❚❤❡ str❡t❝❤✐♥❣ ♣r♦❝❡ss r❡s✉❧ts ✐♥ ❛ r❛♣✐❞ ❡✈❛♣♦r❛t✐♦♥ ♦❢ t❤❡ s♦❧✈❡♥t ❞✉❡ t♦ t❤❡ ✐♥❝r❡❞✐❜❧② ❤✐❣❤ ❛s♣❡❝t r❛t✐♦ ♦❢ t❤❡ ❥❡t✱ ✇❤✐❝❤ ❧❡❛❞s t♦ ❛ ❢✉rt❤❡r ❞✐❛♠❡t❡r r❡❞✉❝t✐♦♥✳ ❚❤❡ ❞r✐❡❞ ✜❜❡rs ❛r❡ ❞❡♣♦s✐t❡❞ ✐♥ ❛ r❛♥❞♦♠ ❝♦♥✜❣✉r❛t✐♦♥ ♦♥ t❤❡ ❝♦❧❧❡❝t✐♦♥ ♣❧❛t❡ ✭❛❧t❤♦✉❣❤ ♠❡t❤♦❞s ♦❢ ❛❧✐❣♥✐♥❣ ✜❜❡rs ❞♦ ❡①✐st✮ r❡s✉❧t✐♥❣ ✐♥ ❛ t❤✐♥ ♠❛t ♦❢ ♥❛♥♦✜❜❡rs✳ ❚❤❡ ✜❜❡r ❞✐❛♠❡t❡r ❛♥❞ t❤❡ t❤✐❝❦♥❡ss ♦❢ t❤❡ s❝❛✛♦❧❞s ❝❛♥ ❜❡ ❝♦♥tr♦❧❧❡❞ ❜② ✈❛r②✐♥❣ ❛ ✇✐❞❡ ✈❛r✐❡t② ♦❢ ♣❛r❛♠❡t❡rs ✐♥❝❧✉❞✐♥❣✱ ❜✉t ♥♦t ❧✐♠✐t❡❞ t♦✱ t❤❡ s♦❧✉t✐♦♥ ❝♦♥❝❡♥tr❛t✐♦♥✱ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♥❡❡❞❧❡ ❛♥❞ t❤❡ ♣❧❛t❡✱ ❛♥❞ t❤❡ ✈♦❧t❛❣❡ ❬✶✵❪✳  ✷✾  ❈❤❛♣t❡r ✹  Pr♦t❡✐♥s  ❱✐rt✉❛❧❧② ❡✈❡r② ♣❤②s✐❝❛❧ ❛s♣❡❝t ♦❢ ❧✐✈✐♥❣ ♦r❣❛♥✐s♠s ❛r❡ ❡✐t❤❡r ❛✛❡❝t❡❞ ♦r ❞❡t❡r♠✐♥❡❞ ❜② ♣r♦t❡✐♥s✳ ❲❤✐❧❡ ♥✉❝❧❡✐❝ ❛❝✐❞s s✉❝❤ ❛s ❉◆❆ ❛♥❞ ❘◆❆ ♠❛② ❜❡ t❤❡ ❜❧✉❡♣r✐♥ts ❢♦r ❧✐❢❡✱ ♣r♦t❡✐♥s ❛r❡ t❤❡ ♠❛t❡r✐❛❧s t❤❛t ❜✉✐❧❞ ✐t ❛♥❞ t❤❡ ❡♥❣✐♥❡s t❤❛t r✉♥ ✐t✳ ❚❤❡ r♦❧❡ ♦❢ ♣r♦t❡✐♥s ✐s ❛s ❞✐✈❡rs❡ ❛s ❧✐❢❡ ✐ts❡❧❢✳ ❚❤❡② ✜❧❧ t❤❡ ❛❞♠✐♥✐str❛t✐✈❡ r♦❧❡s ✐♥ ♦r❣❛♥✐s♠s s✉❝❤ ❛s tr❛♥s♣♦rt✐♥❣ ♠❛t❡r✐❛❧s ❛♥❞ ❝♦♥tr♦❧❧✐♥❣ ❝❡❧❧✉❧❛r ❢✉♥❝t✐♦♥s✱ t❤❡② ♣r♦✈✐❞❡ str✉❝t✉r❛❧ r♦❧❡s s✉❝❤ ❛s ❤❛✐r✱ ♥❛✐❧s✱ t❡♥❞♦♥s✱ ❛♥❞ ♠✉s❝❧❡✳ ❚❤❡② ❛❧s♦ s❡r✈❡ ❛s t❤❡ ♠❛t❡r✐❛❧ ❢♦r ♥♦✈❡❧ ❜✐♦❧♦❣✐❝❛❧ ❢✉♥❝t✐♦♥s✱ s✉❝❤ ❛s s✐❧❦s✱ ❤❛❣✜s❤ s❧✐♠❡✱ ❛♥❞ ♣❤❡r♦♠♦♥❡ ❞❡t❡❝t♦rs✳  ✹✳✶  ■♥tr♦❞✉❝t✐♦♥  ❚❤❡ ❞✐✈❡rs❡ ♣r♦♣❡rt✐❡s ♦❢ ♣r♦t❡✐♥s ❛r❡ ❢❛❝✐❧✐t❛t❡❞ t❤r♦✉❣❤ ✷✵ ❛♠✐♥♦ ❛❝✐❞s✱ t❤❡ ❜✉✐❧❞✐♥❣ ❜❧♦❝❦s ♦❢ ♣r♦t❡✐♥s✳ ❚❤❡s❡ ✷✵ ❛♠✐♥♦ ❛❝✐❞s ❢♦r♠ ❧✐♥❡❛r ♣♦❧②♠❡r✐❝ ❝❤❛✐♥s t❤❛t ❢♦r♠ t❤❡ s❡❝♦♥❞❛r② ❛♥❞ t❡rt✐❛r② str✉❝t✉r❡ ❛♥❞ t❤✉s ❞✐❝t❛t❡ t❤❡ ♣r♦♣❡rt✐❡s ❛♥❞ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♣r♦t❡✐♥s✳ ❆♠✐♥♦ ❛❝✐❞s s❤❛r❡ ❛ ❝♦♠♠♦♥ ❜❛❝❦❜♦♥❡ ✇✐t❤ t✇♦ ♣♦❧②♠❡r✐③❛t✐♦♥ s✐t❡s✱ r❡s✉❧t✐♥❣ ✐♥ t❤❡ ❢♦r♠❛✲ t✐♦♥ ♦❢ str✐❝t❧② ❧✐♥❡❛r ♠♦❧❡❝✉❧❡s✳ ❚❤✐s ❜❛❝❦❜♦♥❡ ❝♦♥t❛✐♥s t✇♦ ❝❛r❜♦♥s✱ ❦♥♦✇♥ ❛s t❤❡ ❝❛r❜♦♥②❧✱ ✇❤✐❝❤ s❡r✈❡s ❛s ♦♥❡ ♦❢ t❤❡ ♣♦❧②♠❡r✐③❛t✐♦♥ s✐t❡s✱ ❛♥❞ t❤❡ α ❝❛r❜♦♥✳ ❲❤❛t ♠❛❦❡s t❤❡ ❛♠✐♥♦ ❛❝✐❞s ❞✐✛❡r❡♥t ✐s ❛ s✐❞❡ ❝❤❛✐♥ ✇❤✐❝❤ ✐s ❜♦♥❞❡❞ t♦ t❤❡ α ❝❛r❜♦♥✱ ❛s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✹✳✶✱ ❛♥❞ ✐♥ t❤❡ ❝❛s❡ ♦❢ ♣r♦❧✐♥❡ ✐s ❛❧s♦ ❜♦♥❞❡❞ t♦ t❤❡ ♥✐tr♦❣❡♥ ❛s ✇❡❧❧✳ ❲✐t❤ t❤❡ ❡①❝❡♣t✐♦♥ ♦❢ ❣❧②❝✐♥❡✱ ❛❧❧ ♦❢ t❤❡ ✷✵ ❝♦♠♠♦♥ ❛♠✐♥♦ ❛❝✐❞s ❝♦♥t❛✐♥ ❛❞❞✐t✐♦♥❛❧ ❝❛r❜♦♥s ♦♥ t❤❡✐r s✐❞❡ ❝❤❛✐♥s✳ ❚❤❡s❡ ❛r❡ ❧❛❜❡❧❡❞ ❛s β, γ, δ, , ζ ✇❤❡r❡ t❤❡ ❝❧♦s❡st ❝❛r❜♦♥ t♦ t❤❡ α ✇✐❧❧ ❜❡ t❤❡ β ❝❛r❜♦♥✱ ❛♥❞ s♦ ♦♥✳ ✸✵  ❋✐❣✉r❡ ✹✳✶✿ ❆ ❝♦♥♥❡❝t✐♦♥ ♦❢ t✇♦ ✉♥s♣❡❝✐✜❡❞ ❛♠✐♥♦ ❛❝✐❞s✳ ❆♠✐♥♦ ❛❝✐❞s ❛r❡ ❞✐✛❡r❡♥t✐❛t❡❞ ❜② t❤❡✐r s✐❞❡❝❤❛✐♥s ✭❘✮✳ ■♠❛❣❡ ❢r♦♠ ❉❡♣❡✇ ❬✽❪✳ ❙♦♠❡ ❛♠✐♥♦ ❛❝✐❞s ❤❛✈❡ ♥♦♥ ❧✐♥❡❛r s✐❞❡ ❝❤❛✐♥s✳ ■♥ t❤✐s ❝❛s❡ ❛ ❞✐✛❡r❡♥t ♥❛♠✐♥❣ ❝♦♥✈❡♥t✐♦♥ ✐s ✉s❡❞✳ ❚❤❡ ♣❡♣t✐❞❡ ❜♦♥❞ ✐s t❤❡ ❜♦♥❞ ❜❡t✇❡❡♥ t❤❡ ❝❛r❜♦♥②❧ ❝❛r❜♦♥ ❛♥❞ t❤❡ ♥✐tr♦❣❡♥ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛♠✐♥♦ ❛❝✐❞ ❛♥❞ s❡r✈❡s t♦ ❧✐♥❦ ❛♠✐♥♦ ❛❝✐❞s t♦❣❡t❤❡r✳ ❚❤✐s ✐s ❛ ✉♥✐q✉❡ ❜♦♥❞ ❜❡❝❛✉s❡ ♦❢ ✐ts ♣❛rt✐❛❧ ❞♦✉❜❧❡ ❝❤❛r❛❝t❡r✐st✐❝ t❤❛t ❤❛s ❛ r❡s♦♥❛♥❝❡ ✇✐t❤ t❤❡ ❝❛r❜♦♥②❧ ✈❛❧❡♥❝② C O ✳ ❚❤✐s r❡s✉❧ts ✐♥ t❤❡ ♣❡♣t✐❞❡ ✉♥✐t ❜❡✐♥❣ ♣❧❛♥❛r✳ ❚❤✐s r❡str✐❝ts t❤❡ ❛t♦♠s ❢♦r♠✐♥❣ t❤❡ ♣❡♣t✐❞❡ ✉♥✐t✱ ❛♥❞ ❛❝ts ❛s ❛ r✐❣✐❞ str✉❝t✉r❡✳ ■♥ ♠♦st ❝❛s❡s✱ t❤❡ ♠♦st ❢❛✈♦r❛❜❧❡ st❛t❡ ✐s trans ✇❤❡r❡ t❤❡ ❜♦♥❞ r♦t❛t✐♦♥ ✐s ω = 180✳ ❍♦✇❡✈❡r✱ ✐♥ s♦♠❡ ❝❛s❡s ♣❛rt✐❝✉❧❛r❧② ✐♥ t❤❛t ♦❢ ❜♦♥❞s ✇✐t❤ t❤❡ ♣r♦❧✐♥❡ r❡s✐❞✉❡✱ t❤❡ ♠♦st ❢❛✈♦r❛❜❧❡ ❡♥❡r❣② st❛t❡ ✐s t❤❛t ♦❢ t❤❡ cis ❝♦♥❢♦r♠❛t✐♦♥✱ ✇❤❡r❡ t❤❡ ❜♦♥❞ r♦t❛t✐♦♥ ✐s  ω = 0✳  ✹✳✷  ❙❡❝♦♥❞❛r② ❙tr✉❝t✉r❡  ❚❤❡ ♣r✐♠❛r② str✉❝t✉r❡ ♦❢ ❛ ♣r♦t❡✐♥ ✐s t❤❡ ❝♦✈❛❧❡♥t str✉❝t✉r❡ ❞❡✜♥❡❞ ❜② t❤❡ ❛♠✐♥♦ ❛❝✐❞ s❡✲ q✉❡♥❝❡✳ ■♥ ♣r♦t❡✐♥s✱ ❝❡rt❛✐♥ ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❛♠✐♥♦ ❛❝✐❞s ❝❛♥ ❛❧❧♦✇ ✢❡①✐❜✐❧✐t② ♦❢ t❤❡ ❝❤❛✐♥✱ ✇❤✐❧❡ ♦t❤❡r ❝♦♠❜✐♥❛t✐♦♥s ❝❛♥ r❡str✐❝t ♠♦t✐♦♥ r❡s✉❧t✐♥❣ ✐♥ r✐❣✐❞ s❡❝t✐♦♥s✳ ❚❤✐s ❤❛♣♣❡♥s ❜❡✲ ❝❛✉s❡ t❤❡r❡ ✐s s♦♠❡ ❢r❡❡❞♦♠ ✐♥ t❤❡ ♠❛♥♥❡r ✐♥ ✇❤✐❝❤ t❤❡ ❝♦✈❛❧❡♥t ❜♦♥❞s ❛r❡ ❝♦♥✜❣✉r❡❞✳ ❆t r♦♦♠ t❡♠♣❡r❛t✉r❡ ❜♦♥❞ ❧❡♥❣t❤s ❛♥❞ ❜♦♥❞ ❛♥❣❧❡s ❝❛♥ ✈❛r② ❜② ±0.05 ❛♥❞ ±5o r❡s♣❡❝t✐✈❡❧②✳ ❍♦✇❡✈❡r✱ ❡♥❡r❣❡t✐❝❛❧❧② ❢❛✈♦r❛❜❧❡ ❝♦♥❢♦r♠❛t✐♦♥s ❡①✐st ❞✉❡ t♦ ♥♦♥✲❝♦✈❛❧❡♥t ✐♥t❡r❛❝t✐♦♥s ❛s ✇❡❧❧✳ ❚❤❡s❡ ❝♦♥❢♦r♠❛t✐♦♥s✱ s❤♦✇♥ ✐♥ ❚❛❜❧❡ ✹✳✶✱ ❣♦✈❡r♥ ✇❤❛t ✐s ❦♥♦✇♥ ❛s t❤❡ s❡❝♦♥❞❛r② str✉❝t✉r❡ ♦❢ ♣r♦t❡✐♥s✳ ❚❤✐s ❧❡✈❡❧ ♦❢ str✉❝t✉r❡ ❝❛♥ ❤❛✈❡ ❛ ❞r❛♠❛t✐❝ ❡✛❡❝t ♦♥ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝s ❛♥❞ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ♣r♦t❡✐♥✱ ♠♦r❡ t❤❛♥ t❤❡ s♣❡❝✐✜❝ s❡q✉❡♥❝❡ ✐ts❡❧❢ ✐♥ s♦♠❡ ❝❛s❡s✳ ❈♦♠♣✉t❛t✐♦♥❛❧ ✸✶  ❙tr✉❝t✉r❡ r✐❣❤t ❤❛♥❞❡❞ α ❤❡❧✐① 310 ❤❡❧✐① 31 ❤❡❧✐① ❛♥t✐ ♣❛r❛❧❧❡❧ β ✲s❤❡❡t ♣❛r❛❧❧❡❧ β s❤❡❡t  φ(◦ ) ✲✺✼ ✲✹✾ ✲✽✵ ✲✶✸✾ ✲✶✶✾  ψ(◦ ) ✲✹✼ ✲✷✻ ✶✺✵ ✶✸✺ ✶✶✸  ω(◦ ) ✶✽✵ ✶✽✵ ✶✽✵ ✲✶✼✽ ✶✽✵  ❚❛❜❧❡ ✹✳✶✿ ❚♦rs✐♦♥ ❛♥❣❧❡s ❢♦r ✈❛r✐♦✉s s❡❝♦♥❞❛r② str✉❝t✉r❡s ❬✶✶❪✳ s✐♠✉❧❛t✐♦♥s ♦❢ ♣r♦t❡✐♥ ❢♦❧❞✐♥❣ ❛r❡ ♠❛❞❡ ✐♥ ❛♥ ❛tt❡♠♣t t♦ ✜♥❞ t❤❡ ❡①❛❝t s❡❝♦♥❞❛r② str✉❝t✉r❡s t❤❛t ♣r♦t❡✐♥s ❢❛✈♦r✳ ◆▼❘ ✐s ❡①tr❡♠❡❧② s❡♥s✐t✐✈❡ t♦ ♠♦❧❡❝✉❧❛r ❝♦♥✜❣✉r❛t✐♦♥s✳ ❚❤✐s ♠❛❦❡s ✐t ❛ ♣❡r❢❡❝t t♦♦❧ ❢♦r st✉❞②✐♥❣ ♣r♦t❡✐♥s✳ ◆♦t ♦♥❧② ✇✐❧❧ ❡❛❝❤ ❝❛r❜♦♥ ❤❛✈❡ ❛ ✇❡❧❧ ❞❡✜♥❡❞ ♣❡❛❦ ✭✐❞❡❛❧❧②✮✱ ❜✉t t❤❡ ❝❤❡♠✐❝❛❧ s❤✐❢t ❝❛♥ ❜❡ ✉s❡❞ t♦ ❤❡❧♣ ✜♥❞ t❤❡ t♦rs✐♦♥ ❛♥❣❧❡s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❛♠✐♥♦ ❛❝✐❞s✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❣❡t ❛ ❝❧❡❛r ♣✐❝t✉r❡ ♦❢ ✇❤❛t s❡❝♦♥❞❛r② str✉❝t✉r❡s ❛r❡ ♣r❡s❡♥t ✐♥ t❤❡ ♣r♦t❡✐♥✳  ✹✳✷✳✶  α✲❍❡❧✐①  ❍❡❧✐❝❡s ❛r❡ t❤❡ ♠♦st ❝♦♠♠♦♥ s❡❝♦♥❞❛r② str✉❝t✉r❡✳ ❚❤❡r❡ ❛r❡ ❛ ✇✐❞❡ ✈❛r✐❡t② ♦❢ t❤❡♠✱ ❝❛t❡❣♦✲ r✐③❡❞ ❜② t❤❡✐r r❡s✐❞✉❡s ♣❡r t✉r♥✳ ❚❤❡ α✲❤❡❧✐① ✐s ❝♦♥s✐❞❡r❡❞ t♦ ❜❡ t❤❡ ♠♦st ❝♦♠♠♦♥✳ ❚❤✐s ❤❡❧✐① ✐s ❛ r✐❣❤t ❤❛♥❞❡❞ s♣✐r❛❧ ✐♥ ✇❤✐❝❤ ❡✈❡r② ❜❛❝❦❜♦♥❡ ❝❛r❜♦♥②❧ ❣r♦✉♣ r❡❝❡✐✈❡s ❛♥ ❡①tr❛ ❤②❞r♦❣❡♥ ❜♦♥❞ ❞♦♥❛t❡❞ ❢r♦♠ ❛ ❜❛❝❦❜♦♥❡ ◆✲❍ ❣r♦✉♣ ❢♦✉r r❡s✐❞✉❡s ❧❛t❡r ✐♥ t❤❡ s❡q✉❡♥❝❡✳ ❊❛❝❤ ❛♠✐♥♦ ❛❝✐❞ r❡s✐❞✉❡ ♠❛❦❡s ❛ t✉r♥ ♦❢ 100o ❛♥❞ ❛ tr❛♥s❧❛t✐♦♥ ♦❢ ✶✳✺➴✳ ❚❤✐s ♠❡❛♥s t❤❛t ❡❛❝❤ ❢✉❧❧ t✉r♥ ❝♦♥t❛✐♥s ✸✳✻ r❡s✐❞✉❡s ❛♥❞ tr❛♥s❧❛t❡s ✺✳✹ ➴✳ ❚❤❡ ❞②♥❛♠✐❝s ♦❢ α✲❤❡❧✐❝❡s ❛r❡ ❧♦✇ ❢r❡q✉❡♥❝② ✶✲ ❞✐♠❡♥s✐♦♥❛❧ ♦s❝✐❧❧❛t✐♦♥s ✐♥ ❛♥ ❛❝❝♦r❞✐♦♥ ❧✐❦❡ ♠❛♥♥❡r❬✶✷❪✳ ❚❤✐s ❤❡❧✐① ✐s ♦❢t❡♥ ♦❜s❡r✈❡❞ t♦ ❤❛✈❡ ❛ ❝❤❛r❛❝t❡r✐st✐❝ t❤r❡❡ ♣❤❛s❡ ❞❡❢♦r♠❛t✐♦♥ ✉♥❞❡r ♠❡❝❤❛♥✐❝❛❧ ❞✉r❡ss✳ ❉✉r✐♥❣ t❤❡ ✜rst ♣❤❛s❡ t❤❡ ❤❡❧✐① ✐s str❡t❝❤❡❞ ✉♥✐❢♦r♠❧② r❡s✉❧t✐♥❣ ✐♥ ❛ s❤♦rt ❛♥❞ st✐✛ ❞❡❢♦r♠❛t✐♦♥✳ ■♥ t❤❡ s❡❝♦♥❞ ♣❤❛s❡✱ t❤❡ t✉r♥s ❜r❡❛❦ ❛♥❞ r✉♣t✉r❡ t❤❡ ❤②❞r♦❣❡♥ ❜♦♥❞s r❡s✉❧t✐♥❣ ✐♥ ❛ ❞❡❝r❡❛s❡ ✐♥ ♠♦❞✉❧✉s✳ ❉✉r✐♥❣ t❤❡ ✜♥❛❧ ♣❤❛s❡✱ t❤❡ st✐✛ ❝♦✈❛❧❡♥t ❜♦♥❞s ❛r❡ str❡t❝❤❡❞ ✉♥t✐❧ t❤❡ ❤❡❧✐① r✐♣s ❬✶✸❪✳  ✹✳✷✳✷  β ✲❙❤❡❡ts  ❲❤✐❧❡ t❤❡ α✲❤❡❧✐① ♠❛② ❜❡ t❤❡ ♠♦st ❝♦♠♠♦♥ ❝♦♥❢♦r♠❛t✐♦♥ ✐t ✐s ❞♦❡s ♥♦t ♣❧❛② ❛♥ ✐♠♣♦rt❛♥t r♦❧❡ ✐♥ ♠❛❥♦r ❛♠♣✉❧❧❛t❡ s♣✐❞❡r s✐❧❦✳ ❚❤❡ s❡❝♦♥❞ ♠♦st ❝♦♠♠♦♥ ♣r♦t❡✐♥ s❡❝♦♥❞❛r② str✉❝t✉r❡ ✐s t❤❡ β s❤❡❡t✱ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✹✳✷✳ β s❤❡❡ts ♣❧❛②s ❛ ❝r✉❝✐❛❧ r♦❧❡ ✐♥ t❤❡ ♣❤②s✐❝❛❧ ♣r♦♣❡rt✐❡s ✸✷  ❋✐❣✉r❡ ✹✳✷✿ ❚❤❡ s✐♥❣❧❡ β str❛♥❞ ❛♥❞ t❤❡ β s❤❡❡t✱ s❤♦✇✐♥❣ ❤②❞r♦❣❡♥ ❜♦♥❞✐♥❣ ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t β str❛♥❞s ✈✐❛ t❤❡ ♣❡♣t✐❞❡ ✉♥✐t✳ ■♠❛❣❡ ❢r♦♠ ❈r❡✐❣❤t♦♥ ❬✶✷❪✳ ♦❢ s✐❧❦s✱ ❛♥❞ ✇✐❧❧ ❜❡ ♦❢ ✐♠♣♦rt ❢♦r ♠✉❝❤ ♦❢ t❤✐s t❤❡s✐s✳ ❚❤❡ ❜❛s✐❝ ✉♥✐t ♦❢ t❤❡ β ✲s❤❡❡t ✐s t❤❡  β str❛♥❞✱ ✇✐t❤ t❤❡ ♣♦❧②♣❡♣t✐❞❡ ❛❧♠♦st ❡①t❡♥❞❡❞ ❢✉❧❧②✱ t❤✐s ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛ s♣❡❝✐❛❧ ❤❡❧✐① ✇✐t❤ ✷ r❡s✐❞✉❡s ♣❡r t✉r♥✱ ❛♥❞ ❛ ✸✳✹ ➴ ♣❡r r❡s✐❞✉❡ tr❛♥s❧❛t✐♦♥✳ ❖♥ ✐ts ♦✇♥✱ t❤❡ β str❛♥❞ ✐s ♥♦t ❡♥❡r❣❡t✐❝❛❧❧② st❛❜❧❡✳ ❚♦ ♠❛✐♥t❛✐♥ str✉❝t✉r❛❧ ✐♥t❡❣r✐t②✱ ✐t ♠✉st ❜❡ ✐♥❝♦r♣♦r❛t❡❞ ✐♥t♦ ❛ β s❤❡❡t✱ ✇❤❡r❡ ❤②❞r♦❣❡♥ ❜♦♥❞s ❢♦r♠ ❜❡t✇❡❡♥ ♣❡♣t✐❞❡ ❣r♦✉♣s ♦♥ ❛❞❥❛❝❡♥t β ✲str❛♥❞s✳ ❚❤❡s❡ ❛❞❥❛❝❡♥t str❛♥❞s ❝❛♥ ❛❧✐❣♥ t♦ ❜❡ ❡✐t❤❡r ♣❛r❛❧❧❡❧ ♦r ❛♥t✐✲♣❛r❛❧❧❡❧✳ ❚❤❡ ❧❛tt❡r ❛r❡ t❤♦✉❣❤t t♦ ❜❡ ❛ ♠♦r❡ st❛❜❧❡ str✉❝t✉r❡✱ ❤♦✇❡✈❡r✱ t❤✐s ♠❛② ❛❧s♦ ❞❡♣❡♥❞ ♦♥ ✇❤✐❝❤ ❛♠✐♥♦ ❛❝✐❞s ❛r❡ ♣r❡s❡♥t ✐♥ t❤❡ str❛♥❞s✳ ❆❞❥❛❝❡♥t s✐❞❡ ❝❤❛✐♥s ❢r♦♠ t❤❡ s❛♠❡ str❛♥❞s ♣r♦tr✉❞❡ ❢r♦♠ ♦♣♣♦s✐t❡ s✐❞❡s ♦❢ t❤❡ s❤❡❡t✱ ❛♥❞ ❞♦ ♥♦t ✐♥t❡r❛❝t✳ ❍♦✇❡✈❡r✱ s✐❞❡ ❝❤❛✐♥s ❞♦ ❤❛✈❡ s✐❣♥✐✜❝❛♥t ✐♥t❡r❛❝t✐♦♥s ✇✐t❤ t❤❡ ❜❛❝❦❜♦♥❡ ❛♥❞ s✐❞❡ ❝❤❛✐♥s ❢r♦♠ ♥❡❛r❜② str❛♥❞s✳ ❚❤✐s t✇♦ ❞✐♠❡♥s✐♦♥❛❧ str✉❝t✉r❡ ♠❡❛♥s t❤❛t ✐♥ β s❤❡❡ts t❤❡ ✐♥t❡r❛❝t✐♦♥s ♦❝❝✉r ❜❡t✇❡❡♥ r❡s✐❞✉❡s ♦♥ ❞✐st❛♥t ♣❛rts ♦❢ t❤❡ ♣r♦t❡✐♥✱ ✇❤✐❝❤ ✐s ❛ st❛r❦ ❝♦♥tr❛st t♦ t❤❛t ♦❢ α ❤❡❧✐❝❡s ✇❤❡r❡ ✐♥t❡r❛❝t✐♦♥s ♦❝❝✉r ❜❡t✇❡❡♥ ♥❡✐❣❤❜♦r✐♥❣ r❡s✐❞✉❡s ❬✶✷❪✳  ✹✳✸  ❙✐❧❦  ❋♦r t❤♦✉s❛♥❞s ♦❢ ②❡❛rs✱ s✐❧❦s ❤❛✈❡ ❜❡❡♥ ❛ s②♠❜♦❧ ♦❢ ✇❡❛❧t❤✱ ♣♦✇❡r✱ ❛♥❞ st❛t✉s✳ ❚❤❡ t❡❝❤♥✐q✉❡s ❢♦r ♣r♦❝❡ss✐♥❣ t❤❡ t❤✐♥ ✜❜❡rs ❡①tr✉❞❡❞ ❜② t❤❡ s✐❧❦ ✇♦r♠ ✇❡r❡ ✜rst ♠❛st❡r❡❞ ✐♥ ❆s✐❛✱ ❛♥❞ ❢♦r ✸✸  ♠❛♥② ②❡❛rs t❤✐s ❦♥♦✇❧❡❞❣❡ ✇❛s ❛ s♦✉r❝❡ ♦❢ ❣r❡❛t ❡❝♦♥♦♠✐❝ ♣♦✇❡r ♦✈❡r t❤❡ ♥❛t✐♦♥s ♦❢ ❊✉r♦♣❡✳ ❊✈❡♥ t♦❞❛② s✐❧❦ ❝❧♦t❤✐♥❣ r❡♣r❡s❡♥ts t❤❡ ♣✐♥♥❛❝❧❡ ♦❢ ❞❡❝❛❞❡♥❝❡ ❛♥❞ ❧✉①✉r②✳ ❚❤❡ s♦❢t♥❡ss ♦❢ s✐❧❦ ✐s ♦♥❧② ♣❛rt ♦❢ t❤✐s ❡q✉❛t✐♦♥✳ ❉✉❡ t♦ t❤❡ ❛s♣❡❝t r❛t✐♦ ❛♥❞ str❡♥❣t❤ ♦❢ t❤❡s❡ ✜❜❡rs✱ ❝❧♦t❤✐♥❣ ♠❛❞❡ ❢r♦♠ t❤❡♠ ✇❛s ❛❜❧❡ t♦ ❤❛✈❡ ❛ ✜♥❡r ✇❡❛✈❡ ❛♥❞ ❜❡ ❜♦t❤ str♦♥❣❡r ❛♥❞ t❤✐♥♥❡r t❤❛♥ ❛♥② ♦t❤❡r ♠❛t❡r✐❛❧✳ ■♥ ❢❛❝t✱ ❤✐st♦r✐❝❛❧ ❡✈✐❞❡♥❝❡ s✉❣❣❡sts t❤❛t ♦♥❡ ♦❢ t❤❡ r❡❛s♦♥s ❢♦r t❤❡ ▼♦♥❣♦❧s ❣r❡❛t s✉❝❝❡ss ✐♥ t❤❡✐r ✇❡st✇❛r❞ ❝♦♥q✉❡sts ✇❛s ❞✉❡ t♦ t❤❡ s✐❧❦ ✈❡sts t❤❛t t❤❡ ❤♦rs❡♠❡♥ ✇♦r❡ ✉♥❞❡r t❤❡ ❛r♠♦r✳ ❚❤❡s❡ ✈❡sts ✇❡r❡ ❛❜❧❡ t♦ ♣r♦t❡❝t t❤❡♠ ❜❡tt❡r t❤❛♥ ❛r♠♦r ❢r♦♠ ♣✐❡r❝✐♥❣ ✇♦✉♥❞s✱ s✉❝❤ ❛s ❛rr♦✇s✳ ❲❤✐❧❡ t❤❡r❡ ✇♦✉❧❞ st✐❧❧ ❜❡ ❛ ✇♦✉♥❞✱ ♦❢t❡♥ t❤❡ ❛rr♦✇ ✇♦✉❧❞ ♥♦t ❛❝t✉❛❧❧② ♣❡♥❡tr❛t❡ t❤❡ s✐❧❦ r❡s✉❧t✐♥❣ ✐♥ ❛ ❧❡ss ❞❡❡♣ ✇♦✉♥❞✱ ❛♥❞ ❛❧❧♦✇ ✐ts r❡♠♦✈❛❧ t♦ ❜❡ ❛ ❢❛r ❣❡♥t❧❡r ♣r♦❝❡ss ❬✶✹❪✳ ❙✐❧❦ ❢r♦♠ t❤❡ ❝❧❛ss✐❝❛❧ s✐❧❦✇♦r♠ ✭♠♦st ♦❢t❡♥ Bombyx mori✮ ✐s ♥♦t t❤❡ ♦♥❧② s✐❧❦ t❤❡r❡ ✐s✳ ❖t❤❡r s♣❡❝✐❡s ♣r♦❞✉❝❡ s✐❧❦s✱ ✐♥❝❧✉❞✐♥❣ ❡✈❡r② ❦✐♥❞ ♦❢ t❤❡ ♦✈❡r ✸✵ ✵✵✵ ✈❛r✐❡t✐❡s ♦❢ s♣✐❞❡r✳ ❙♣✐❞❡r s✐❧❦ ✐s ♦❢ ♣❛rt✐❝✉❧❛r s✐❣♥✐✜❝❛♥❝❡ ❜❡❝❛✉s❡ ♦❢ ✐ts r❡♠❛r❦❛❜❧❡ ♣r♦♣❡rt✐❡s ❛♥❞ ✇✐❧❧ ❜❡ t❤❡ ❢♦❝✉s❡❞ ♦♥ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳ ❉❡s♣✐t❡ t❤✐s ❣r❡❛t ✈❛r✐❛♥❝❡ ✐♥ ❦✐♥❞s ♦❢ s✐❧❦s✱ ♠❛♥② ♦❢ t❤❡♠ s❤❛r❡ s❡✈❡r❛❧ ❢❡❛t✉r❡s ✐♥❝❧✉❞✐♥❣❀ ❛♥ ✉♥r✐✈❛❧❡❞ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♠❡❝❤❛♥✐❝❛❧ ♣r♦♣❡rt✐❡s✱ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ✇❛t❡r s♦❧✉❜❧❡ st❛t❡✱ ❜❡✐♥❣ ♠❛❞❡ ♣r✐♠❛r✐❧② ♦✉t ♦❢ ♣r♦t❡✐♥s✱ ❛♥❞ t❤❡ ❞✉r❛❜✐❧✐t② ♦❢ t❤❡ ✜♥✐s❤❡❞ ♣r♦❞✉❝t✳ ❲✐t❤ r❡❣❛r❞s t♦ t❤❡ s❡❝♦♥❞ ♣♦✐♥t✱ ✐t ♠✉st ❜❡ ♥♦t❡❞ t❤❛t t❤❡ ✜♥✐s❤❡❞ s✐❧❦ ✐ts❡❧❢ ✐s ♥♦t ✇❛t❡r s♦❧✉❜❧❡ ✉♥❞❡r ♥♦r♠❛❧ ❝♦♥❞✐t✐♦♥s✱ ❛♥❞ ✐♥ ❢❛❝t ✐s q✉✐t❡ r❡s✐st❛♥t t♦ ♠♦st ❢♦r♠s ♦❢ ❛❝✐❞ ❛♥❞ ♦r❣❛♥✐❝ s♦❧✈❡♥ts✳ ❚❤❡ ✇❛t❡r s♦❧✉❜❧❡ st❛t❡ ❡①✐sts ♣r✐♦r t♦ t❤❡ s♣✐♥♥✐♥❣ ♦❢ t❤❡ s✐❧❦✱ ❛♥❞ r❡q✉✐r❡s ❛ ❣r❡❛t ❞❡❛❧ ♦❢ tr❡❛t♠❡♥t ❢♦r t❤❡ s✐❧❦ ✜❜❡rs t♦ ❜❡ ♠❛❞❡ ✐♥t♦ ❛ ✇❛t❡r s♦❧✉❜❧❡ ♣r♦❞✉❝t✳  ✹✳✸✳✶  ❙♣✐❞❡r ❙✐❧❦  ❲❤✐❧❡ ♠❛♥② s✐❧❦s ❤❛✈❡ r❡♠❛r❦❛❜❧❡ ♠❡❝❤❛♥✐❝❛❧ ♣r♦♣❡rt✐❡s✱ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ s♣✐❞❡r s✐❧❦ ❣r❡❛t❧② s✉r♣❛ss t❤♦s❡ ♦❢ t❡①t✐❧❡ s✐❧❦s✳ ▼✐❧❧✐♦♥s ♦❢ ②❡❛rs ♦❢ ♣r❡❞❛t♦r② ❡✈♦❧✉t✐♦♥ ❤❛s ❡✈♦❧✈❡❞ s♣✐❞❡r s✐❧❦ ✐♥t♦ ❛ ♠❛t❡r✐❛❧ ✇✐t❤ ♠❡❝❤❛♥✐❝❛❧ ♣r♦♣❡rt✐❡s ❝♦♠♣❛r❛❜❧❡ t♦ ❛❞✈❛♥❝❡❞ s②♥t❤❡t✐❝ ♠❛t❡r✐❛❧s✳ ❙♣✐❞❡r s✐❧❦ ✐s t♦✉❣❤❡r t❤❛♥ st❡❡❧ ❛♥❞ ❑❡✈❧❛r✳ ❲❤❛t ♠❛❦❡s s♣✐❞❡r s✐❧❦ s♦ ❞✉r❛❜❧❡ ✐s t❤❛t ✐t ✐s ❡①❤✐❜✐ts ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❛ ❤✐❣❤ str❡♥❣t❤✱ ♠♦❞✉❧✉s✱ ❛♥❞ ❡①t❡♥❞❛❜✐❧✐t②✳ ❚❤✐s ❝♦♠❜✐♥❛t✐♦♥ ♠❡❛♥s t❤❛t t❤❡ ❛♠♦✉♥t ♦❢ ❡♥❡r❣② r❡q✉✐r❡❞ t♦ ❜r❡❛❦ s♣✐❞❡r s✐❧❦ ✐s ❝♦♠♣❛r❛❜❧❡ t♦✱ ❛♥❞ ✐♥ ♠♦st ❝❛s❡s s✉r♣❛ss❡s✱ s♦♠❡ ♦❢ t❤❡ ♠♦st s✉❝❝❡ss❢✉❧ ♠❛♥ ♠❛❞❡ ♠❛t❡r✐❛❧s✱ ❛s ❝❛♥ ❜❡ s❡❡♥ ✐♥ ❚❛❜❧❡ ✹✳✷✳ ■♥ ❛❞❞✐t✐♦♥ t♦ ✐ts ✐♠♣r❡ss✐✈❡ ♠❡❝❤❛♥✐❝❛❧ ❝❤❛r❛❝t❡r✐st✐❝s✱ s♣✐❞❡r s✐❧❦ ❝❛♥ ❡①❤✐❜✐t ❛ ♣❤❡♥♦♠❡♥♦♥ ❦♥♦✇♥ ❛s s✉♣❡r❝♦♥tr❛❝t✐♦♥✳ ❲❤❡♥ ❡①♣♦s❡❞ t♦ ✇❛t❡r✱ s♣✐❞❡r s✐❧❦ ❝❛♥ ❝♦♥tr❛❝t ❜② ✉♣ t♦ ✺✵%✱ t❤✐s ♣❤❡♥♦♠❡♥♦♥ ❛❧❧♦✇s t❤❡ ♠♦r♥✐♥❣ ❞❡✇ t♦ ❦❡❡♣ s♣✐❞❡r ✇❡❜s t✐❣❤t ❬✶✺❪✳ ❙✉♣❡r❝♦♥tr❛❝t✐♦♥ ✸✹  ▼❛t❡r✐❛❧ ❍✐❣❤✲t❡♥s✐❧❡ st❡❡❧ ❑❡✈❧❛r ◆②❧♦♥ ❉r❛❣❧✐♥❡ ❙✐❧❦ ✭◆❡♣❤✐❧❛ ✮  ❙tr❡♥❣t❤ ✭●P❛✮ ✶✳✺ ✹✼ ✵✳✾✺ ✶✳✶  ▼♦❞✉❧✉s ✭●P❛✮ ✷✵✵ ✷✼ ✹ ✷✵  ❊①t❡♥❞❛❜✐❧✐t② ✭✪ ♦❢ ❧❡♥❣t❤✮ ✵✳✽ ✸✳✻ ✶✽ ✸✵  ❚♦✉❣❤♥❡ss ✭▼❏✴♠3 ✮ ✻ ✶✸✵ ✽✵ ✶✼✵  ❚❛❜❧❡ ✹✳✷✿ ▼❡❝❤❛♥✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ ✈❛r✐♦✉s ♠❛♥ ♠❛❞❡ ♠❛t❡r✐❛❧s ❛♥❞ ❞r❛❣❧✐♥❡ s✐❧❦ ❬✶✻❪✳  ❋✐❣✉r❡ ✹✳✸✿ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ ▼❛❙♣✶ ♣r♦t❡✐♥✳ ❚❤❡ r❡♣❡❛t✐♥❣ ❛♠✐♥♦ ❛❝✐❞ s❡q✉❡♥❝❡s ❤❛✈❡ ❜❡❡♥ ❛rr❛♥❣❡❞ ❢r♦♠ ❛♠✐♥♦✲t❡r♠✐♥❛❧ ❡♥❞ t♦ t❤❡ ❡♥❞ ♦❢ t❤❡ r❡♣❡❛t✐♥❣ s❡❣♠❡♥t✱ ✇❤✐❝❤ ✐s ✻✾ ❛♠✐♥♦ ❛❝✐❞s ❢r♦♠ t❤❡ ❝❛r❜♦①②❧ t❡r♠✐♥✉s✳ ❚❤❡ ✉♥✐ts ❤❛✈❡ ❜❡❡♥ ❛rr❛♥❣❡❞ ❢♦r ♠❛①✐♠✉♠ ✐❞❡♥t✐t②✳ ❚❤❡ ❞❛s❤❡s ❛r❡ ❞❡❧❡t✐♦♥s✳ ■♠❛❣❡ ❢r♦♠ ❳✉ ❛♥❞ ▲❡✇✐s ❬✶✼❪✳ ❤❛s ❜❡❡♥ ♦❜s❡r✈❡❞ ✐♥ s②♥t❤❡t✐❝ ♠❛t❡r✐❛❧s✱ ❤♦✇❡✈❡r✱ t❤✐s ❤❛s ♦♥❧② ♦❝❝✉rr❡❞ ❛t ❡①tr❡♠❡❧② ❤✐❣❤ t❡♠♣❡r❛t✉r❡s ♦r ✇✐t❤ t❤❡ ✉s❡ ♦❢ ❝❛✉st✐❝ s♦❧✈❡♥ts✳ ❆ ❝♦♠♣❧❡t❡ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ ✇❤❡r❡ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ s♣✐❞❡r s✐❧❦ ❝♦♠❡ ❢r♦♠ ✐s st✐❧❧ ❛ ♣❛rt✐❛❧ ♠②st❡r②✳ ❚❤❡ ❛❝❝❡♣t❡❞ ♠♦❞❡❧ ✐s t❤❛t ✐ts ♣r♦♣❡rt✐❡s ♦r✐❣✐♥❛t❡ ❢r♦♠ t❤❡ ♣r♦t❡✐♥s t❤❡♠s❡❧✈❡s✱ ❜✉t ✐t ✐s t❤❡ s❡❝♦♥❞❛r② str✉❝t✉r❡s t❤❛t ❛❧❧♦✇ ❢♦r t❤❡♠✳  ◆❡♣❤✐❧❛ ❝❧❛✈✐♣❡s  ❞r❛❣❧✐♥❡ s✐❧❦ ✐s  ❝♦♠♣♦s❡❞ ♦❢ t✇♦ ♣r♦t❡✐♥s✱ ❙♣✐❞r♦✐♥ ■ ✭▼❛❙♣✶✮ ❛♥❞ ❙♣✐❞r♦✐♥ ■■ ✭▼❛❙♣✷✮ ✇❤♦s❡ s❡q✉❡♥❝❡s ❛r❡ s❤♦✇♥ ✐♥ ❋✐❣✉r❡s ✹✳✸ ❛♥❞ ✹✳✹✳ ❙♣■ ✐s ❞♦♠✐♥❛t❡❞ ❜② r✉♥s ♦❢ ✺✲✼ r❡s✐❞✉❡s ♦❢ ♣♦❧②❛❧❛♥✐♥❡ ✇✐t❤ ❣❡♥❡r❛❧❧② ✜✈❡ ✲●❧②✲●❧②✲❳✲ s❡ts s❡♣❛r❛t✐♥❣ t❤❡♠✱ ✇❤❡r❡ ❳ ✐s ♣r❡❞♦♠✐♥❛♥t❧② ●❧♥✱ ❚②r✱ ❛♥❞ ▲❡✉✳ ■t ✐s t❤♦✉❣❤t t❤❛t t❤❡ ♣♦❧②❛❧❛♥✐♥❡ s❡ts ♣❛rt✐❝✐♣❛t❡ ✐♥ β ✲s❤❡❡t ❢♦r♠❛t✐♦♥✳ ❙♣✷ ❤❛s r✉♥s ♦❢ ♣♦❧②❛❧❛♥✐♥❡ ✇✐t❤ ❧❡♥❣t❤ ♦❢  ✸✺  ❋✐❣✉r❡ ✹✳✹✿ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ ▼❛❙♣✷ ♣r♦t❡✐♥✳ ❚❤❡ r❡♣❡❛t✐♥❣ ❛♠✐♥♦ ❛❝✐❞ s❡q✉❡♥❝❡s ❤❛✈❡ ❜❡❡♥ ❛rr❛♥❣❡❞ ❢r♦♠ t❤❡ ❛♠✐♥♦ t❡r♠✐♥❛❧ t❤r♦✉❣❤ t❤❡ ❤✐❣❤❧② ❝♦♥s❡r✈❡❞ r❡❣✐♦♥✱ ❢♦❧❧♦✇❡❞ ❜② t❤❡ ❧❡ss ❝♦♥s❡r✈❡❞ r❡❣✐♦♥ ❛♥❞ ❞✐✈❡r❣❡♥t ❈❖❖❍✲t❡r♠✐♥❛❧ t❛✐❧✳ ❚❤❡ ❞❛s❤❡s ❛r❡ ❞❡❧❡t✐♦♥s✳ ■♠❛❣❡ ❢r♦♠ ❍✐♥♠❛♥ ❛♥❞ ▲❡✇✐s ❬✶✽❪✳ ✻✲✶✵ r❡s✐❞✉❡s✳ ❆ s✐❣♥✐✜❝❛♥t ❢❡❛t✉r❡ ♦❢ ❙♣✷ ❛r❡ s❡✈❡r❛❧ ♣r♦❧✐♥❡ ❝♦♥t❛✐♥✐♥❣ ♣❡♥t❛♣❡♣t✐❞❡s s✉❝❤ ❛s❀ ✲●❧②✲❚②r✲●❧②✲Pr♦✲●❧②✲✱ ●❧②✲Pr♦✲●❧②✲●❧②✲❚②r✲✱ ❛♥❞ ✲●❧②✲Pr♦✲●❧②✲●❧♥✲●❧♥✲✳ ❚❤❡ ♣r♦❧✐♥❡ ✐♥ t❤❡s❡ ♣❡♥t❛♣❡♣t✐❞❡s ❢♦r❝❡s ❛♥ ❛❜r✉♣t ❦✐♥❦ ✐♥ t❤❡ ♣♦❧②♠❡r ❜❛❝❦❜♦♥❡✳ ■t ✐s t❤♦✉❣❤t t❤❛t t❤✐s ♠❛❦❡s ✐t ❞✐✣❝✉❧t ❢♦r β ✲s❤❡❡t ❝r②st❛❧s t♦ ❢♦r♠ ♦✉t ♦❢ t❤❡s❡ s❡❣♠❡♥ts ♦❢ ▼❛❙♣✷✱ t❤✉s ✐❢ ❛♥ ❛tt❡♠♣t ✇❡r❡ ♠❛❞❡ t♦ ❝♦♥str✉❝t β s❤❡❡t ❝r②st❛❧s ♦✉t ♦❢ ▼❛❙♣✷✱ t❤❡ ♣r♦❧✐♥❡ ✇♦✉❧❞ ❞✐sr✉♣t t❤❡✐r str✉❝t✉r❛❧ ✐♥t❡❣r✐t② ❡✛❡❝t✐✈❡❧② t✉r♥✐♥❣ t❤❡ ❝r②st❛❧s ✐♥t♦ ❞✐s♦r❞❡r❡❞ ♠❛tr✐❝❡s ❬✶✾❪✳ ❚❤❡ ❜r❡❛❦❞♦✇♥ ❜② ❛♠✐♥♦ ❛❝✐❞ ❝♦♥tr✐❜✉t✐♦♥ ♦❢ ❜♦t❤ ▼❛❙♣✶ ❛♥❞ ▼❛❙♣✷ ❛r❡ s❤♦✇♥ ✐♥ ❚❛❜❧❡ ✹✳✸✳ ❲❤✐❧❡ β s❤❡❡ts ❞♦ ❢♦r♠ ❢r♦♠ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ♣r♦t❡✐♥s✱ t❤❡ s❡q✉❡♥❝❡ ✐ts❡❧❢ ❞♦❡s ♥♦t ❤❛✈❡ ❛♥ ✐♥tr✐♥s✐❝ t❡♥❞❡♥❝② t♦ ❞♦ s♦✳ ❚❤❡ ❢❛❝t t❤❛t β s❤❡❡ts ❢♦r♠ ❛t ❛❧❧ ✐s t❤♦✉❣❤t t♦ ❜❡ ❛ r❡s✉❧t ♦❢ ❡①t❡r♥❛❧ ❝♦♥❞✐t✐♦♥s✱ ♥♦t s♦♠❡ ✐♥❤❡r❡♥t ♣r♦♣❡rt② ♦❢ t❤❡ ♣r♦t❡✐♥ ❬✷✵✱ ✶✾✱ ✷✶❪✳ ■t ✐s t❤♦✉❣❤t t❤❛t ❛s t❤❡ ♣r♦t❡✐♥ s♦❧✉t✐♦♥ ✐s ❞r❛✇♥ t❤r♦✉❣❤ t❤❡ s♣✐♥♥❡r❡t✱ ✐t ✐s ❡①♣♦s❡❞ t♦ ❛ ❣r❡❛t❡r✲t❤❛♥✲❝r✐t✐❝❛❧ s❤❡❛r r❛t❡✱ ✇❤♦s❡ ♠❛❣♥✐t✉❞❡ ✐s ❞❡♣❡♥❞❡♥t ♦♥ t❤❡ ❝♦♥❝❡♥tr❛t✐♦♥✱ s♦♠❡ ♦❢ t❤❡ β str❛♥❞s ❛r❡ st❛❜✐❧✐③❡❞ ❜② ❢♦r♠✐♥❣ ❤②❞r♦❣❡♥ ❜♦♥❞s ❛♥❞ β s❤❡❡ts✳ ❚❤❡s❡ s❤❡❡ts st❛❝❦ ✐♥t♦ ✸✲❞✐♠❡♥s✐♦♥❛❧ ❝r②st❛❧s ❝❛❧❧❡❞ β ✲s❤❡❡t ❝r②st❛❧s✳ ■t ✐s t❤❡s❡ ❝r②st❛❧s t❤❛t ❣✐✈❡ s♣✐❞❡r s✐❧❦ ♠✉❝❤ ♦❢ ✐ts str❡♥❣t❤ ❬✶✾❪✳ ▼✉❝❤ ♦❢ t❤❡ t❤❡ ♥♦♥ ❝r②st❛❧❧✐♥❡ r❡❣✐♦♥ ✐s ❜❡❧✐❡✈❡❞ t♦ t❛❦❡ t❤❡ ❢♦r♠ ♦❢ 31 ❤❡❧✐❝❡s ❬✺❪✳ ❚❤✐s ♣❤❛s❡ ❝♦♥♥❡❝ts t❤❡ β ❝r②st❛❧s t♦❣❡t❤❡r ❛♥❞ ❜❡❝♦♠❡s ❤✐❣❤❧② ❡♥t❛♥❣❧❡❞ ✇✐t❤ ❡❛❝❤ ♦t❤❡r ❢♦r♠✐♥❣ ❤②❞r♦❣❡♥ ❜♦♥❞s✳ ❚❤✐s r❡❣✐♦♥ ❣✐✈❡s s✐❧❦ ❛ r✉❜❜❡r② ♣r♦♣❡rt②✱ r❡s✉❧t✐♥❣ ✐♥ t❤❡ s✐❧❦ ❤❛✈✐♥❣ ❛ ❤✐❣❤ ♠♦❞✉❧✉s ❛♥❞ ❡①t❡♥❞❛❜✐❧✐t②✳ ❚❤✐s ♠♦❞❡❧ ✐s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✹✳✺✳ ■t ✐s ❜❡❧✐❡✈❡❞ t❤❛t ✐t ✐s t❤✐s ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡s❡ t✇♦ r❡❣✐♦♥s t❤❛t ❣✐✈❡s s♣✐❞❡r s✐❧❦ ✐ts r❡♠❛r❦❛❜❧❡ ♠❡❝❤❛♥✐❝❛❧ ♣r♦♣❡rt✐❡s✳  ✸✻  ❋✐❣✉r❡ ✹✳✺✿ ❆ ♣♦ss✐❜❧❡ str✉❝t✉r❡ ♦❢ s♣✐❞❡r ❞r❛❣❧✐♥❡ s✐❧❦ ♣r♦♣♦s❡❞ ❜② ✈❛♥ ❇❡❡❦ ✇✐t❤ ✭❆✮ ❛ s❦✐♥✲❝♦r❡ ♦r❣❛♥✐③❛t✐♦♥✱ ✇✐t❤ ❛ ♠✉❧t✐t✉❞❡ ♦❢ ✜❜r✐❧❧❛r s✉❜str✉❝t✉r❡s ❛♥❞ ❝♦✈❡r❡❞ ❜② ❛ ❤❛r❞ s❦✐♥✱ ❢♦r♠s t❤❡ ✜❜❡r ❚❤❡ ♠♦❧❡❝✉❧❛r str✉❝t✉r❡ ❝♦♥s✐sts ♦❢ s❤❡❡t r❡❣✐♦♥s✱ ❝♦♥t❛✐♥✐♥❣ ❛❧❛♥✐♥❡ ✭r❡❞ ❧✐♥❡s✮ ❛♥❞ ❣❧②❝✐♥❡ ✭❜❧✉❡ ❧✐♥❡s✮ ✇✐t❤ t❤❡ ❣r❡❡♥ ❜♦①❡s ❜❡✐♥❣ t❤❡ ❝r②st❛❧❧✐♥❡ ❞♦♠❛✐♥s✳ ❚❤❡s❡ r❡❣✐♦♥s ❛r❡ ✐♥t❡r❧❡❛✈❡❞ ✇✐t❤ ❛ ♣r❡❞♦♠✐♥❛♥t❧② 31 ❤❡❧✐❝❛❧ ✭❜❧✉❡ ❝✉r❧s✮ ❛♠♦r♣❤♦✉s r❡❣✐♦♥✱ ✇❤✐❝❤ ❞♦ ♥♦t ❝♦♥t❛✐♥ ❛❧❛♥✐♥❡✳ ❆❧❧ ❝❤❛✐♥s t❡♥❞ t♦ ❜❡ ♣❛r❛❧❧❡❧✳ ✭❇ ❛♥❞ ❈✮ ❙✐❞❡ ❛♥❞ t♦♣ ♣r♦❥❡❝t✐♦♥s ♦❢ ❛ r❡♣❡t✐t✐✈❡ ♠♦❞❡❧ ♣❡♣t✐❞❡ ✇✐t❤ ❞✐❤❡❞r❛❧ ❛♥❣❧❡s t♦ s❤♦✇ t❤❡ ❛♣♣r♦①✐♠❛t❡ ✸✲❢♦❧❞ s②♠♠❡tr② ♦❢ t❤❡ 31 ❤❡❧✐❝❛❧ str✉❝t✉r❡✳ ■♠❛❣❡ ❢r♦♠ ✈❛♥ ❇❡❡❦ ❡t ❛❧✳ ❬✺❪✳  ✸✼  ❆♠✐♥♦ ❆❝✐❞ ❣❧②❝✐♥❡ ❛❧❛♥✐♥❡ ❣❧✉t❛♠✐♥❡ ❧❡✉❝✐♥❡ t②rs✐♥❡ s❡r✐♥❡ ❛r❣✐♥✐♥❡ ♣r♦❧✐♥❡ ❛s♣❛r❣✐♥❡ ✈❛❧✐♥❡ ❤✐st✐❞✐♥❡ ♣❤❡♥②❧❛❧❛♥✐♥❡ ❧②s✐♥❡ ✐s♦❧❡✉❝✐♥❡ ❝②st❡✐♥❡ ♠❡t❤✐♦♥✐♥❡  ❙✐❧❦ ♠❛❥♦r ❞r❛❣❧✐♥❡ ❬✷✷❪ ✹✺✳✽ ✷✷✳✷ ✶✵✳✾ ✶✳✺ ✹✳✸ ✼✳✹ ✶✳✼ ✸✳✺ ✶✳✷ ✵✳✽ ✵ ✵✳✸ ✵✳✸ ✵✳✶ ✵ ✵  ❚❛❜❧❡ ✹✳✸✿ ❇r❡❛❦❞♦✇♥ ❜② ❛♠✐♥♦ ❛❝✐❞ ♦❢  ❙❡q✉❡♥❝❡❞ Pr♦t❡✐♥ ▼❛❙♣✶ ❬✶✼❪ ❙♣✷ ❬✶✽❪ ✹✾✳✺ ✷✾✳✻✸ ✷✾✳✹ ✷✶✳✸ ✶✵✳✻ ✶✶✳✾ ✸✳✹ ✶✳✵ ✸✳✷ ✹✳✽ ✶✳✽ ✶✵✳✶ ✵✳✸ ✶✳✸ ✵ ✶✸✳✾ ✵✳✺ ✵✳✻ ✵✳✽ ✷✳✶ ✵ ✵ ✵ ✵✳✸ ✵ ✵ ✵✳✷ ✶✳✶ ✵ ✳✸ ✵ ✵  ◆✳ ❝❧❛✈✐♣❡s  ❞r❛❣❧✐♥❡ s✐❧❦✱ ▼❛❙♣✶✱ ❛♥❞ ▼❛❙♣✷✳  ▼❡❛s✉r❡♠❡♥ts ❤❛✈❡ ❞❡t❡r♠✐♥❡❞ t❤❛t ✷✽✪ ♦❢ t❤❡ ❣❧②❝✐♥❡ ❛♥❞ ✽✷✪ ❬✷✸❪ ♦❢ t❤❡ ❛❧❛♥✐♥❡s t❛❦❡ t❤❡ ❢♦r♠ ♦❢ β s❤❡❡ts✳ ■♥ t♦t❛❧ β s❤❡❡ts ♠❛❦❡ ❧❡ss t❤❛♥ ✹✵✪ ♦❢ s♣✐❞❡r s✐❧❦ ❬✶✺❪✳  ✹✳✸✳✷  ❚r❛♥s❣❡♥✐❝ ❙♣✐❞❡r ❙✐❧❦  ■♥ ✶✾✾✾✱ ◆❡①✐❛ ❇✐♦t❡❝❤♥♦❧♦❣✐❡s ■♥❝✳✱ ❛ ◗✉❡❜❡❝ ❜✐♦t❡❝❤♥♦❧♦❣② ❝♦♠♣❛♥② ❞❡✈❡❧♦♣❡❞ ❛ ♥♦✈❡❧ ❛♣♣r♦❛❝❤ ❢♦r t❤❡ s②♥t❤❡t✐❝ ♣r♦❞✉❝t✐♦♥ ♦❢ s♣✐❞❡r s✐❧❦✳ ❚❤❡ ✇❡r❡ ❛❜❧❡ t♦ ♣r♦❞✉❝❡ r❡❝♦♠❜✐✲ ♥❛♥t s♣✐❞❡r s✐❧❦✱ ❞✉❜❜❡❞ ❇✐♦st❡❡❧✱ ✉s✐♥❣ ❇❊▲❊ ✭❇r❡❡❞ ❊❛r❧② ▲❛❝t❛t❡ ❊❛r❧②✮ ❣♦❛ts t❤❛t ✇❡r❡ tr❡❛t❡❞ ✉s✐♥❣ ♣r♦♥✉❝❧❡❛r ♠✐❝r♦ ✐♥❥❡❝t✐♦♥s ❛♥❞ ♥✉❝❧❡❛r tr❛♥s❢❡r t❡❝❤♥♦❧♦❣✐❡s ❬✷✹❪✳ ◆❡①✐❛ ❤❛s s✉❝❝❡ss❢✉❧❧② ❣❡♥❡r❛t❡❞ ❜♦t❤ ▼❛❙♣✶ ❛♥❞ ▼❛❙♣✷ ✐♥ t❤❡✐r tr❛♥s❣❡♥✐❝ ❣♦❛t ♠✐❧❦✳ ❚❤❡ ♠✐❧❦ ❤❛s ❤❛❞ ②✐❡❧❞s ♦❢ ✉♣ t♦ ✷ ❣r❛♠s ♦❢ ♣r♦t❡✐♥ ♣❡r ❧✐t❡r ❛♥❞ t❤❡ ♣r♦t❡✐♥s ❝❛♥ ❜❡ ✐s♦❧❛t❡❞ ❛♥❞ ♣✉r✐✜❡❞ t♦ ❤♦♠♦❣❡♥❡✐t② ❬✶✵❪✳ ❯♥❢♦rt✉♥❛t❡❧②✱ ❞❡s♣✐t❡ t❤❡ ❛❜✐❧✐t② t♦ ♣r♦❞✉❝❡ t❤❡ s♣✐❞❡r s✐❧❦ ♣r♦t❡✐♥s ❛rt✐✜❝✐❛❧❧②✱ r❡s❡❛r❝❤❡rs ❤❛✈❡ ❜❡❡♥ ✉♥❛❜❧❡ t♦ ♣r♦❞✉❝❡ ✜❜❡rs ✇✐t❤ ❝♦♠♣❛r❛❜❧❡ ♠❡❝❤❛♥✐❝❛❧ ♣r♦♣❡rt✐❡s t♦ t❤♦s❡ ♣r♦❞✉❝❡❞ ❜② s♣✐❞❡rs✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ♠❡❝❤❛♥✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✐❧❦ ♥❛♥♦✜❜❡rs ♣r♦❞✉❝❡❞ ✈✐❛ ❡❧❡❝tr♦✲ s♣✐♥♥✐♥❣ ❜② ▼✳ ●❛♥❞❤✐ ❤❛❞ ♠♦❞✉❧✐ ❛♥❞ t❡♥s✐❧❡ str❡♥❣t❤s ♦✈❡r t✇♦ ♦r❞❡rs ♦❢ ♠❛❣♥✐t✉❞❡ ❜❡❧♦✇ t❤❛t ♦❢ ♥❛t✉r❛❧❧② ♣r♦❞✉❝❡❞ s♣✐❞❡r s✐❧❦✱ ❛♥❞ ❛t ❜❡st✱ t✇♦ t❤✐r❞s ♦❢ t❤❡ ♠❛①✐♠✉♠ ❡❧♦♥❣❛t✐♦♥ ✸✽  ❙✐❧❦ t②♣❡ ❉r❛❣❧✐♥❡ ❙✐❧❦ ✸✿✶ ▼❛❙♣✶✿▼❛❙♣✷ ✶✿✶ ▼❛❙♣✶✿▼❛❙♣✷ ✶✿✸ ▼❛❙♣✶✿▼❛❙♣✷ ✶✿✵ ▼❛❙♣✶✿▼❛❙♣✷ ✶✿✵ ▼❛❙♣✶✿▼❛❙♣✷+1%❈◆❚  ❙tr❡♥❣t❤ ✭▼P❛✮ ✶✶✵✵ ✼✳✹±✵✳✸✻ ✺✳✵±✶✳✷ ✷✳✹±✵✳✽ ✾✳✻±✷✳✵ ✹✵✳✼±✻✳✸  ▼♦❞✉❧✉s ✭▼P❛✮ ✭▼P❛✮ ✷✵✵✵✵ ✶✵✹±✶✺ ✼✼±✹ ✷✶✳✸±✺✳✷ ✶✷✸±✷✺ ✶✵✵✹±✺✸  ❊①t❡♥❞❛❜✐❧✐t② ✭✪ ♦❢ ❧❡♥❣t❤✮ ✸✵ ✶✹✳✵±✵✳✶ ✶✷✳✻±✵✳✷ ✷✷✳✷±✶✳✷ ✶✹✳✸±✵✳✷ ✼✳✸✾±✶✳✹  ❚❛❜❧❡ ✹✳✹✿ ▼❡❝❤❛♥✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ ♥❛t✉r❛❧ s♣✐❞❡r s✐❧❦ ❬✶✻❪ ❛♥❞ ✈❛r✐♦✉s ❡❧❡❝tr♦s♣✉♥ s✐❧❦s ♠❛❞❡ ❢r♦♠ ▼❛❙♣✶ ❛♥❞ ▼❛❙♣✷ ❬✶✵❪✳ ❙tr✉❝t✉r❡ ❛♥t✐ ♣❛r❛❧❧❡❧ β s❤❡❡ts✴❛❣❣r❡❣❛t❡❞ str❛♥❞s 310 ❤❡❧✐① α ❤❡❧✐① ✉♥♦r❞❡r❡❞ β s❤❡❡t ❛❣❣r❡❣❛t❡❞ str❛♥❞s  ❆♠✐❞❡ ✶ ❢r❡q✉❡♥❝② ✭❝♠−1 ✮ ✶✻✼✺✲✶✻✾✺ ✶✻✻✵✲✶✻✼✵ ✶✻✹✽✲✶✻✻✵ ✶✻✹✵✲✶✻✹✽ ✶✻✷✺✲✶✻✹✵ ✶✻✶✵✲✶✻✷✽  ❚❛❜❧❡ ✹✳✺✿ ❙❡❝♦♥❞❛r② str✉❝t✉r❡ ❛ss✐❣♥♠❡♥ts ❢r♦♠ ❋❚■❘ ✇❛✈❡♥✉♠❜❡rs ❬✷✺❪✳ ❛s s❤♦✇♥ ✐♥ ❚❛❜❧❡ ✹✳✹✳ ❚❤✐s ❞❛r❦ ❝❧♦✉❞ ❞♦❡s ❤❛✈❡ ❛ s✐❧✈❡r ❧✐♥✐♥❣ ❤♦✇❡✈❡r✳ ❇② ❛❞❞✐♥❣ ❝❛r❜♦♥ ♥❛♥♦t✉❜❡s ✐♥t♦ t❤❡ ✜❜❡rs ❞✉r✐♥❣ s♣✐♥♥✐♥❣ ♣r♦❝❡ss✱ t❤❡ ♠♦❞✉❧✉s ❛♥❞ str❡♥❣t❤ ✇❡r❡ ❣r❡❛t❧② ✐♥❝r❡❛s❡❞✳ ❚❤❡s❡ ✐♠♣r♦✈❡♠❡♥ts ❤♦✇❡✈❡r✱ ❝❛♠❡ ❛t t❤❡ ❝♦st ♦❢ ❞❡❝r❡❛s✐♥❣ t❤❡ ❡❧♦♥❣❛t✐♦♥ ❜② ❝❧♦s❡ t♦ ✺✵%✳  ✹✳✸✳✷✳✶  ❋❚■❘ ❘❡s✉❧ts ♦❢ t❤❡ ❚r❛♥s❣❡♥✐❝ ▼❛❙♣✶ ❛♥❞ ▼❛❙♣✷  ❋❚■❘ ❡①♣❡r✐♠❡♥ts ❝♦♥❞✉❝t❡❞ ♦♥ ❡❧❡❝tr♦s♣✉♥ s✐❧❦ ❛♥❞ t❤❡ ❧②♦♣❤✐❧✐③❡❞ ▼❛❙♣✶ ❛♥❞ ▼❛❙♣✷ ♣♦✇❞❡rs ❜② ●❛♥❞❤✐ ❬✶✵❪ ❢♦✉♥❞ t❤❛t t❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ❝♦♥t❛✐♥❡❞ ❛r♦✉♥❞ ✹✵✪ β s❤❡❡ts✱ t❤❡ ▼❛❙♣✶ ❛♥❞ ▼❛❙♣✷ ♣♦✇❞❡rs r❡s♣❡❝t✐✈❡❧② ❝♦♥t❛✐♥ ❛r♦✉♥❞ ✶✷✪ ❛♥❞ ✸✲✹✪ β s❤❡❡ts✳ ❚❤❡s❡ r❡s✉❧ts ✇❡r❡ ❢♦✉♥❞ ❜② ❛♥❛❧②③✐♥❣ t❤❡ ✐♥t❡♥s✐t② ♦❢ t❤❡ ❋❚■❘ ♣❡❛❦ ❢r♦♠ t❤❡ ❛♠✐❞❡ ✶ r❛♥❣❡ ✇❤♦s❡ ✇❛✈❡♥✉♠❜❡r ✇❛s ✶✻✸✵❝♠−1 ✱ ✇❤✐❝❤ ❝♦rr❡s♣♦♥❞s t♦ β s❤❡❡ts str✉❝t✉r❡✱ ❛♥❞ ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ♣❡r❝❡♥t ♦❢ t❤❡ s✐❣♥❛❧ t❤❛t t❤❡② r❡♣r❡s❡♥t❡❞ ❜② ❝♦♠♣❛r✐♥❣ t❤❡ ❛r❡❛ ✉♥❞❡r t❤❡ ♣❡❛❦ t♦ t❤❡ ❛r❡❛ ✉♥❞❡r t❤❡ ❤❡❧✐❝❛❧ r❡❣✐♦♥s✳ ❆ s❡❧❡❝t✐♦♥ ♦❢ s❡❝♦♥❞❛r② str✉❝t✉r❡s ❛♥❞ ❝♦rr❡s♣♦♥❞✐♥❣ ✇❛✈❡ ♥✉♠❜❡rs ❛r❡ s❤♦✇♥ ✐♥ ❚❛❜❧❡ ✹✳✺✳ ❍♦✇❡✈❡r✱ ❝❧♦s❡ ❡①❛♠✐♥❛t✐♦♥ ♦❢ t❤❡ ❋❚■❘ ❞❛t❛ ✐♥ ●❛♥❞❤✐✬s ✇♦r❦ r❛✐s❡s s♦♠❡ q✉❡st✐♦♥s✳ ●❛♥❞❤✐ st❛t❡s t❤❛t t❤❡ ♣❡❛❦s ❢♦r β s❤❡❡ts ❛♣♣❡❛rs ❛t ✶✻✸✵❝♠−1 ✱ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✹✳✻✱ ❜✉t ✸✾  ❘❡♣♦rt❡❞ ✈❛❧✉❡ ✭❝♠−1 ✮ ✶✻✺✵ ✶✻✸✵ ✶✺✹✺ ✶✺✷✵ ✶✷✼✵ ✶✷✸✵  ❆♥❛❧②③❡❞ ✈❛❧✉❡ ✭❝♠−1 ✮ ✶✻✺✵ ✶✻✶✵ ✶✺✹✵ ✶✺✵✵ ✶✷✾✵ ✶✷✸✵  ❚❛❜❧❡ ✹✳✻✿ ❘❡♣♦rt❡❞ ✇❛✈❡♥✉♠❜❡rs ❢r♦♠ ❋✐❣✉r❡ ✹✳✻ ❛♥❞ ❡st✐♠❛t❡s ♦❢ ❛❝t✉❛❧ ✇❛✈❡♥✉♠❜❡rs ✇❤✐❝❤ ❢♦✉♥❞ ✉s✐♥❣ ✐♠❛❣❡ ❛♥❛❧②s✐s s♦❢t✇❛r❡ ✭❆❞♦❜❡ ■❧❧✉str❛t♦r✮✳ ✐t ❛♣♣❡❛rs t❤❛t s♦♠❡ ♦❢ t❤❡ ❧✐♥❡s ❛r❡ ♥♦t ❛❝t✉❛❧❧② ✇❤❡r❡ t❤❡② ❛r❡ s❛✐❞ t♦ ❜❡✳ ❚❤✐s ❝❛♥ ❜❡ s❡❡♥ ❜② ♥♦t✐♥❣ t❤❛t t❤❡ s❡♣❛r❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❧✐♥❡s ❛t ✶✻✺✵ ❝♠−1 ❛♥❞ ✶✻✸✵ ❝♠−1 ✐s ❜❛r❡❧② ❤❛❧❢ ♦❢ t❤❡ s❡♣❛r❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❧✐♥❡s ❛t ✶✻✸✵ ❝♠−1 ❛♥❞ ✶✺✹✵ ❝♠−1 ✇❤❡♥ ✐t s❤♦✉❧❞ ❜❡ ❧❡ss t❤❛♥ ❛ q✉❛rt❡r✳ ❋✉rt❤❡r♠♦r❡✱ s❝❛❧✐♥❣ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ✶✺✸✵ ❝♠−1 ❛♥❞ ✶✻✸✵ ❝♠−1 ♣❡❛❦s ❜② t❡♥ r❡s✉❧ts ✐♥ ❛ ❧✐♥❡ t❤❛t ✐s ♥❡❛r❧② ✹✵✵ ✇❛✈❡♥✉♠❜❡rs ❧♦♥❣✳ ❆tt❛❝❤✐♥❣ ❛ s❝❛❧❡ t♦ t❤❡ ❞❛t❛ ✭t❤❡r❡ ✇❡r❡ ♥♦ ① ❛①✐s t✐❝s ✐♥✐t✐❛❧❧②✱ r❡✈❡❛❧s t❤❛t s❡✈❡r❛❧ ♦❢ t❤❡ ♣❡❛❦s ❛r❡ ❧♦❝❛t❡❞ ❛t ❞✐✛❡r❡♥t ✇❛✈❡♥✉♠❜❡rs✱ ❛s s❤♦✇♥ ✐♥ ❚❛❜❧❡ ✹✳✻✳ ❆ss✉♠✐♥❣ t❤✐s ✐s ❝♦rr❡❝t✱ t❤✐s ✐♠♣❧✐❡s t❤❛t t❤❡ ❞♦♠✐♥❛♥t str✉❝t✉r❡ ♦❜s❡r✈❡❞ ❛t t❤❡ ❧✐♥❡ ✶✻✸✵❝♠−1 ✐s ♥♦t β s❤❡❡ts✱ ❜✉t ✐s ✐♥st❡❛❞ ❛❣❣r❡✲ ❣❛t❡❞ str❛♥❞s✱ ❛ str✉❝t✉r❡ t❤❛t ✐s ❝♦♥s✐st❡♥t ✇✐t❤ ❞❡♥❛t✉r❡❞ ♣r♦t❡✐♥s✳ ❋✉rt❤❡r♠♦r❡✱ ✐♥ t❤❡ r❛♥❣❡ ♦❢ ✇❤❡r❡ β s❤❡❡ts ❛❝t✉❛❧❧② ♦❝❝✉r✱ t❤❡r❡ ✐s ❛ ♠✐♥✐♠✉♠ ✐♥ t❤❡ ❡❧❡❝tr♦s♣✉♥ s✐❧❦ s✉❣❣❡st✐♥❣ t❤❛t β s❤❡❡ts ❛r❡ ♥♦t ❛ ❞♦♠✐♥❛♥t str✉❝t✉r❡ ✐♥ t❤❡ ♠❛t❡r✐❛❧✳ ❲❤❛t t❤✐s ✐♠♣❧✐❡s ✐s t❤❛t t❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ♣r♦t❡✐♥s ❞♦ ♥♦t ❢♦r♠ ❛ r❡❣✉❧❛r s❡❝♦♥❞❛r② str✉❝t✉r❡✳ ❆❧s♦✱ ✇❤✐❧❡ t❤❡r❡ ✐s ♥♦ ♣❡❛❦ ✐♥ t❤❡ ▼❛❙♣✶ ♣♦✇❞❡r ❛t ✶✻✸✵✱ t❤❡r❡ ✐s s✐❣♥✐✜❝❛♥t❧② ♠♦r❡ s✐❣♥❛❧ ✐♥ t❤❛t r❡❣✐♦♥ t❤❛♥ t❤❡ ❡❧❡❝tr♦s♣✉♥ ✜❜❡rs✱ ♠❡❛♥✐♥❣ t❤❛t t❤❡ ♣r❡s❡♥❝❡ ♦❢ β s❤❡❡ts ✐s ❧✐❦❡❧②✳ ■♥t❡r❡st✐♥❣❧② ❤♦✇❡✈❡r✱ t❤❡r❡ ✐s ❛ s♠❛❧❧ ♣❡❛❦ ✐♥ t❤❡ β s❤❡❡t r❡❣✐♠❡ ✐♥ ❜♦t❤ t❤❡ ▼❛❙♣✷ ♣♦✇❞❡r ❛♥❞ ✜❜❡rs✳ ■t s❤♦✉❧❞ ❛❧s♦ ❜❡ ❛❞❞❡❞ t❤❛t ✇❤✐❧❡ t❤❡ ❛❝t✉❛❧ str✉❝t✉r❡s ♠❛② ❜❡ ❞✐✛❡r❡♥t t❤❛♥ r❡♣♦rt❡❞✱ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞❛r② str✉❝t✉r❡ t❤❛t ✇❛s ❝❛❧❝✉❧❛t❡❞ ❢r♦♠ t❤❡s❡ ♣❡❛❦s ✐s ❛❧s♦ s✉s♣❡❝t ❛s s❡✈❡r❛❧ ❝r✐t✐❝❛❧ r❡✈✐❡✇s ♦❢ ❋❚■❘ ❤❛✈❡ ❜❡❡♥ s❦❡♣t✐❝❛❧ ♦❢ t❤❡ ✉s❡ ♦❢ ❋❚■❘ ✐♥ ❞❡t❡r♠✐♥✐♥❣ s❡❝♦♥❞❛r② ❡①❛❝t ♣r♦♣♦rt✐♦♥s ❬✷✺✱ ✷✻❪✳  ✹✵  90 cm-1 20 cm-1  20 cm-1 20 cm-1  20 cm-1 X10 400 cm-1  ❋✐❣✉r❡ ✹✳✻✿ ❆♥❛❧②s✐s ♦❢ ●❛♥❞❤✐✬s ▼❛❙♣✶ ❋❚■❘ r❡s✉❧ts✳ ❚❤❡ ✷✵ ❝♠−1 s❡♣❡r❛t✐♦♥ ✐s ♠♦r❡ t❤❛♥ ❤❛❧❢ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✾✵ ❝♠−1 s❡♣❡r❛t✐♦♥✱ ❛♥❞ t❤❛t ✇❤❡♥ s❝❛❧❡❞ ❜② ❛ ✶✵✱ ✐t r❡s✉❧ts ✐♥ ❛ ❧❡♥❣t❤ ♦❢ ✹✵✵ ❝♠−1 ✳ ◆♦t❡ t❤❛t t❤❡ t✐❝ ❛❧♦♥❣ t❤❡ ① ❛①✐s ✇❡r❡ ❛❞❞❡❞ t♦ t❤❡ ✐♠❛❣❡ ❛s t❤❡② ✇❡r❡ ♥♦t ✐♥❝❧✉❞❡❞ ✐♥ t❤❡ ♦r✐❣✐♥❛❧✳ ❚❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ✶✻✺✵ ❝♠−1 ❧✐♥❡ t♦ t❤❡ ❜♦✉♥❞❛r② ❛t ✶✽✵✵ ❝♠−1 ✇❛s ✉s❡❞ ❛s t❤❡ r❡❢❡r❡♥❝❡ ❢♦r t❤❡s❡ t✐❝s✳  ✹✶  ❈❤❛♣t❡r ✺  ❊①♣❡r✐♠❡♥t❛❧ ▼❛t❡r✐❛❧s✱ ▼❡t❤♦❞s✱ ❛♥❞ ❘❡s✉❧ts  ■♥ t❤✐s ❝❤❛♣t❡r ✇❡ ❞❡s❝r✐❜❡ s❛♠♣❧❡s ❛♥❞ t❤❡✐r ♣r❡♣❡r❛t✐♦♥✱ t❤❡ ◆▼❘ ❡①♣❡r✐♠❡♥ts ❛♥❞ ♠❡t❤✲ ♦❞s ♦❢ ❛♥❛❧②s✐s✱ ❛♥❞ ♣r❡s❡♥t t❤❡ r❛✇ r❡s✉❧ts ❢r♦♠ t❤❡♠✳  ✺✳✶  ❙❛♠♣❧❡s Pr❡♣❛r❛t✐♦♥  ❋♦✉r ❞✐✛❡r❡♥t ♣r♦t❡✐♥ s❛♠♣❧❡s ✇❡r❡ ❛♥❛❧②③❡❞ ✉s✐♥❣ ◆▼❘❀ s♣✐❞❡r ✭N. Clavipes✮ s✐❧❦✱ ▼❛❙♣✶✱ ▼❛❙♣✷✱ ❛♥❞ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶✳ ❚❤❡ s❛♠♣❧❡ ♠❛ss❡s ✇❡r❡ ✸✹✳✾♠❣ ♦❢ s♣✐❞❡r s✐❧❦✱ ✹✷♠❣ ♦❢ ▼❛❙♣✶✱ ✹✹♠❣ ♦❢ ▼❛❙♣✷✱ ❛♥❞ ✼♠❣ ♦❢ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶✳ ❚❤❡ ▼❛❙♣✶ ❛♥❞ ▼❛❙♣✷ ✇❡r❡ ♣r♦✈✐❞❡❞ t♦ t❤❡ ❆❞✈❛♥❝❡❞ ❋✐❜r♦✉s ▼❛t❡r✐❛❧s ▲❛❜♦r❛t♦r② ❛t ❯❇❈ ❜② ◆❡①✐❛ ❇✐♦t❡❝❤♥♦❧♦❣✐❡s ■♥❝✳ ❚❤❡ ♣r♦t❡✐♥s ✇❡r❡ ♣r♦❞✉❝❡❞ ✉s✐♥❣ ❣❡♥❡t✐❝❛❧❧② ❡♥❣✐♥❡❡r❡❞ ❣♦❛ts ✇❤♦ ♣r♦❞✉❝❡❞ t❤❡ ♣r♦t❡✐♥ ✈✐❛ ❧❛❝t❛t✐♦♥ ✭♠✐❧❦✮ ✉s✐♥❣ ❛ ♣r♦❝❡ss ❝r❡❛t❡❞ ❜② ▲❛③❛r✐s ❡t ❛❧✳ ❬✷✹❪ ❛♥❞ ✇❡r❡ ♣r♦✈✐❞❡❞ ❛s ❧②♦♣❤✐❧✐③❡❞ ♣♦✇❞❡rs ❛♥❞ ✉s❡❞ ✇✐t❤♦✉t ❢✉rt❤❡r ♣r♦❝❡ss✐♥❣✳ ❚❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ✇❛s ❡❧❡❝tr♦s♣✉♥ ✐♥ ❛ ❝✉st♦♠ ❜✉✐❧t ❡❧❡❝tr♦s♣✐♥♥✐♥❣ s❡t ✉♣✱ ✇✐t❤ ♠✐♥✲ ✐♠❛❧ ♠❡t❛❧ ♣❛rts t❤❛t ❝❛✉st✐❝ s♦❧✉t✐♦♥s ❝♦✉❧❞ ❝❛✉s❡ ❞❛♠❛❣❡ t♦✳ ❚❤❡ ♣♦❧②♠❡r s♦❧✉t✐♦♥ ✇❛s ♠❛❞❡ ❢r♦♠ ✽% ▼❛❙♣✶ ♣♦✇❞❡r ✐♥ ✾✾% ❢♦r♠✐❝ ❛❝✐❞ ❛♥❞ ❡❧❡❝tr♦s♣✉♥ ❢r♦♠ ❛ ❋✐s❝❤❡r ✶♠❧ ❣❧❛ss ♣✐♣❡tt❡✱ ✇❤✐❝❤ ❤❛❞ ❛ ❝♦♣♣❡r ✇✐r❡ ❝♦♥♥❡❝t❡❞ t♦ ❛ ♣♦✇❡r s✉♣♣❧②✱ ❝❤❛r❣❡❞ ✐t t♦ ✷✵ ❦❱✱ ✐♥s❡rt❡❞ ✐♥t♦ t❤❡ s♦❧✉t✐♦♥✳ ❚❤❡ s♦❧✉t✐♦♥ ✇❛s s♣✉♥ ❛t ❛♥ ❛♥❣❧❡ ♦❢ ✹✺ ❞❡❣r❡❡s ❜❡❧♦✇ t❤❡ ❤♦r✐③♦♥t❛❧ ✇✐t❤ t❤❡ ♣✐♣❡tt❡ t✐♣ ✶✷❝♠ ❢r♦♠ t❤❡ ❣r♦✉♥❞❡❞ ♣❧❛t❡✳ ❚❤❡ s❛♠♣❧❡ ❞✐❞ ♥♦t ✐♥✐t✐❛❧❧② s♣✐♥ ❛♥❞ ✐♥st❡❛❞ ✹✷  s♣✉tt❡r❡❞✳ ❍✐❣❤❡r ✈♦❧t❛❣❡ r❡s✉❧t❡❞ ✐♥ ❛r❝✐♥❣✳ ❚♦ ❝♦♠♣❡♥s❛t❡ ❢♦r t❤✐s t❤❡ ❝♦♥❝❡♥tr❛t✐♦♥ ✇❛s ✐♥❝r❡❛s❡❞ ❜② ✷% t♦ ✶✵% ✇❤❡r❡ ✐t s♣✉♥ ✐♥t♦ ✜❜❡rs✳  ✺✳✷  ❈r♦ss P♦❧❛r✐③❛t✐♦♥ ❊①♣❡r✐♠❡♥ts ❛♥❞ ❈❤❡♠✐❝❛❧ ❙❤✐❢t  ❈r♦ss ♣♦❧❛r✐③❛t✐♦♥ ♠❛❣✐❝ ❛♥❣❧❡ s♣✐♥♥✐♥❣ ❡①♣❡r✐♠❡♥ts ✇❡r❡ ♣❡r❢♦r♠❡❞ ♦♥ ❛ ❱❛r✐❛♥ ✹✵✵ ▼❍③ ✭✾✳✹❚✮ ◆▼❘ s♣❡❝tr♦♠❡t❡r ✇✐t❤ ❛♥ ❍❳❨ ▼❆❙ ♣r♦❜❡ ♦♥ t❤❡ ❢♦✉r s❛♠♣❧❡s✳ ❚❤❡ s❛♠♣❧❡s ✇❡r❡ ♣❛❝❦❡❞ ✐♥t♦ ❛ ③✐r❝♦♥✐❛ r♦t♦r ❛♥❞ t❤❡♥ s♣✉♥ ❛t ✺❦❍③✳ ❚❤❡ ❣♦❛❧ ♦❢ t❤❡ ❝r♦ss ♣♦❧❛r✐③❛t✐♦♥ ❡①♣❡r✐♠❡♥ts ✇❡r❡ t♦ ✜♥❞ ❝❤❡♠✐❝❛❧ s❤✐❢t ❛ss✐❣♥♠❡♥ts ❢♦r t❤❡ ❛♠✐♥♦ ❛❝✐❞ ❝❛r❜♦♥s s♦ s❡❝♦♥❞❛r② str✉❝t✉r❡s ❝♦✉❧❞ ❜❡ ❛ss✐❣♥❡❞✳ ✹✵✵✵ s❝❛♥s ✇❡r❡ t❛❦❡♥ ❢♦r ❡❛❝❤✱ ❡①❝❡♣t ❢♦r s♣✐❞❡r s✐❧❦✱ ✇❤✐❝❤ ✇❛s ✷✵✵✵✳ ❚❤❡ s♣❡❝tr❛ ✇❡r❡ r❡❢❡r❡♥❝❡❞ t♦ ❛❞❛♠❛♥t❛♥❡ s♣❡❝tr❛ t❛❦❡♥ ❡✐t❤❡r ❜❡❢♦r❡ ♦r ❛❢t❡r t❤❡ ❡①♣❡r✐✲ ♠❡♥ts t♦ ♦❜t❛✐♥ t❤❡ ❝❤❡♠✐❝❛❧ s❤✐❢ts ♦❢ ❡❛❝❤ ♣❡❛❦✳ ❚❤❡s❡ ❝❤❡♠✐❝❛❧ s❤✐❢ts ✇❡r❡ t❤❡♥ ❝♦♠♣❛r❡❞ t♦ t❤❡ ✈❛❧✉❡s ❢r♦♠ ❚❛❜❧❡ ✺✳✷ t♦ ❜♦t❤ ❞❡t❡r♠✐♥❡ ✇❤❛t ❛♠✐♥♦ ❛❝✐❞ ❝❛r❜♦♥ t❤❡ ♣❡❛❦ r❡♣r❡s❡♥t❡❞ ❛♥❞ t❤❡ r❛♥❣❡ ♦❢ str✉❝t✉r❡s t❤❛t t❤❡② r❡♣r❡s❡♥t✳ ❚❤❡ ♠❡❛s✉r❡❞ ❝❤❡♠✐❝❛❧ s❤✐❢ts ❢♦r ❡❛❝❤ s❛♠✲ ♣❧❡ ❛r❡ ❧✐st❡❞ ✐♥ ❚❛❜❧❡ ✺✳✶✳ ❚❤❡ s❡❝♦♥❞❛r② str✉❝t✉r❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡s❡ ❝❤❡♠✐❝❛❧ s❤✐❢ts ✇✐❧❧ ❜❡ ❞✐s❝✉ss❡❞ ✐♥ ❞❡t❛✐❧ ✐♥ t❤❡ ❢✉t✉r❡ s❡❝t✐♦♥s✳ ❖♥❡ ✐ss✉❡ ✇✐t❤ ❝r♦ss ♣♦❧❛r✐③❛t✐♦♥ ❡①♣❡r✐♠❡♥ts ✐s t❤❛t t❤❡ ✐♥t❡♥s✐t✐❡s ❛r❡ ♥♦t q✉❛♥t✐t❛t✐✈❡✳ ❱❡r② r✐❣✐❞ str✉❝t✉r❡s t❡♥❞ t♦ ❝r♦ss ♣♦❧❛r✐③❡ ✇❡❧❧ ❛♥❞ ❝❛♥ s❤♦✇ s✐❣♥❛❧ ✐♥t❡♥s✐t✐❡s ❛♣♣r♦❛❝❤✐♥❣ t❤❡ t❤❡♦r❡t✐❝❛❧ ♠❛①✐♠✉♠✳ ▼♦r❡ ♠♦❜✐❧❡ str✉❝t✉r❡s t❡♥❞ t♦ ❝r♦ss ♣♦❧❛r✐③❡ ❧❡ss ✇❡❧❧ ❞✉❡ t♦ ♣❛rt✐❛❧ ❛✈❡r❛❣✐♥❣ ♦❢ t❤❡ ❈✲❍ ❞✐♣♦❧❛r ❝♦✉♣❧✐♥❣s t❤❛t ❛r❡ ♥❡❝❡ss❛r② ❢♦r ❝r♦ss ♣♦❧❛r✐③❛t✐♦♥✱ s✉❝❤ ♣❛rt✐❛❧ ❛✈❡r❛❣✐♥❣ ❝❛♥ r❡s✉❧t ✐♥ r❡❞✉❝❡❞ ✐♥t❡♥s✐t✐❡s ❢♦r ♣❡❛❦s ❛r✐s✐♥❣ ❢r♦♠ ❧❡ss r✐❣✐❞ str✉❝t✉r❡s ❬✹❪✳  ✺✳✸  13  ❘❡❧❛①❛t✐♦♥ ▼❡❛s✉r❡♠❡♥ts  ❈ r❡❧❛①❛t✐♦♥ t✐♠❡s ✇❡r❡ ♠❡❛s✉r❡❞ ✉s✐♥❣ ❛ ❝r♦ss ♣♦❧❛r✐③❡❞ ♣✉❧s❡ s❡q✉❡♥❝❡ s❤♦✇♥ ✐♥ ❋✐❣✲  ✉r❡ ✺✳✶✳ ❚❤✐s ♣✉❧s❡ s❡q✉❡♥❝❡ ✇❛s ❞❡✈❡❧♦♣❡❞ ❜② ❚♦r❝❤✐❛ ❬✷✼❪✱ ❛♥❞ ❛♠♣❧✐✜❡s t❤❡  13  ❈ s✐❣♥❛❧  ✉s✐♥❣ ❝r♦ss ♣♦❧❛r✐③❛t✐♦♥✳ ❚❤❡ ✜rst ♣❛rt ♦❢ t❤❡ s❡q✉❡♥❝❡ ✐s t❤❡ s❛♠❡ ❛s ❛ s✐♠♣❧❡ ❝r♦ss✲ ♣♦❧❛r✐③❛t✐♦♥ ❡①♣❡r✐♠❡♥t ❡♠♣❧♦②❡❞ ✐♥ ❙❡❝t✐♦♥ ✺✳✷ ✇❤❡r❡ ②✲❛①✐s✳ ❚❤❡♥✱ ❛  π 2  13  ❈ ♠❛❣♥❡t✐③❡s ❞✐r❡❝t❧② ❛❧♦♥❣ t❤❡  ♣✉❧s❡ r♦t❛t❡s t❤❡ ♠❛❣♥❡t✐③❛t✐♦♥ ❛❧t❡r♥❛t❡❧② t♦ t❤❡ ③ ♦r ✲③ ❛①✐s✱ ✇❤❡r❡ ✐t ✹✸  π 2 1  H  CP  π 2  π 2 D2  13  C  CP  ❋✐❣✉r❡ ✺✳✶✿ T1 ❡①♣❡r✐♠❡♥t ♣✉❧s❡ s❡q✉❡♥❝❡ ❡♠♣❧♦②✐♥❣ ❝r♦ss✲♣♦❧❛r✐③❛t✐♦♥ t♦ st✉❞② t❤❡ ❧♦♥❣✐t✉✲ ❞✐♥❛❧ r❡❧❛①❛t✐♦♥ ♦❢ t❤❡ 13 C ♥✉❝❧❡✐✳ r❡❧❛①❡s ❜❛❝❦ t♦ ❡q✉✐❧✐❜r✐✉♠ ❛❧♦♥❣ t❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ ❛①✐s ❢♦r ❛ t✐♠❡ ♦❢ ❉✷✳ ❆❢t❡r t❤✐s ❞❡❧❛② t✐♠❡✱ ❛ t❤✐r❞  π 2  ♣✉❧s❡ r❡t✉r♥s ❛♥② r❡♠❛✐♥✐♥❣ ♠❛❣♥❡t✐③❛t✐♦♥ t♦ t❤❡ ①②✲♣❧❛♥❡ ✇❤❡r❡ ✐t ♣r♦✲  ✈✐❞❡s t❤❡ ❛❝q✉✐s✐t✐♦♥ s✐❣♥❛❧✳ ❚❤✐s s❡q✉❡♥❝❡ ✐s r✉♥ ♠✉❧t✐♣❧❡ t✐♠❡s ✇✐t❤ ❛ ✈❛r✐❡t② ♦❢ ❉✷✬s✱ t❤❛t r❛♥❣❡❞✱ ❞❡♣❡♥❞✐♥❣ ♦♥ ✇❤✐❝❤ s❛♠♣❧❡✱ ❢r♦♠ ✶♠s t♦ ✶✵✵s✳ ❚❤❡ r❛♥❣❡ t❤❛t ❛❧❧ ✜ts ❝♦✈❡r❡❞ ✇❛s ❉✷❂✭✵✳✶s✱✵✳✷s✱✵✳✺s✱✶s✱✷s✱✺s✱✶✵s✱✷✵s✱✺✵s✮✳ ❖t❤❡r t✐♠❡s ✇❡r❡ ✉s❡❞ ❛s ✇❡❧❧✱ ❜✉t t❤✐s r❛♥❣❡ ✇❛s ❢♦✉♥❞ t♦ ❜❡ ❝r✐t✐❝❛❧✳ ❚❤❡s❡ s♣❡❝tr❛ ✇❡r❡ ❛♥❛❧②③❡❞ ❜② ❡✐t❤❡r ✐♥t❡❣r❛t✐♥❣ ❛❝r♦ss t❤❡ ♣❡❛❦ t♦ ✜♥❞ t❤❡ ♣❡❛❦ ✐♥t❡♥s✐t② ✉s✐♥❣ ❛ ❝✉st♦♠ ◆▼❘ ❛♥❛❧②s✐s ♣r♦❣r❛♠ ❝❛❧❧❡❞ ❳◆▼❘✳ ❚❤❡ ❡rr♦r ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ s❛♠♣❧❡ ♥♦✐s❡ t✐♠❡s t❤❡ sq✉❛r❡ r♦♦t ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❞❛t❛ ♣♦✐♥ts t❤❛t t❤❡ ♣❡❛❦ ✇❛s ✐♥t❡❣r❛t❡❞ ♦✈❡r✳ ❚❤❡ ✐♥t❡♥s✐t✐❡s ♦❢ t❤❡ ♣❡❛❦s ❞❡❝❛② ✇✐t❤ ✐♥❝r❡❛s✐♥❣ ❞❡❧❛② t✐♠❡✳ ❚❤❡ ❡①❛❝t ❞❡♣❡♥❞❡♥❝❡ ♦♥ ❉✷ ✐s ❞❡♣❡♥❞❡♥t ♦♥ t❤❡ ✉♥❞❡r❧②✐♥❣ ♠♦❧❡❝✉❧❛r ♠♦t✐♦♥s t❤❛t ❝❛✉s❡ r❡❧❛①❛t✐♦♥✳ ❋♦r ❛ s✐♠♣❧❡ s②st❡♠ ✇✐t❤ ♦♥❧② ❛ s✐♥❣❧❡✱ ♦r ✈❡r② t✐❣❤t ❞✐str✐❜✉t✐♦♥ ♦❢ ♠♦t✐♦♥❛❧ ❝♦rr❡❧❛t✐♦♥ t✐♠❡s t❤❡ ❢✉♥❝t✐♦♥ ✇✐❧❧ ❧♦♦❦ ❧✐❦❡  I(D2) = I0 e  −D2 T1  ✭✺✳✶✮  ❋♦r s②st❡♠s t❤❛t ❤❛✈❡ t✇♦ ❞✐st✐♥❝t s❡ts ♦❢ ♠♦t✐♦♥✱ t②♣✐❝❛❧❧② ✐♥❞✐❝❛t✐✈❡ ♦❢ t✇♦ str✉❝t✉r❡s✱ t❤❡ ✐♥t❡♥s✐t② ❞❡❝❛② ✇✐❧❧ t❛❦❡ t❤❡ ❢♦r♠ ♦❢ ❛ ❞♦✉❜❧❡ ❡①♣♦♥❡♥t✐❛❧ ✇❤✐❝❤ ✇✐❧❧ ❜❡ ✹✹  −D2 f ast  I(D2) = I0f ast e T1  −D2 slow  + I0slow e T1  ,  ✭✺✳✷✮  ✇❤❡r❡ T1f ast ❛♥❞ I1f ast ❛r❡ t❤❡ r❡❧❛①❛t✐♦♥ t✐♠❡ ❛♥❞ ✐♥✐t✐❛❧ ✐♥t❡♥s✐t② ♦❢ t❤❡ ❢❛st r❡❧❛①✐♥❣ s②st❡♠s✱ ❛♥❞ T1slow ❛♥❞ I0slow ❛r❡ t❤❡ r❡❧❛①❛t✐♦♥ t✐♠❡ ❛♥❞ ✐♥✐t✐❛❧ ✐♥t❡♥s✐t② ♦❢ t❤❡ s❧♦✇❡r r❡❧❛①✐♥❣ s②st❡♠✳ ❆♥ ✐♠♣♦rt❛♥t ♣♦✐♥t ♦❢ ❝❧❛r✐✜❝❛t✐♦♥ ♠✉st ❜❡ ♠❛❞❡ ❤❡r❡ t❤❛t t❤❡ ❢❛st ❛♥❞ s❧♦✇ t❡r♠✐♥♦❧♦❣② ✐s ♥♦t ❝♦♥♥❡❝t❡❞ t♦ t❤❡ s♣❡❡❞ ♦❢ t❤❡ ♠♦❧❡❝✉❧❛r ♠♦t✐♦♥✱ ❛s t❤❡ t✇♦ ♠♦t✐♦♥s ❝♦✉❧❞ ❜❡ ♦♥ ❡✐t❤❡r s✐❞❡ ♦❢ t❤❡ ❢❛st✴s❧♦✇ r❡❣✐♠❡ ♦❢ t❤❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡✳ ❋♦r s②st❡♠s t❤❛t ❤❛✈❡ ❛ ❞✐str✐❜✉t✐♦♥ ♦❢ r❡❧❛①❛t✐♦♥ t✐♠❡s✱ t❤❡ ❝✉r✈❡ ✇✐❧❧ ♦❢t❡♥ r❡❧❛① ✐♥ ❛ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧✳ ❚❤❡ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ r❡♣r❡s❡♥ts t❤❡ ❲✐❧❧✐❛♠✲❲❛tts ❞✐str✐❜✉t✐♦♥✱ ❛♥ ❛ss②♠❡tr✐❝ ❞✐str✐❜✉t✐♦♥ t❤❛t ❢r♦♠ ❛♥ ❡①♣❡r✐♠❡♥t❛❧ ♣❡rs♣❡❝t✐✈❡✱ ✐s ♥❡❛r❧② ✐♥❞✐st✐♥❣✉✐s❤❛❜❧❡ ❢r♦♠ t❤❡ ❉❛✈✐❞s♦♥✲❈♦❧❡ ❞✐str✐❜✉t✐♦♥ ❬✷✽❪✳ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ r❡❧❛①❛t✐♦♥ t✐♠❡s ✐♥ t❤❡ ❲✐❧❧✐❛♠✲ ❲❛tts ❞✐str✐❜✉t✐♦♥ ✐s  1 T1 ρ( ∗ ) = − T1 πT1  ∞  k=0  T1 βk (−1)k sin(πβk)Γ(βk + 1) ∗ , k! T1  ✭✺✳✸✮  ✇❤❡r❡ β ✐s r❡❧❛t❡❞ t♦ t❤❡ ❞✐str✐❜✉t✐♦♥ ✇✐❞t❤ ♦❢ r❡❧❛①❛t✐♦♥ t✐♠❡s ❛♥❞ r❛♥❣❡s ❢r♦♠ ✵✱ r❡♣✲ r❡s❡♥t✐♥❣ ❛ ❧❛r❣❡ ❞✐str✐❜✉t✐♦♥✱ ❛♥❞ ✶✱ ✇❤✐❝❤ r❡♣r❡s❡♥ts ❛ s✐♥❣❧❡ r❡❧❛①❛t✐♦♥ t✐♠❡✳ T1∗ ✐s t❤❡ ❞✐str✐❜✉t✐♦♥ ❝❡♥t❡r ❢♦r β = 1✳ ❋♦r ✈❛❧✉❡s ♦❢ β ❧❡ss t❤❛♥ ♦♥❡✱ t❤❡ ❞✐str✐❜✉t✐♦♥ ♠❛①✐♠✉♠ ♣♦✐♥t ✐s ❞❡♣❡♥❞❡♥t ♦♥ ❜♦t❤ T1∗ ❛♥❞ β ♣❛r❛♠❡t❡rs✳ ❚❤❡ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ②✐❡❧❞s ❛ r❡❧❛①❛t✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❢♦r♠✿  β  I(D2) = I0 e  −  D2 T1∗  .  ✭✺✳✹✮  ❚❤❡ ♠❡❛♥ r❡❧❛①❛t✐♦♥ t✐♠❡ ❢♦r ❛ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✐s ❣✐✈❡♥ ❜②✿  < T1 >=  T1∗ Γ β  1 β  .  ✭✺✳✺✮  ❙❡❧❡❝t✐♦♥ ❛♠♦♥❣st ✜ts t♦ ❛ s✐♥❣❧❡ ❡①♣♦♥❡♥t✐❛❧✱ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧✱ ❛♥❞ ❞♦✉❜❧❡ ❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛②s ✐s ❞♦♥❡ ❜❛s❡❞ ♦♥ t❤❡ t❡st ♦❢ ❛❞❞✐t✐♦♥❛❧ ✜ts✿  Fχ =  χ2 (m) − χ2 (m + 1) χ2 (m + 1)/(N − m − 1) ✹✺  ✭✺✳✻✮  α  α  Q  G  Electrospun MaSp1 Spider Silk MaSp1 MaSp2  β  Intensity  S  β,γ  Q  α  α*  S  α  P 70  60  A  50  40  30  β  A 20  10  0  Chemical Shift (ppm)  ❋✐❣✉r❡ ✺✳✷✿ ❚❤❡ ❝r♦ss ♣♦❧❛r✐③❛t✐♦♥ s♣❡❝tr❛ ❢♦r ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶✱ ▼❛❙♣✶✱ ▼❛❙♣✷✱ ❛♥❞ s♣✐❞❡r s✐❧❦✳ ❚❤❡ ♣r♦❧✐♥❡ r❡s✐❞✉❡s ❝❛r❜♦♥s ❜❡t✇❡❡♥ ✷✵♣♣♠✱ ❛♥❞ ✹✵♣♣♠ ❛r❡ ✉♥❧❛❜❡❧❡❞ ❜❡❝❛✉s❡ ♦❢ t❤❡ ♦✈❡r❧❛♣ ✇✐t❤ ♦t❤❡r r❡s✐❞✉❡s✱ ❛♥❞ t❤❡ ❢❛❝t t❤❛t ♦♥❧② ▼❛❙♣✷ ❝♦♥t❛✐♥s ❛ s✐❣♥✐✜❝❛♥t ❛♠♦✉♥t ♦❢ ♣r♦❧✐♥❡✳ ✇❤❡r❡ χ2 ✐s t❤❡ ❣♦♦❞♥❡ss ♦❢ ✜t✱ ◆ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❞❛t❛ ♣♦✐♥ts✱ ❛♥❞ ♠ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ♣❛r❛♠❡t❡rs ✉s❡❞ ✐♥ t❤❡ ✜t✳ Fχ r❡♣r❡s❡♥ts t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ✜t ❜❡✐♥❣ ❛♥ ✐♠♣r♦✈❡♠❡♥t✱ ✇❤✐❝❤ ♠✉st ❜❡ ❛❜♦✈❡ ❛ t❤r❡s❤♦❧❞ ✈❛❧✉❡ ❢♦r t❤❡ ✜t t♦ ❤❛✈❡ ❛ ✾✺✪ ❝♦♥✜❞❡♥❝❡ ❢♦r t❤❡ ❤✐❣❤❡r ♣❛r❛♠❡t❡r ✜t t♦ ❜❡ ❛❝❝❡♣t❡❞✳ ❚❤✐s t❤r❡s❤♦❧❞ ✈❛❧✉❡ ✐s ❞❡♣❡♥❞❡♥t ♦♥ t❤❡ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ✐♥ t❤❡ t✇♦ ✜ts ❛♥❞ ❝❛♥ ✇❛s ❢♦✉♥❞ ✐♥ ❛ r❡❢❡r❡♥❝❡ t❛❜❧❡ ❬✷✾❪✳ ■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡s❡ r❡❧❛①❛t✐♦♥ ❡①♣❡r✐♠❡♥ts✱ ❛♥ ❛❞❞✐t✐♦♥❛❧ ❡①♣❡r✐♠❡♥t ✇❛s ♣❡r❢♦r♠❡❞ ♦♥ t❤❡ s♣✐❞❡r s✐❧❦✱ ♠❡❛s✉r✐♥❣ t❤❡ r❡❧❛①❛t✐♦♥ t✐♠❡ ❛t 40◦ C ✳ ❚❤✐s ♠❡❛s✉r❡♠❡♥t ✇❛s ❝♦♥❞✉❝t❡❞ t♦ ✐♥❝r❡❛s❡ t❤❡ t❤❡r♠❛❧ ♠♦t✐♦♥ t❤✉s ❧♦✇❡r✐♥❣ t❤❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡✱ ❛❧❧♦✇✐♥❣ ❛ ❣❧✐♠♣s❡ ♦❢ ✇❤❡t❤❡r t❤❡ ♣r♦t❡✐♥ ♠♦t✐♦♥s ✇❡r❡ ✐♥ t❤❡ ❢❛st ♦r s❧♦✇ r❡❣✐♠❡✳ ▲❛st❧②✱ ✐t ✐s ✐♠♣♦rt❛♥t t♦ ♥♦t❡ t❤❛t ✇✐t❤ t❤❡ ❡①❝❡♣t✐♦♥ ♦❢ t❤❡ ❝❛r❜♦♥②❧✱ ❛❧❧ ♦❢ t❤❡ ❜❛❝❦❜♦♥❡ ❛♥❞ s✐❞❡ ❝❤❛✐♥ ❝❛r❜♦♥s ❛r❡ ❛ss✉♠❡❞ t♦ ♣r✐♠❛r✐❧② r❡❧❛① t❤r♦✉❣❤ t❤❡ ❞✐♣♦❧❛r ✐♥t❡r❛❝t✐♦♥ ❞✉❡ t♦ ❝♦✈❛❧❡♥t❧② ❜♦♥❞❡❞ ❤②❞r♦❣❡♥✳  ✺✳✹  ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❙❡❝♦♥❞❛r② ❙tr✉❝t✉r❡s  ❚❤❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞❛r② str✉❝t✉r❡ ✐s ❞♦♥❡ t❤r♦✉❣❤ ❝❛r❡❢✉❧ ❛♥❛❧②s✐s ♦❢ t❤❡ ❞✐❢✲ ❢❡r❡♥t ❜❛❝❦❜♦♥❡ ❛♥❞ s✐❞❡ ❝❤❛✐♥ ❝❛r❜♦♥s ♣❡❛❦s ✐♥ t❤❡ ◆▼❘ s♣❡❝tr✉♠✳ ❆❢t❡r t❤❡ ♠♦❧❡❝✉❧❛r  ✹✻  ❈ ❝❤❡♠✐❝❛❧ s❤✐❢t ✭✐♥ ♣♣♠✮ ❉r❛❣❧✐♥❡ ▼❛❙♣✶ ▼❛❙♣✷ ❊❧❡❝tr♦s♣✉♥ ❙✐❧❦ ▼❛❙♣✶ ✹✾✳✷ ✹✾✳✻ ✹✾✳✷ ✺✷✳✸ ✷✵✳✺ ✷✶✳✶ ✷✵✳✽ ✶✺✳✽✲✶✻✳✽ ✷✽✳✼ ✷✽✳✺ ✷✽✳✹ ✸✺✳✺✲✸✻✳✺ ✹✷✳✽ ✹✸✳✹ ✹✷✳✾ ✹✷✳✹ ✷✾✲✸✸✱ ✷✺ ✷✽✲✸✸✱ ✷✺ ✷✽✲✸✶✱ ✷✺ ✷✽✲✸✸ ✺✹✳✶ ✻✵✳✺ ✻✵✳✽ ✻✵✳✼ ✺✹✳✺ 13  ❘❡s✐❞✉❡ ❆❧❛♥✐♥❡ α ❈❛r❜♦♥ ❆❧❛♥✐♥❡ β ❈❛r❜♦♥ αβ ∆CSala ●❧②❝✐♥❡ α ❈❛r❜♦♥ ●❧✉t❛♠✐♥❡ β, γ ❈❛r❜♦♥ ●❧✉t❛♠✐♥❡ α ❈❛r❜♦♥ Pr♦❧✐♥❡ α❈❛r❜♦♥ ❙❡r✐♥❡ β ❈❛r❜♦♥ ❙❡r✐♥❡ α ❈❛r❜♦♥  ❉◆ s✐❧❦1  ●❧❛♥❞ s✐❧❦1  ✺✷✳✹ ✶✼✳✷ ✸✺  ✺✶✳✼ ✶✻✳✶ ✸✺✳✻  ❚❛❜❧❡ ✺✳✶✿ ❈❤❡♠✐❝❛❧ s❤✐❢ts ❢♦r s♣✐❞❡r s✐❧❦✱ ▼❛❙♣✶✱ ▼❛❙♣✷✱ ❛♥❞ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶✳ 1 ❚❛❦❡♥ ❢r♦♠ ❬✷✵❪✳ str✉❝t✉r❡✱ ✇❤✐❝❤ ❛♠✐♥♦ ❛❝✐❞ t❤❡ ❝❛r❜♦♥ ✐s ✐♥✱ ❛♥❞ ✇❤❛t ♣♦s✐t✐♦♥ ✐t ♦❝❝✉♣✐❡s ✐♥ t❤❡ ❛♠✐♥♦ ❛❝✐❞ ✭❈❂❖✱ α,β,❡t❝✱✮ t❤❡ s❡❝♦♥❞❛r② str✉❝t✉r❡ ♠❛❦❡s t❤❡ ❧❛r❣❡st ❝♦♥tr✐❜✉t✐♦♥ t♦ ❝❤❡♠✐❝❛❧ s❤✐❢t✳ ❚❤✐s ❛❧❧♦✇s t❤❡ ❝❤❡♠✐❝❛❧ s❤✐❢ts ❢r♦♠ t❤❡ ❝r♦ss ♣♦❧❛r✐③❛t✐♦♥ s♣❡❝tr❛ ✐♥ ❋✐❣✉r❡ ✺✳✷✱ ❛♥❞ ❧✐st❡❞ ✐♥ ❚❛❜❧❡ ✺✳✶✱ t♦ ❜❡ ❝♦♠♣❛r❡❞ t♦ ❦♥♦✇♥ ❝❤❡♠✐❝❛❧ s❤✐❢ts ♦❢ ❝❛r❜♦♥s ✐♥ ❦♥♦✇♥ s❡❝♦♥❞❛r② str✉❝t✉r❡s✱ ✇❤✐❝❤ ❝❛♥ ❜❡ s❡❡♥ ✐♥ ❚❛❜❧❡ ✺✳✷✳ ❋♦r s♣✐❞❡r s✐❧❦ ❛♥❞ ▼❛❙♣✶ t❤❡ t❤r❡❡ ❛♠✐♥♦ ❛❝✐❞s✱ ❣❧②❝✐♥❡✱ ❛❧❛♥✐♥❡✱ ❛♥❞ ❣❧✉t❛♠✐♥❡✱ ♠❛❦❡ ✉♣ ❛❧♠♦st ✾✵✪ ♦❢ t❤❡ r❡s✐❞✉❡s ❛s ❝❛♥ ❜❡ s❡❡♥ ✐♥ ❚❛❜❧❡ ✹✳✸✳ ■♥ ▼❛❙♣✷ ♣r♦❧✐♥❡ ❛♥❞ s❡r✐♥❡ ❛r❡ ❛❧s♦ ❛❜✉♥❞❛♥t✱ ♠❛❦✐♥❣ t❤❡♠ ✐♠♣♦rt❛♥t t♦ st✉❞② ❛s ✇❡❧❧✳ ❚❤❡  13  ❈ s♣❡❝tr❛ ❝♦♥s✐st ♦❢ ♣❡❛❦s ❞✉❡ t♦ ❛❧❧ ♦❢ t❤❡ ❝❛r❜♦♥ ♥✉❝❧❡✐ ✐♥ ❛❧❧ ♦❢ t❤❡ ❛♠✐♥♦  ❛❝✐❞s✳ ❊❛❝❤ ❛♠✐♥♦ ❛❝✐❞ ❤❛s t✇♦ ❜❛❝❦❜♦♥❡ ❝❛r❜♦♥s✱ ❈α ❛♥❞ ❝❛r❜♦♥②❧✳ ❇❡❝❛✉s❡ ♦❢ s❡✈❡r❡ ♦✈❡r❧❛♣ ❛♠♦♥❣st t❤❡ ✈❛r✐♦✉s ❝❛r❜♦♥②❧ ❝❛r❜♦♥s✱ ♥♦ ❛tt❡♠♣t ❛s ❜❡❡♥ ♠❛❞❡ t♦ ❞r❛✇ str✉❝t✉r❛❧ ✐♥❢♦r♠❛t✐♦♥ ❢r♦♠ t❤❡ ❝❛r❜♦♥②❧ ♣❡❛❦s  ✺✳✹✳✶  ❆❧❛♥✐♥❡  ❚❤❡ ❛❧❛♥✐♥❡ r❡s✐❞✉❡ ❤❛s ❛ ❝❤❡♠✐❝❛❧ s❤✐❢t r❛♥❣❡ ♦❢ ❛r♦✉♥❞ ✹✳✺ ♣♣♠✱ st❛rt✐♥❣ ❛t ✹✽✳✷✲✹✾✳✸ ♣♣♠ ❢♦r β s❤❡❡ts t♦ ✺✷✳✸✲✺✷✳✽ ♣♣♠ ❢♦r α ❤❡❧✐❝❡s✳ ■t ✐s ✉♥❢♦rt✉♥❛t❡ t❤❛t t❤❡ 31 ❤❡❧✐① ❝❤❡♠✐❝❛❧ s❤✐❢t ✐s ✹✽✳✾ ♣♣♠✱ r✐❣❤t ✐♥ t❤❡ ♠✐❞❞❧❡ ♦❢ t❤❡ β s❤❡❡t r❛♥❣❡ ♠❛❦✐♥❣ ✐t ❞✐✣❝✉❧t t♦ s❡♣❛r❛t❡ t❤❡ t✇♦ ❢r♦♠ t❤❡ ❝❤❡♠✐❝❛❧ s❤✐❢t ❞❛t❛✳ ❇❡❝❛✉s❡ s♣✐❞❡r s✐❧❦ ❛❧❛♥✐♥❡ ✐s ❛❧♠♦st ❡♥t✐r❡❧② β s❤❡❡ts✱ ❝♦♠♣❛r✐s♦♥s t♦ t❤❡ r❡❧❛①❛t✐♦♥ ❞❛t❛ s❤♦✉❧❞ ❜❡ ❛❜❧❡ t♦ ❤❡❧♣ s❤♦✇ t❤❡ ♣r❡s❡♥❝❡ ♦❢ β s❤❡❡ts✳ ❚❤❡ r❡❧❛①❛t✐♦♥ ♣❛r❛♠❡t❡rs ♦❢ ❛❧❧ t❤❡ s❛♠♣❧❡s✬ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❚❛❜❧❡ ✺✳✸✳ ❚❤❡ ❛❧❛♥✐♥❡ β ❝❛r❜♦♥ ✐s ❛ ❈❍3 ❣r♦✉♣ ❦♥♦✇♥ ❛s ❛ ♠❡t❤②❧ ❣r♦✉♣✳ ❚❤❡ ❛❧❛♥✐♥❡ β ❝❛r❜♦♥ ✹✼  ❈ ❝❤❡♠✐❝❛❧ s❤✐❢t ✭✐♥ ♣♣♠✮ α✲❤❡❧✐① β ✲s❤❡❡t r✳ ❝♦✐❧ 31 ✲❤❡❧✐① ✺✷✳✸✲✺✷✳✽ ✹✽✳✷✲✹✾✳✸ ✺✵✳✺ ✹✽✳✾ ✶✹✳✽✲✶✻✳✵ ✶✾✳✾✲✷✵✳✼ ✶✼✳✶ ✶✼✳✹ ✸✻✳✸✲✸✽ ✷✼✳✺✲✷✾✳✹ ✸✸✳✹ ✸✶✳✺ ✷✺✳✻✲✷✻✳✸ ✷✾✳✵✲✷✾✳✸ ✷✼✳✹ ✷✻✳✾ ✷✾✳✼✲✷✾✳✽ ✺✶✳✵✲✺✶✳✹ ✺✷✳✷ ✺✶✳✷✲✺✶✳✼ ✺✻✳✹✲✺✼✳✵ ✺✶✳✵✲✺✶✳✹ ✺✹✳✷ ✺✸✳✼✲✺✹✳✼2 ✹✸✳✷✲✹✹✳✸ ✹✸✳✶ ✹✶✳✹✲✹✷✳✺ ✻✻✳✸2 ✻✸✳✸1 ✻✷✳✸✲✻✷✳✽2 ✸✵✳✻✲✸✶✳✶2 ✸✷✳✶1 ✸✶✳✻✲✸✷✳✶2 ✺✾✳✷ ✺✹✳✹✲✺✺ ✺✽✳✸ ✺✻✳✽✲✺✼✳✸2 ✻✵✳✼ ✻✷✳✸✲✻✸✳✾ ✻✸✳✽ ✻✸✳✽2 13  ❘❡s✐❞✉❡ ❆❧❛ ❈α ❆❧❛ ❈β αβ ∆CSala ●❧♥ ❈β ●❧♥ ❈γ ●❧♥ ❈α ●❧② ❈α Pr♦ ❈α Pr♦ ❈β ❙❡r ❈α ❙❡r ❈β  ▼❛❥♦r ❙✐❧❦ ✹✾✱✹✾✳✷✱✺✵ ✶✼✳✺✱✷✶✱✷✸✳✸ ✸✶✳✺✱ ✸✷ ✸✸✳✷ ✺✷✳✾ ✹✸✳✸ ✺✺✳✺ ✻✶✳✻  β ✲t✉r♥ ✺✵✳✽✲✺✶✳✼ ✶✻✳✺✲✶✼✳✹ ✸✸✳✹✲✸✺✳✷  ✹✸✳✽✲✹✹ ✺✽✳✵ ✻✵✳✼  ❚❛❜❧❡ ✺✳✷✿ ❑♥♦✇♥ ❝❤❡♠✐❝❛❧ s❤✐❢ts ❢♦r ❛♠✐♥♦ ❛❝✐❞s ✐♠♣♦rt❛♥t ✐♥ s♣✐❞❡r s✐❧❦ ❛♥❞ ✈❛r✐♦✉s s❡❝✲ ♦♥❞❛r② str✉❝t✉r❡s✳ ❆❧❧ ✉♥r❡❢❡r❡♥❝❡❞ ❝❤❡♠✐❝❛❧ s❤✐❢ts ❛r❡ ❢r♦♠ ❬✸✵❪✳ 1 ❘❛♥❞♦♠ ❝♦✐❧ ❝❤❡♠✐❝❛❧ s❤✐❢ts ❢r♦♠ ❬✸✶❪.2 ❊st✐♠❛t❡❞ ✉s✐♥❣ ❝♦♥t♦✉r ♣❧♦ts ❢r♦♠ ❬✸✷❪ ✉s✐♥❣ t❤❡ r❛♥❞♦♠ ❝♦✐❧ ✈❛❧✉❡ ❛♥❞ t❤❡ ✈❛❧✉❡s ❢r♦♠ ❚❛❜❧❡ ✹✳✶ ❢♦r t❤❡ t♦rs✐♦♥ ❛♥❣❧❡s✳  ▼❛t❡r✐❛❧ ❙♣✐❞❡r ❙✐❧❦ ✭✷✵✵ ▼❍③✮ ❙♣✐❞❡r ❙✐❧❦ ✭✹✵✵ ▼❍③✮ ❙♣✐❞❡r ❙✐❧❦ ✭✹✵ o ❈✮ ▼❛❙♣✶ ▼❛❙♣✷ ❊❧❡❝tr♦s♣✉♥ ▼❛❙♣✶  ❇❡st ❋✐t ❙✐♥❣❧❡ ❙tr❡t❝❤❡❞ ❙tr❡t❝❤❡❞ ❙tr❡t❝❤❡❞ ❙tr❡t❝❤❡❞ ❙tr❡t❝❤❡❞  T1 ✭s✮ T1∗ ✭s✮ ✶✷ 20.2 ± 0.8 19.7 ± 0.5 16.3 ± 0.6 15.4 ± 0.4 21.1 ± .3.7  β 0.77 ± .04 0.86 ± 0.03 0.70 ± 0.02 0.71 ± 0.2 0.69 ± 0.13  T1 ✭s✮ 23.6 ± 1.3 21.3 ± .7 21.0 ± 1.0 19.2 ± 0.8 27.0 ± 6.9  ❚❛❜❧❡ ✺✳✸✿ ❘❡❧❛①❛t✐♦♥ ♣❛r❛♠❡t❡rs ❢♦r ❛❧❛♥✐♥❡ α ❝❛r❜♦♥s✳ T1∗ ✱ β ✱ ❛♥❞ T1 ❛r❡ t❤❡ ♣❛r❛♠❡t❡rs ❢♦r t❤❡ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧✳ ❚❤❡ ✷✵✵ ▼❍③ ❞❛t❛ ✐s ❢r♦♠ ❬✸✸❪✳  ✹✽  ▼❛t❡r✐❛❧ ❙♣✐❞❡r ❙✐❧❦ ✭✷✵✵ ▼❍③✮ ❙♣✐❞❡r ❙✐❧❦ ✭✹✵✵ ▼❍③✮ ❙♣✐❞❡r ❙✐❧❦ ✭✹✵ ◦ ❈ ▼❛❙♣✶ ▼❛❙♣✷ ❊❧❡❝tr♦s♣✉♥ ▼❛❙♣✶  ❇❡st ❋✐t ❉♦✉❜❧❡ ❉♦✉❜❧❡ ❉♦✉❜❧❡ ❉♦✉❜❧❡ ❉♦✉❜❧❡ ❙✐♥❣❧❡  T1f ast ✭s✮ T1slow ✭s✮ ✪ ❢❛st ✵✳✶✽ ✷ 40 0.23 ± 0.04 1.2 ± 0.2 56 ± 10 0.29 ± 0.08 1.3 ± 0.2 43 ± 12 0.40 ± 0.09 1.8 ± 0.7 70 ± 16 0.26 ± 0.07 1.7 ± 0.4 53 ± 12  T1  0.77 ± 0.02  ❚❛❜❧❡ ✺✳✹✿ ❘❡❧❛①❛t✐♦♥ ♣❛r❛♠❡t❡rs ❢♦r ❛❧❛♥✐♥❡ β ❝❛r❜♦♥s✳ ❚❤❡ s✐♥❣❧❡ ❡①♣♦♥❡♥t✐❛❧ ✜ts ✇❡r❡ ✐♥❝❧✉❞❡❞ ❢♦r ❛❧❧ ♦❢ t❤❡ s❛♠♣❧❡s ❢♦r ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ ✷✵✵ ▼❍③ ❞❛t❛ ❬✸✸❪✳ ✐s ♣❛rt✐❝✉❧❛r❧② ✉s❡❢✉❧ ❢♦r ❝❤❡♠✐❝❛❧ s❤✐❢t ❛ss✐❣♥♠❡♥t ❛s t❤❡r❡ ✐s ✈❡r② ❧✐tt❧❡ ♦✈❡r❧❛♣ ❜❡t✇❡❡♥ str✉❝t✉r❡s✳ ❚❤❡ ❝❤❡♠✐❝❛❧ s❤✐❢t r❛♥❣❡ ✐s ❛r♦✉♥❞ ✻ ♣♣♠ st❛rt✐♥❣ ❛t ✶✹✳✻✲✶✻✳✵ ♣♣♠ ❢♦r α ❤❡❧✐❝❡s t♦ ✶✾✳✾✲✷✵✳✼ ♣♣♠ ❢♦r β s❤❡❡ts✳ ❚❤❡ ♠♦t✐♦♥ ❢♦r ♠❡t❤②❧ ❣r♦✉♣s ❤❛s ❜❡❡♥ ♠♦❞❡❧❡❞ ❛s ❛ ❢❛st ♠♦✈✐♥❣ r♦t♦r ❬✸✹❪✳ ❇❡❝❛✉s❡ ♠❡t❤②❧ ❣r♦✉♣ r♦t❛t✐♦♥ ✐s ✉s✉❛❧❧② ✐♥ t❤❡ ❢❛st ♠♦t✐♦♥ ❧✐♠✐t✱ ❧♦♥❣❡r r❡❧❛①❛t✐♦♥ t✐♠❡s ❛t r♦♦♠ t❡♠♣❡r❛t✉r❡ ❝♦rr❡s♣♦♥❞ t♦ ❢❛st❡r ♠♦t✐♦♥✳ ■♥ s♣✐❞❡r s✐❧❦ t❤❡ ❛❧❛♥✐♥❡ β ❝❛r❜♦♥ ❤❛s ❜❡❡♥ ✇❡❧❧ ❞♦❝✉♠❡♥t❡❞ t♦ ❤❛✈❡ t✇♦ r❡❧❛①❛t✐♦♥ r❡❣✐♠❡s✱ ❛ s❤♦rt T1f ast ✱ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ s❧♦✇ ♠♦✈✐♥❣ r✐❣✐❞ ❛r❡❛✱ ❛♥❞ ❛ ❧♦♥❣ T1slow ✱ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❛ ❧♦♦s❡r r❡❣✐♠❡✳ ❚❤❡ ❞✐✛❡r❡♥t r❡❧❛①❛t✐♦♥ t✐♠❡s ❛r❡ s❤♦✇♥ ✐♥ ❚❛❜❧❡ ✺✳✹✳ ❇❡❝❛✉s❡ ❛❧❛♥✐♥❡✬s α ❛♥❞ β ❝❛r❜♦♥s ❝❤❡♠✐❝❛❧ s❤✐❢t ♠♦✈❡ ✐♥ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥s ❢r♦♠ t❤❡ r❛♥❞♦♠ ❝♦✐❧ s❤✐❢ts ❢♦r ❞✐✛❡r❡♥t ❝♦♥❢♦r♠❛t✐♦♥s✱ ✉s✐♥❣ t❤❡ ❞✐✛❡r❡♥❝❡ ✐♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ ♣❡❛❦s ♣r♦✈✐❞❡s ❛ str✉❝t✉r❛❧ ♠❡tr✐❝ t❤❛t ✐s ❜♦t❤ ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ r❡❢❡r❡♥❝❡ ♠❡❛s✉r❡♠❡♥t ❛♥❞ ❤❛s ♠✉❝❤ ❧❛r❣❡r s❡♣❛r❛t✐♦♥s ❜❡t✇❡❡♥ str✉❝t✉r❡s ✇❤✐❝❤ ♠❛❦❡s str✉❝t✉r❛❧ ✐❞❡♥t✐✜❝❛t✐♦♥ ❡❛s✐❡r✳ ❚♦ αβ ❢❛❝✐❧✐t❛t❡ t❤✐s✱ t❤❡ ♣❛r❛♠❡t❡r ∆CSala ✇✐❧❧ ❜❡ ❞❡✜♥❡❞ ❛s  β αβ α = CSala − CSala ∆CSala  β α ❛r❡ t❤❡ ❝❤❡♠✐❝❛❧ s❤✐❢ts ♦❢ ❛❧❛♥✐♥❡ α ❛♥❞ β ❝❛r❜♦♥ ♣❡❛❦s✳ ✇❤❡r❡ CSala ❛♥❞CSala  ✺✳✹✳✷  ●❧②❝✐♥❡  α  ❈❛r❜♦♥  ❚❤❡ ❣❧②❝✐♥❡ α ❝❛r❜♦♥ ❤❛s ❛ ✈❡r② ♥❛rr♦✇ r❛♥❣❡ ♦❢ ❝❤❡♠✐❝❛❧ s❤✐❢ts ♦❢ ❛❜♦✉t ✷✳✼ ♣♣♠✳ ❚❤✐s st❛rts ❢r♦♠ ✹✶✳✹✲✹✷✳✺ ♣♣♠ ❢♦r t❤❡ 31 ❤❡❧✐①✱ t♦ ✹✸✳✽✲✹✹✳✶ ♣♣♠ ❢♦r t❤❡ β t✉r♥✳ P♦❧②✲❣❧②❝✐♥❡ ✹✾  ▼❛t❡r✐❛❧ ❙♣✐❞❡r ❙✐❧❦ ✭✷✵✵ ▼❍③✮ ❙♣✐❞❡r ❙✐❧❦ ✭✹✵✵ ▼❍③✮ ❙♣✐❞❡r ❙✐❧❦ ✭✹✵ o ❈✮ ▼❛❙♣✶ ▼❛❙♣✷ ❊❧❡❝tr♦s♣✉♥ ▼❛❙♣✶  ❇❡st ❋✐t ❙✐♥❣❧❡  T1 ✭s✮ T1∗ ✭s✮ ✾  β  T1 ✭s✮ T1f ast ✭s✮  T1slow ✭s✮  ✪ ❢❛st  ❙tr❡t❝❤❡❞  20.8 ± 1.0 .66 ± .03  28 ± 2  5.5 ± 1.8  39 ± 6  37 ± 6  ❉♦✉❜❧❡  23.0 ± 0.8 .70 ± .03  29 ± 1  6.5 ± 1.3  42 ± 4  37 ± 4  ❙tr❡t❝❤❡❞ ❙tr❡t❝❤❡❞ ❙tr❡t❝❤❡❞  15.3 ± 2.2 .64 ± .10 18.2 ± 1.2 .70 ± .05 15.6 ± 2.3 .63 ± .09  21 ± 5 23 ± 2 22 ± 4  ❚❛❜❧❡ ✺✳✺✿ ❘❡❧❛①❛t✐♦♥ ♣❛r❛♠❡t❡rs ❢♦r ❣❧②❝✐♥❡ α ❝❛r❜♦♥s✳ T1∗ ✱ β ✱ ❛♥❞ T1 ❛r❡ t❤❡ ❝♦♠♣♦♥❡♥ts ❢♦r t❤❡ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧✳ ❚❤❡ ✷✵✵ ▼❍③ ❞❛t❛ ✐s ❢r♦♠ ❬✸✸❪✳ ❚❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡ ❛♥❞ ✹✵ ◦ ❈ ♠❡❛s✉r❡♠❡♥ts ❜❡st ✜t t♦ ❞✐✛❡r❡♥t ❢✉♥❝t✐♦♥s✱ s♦ ❜♦t❤ s❡ts ♦❢ ♣❛r❛♠❡t❡rs ✇❡r❡ ✐♥❝❧✉❞❡❞✳ t❡♥❞s ♥♦t t♦ ❢♦r♠ α ❤❡❧✐❝❡s ❜✉t ✐♥st❡❛❞ ❢♦r♠s ❛ str✉❝t✉r❡ ❦♥♦✇♥ ❛s t❤❡ 31 ❤❡❧✐①✱ ✇❤✐❝❤ ✐s ❜❡❧✐❡✈❡❞ t♦ ❜❡ ❛♥ ✐♠♣♦rt❛♥t ❡❧❡♠❡♥t ♦❢ s♣✐❞❡r s✐❧❦✬s ❛♠♦r♣❤♦✉s r❡❣✐♦♥ ❬✺❪✳ ❚❤❡ ❞✐✛❡r❡♥t r❡❧❛①❛t✐♦♥ t✐♠❡s ❛r❡ s❤♦✇♥ ✐♥ ❚❛❜❧❡ ✺✳✺✳  ✺✳✹✳✸  ●❧✉t❛♠✐♥❡  α/β/γ  ❈❛r❜♦♥s  ❆s t❤❡r❡ ✐s ♦♥❧② ♦♥❡ t❤✐r❞ t❤❡ ❛♠♦✉♥t ♦❢ ❣❧✉t❛♠✐♥❡ ❝♦♠♣❛r❡❞ t♦ ❛❧❛♥✐♥❡ ✐♥ ▼❛❙♣✶ ❛♥❞ s♣✐❞❡r s✐❧❦✱ ❛♥❞ ❜❡❝❛✉s❡ t❤❡ ❣❧✉t❛♠✐♥❡ α ❝❛r❜♦♥✬s ❝❤❡♠✐❝❛❧ s❤✐❢t ♦✈❡r❧❛♣s ✇✐t❤ t❤❡ ❛❧❛♥✐♥❡  α ❝❛r❜♦♥s✱ ✐t ✐s ❞♦♠✐♥❛t❡❞ ✐♥ t❤❡s❡ s♣❡❝tr❛ ❛♥❞ ♦♥❧② ✈✐s✐❜❧❡ ❛❢t❡r r❡❧❛①❛t✐♦♥✳ ❍♦✇❡✈❡r✱ ✐♥ ▼❛❙♣✷ t❤❡r❡ ✐s ❛ str♦♥❣ ❡♥♦✉❣❤ ❣❧✉t❛♠✐♥❡ α ❝❛r❜♦♥ s✐❣♥❛❧ ❝♦♠♣❛r❡❞ t♦ t❤❡ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥✬s t♦ ❞✐✛❡r❡♥t✐❛t❡ ✐ts ♣❡❛❦✳ ■ts ❝❤❡♠✐❝❛❧ s❤✐❢t r❛♥❣❡s ❢r♦♠ ✺✶ ♣♣♠ ✐♥ β s❤❡❡ts t♦ ✺✼ ♣♣♠ ✐♥ α ❤❡❧✐❝❡s✳ ❚❤❡ ❣❧✉t❛♠✐♥❡ β ❛♥❞ γ ❝❛r❜♦♥s ♦❝❝✉♣② ❛ r❛t❤❡r ✇✐❞❡ r❛♥❣❡✱ ❢r♦♠ ✷✺✲✸✷ ♣♣♠✳ ❚❤❡ γ ❝❛r❜♦♥ r❛♥❣❡s ❢r♦♠ ✷✾✲✸✷ ♣♣♠✱ ❛♥❞ t❤❡ β ❝❛r❜♦♥ r❛♥❣❡s ❢r♦♠ ✷✺✲✸✵ ♣♣♠✳ ❚❤✐s ♠❡❛♥s t❤❛t ❢♦r ♣❡❛❦s ✐♥ t❤❡✐r s❤❛r❡❞ r❛♥❣❡ t❤❡② ❛r❡ ❡✛❡❝t✐✈❡❧② ✐♥❞✐st✐♥❣✉✐s❤❛❜❧❡✳ ❉✉❡ t♦ s✐❣♥✐✜❝❛♥t ♦✈❡r❧❛♣ ✇✐t❤ t❤❡ ❛❧❛♥✐♥❡ β ✱ ✐t ✇❛s ♥♦t ♣♦ss✐❜❧❡ t♦ ❣❡♥❡r❛t❡ ❛ ❝♦♥s✐st❡♥t ❛♥❞ r❡♣r♦❞✉❝✐❜❧❡ r❡❧❛①❛t✐♦♥ t✐♠❡ ❢♦r t❤❡ ❣❧✉t❛♠✐♥❡ ✐♥ t❤❡ r❛♥❣❡ ♦❢ ✷✺ ♣♣♠✳ ■♥ t❤❡ ❧✐t❡r❛t✉r❡ t❤❡ ❣❧✉t❛♠✐♥❡ β ❛♥❞ γ ♣❡❛❦ ♥❡❛r ✸✵ ♣♣♠ ✇❛s r❡♣♦rt❡❞ ❛s ❜❡✐♥❣ ✜t t♦ ❛ s✐♥❣❧❡ ❡①♣♦♥❡♥t✐❛❧✱ ❤♦✇❡✈❡r ❛❧❧ ♦❢ t❤❡ ❡①♣❡r✐♠❡♥ts ✇❡ ❝♦♥❞✉❝t❡❞ ❜❡st ✜t ❛ ❞♦✉❜❧❡ ❡①♣♦♥❡♥t✐❛❧✳ ❚❤❡ ❞✐✛❡r❡♥t r❡❧❛①❛t✐♦♥ ♣❛r❛♠❡t❡rs t✐♠❡s ❛r❡ s❤♦✇♥ ✐♥ ❚❛❜❧❡ ✺✳✻✳  ✺✵  ●❧✉t❛♠✐♥❡ β/γ ❈❛r❜♦♥s ▼❛t❡r✐❛❧ ❙♣✐❞❡r ❙✐❧❦ ✭✷✵✵ ▼❍③✮ ❙♣✐❞❡r ❙✐❧❦ ✭✹✵✵ ▼❍③✮ ❙♣✐❞❡r ❙✐❧❦ ✭✹✵ ◦ C ▼❛❙♣✶ ▼❛❙♣✷ ❊❧❡❝tr♦s♣✉♥ ▼❛❙♣✶  ❇❡st ❋✐t ❙✐♥❣❧❡ ❉♦✉❜❧❡ ❉♦✉❜❧❡ ❉♦✉❜❧❡ ❉♦✉❜❧❡ ❉♦✉❜❧❡  T1 ✭s✮ T1f ast T1slow ✭✪✮ ✪ ❢❛st ✹ ✶✵ 1.7 ± .6 26 ± 3 50 ± 4 ✶✶ 0.89 ± .3 21.0 ± 3.7 44 ± 7 ✶✻ 0.79 ± 0.38 18.2 ± 1.2 18 ± 4 ✶✵ 1.7 ± 0.2 23.1 ± 2.3 56 ± 4 ✶✸ 3.4 ± 1.1 36 ± 12 41 ± 12  ❚❛❜❧❡ ✺✳✻✿ ❘❡❧❛①❛t✐♦♥ ♣❛r❛♠❡t❡rs ❢♦r ❣❧✉t❛♠✐♥❡ β ✴γ ❝❛r❜♦♥s✳ ❚❤❡ ✷✵✵ ▼❍③ r❡s✉❧ts ❛r❡ ❢r♦♠ ❬✸✸❪✳ ❚❤❡ s✐♥❣❧❡ ❡①♣♦♥❡♥t✐❛❧ ✜ts ✇❡r❡ ✐♥❝❧✉❞❡❞ ❢♦r ❛❧❧ ♦❢ t❤❡ s❛♠♣❧❡s ❢♦r ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ ✷✵✵ ▼❍③ ❞❛t❛✳  ✺✶  ❈❤❛♣t❡r ✻  ❙♣✐❞❡r ❙✐❧❦ ◆▼❘ ❘❡s✉❧ts ❛♥❞ ❆♥❛❧②s✐s  ❚❤✐s ❝❤❛♣t❡r ❢♦❝✉s❡s ♦♥ ❛♥❛❧②③✐♥❣ t❤❡ r❡❧❛①❛t✐♦♥ ❝❤❛r❛❝t❡rs✐t✐❝s ♦❢ s♣✐❞❡r s✐❧❦✳ ❚❤❡ ✜rst s❡❝t✐♦♥ ❞✐s❝✉ss❡s t❤❡ ❛❧❛♥✐♥❡ r❡s✐❞✉❡✱ ❢♦❧❧♦✇❡❞ ❜② ❣❧②❝✐♥❡✱ ❛♥❞ ❡♥❞✐♥❣ ✇✐t❤ ❣❧✉t❛♠✐♥❡✳  ✻✳✶  ❆❧❛♥✐♥❡  ❚❤❡ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥ ✇❛s ♦❜s❡r✈❡❞ t♦ ❤❛✈❡ ❛ ❝❤❡♠✐❝❛❧ s❤✐❢t ♦❢ ✹✾✳✷ ❛♥❞ ❛ ♣❡❛❦ ✇✐❞t❤ ♦❢ ✷✳✽ ♣♣♠✳ ❚❤✐s ✐s ❝♦♥s✐st❡♥t ✇✐t❤ t❤❡ r❡s✉❧ts ♦❜s❡r✈❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ❬✸✵✱ ✸✸❪✳ ▼✉❝❤ ♦❢ t❤❡ ❛❧❛♥✐♥❡ α ♣❡❛❦ ♦✈❡r❧❛♣♣❡❞ ✇✐t❤ t❤❡ s❧♦✇❡r r❡❧❛①✐♥❣ ❣❧✉t❛♠✐♥❡ α ❝❛r❜♦♥ ♣❡❛❦✳ ❚❤✐s ♠❡❛♥t t❤❛t ✉s✐♥❣ t❤❡ ❡♥t✐r❡ ♣❡❛❦ ✇♦✉❧❞ ♥♦t ❜❡ ❛♥ ❛❝❝✉r❛t❡ r❡✢❡❝t✐♦♥ ♦❢ t❤❡ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥✳ ❚♦ ❝♦♠♣❡♥s❛t❡ ❢♦r t❤✐s✱ t❤❡ ✐♥t❡❣r❛t✐♦♥ ❜❡❣❛♥ ❛t ✵✳✺ ♣♣♠ ❞♦✇♥✜❡❧❞ ❢r♦♠ t❤❡ ♣❡❛❦ ❝❡♥t❡r✳ ❈♦♥tr❛r② t♦ t❤❡ r❡s✉❧ts r❡♣♦rt❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✱ t❤❡ ❜❡st ✜t ♦❜s❡r✈❡❞ ❢♦r s♣✐❞❡r s✐❧❦✬s ❛❧❛♥✐♥❡ α ❝❛r❜♦♥ ✇❛s ♥♦t t♦ ❛ s✐♥❣❧❡ ❡①♣♦♥❡♥t✐❛❧ ❛s r❡♣♦rt❡❞ ❜② ❙✐♠♠♦♥s ❡t ❛❧✳ ❬✸✸❪✱ ❜✉t t♦ ❛ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧✱ ❛s s❡❡♥ ✐♥ ❋✐❣✉r❡ ✻✳✶✳ ❲❤✐❧❡ t❤✐s ✐s ❞✐✛❡r❡♥t t♦ t❤❡ ❧✐t❡r❛t✉r❡✱ ✐t s❤♦✉❧❞ ♥♦t ❝♦♠❡ ❛s ❛ s✉r♣r✐s❡ ❛s t❤✐s ✐♠♣❧✐❡s t❤❛t t❤❡r❡ ✐s ❛ ❞✐str✐❜✉t✐♦♥ ♦❢ ❝♦rr❡❧❛t✐♦♥ t✐♠❡s ✇✐t❤✐♥ t❤❡ ❝r②st❛❧s✳ ❚❤❡s❡ ✈❛❧✉❡s ✇❡r❡ ❢♦✉♥❞ t♦ ❜❡ T1 = 20.2 ± 0.8 s ❛♥❞  β = 0.77 ± 0.07✳ ❚❤✐s ❣✐✈❡s ❛ ♠❡❛♥ r❡❧❛①❛t✐♦♥ t✐♠❡ ♦❢ < T1 >= 23.6 ± 0.7 s✳ ▼❛❦✐♥❣ ✉s❡ ♦❢ ♦✉r ♠❡❛s✉r❡♠❡♥ts ❛♥❞ t❤❡ ♣r❡✈✐♦✉s ♠❡❛s✉r❡♠❡♥ts ❛t ❧♦✇❡r ✜❡❧❞ ❬✸✸❪✱ ✇❡ ❝❛♥ ❝❤❛r❛❝t❡r✐③❡ t❤❡ ♠♦❧❡❝✉❧❛r ♠♦t✐♦♥ ❛♥❞ t❤❡ ♦r❞❡r ♣❛r❛♠❡t❡r ♦❢ t❤❡ ❛❧❛♥✐♥❡s✳ ❯s✐♥❣ t❤❡ ♠❡❛♥  T1 ✱ ❛s r❡♣r❡s❡♥t❛t✐✈❡ ♦❢ ❛ ♠❡❛♥ s✐♥❣❧❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡✱ ✇❡ ❝❛♥ ♣✉t t❤❡ ♠♦❞❡❧ ✐♥❞❡♣❡♥❞❡♥t s♣❡❝tr❛❧ ❞❡♥s✐t② ❢r♦♠ ❊q✉❛t✐♦♥ ✭✷✳✾✼✮ ✐♥t♦ ❊q✉❛t✐♦♥ ✭✷✳✽✾✮✳ ❇❡❝❛✉s❡ t❤❡ β s❤❡❡ts ❛r❡ ❧❛r❣❡ ✺✷  Intensity  Spider Silk Alanine C-alpha Relaxation Data Single exponential fit Stretched exponential fit Double exponential fit  0  5  10  15  20  25  30  35  40  45  50  D2 (s)  ❋✐❣✉r❡ ✻✳✶✿ ❚❤❡ r❡❧❛①❛t✐♦♥ ❝✉r✈❡ ♦❢ t❤❡ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥s ✐♥ s♣✐❞❡r s✐❧❦✳ ❝r②st❛❧❧✐♥❡ ❜❧♦❝❦s ✐♥ ❛ s♦❧✐❞✱ ✇❡ ❝❛♥ ♠❛❦❡ t❤❡ ❛ss✉♠♣t✐♦♥s ❢♦r ✉❧tr❛ s❧♦✇ ❜✉❧❦ ♠♦t✐♦♥ ❛♥❞ ✉s❡ t❤❡ s✐♠♣❧✐✜❡❞ s♣❡❝tr❛❧ ❞❡♥s✐t② ❢r♦♠ ❊q✉❛t✐♦♥ ✭✷✳✶✵✵✮✳ ❚❤✐s ❧❡❛✈❡s ✉s ✇✐t❤ ❛ r❡❧❛t✐✈❡❧② s✐♠♣❧❡ ❡q✉❛t✐♦♥ ❢♦r T1 ✳ ◆♦t✐♥❣ t❤❛t t❤❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❡①t❡r♥❛❧ ♠❛❣♥❡t✐❝ ✜❡❧❞ ✇❤✐❝❤ ❢♦r t✇♦ ❞✐✛❡r❡♥t ♠❛❣♥❡t✐❝ ✜❡❧❞s✱ B1 ❛♥❞ B2 ❣✐✈❡s❀  RIB2 RIB1  =  T1B1 T1B2  ≈  3 B 1+(ωI 2 τc )2 3 B  1+(ωI 1 τc )2  +  1 B B 1+((ωI 2 −ωS 2 )τc )2  +  1 B B 1+((ωI 1 −ωS 1 )τc )2  +  6 B B 1+((ωI 2 +ωS 2 )τc )2  +  6 B B 1+((ωI 1 +ωS 1 )τc )2  ✭✻✳✶✮  ❈♦♠♣❛r✐♥❣ t❤✐s t♦ t❤❡ ✈❛❧✉❡s ♣r♦❞✉❝❡❞ ❜② ❙✐♠♠♦♥s ❛t ✷✵✵ ▼❍③ ❛❧❧♦✇s ✉s t♦ s♦❧✈❡ ❢♦r t❤❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡✱ ✇❤✐❝❤ ✇❛s ❢♦✉♥❞ t♦ ❜❡ 5.06 × 10−10 s✱ ✇❤✐❝❤ ✐s ❥✉st ♦♥ t❤❡ s❧♦✇ ♠♦t✐♦♥ s✐❞❡ ♦❢ t❤❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡✱ ❛s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✻✳✷✳ ❲❡ ❝❛♥ t❤❡♥ ✉s❡ t❤✐s t♦ s♦❧✈❡ ❢♦r t❤❡ ♦r❞❡r ♣❛r❛♠❡t❡r✱ ✇❤✐❝❤ ✇❛s ❢♦✉♥❞ t♦ ❜❡ S = 0.994✱ s❤♦✇✐♥❣ ❛ ❤✐❣❤❧② r❡str✐❝t✐✈❡ str✉❝t✉r❡ t❤❛t ♦♥❡ ✇♦✉❧❞ ❡①♣❡❝t ✐♥ ❛ β s❤❡❡t✳ ❯♥❝❡rt❛✐♥t✐❡s ❛r❡ ♥♦t ❣✐✈❡♥ ❢♦r ❡✐t❤❡r t❤❡ ♦r❞❡r ♣❛r❛♠❡t❡r ♦r ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ❜❡❝❛✉s❡ t❤❡ ❡rr♦r ✇❛s ✉♥❦♥♦✇♥ ✐♥ t❤❡ r❡❧❛①❛t✐♦♥ ❞❛t❛ ❢r♦♠ t❤❡ ❧♦✇ ✜❡❧❞ T1 ✳ ■t ♠✉st ❜❡ ♠❛❞❡ ❝❧❡❛r t❤❛t t❤❡r❡ ❛r❡ s❡✈❡r❛❧ ❛ss✉♠♣t✐♦♥s t❤❛t ❤❛✈❡ ❜❡❡♥ ♠❛❞❡ t❤❛t ❝♦✉❧❞ ❤❛✈❡ ②✐❡❧❞❡❞ ✐♥❝♦rr❡❝t ✈❛❧✉❡s✳ ❚❤❡ ✜rst ✐s t❤❡ ✉s❡ ♦❢ t❤❡ ♠❡❛♥ T1 ✳ ❚❤✐s ❜② ♥♦ ♠❡❛♥s ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ♠❡❛♥ ❝♦rr❡❧❛t✐♦♥ t✐♠❡✳ ❲❤✐❧❡ t②♣✐❝❛❧❧② t❤✐s ✇♦✉❧❞ ♦♥❧② ♠❡❛♥ ❛ s♠❛❧❧ s②st❡♠❛t✐❝ ❡rr♦r✱ t❤❡ ♣r♦①✐♠✐t② ♦❢ t❤❡ r❡s✉❧t✐♥❣ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ t♦ t❤❡ ❢❛st ♠♦t✐♦♥✴s❧♦✇ ♠♦t✐♦♥ tr❛♥s✐t✐♦♥ ♠❡❛♥s t❤❛t t❤❡ ♠❡❛♥ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ♠✐❣❤t ❜❡ q✉✐t❡ ❞✐✛❡r❡♥t✳ ❚❤❡ s❡❝♦♥❞ ❛♥❞ ♠✉❝❤ ❧❛r❣❡r ❛ss✉♠♣t✐♦♥ ✇❛s t❤❡ ❧♦✇ ✜❡❧❞ T1 ❢r♦♠ t❤❡ ❧✐t❡r❛t✉r❡ t❤❛t ✇❛s ✉s❡❞✳ ❆s t❤❡ ✈❛❧✉❡ ♣r♦✈✐❞❡❞ ✇❛s t❤❛t ❢r♦♠ ❛ s✐♥❣❧❡ ❡①♣♦♥❡♥t✐❛❧ ✜t✱ ✐t ✐s ♠♦st ❧✐❦❡❧② ♥♦t t❤❡ ♠♦st ❛❝❝✉r❛t❡ ✈❛❧✉❡ t♦ ✉s❡✳ ❍♦✇❡✈❡r✱ ❛s t❤❛t r❛✇ ❞❛t❛ ✐s ♥♦t ❛✈❛✐❧❛❜❧❡✱ t❤❡r❡ ✐s ❧✐tt❧❡ ❡❧s❡ t❤❛t ❝❛♥ ❜❡ ❞♦♥❡✳ ✺✸  1/T1 (s-1)  MaSp2 MaSp1  τ (Spider Silk)  Correlation Time (s) ❋✐❣✉r❡ ✻✳✷✿ P❧♦t ♦❢ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ✈❡rs✉s T11 ❢♦r t❤❡ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥ ✇✐t❤ ❛♥ ♦r❞❡r ♣❛✲ r❛♠❡t❡r ♦❢ S = 0.994 ✉s✐♥❣ t❤❡ ✉❧tr❛ s❧♦✇ t✉♠❜❧✐♥❣ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❙③❛❜♦ ♠♦❞❡❧✳ ❚❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ T1 ❢♦r ▼❛❙♣✷ ✇❛s ✐♥❝❧✉❞❡❞ t♦ s❤♦✇ t❤❛t ✐t ❞♦❡s ♥♦t ❢❛❧❧ ✇✐t❤✐♥ t❤❡ r❛♥❣❡ ❛❧✲ ❧♦✇❡❞ ✇✐t❤ t❤✐s ♦r❞❡r ♣❛r❛♠❡t❡r✳ ❚❤❡ ✐♥✈❡rs❡ T1 ❢♦r ▼❛❙♣✶ ❢❛❧❧s ❥✉st ♦✉ts✐❞❡ ♦❢ t❤❡ ❛❧❧♦✇❛❜❧❡ r❛♥❣❡✱ ❤♦✇❡✈❡r✱ ✐ts ✉♥❝❡rt❛✐♥t② ✐s ✇✐t❤✐♥ t❤❡ ❛❧❧♦✇❛❜❧❡ ✈❛❧✉❡s ❢♦r t❤✐s ♦r❞❡r ♣❛r❛♠❡t❡r ●✐✈❡♥ t❤❡ ❝❛✈❡❛ts✱ ✇❡ t❛❦❡ t❤❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ❛♥❞ ♦r❞❡r ♣❛r❛♠❡t❡rs ❛s ❣✉✐❞❡s ♣r♦✈✐❞✐♥❣ t❤❡ ♦r❞❡r ♦❢ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ✐♥❞✐❝❛t❡❞ q✉❛♥t✐t✐❡s✳ ❚❤❡ ❤✐❣❤ t❡♠♣❡r❛t✉r❡ r❡s✉❧ts ❧❡♥❞ s✉♣♣♦rt t♦ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤✐s ❛ss✉♠♣t✐♦♥✳ ❚❤❡ ✹✵ ◦ ❈ r❡❧❛①❛t✐♦♥ ❝✉r✈❡ ✜tt❡❞ t♦ ❛ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✇✐t❤ ❛ T1∗ = 19.7±0.5 s ❛♥❞ β = 0.86±0.03 ②✐❡❧❞✐♥❣ T1 = 21.0 ± 0.92 s✳ ❲❤✐❧❡ t❤❡ ✹✵◦ ❈✱ T1∗ ♠❛② ❜❡ st❛t✐st✐❝❛❧❧② ✐♥❞✐st✐♥❣✉✐s❤❛❜❧❡ ❢r♦♠ t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡ ✈❛❧✉❡✱ t❤❡ ✐♥❝r❡❛s❡ ♦❢ β s❤♦✇s t❤❛t t❤❡ r❡❧❛①❛t✐♦♥ t✐♠❡ ❞✐str✐❜✉t✐♦♥ ✐s t✐❣❤t❡♥✐♥❣ ✉♣✳ ❚❤✐s ✇♦✉❧❞ ❤❛♣♣❡♥ ❛s t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ❝♦rr❡❧❛t✐♦♥ t✐♠❡s ♠♦✈❡s ✐♥t♦ r❛♥❣❡ ♦❢ t❤❡ ♠✐♥✐♠✉♠ T1 ✱ ❛s t❤❡ ❝❤❛♥❣❡ ✐♥ r❡❧❛①❛t✐♦♥ t✐♠❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝❤❛♥❣❡ ✐♥ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ❞❡❝r❡❛s❡s✳ ❚❤✐s s✉❜st❛♥t✐❛❧❧② ❤❡❧♣s t♦ ✈❛❧✐❞❛t❡ t❤❡ ❝❤♦✐❝❡ ♦❢ ♠♦❞❡❧ ✉s❡❞ ❢♦r t❤✐s ❛ss✉♠♣t✐♦♥✳ ❚❤✐s ✐s ❢✉rt❤❡r s✉♣♣♦rt❡❞ ❜② t❤❡ T1 ✇❤✐❝❤ ❤❛s ❛ st❛t✐st✐❝❛❧❧② s✐❣♥✐✜❝❛♥t ❞❡❝r❡❛s❡ ✐♥ t❤❡ ❤✐❣❤ t❡♠♣❡r❛t✉r❡ ❝❛s❡✳ ❚❤❡ ❛❧❛♥✐♥❡ β ❝❛r❜♦♥ ✐♥ s♣✐❞❡r s✐❧❦ ✇❛s ❢♦✉♥❞ t♦ ❤❛✈❡ ❛ ❝❤❡♠✐❝❛❧ s❤✐❢t ♦❢ ✷✵✳✽ ♣♣♠✱ ✇✐t❤✐♥ ❜♦t❤ t❤❡ r❛♥❣❡ ❞❡t❡r♠✐♥❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ❛♥❞ t❤❛t ♦❢ β s❤❡❡ts✳ ❚❤❡r❡ ✐s ❛❧s♦ ❛ s❧✐❣❤t ❧❡❞❣❡ ❛t ✶✺✳✼ ♣♣♠✳ ❚❤❡ ♣❡❛❦ ✇❛s ✐♥t❡❣r❛t❡❞ ♦✈❡r t❤❡ r❛♥❣❡ ✶✺✲✷✶ ♣♣♠✳ ❚❤❡ ❜❡st ✜t✱ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✻✳✸✱ ✺✹  Intensity  Spider Silk Sp1 Alanine C-beta Relaxation Data Single exponential fit Stretched exponential fit Double exponential fit  0  1  2  3  4  5  6  7  8  D2 (s)  ❋✐❣✉r❡ ✻✳✸✿ ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ s♣✐❞❡r s✐❧❦ ❛❧❛♥✐♥❡ β ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛ ✳ ✇❛s ♦❜s❡r✈❡❞ t♦ ❜❡ ❛ ❞♦✉❜❧❡ ❡①♣♦♥❡♥t✐❛❧ ✜t ❝♦♥s✐st✐♥❣ ♦❢ ❛ ❢❛st r❡❧❛①❛t✐♦♥ t✐♠❡ ♦❢ T1f ast =  0.23 ± 0.04 s ✇❤✐❝❤ ♠❛❞❡ ✉♣ 56 ± 10% ❛♥❞ ❛ s❧♦✇ ❝♦♠♣♦♥❡♥t ♦❢ T1slow = 1.2 ± .2 s ✳ ❲❤✐❧❡ t❤❡ r❛t✐♦s ❞✐✛❡r s♦♠❡✇❤❛t ❢r♦♠ t❤♦s❡ ♦❜s❡r✈❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✱ ✐t ✐s ♣♦ss✐❜❧❡ t❤❛t t❤✐s ✐s ❛ r❡s✉❧t ♦❢ ✈❛r✐❛t✐♦♥s ✐♥ t❤❡ ❝r♦ss ♣♦❧❛r✐③❛t✐♦♥ ♣❛r❛♠❡t❡rs ❛♥❞ ❢❛❝t♦r✐♥❣ ✐♥ ❡rr♦r✱ ✐s ♥♦t ♣❛rt✐❝✉❧❛r❧② ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ r❡s✉❧ts r❡♣♦rt❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✳ ❚❤❡ T1slow ✐s s♦♠❡✇❤❛t s♠❛❧❧❡r t❤❛♥ t❤❡ ✈❛❧✉❡s ♦❜s❡r✈❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ♦❢ 2 s✱ ❤♦✇❡✈❡r ❛s t❤❡ ❡rr♦r ✇❛s ✉♥r❡♣♦rt❡❞✱ ✐t ✐s ❞✐✣❝✉❧t t♦ ♠❛❦❡ ❛♥② ❞❡✜♥✐t✐✈❡ st❛t❡♠❡♥ts ❛❜♦✉t t❤❡ ♣♦ss✐❜❧❡ ❝❛✉s❡ ♦❢ t❤✐s ❞✐✛❡r❡♥❝❡✳ ❚❤❡ r❡s✉❧ts ♦❢ t❤❡ ♠❡❛s✉r❡♠❡♥t ❛t ✹✵ o ❈ ❛r❡ ♣r❛❝t✐❝❛❧❧② ✐❞❡♥t✐❝❛❧✱ T1f ast = 0.29 ± 0.07 s ♠❛❦✐♥❣ ✉♣ 43% ♦❢ t❤❡ s✐❣♥❛❧ ❛♥❞ T1slow = 1.3 ± 0.2 s ♠❛❦✐♥❣ ✉♣ 57%✳ ❚❤❡ ✐♥❝r❡❛s❡ ✐♥ r❡❧❛①❛t✐♦♥ t✐♠❡s ❧❡♥❣t❤ ❞♦ ❝♦rr❡s♣♦♥❞ t♦ ❢❛st❡r ♠♦t✐♦♥ ✐♥ t❤❡ ❢❛st ❧✐♠✐t ❛s ❡①♣❡❝t❡❞✳ ❚❤❡s❡ r❡s✉❧ts ❛r❡ ❣❡♥❡r❛❧❧② ❝♦♥s✐st❡♥t ✇✐t❤ ♣r❡✈✐♦✉s st✉❞✐❡s ♦❢ s♣✐❞❡r s✐❧❦ ❛♥❞ s✉❣❣❡st t❤❛t ❛❧❛♥✐♥❡s ♦❝❝✉♣② t✇♦ ❞✐✛❡r❡♥t ❦✐♥❞s ♦❢ ❡♥✈✐r♦♥♠❡♥ts✳ ❆❧❛♥✐♥❡s ❛r❡ ❜❡❧✐❡✈❡❞ t♦ ❣❡♥❡r❛❧❧② ❛❞♦♣t t❤❡ ❜❛❝❦❜♦♥❡ t♦rs✐♦♥ ❛♥❣❧❡s ♦❢ β str❛♥❞s ❬✷✸❪✳ ❚❤❡ β ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛ s✉❣❣❡sts t❤❛t t❤❡  T1f ast ❝♦rr❡s♣♦♥❞s t♦ ❛ t✐❣❤t❧② ♣❛❝❦❡❞ ❝r②st❛❧❧✐♥❡ ❡♥✈✐r♦♥♠❡♥t ✇❤❡r❡ ♠❡t❤②❧ ❣r♦✉♣ r♦t❛t✐♦♥ ✐s ❤✐♥❞❡r❡❞ ❞❡❝r❡❛s✐♥❣ t❤❡ ♠❡t❤②❧ r♦t❛t✐♦♥ ❢r❡q✉❡♥❝② ✇❤✐❧❡ t❤❡ T1slow ✐♠♣❧✐❡s ❛ ❢❛st❡r ♠❡t❤②❧ ❣r♦✉♣ r♦t❛t✐♦♥✱ ✇❤✐❝❤ ❝♦rr❡s♣♦♥❞s t♦ ❛ ❧❡ss t✐❣❤t❧② ♣❛❝❦❡❞ β str❛♥❞ str✉❝t✉r❡✳  ✻✳✷  ●❧②❝✐♥❡  α  ❈❛r❜♦♥  ❚❤❡ ❣❧②❝✐♥❡ α ❝❛r❜♦♥ ✐♥ s♣✐❞❡r s✐❧❦ ✇❛s ❢♦✉♥❞ t♦ ❤❛✈❡ ❛ ❝❤❡♠✐❝❛❧ s❤✐❢t ♦❢ ✹✷✳✽ ♣♣♠✱ ✇✐t❤ ❛ ✇✐❞t❤ ♦❢ ✺ ♣♣♠✳ ❚❤❡ ❝❤❡♠✐❝❛❧ s❤✐❢t ✇❛s ❝♦♥s✐st❡♥t ✇✐t❤ t❤❡ ✈❛❧✉❡s ❢♦✉♥❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✳ ✺✺  Intensity  Spider Silk Glycine C-alpha Relaxation Data Single exponential fit Stretched exponential fit Double exponential fit  0  5  10  15  20  25  30  35  40  45  50  D2 (s)  ❋✐❣✉r❡ ✻✳✹✿ ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ s♣✐❞❡r s✐❧❦ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳ ❚❤❡ r❛♥❣❡ ✉s❡❞ t♦ ✜t t❤❡ r❡❧❛①❛t✐♦♥ s♣❡❝tr❛ ❜❡❣❛♥ ❛t t❤❡ ❤❛❧❢ ❤❡✐❣❤t ♦♥ t❤❡ ✉♣✜❡❧❞ s✐❞❡ ❢♦r t❤❡ ♣❡❛❦✱ ❛♥❞ ✇❡♥t ✉♥t✐❧ ❥✉st ❞♦✇♥✜❡❧❞ ♦❢ t❤❡ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥ ♣❡❛❦ ❛❧❧♦✇✐♥❣ ❢♦r ❛ t✇♦ ❝♦♠♣♦♥❡♥t ✜t✳ ❚❤❡ ❜❡st ✜t t♦ t❤❡ r❡❧❛①❛t✐♦♥ ✇❛s ❢♦✉♥❞ t♦ ❜❡ ❛ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧✱ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✻✳✹✱ ✇✐t❤ ♣❛r❛♠❡t❡rs T1 = 20.8 ± 0.8 s ❛♥❞ β = 0.66 ± 0.03✳ ❚❤✐s ②✐❡❧❞❡❞ ❛ ♠❡❛♥ r❡❧❛①❛t✐♦♥ t✐♠❡ ♦❢ T1 = 28.0 ± 1.0 s✳ ■♥ ❛ ✷✵✵ ▼❍③ ♠❛❣♥❡t✱ t❤✐s t✐♠❡ ✇❛s r❡♣♦rt❡❞ ❛s  T1 = 9 s ❬✸✸❪✳ ▼❛❦✐♥❣ s✐♠✐❧❛r ❛♣♣r♦①✐♠❛t✐♦♥s t♦ t❤♦s❡ ✉s❡❞ ❢♦r t❤❡ ❛❧❛♥✐♥❡ ❝❛r❜♦♥✱ ✇❡ ❢♦✉♥❞ ❛♥ ❛♣♣r♦①✐♠❛t❡ ♠❡❛♥ ❝♦rr❡❧❛t✐♦♥ t✐♠❡✳ ❇❡❝❛✉s❡ t❤❡r❡ ❛r❡ t✇♦ ♣r♦t♦♥s ❜♦♥❞❡❞ t♦ t❤❡ ❣❧②❝✐♥❡ α ❝❛r❜♦♥✱ ❛ ❢❛❝t♦r ♦❢ t✇♦ ✇❛s ❛❞❞❡❞ t♦ t❤❡ ♣r❡❢❛❝t♦r ♦❢ ❊q✉❛t✐♦♥ ✭✷✳✽✾✮✳ ❚❤❡ r❡s✉❧t✐♥❣ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ✇❛s ❢♦✉♥❞ t♦ ❜❡ τc = 1.19 × 10−9 s ✇❤✐❝❤ ✐s ❛❜♦✉t ❛ ❢❛❝t♦r ♦❢ t✇♦ ❧♦♥❣❡r t❤❛♥ t❤❡ ✈❛❧✉❡ ❢♦✉♥❞ ❢♦r t❤❡ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥✳ ❚❤✐s ❧♦♥❣❡r ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ✇❤❡♥ ❝♦♠♣❛r❡❞ t♦ t❤❛t ♦❢ t❤❡ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥ ✐s ❡①♣❡❝t❡❞ ❛s t❤❡ ❣❧②❝✐♥❡✬s r❡❧❛①❛t✐♦♥ t✐♠❡ ❤❛s ❛ str♦♥❣❡r ❞❡♣❡♥❞❡♥❝② ♦♥ t❤❡ ✜❡❧❞ str❡♥❣t❤✳ ❚❤✐s τc ✇❛s t❤❡♥ ✉s❡❞ t♦ ✜♥❞ t❤❡ ♦r❞❡r ♣❛r❛♠❡t❡r ✇❤✐❝❤ ✇❛s ❢♦✉♥❞ t♦ ❜❡ S = 0.996✳ ❚❤❡ ♣❧♦t ♦❢ T11 ✈❡rs✉s τc ❢♦r ❣❧②❝✐♥❡ ✐s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✻✳✺✳ ❚❤✐s ♦r❞❡r ♣❛r❛♠❡t❡r ✐s ✈❡r② ❝❧♦s❡ t♦ t❤❛t ♦❜s❡r✈❡❞ ✐♥ ❛❧❛♥✐♥❡ α✳ ■❢ t❤❡ ❛ss✉♠♣t✐♦♥s ♠❛❞❡ ✇❡r❡ ❛♣♣r♦♣r✐❛t❡✱ t❤✐s ❝♦✉❧❞ ✐♠♣❧② t❤❛t t❤❡ ❣❧②❝✐♥❡s ♦❜s❡r✈❡❞ ❛r❡ ❞♦♠✐♥❛t❡❞ ❜② t❤❡ ♦♥❡s t❤❛t ❛r❡ ♣r❡s❡♥t ✐♥ t❤❡ β s❤❡❡t ❛s t❤❡✐r ♠♦t✐♦♥❛❧ r❡str✐❝t✐♦♥s ❛r❡ s✐♠✐❧❛r✳ ❲❤✐❧❡ ❣❧②❝✐♥❡ ✐s ♦♥❧② ♦❜s❡r✈❡❞ t♦ ❜❡ ✷✽% β s❤❡❡ts ✐♥ s♣✐❞❡r s✐❧❦ ❬✷✸❪✱ ✐t ✇♦✉❧❞ ♥♦t ❜❡ s✉r♣r✐s✐♥❣ ✐❢ ❣❧②❝✐♥❡ ♦❝❝✉♣②✐♥❣ t❤✐s ❝r②st❛❧❧✐♥❡ r❡❣✐♦♥s ❝r♦ss ♠♦r❡ ❡✣❝✐❡♥t❧② t❤❛♥ t❤❡ ♦t❤❡rs ✇❡✐❣❤t✐♥❣ t❤❡ ❝r②st❛❧❧✐♥❡ ❣❧②❝✐♥❡s ♠♦r❡ str♦♥❣❧② ✐♥ t❤❡ s♣❡❝tr❛✳ ❚❤❡ r❡❧❛①❛t✐♦♥ ♠❡❛s✉r❡♠❡♥t ❛t ✹✵ o ❈ ❤❛❞ ❛♥ ✐♥t❡r❡st✐♥❣ r❡s✉❧t✳ ❚❤❡ ❜❡st ✜t ✇❛s ♥♦t t❤❡ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ❛s ❡①♣❡❝t❡❞✱ ❜✉t ✐♥st❡❛❞ ✇❛s ✜t ❜❡st t♦ ❛ ❞♦✉❜❧❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✳ ❚❤❡ t✇♦ r❡❧❛①❛t✐♦♥ t✐♠❡s ✇❡r❡ T1f ast = 6.54 ± 1.35 s ♠❛❦✐♥❣ ✉♣ 38 ± 7% ♦❢ t❤❡ s✐❣♥❛❧ ❛♥❞ ✺✻  τ (Spider Silk)  1/T1 (s-1)  1 (MaSp2) T1  Correlation Time (s) ❋✐❣✉r❡ ✻✳✺✿ P❧♦t ♦❢ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ✈❡rs✉s T11 ❢♦r t❤❡ ❣❧②❝✐♥❡ α ❝❛r❜♦♥ ✇✐t❤ ❛♥ ♦r❞❡r ♣❛r❛♠❡t❡r ♦❢ S = .996 ✉s✐♥❣ t❤❡ ✉❧tr❛ s❧♦✇ t✉♠❜❧✐♥❣ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❙❇❆ ♠♦❞❡❧✳ ❚❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ T1 ❢♦r ▼❛❙♣✷ ✇❛s ✐♥❝❧✉❞❡❞ t♦ s❤♦✇ t❤❛t ✐t ❢❛❧❧s ✇✐t❤✐♥ t❤❡ r❛♥❣❡ ❛❧❧♦✇❡❞ ✇✐t❤ t❤✐s ♦r❞❡r ♣❛r❛♠❡t❡r✳  T1slow = 42.21 ± 4.35 s✳ ❲❤✐❧❡ t❤✐s r❡s✉❧t ❛♣♣❡❛rs t♦ ❜❡ ❛ ❧❛r❣❡ ❞✐s❝r❡♣❛♥❝② ❜❡t✇❡❡♥ t❤❡ t✇♦ r❡s✉❧ts✱ t❤❡② ❛r❡ ♥♦t ✐♥ ❢❛❝t t❤❛t ❞✐✛❡r❡♥t✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s ✐s ✈❡r② s♠❛❧❧ ❛♥❞ ✇♦✉❧❞ r❡q✉✐r❡ ❤✐❣❤ s✐❣♥❛❧ t♦ ♥♦✐s❡ ❞❛t❛ t♦ ❜❡ ❛❜❧❡ t♦ s❡❧❡❝t t❤❡ ❝♦rr❡❝t ❢✉♥❝t✐♦♥✳ ❈♦♥s✐❞❡r✐♥❣ t❤❛t t❤❡ t❡st ❢♦r ❛♥ ❛❞❞✐t✐♦♥❛❧ t❡r♠s ✐s s✐♠♣❧② ❛ ♣r♦❜❛❜✐❧✐st✐❝ t❡st t❤❛t ❛❧❧♦✇s ❛ ✺✪ ❝❤❛♥❝❡ ♦❢ ❡rr♦r✱ ❛♥❞ ❝♦♥s✐❞❡r✐♥❣ ❤♦✇ ♠❛♥② ✜ts ❛r❡ ❞♦♥❡ ✐♥ t❤❡ ❝♦✉rs❡ ♦❢ t❤✐s t❤❡s✐s✱ ✐t ✐s ♠♦r❡ t❤❛♥ ❧✐❦❡❧② t❤❛t ♦♥❡ ✜t ✇♦✉❧❞ ❤❛✈❡ ❛♥ ♦♣t✐♠❛❧ ✜t t❤❛t ✇❛s ♥♦t ❝♦rr❡❝t✳ ■t ✐s ❞✐✣❝✉❧t t♦ ✉s❡ t❤✐s t♦ ❝♦♥✜r♠ t❤❡ ❝❤♦✐❝❡ ♦❢ ♠♦❞❡❧ ❛s t❤❡ ❝❤❛♥❣❡s ✐♥ r❡❧❛①❛t✐♦♥ t✐♠❡s ❛♥❞ t❤❡ ❝❤❛♥❣❡ ✐♥ t❤❡ ❞✐str✐❜✉t✐♦♥✱ β ✱ ❛r❡ ❛❧❧ ✇✐t❤✐♥ t❤❡ r❛♥❣❡ ♦❢ ❡rr♦r✳  ✻✳✸  ●❧✉t❛♠✐♥❡  ❚❤❡r❡ ✐s ✈❡r② ❧✐tt❧❡ ♣✉❜❧✐s❤❡❞ ✇♦r❦ ♦♥ t❤❡  13  ❈ r❡❧❛①❛t✐♦♥ ♦❢ ❣❧✉t❛♠✐♥❡ α ❝❛r❜♦♥s ✐♥ s♣✐❞❡r  s✐❧❦✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ ✐♥ s♣✐❞❡r s✐❧❦ ✐t ✐s ✈❡r② ❞✐✣❝✉❧t t♦ ♠❡❛s✉r❡ t❤❡ ❞❡❝❛② ♦❢ ❛♠♣❧✐t✉❞❡ ♦❢ t❤❡ ❣❧✉t❛♠✐♥❡ α ❝❛r❜♦♥ ❛s ✐t ✐s ♥❡st❧❡❞ ✐♥t♦ t❤❡ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥ ♣❡❛❦✳ ❚❤❡ ✇♦r❦ ❜② ❙✐♠♠♦♥s ❡t ❛❧✳ ❬✸✸❪ ❞✐s❝♦✉♥ts ✐ts r❡❧❡✈❛♥❝❡ ✐♥ ♠❡❛s✉r✐♥❣ t❤❡ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ✺✼  Intensity  D2=0.05s D2=0.15s D2=0.5s D2=2s D2=4s D2=10s D2=15s D2=25s D2=50s  60  50  40  30  20  10  D2 (s)  ❋✐❣✉r❡ ✻✳✻✿ ▼✉❧t✐♣❧❡ s♣❡❝tr❛ ❢r♦♠ t❤❡ s♣✐❞❡r s✐❧❦ r♦♦♠ t❡♠♣❡r❛t✉r❡ T1 r❡❧❛①❛t✐♦♥ ❡①♣❡r✐♠❡♥t✳ ❲❤✐❧❡ ♠♦st ✐♥❢♦r♠❛t✐♦♥ ❢r♦♠ t❤✐s ❞❛t❛ r❡q✉✐r❡❞ ✜tt✐♥❣ ❛♥❞ ❛♥❛❧②s✐s✱ ♦♥❡ r❡s✉❧t t❤❛t ✇❛s ♦❜✈✐♦✉s ❢r♦♠ ♦❜s❡r✈❛t✐♦♥ ✇❛s t❤❡ ❡♠❡r❣❡♥❝❡ ♦❢ t❤❡ ❧❡❞❣❡ ❛t ✷✺✳✸ ♣♣♠ ✐♥t♦ ❛ ❢✉❧❧ ♣❡❛❦✳ ■t s❤♦✉❧❞ ❛❧s♦ ❜❡ ♥♦t❡❞ t❤❛t t❤❡s❡ ❛r❡ ♥♦t ❛❧❧ ♦❢ t❤❡ s♣❡❝tr❛ ❢r♦♠ t❤❡ T1 ❡①♣❡r✐♠❡♥t✳ t✐♠❡ ❛s ✐t ❞❡❝❛②s ❢❛st❡r t❤❛♥ t❤❡ ❛❧❛♥✐♥❡✳ ❲❤✐❧❡ t❤✐s ✐s ❤❛r❞❧② ❛ t❤♦r♦✉❣❤ ❛♥❛❧②s✐s✱ ✐t ❞♦❡s ♣r♦✈✐❞❡ ❛♥ ✐♠♣♦rt❛♥t ✐♥s✐❣❤t✳ ▲✐❦❡ t❤❡ ✇♦r❦ ❜② ❙✐♠♠♦♥s✱ t❤❡ ❣❧✉t❛♠✐♥❡ α ❝❛r❜♦♥ ♦♥❧② ❛♣♣❡❛rs ❛s ❛ ❧❡❞❣❡ ✐♥ t❤❡ s♣✐❞❡r s✐❧❦ ❝r♦ss ♣♦❧❛r✐③❛t✐♦♥ s♣❡❝tr✉♠✳ ■♥ t❤❡ ♣r❡s❡♥t r❡❧❛①❛t✐♦♥ ❞❛t❛ ❤♦✇❡✈❡r✱ t❤✐s ❝❛r❜♦♥ ❛❧s♦ ❜❡❣✐♥s ❛s ❛ ❧❡❞❣❡✱ ❛s ❝❛♥ ❜❡ s❡❡♥ ✐♥ ❋✐❣✉r❡ ✻✳✻✳ ❆s t❤❡ s✐❧❦ ✉♥❞❡r❣♦❡s r❡❧❛①❛t✐♦♥ ❤♦✇❡✈❡r✱ t❤❡ ♠❛❣♥✐t✉❞❡s ♦❢ t❤❡ ❛❧❛♥✐♥❡ ♣❡❛❦ ❛♥❞ t❤❡ ♣❡❛❦ ❛t ✺✹ ♣♣♠ ❡✈❡♥t✉❛❧❧② ❜❡❝♦♠❡ ❡q✉❛❧ ❛t ❛ ❉✷ ♦❢ ✷✺ s❡❝♦♥❞s✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ r❡❧❛①❛t✐♦♥ ♦❢ t❤❡ ❣❧✉t❛♠✐♥❡ α ❝❛r❜♦♥ ✉♥❞❡r ❛ ❧❛r❣❡ ✜❡❧❞ ✐s s❧♦✇❡r✱ ♦r ♠♦r❡ s✐♠♣❧② ♣✉t✱ t❤❡ ❣❧✉t❛♠✐♥❡ α ❝❛r❜♦♥ ✐s ♠✉❝❤ ♠♦r❡ ✜❡❧❞ ❞❡♣❡♥❞❡♥t t❤❛♥ t❤❛t ♦❢ t❤❡ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥ ❛♥❞ t❤❡r❡❢♦r❡ ❤❛s ❛ ❧♦♥❣❡r ❝♦rr❡❧❛t✐♦♥ t✐♠❡✳ ❚❤✐s ✐s ❡①♣❡❝t❡❞ ❛s ✈❡r② ❧✐tt❧❡ ♦❢ t❤❡ ❣❧✉t❛♠✐♥❡ ❝❛♥ ❜❡ ✐♥ β s❤❡❡ts✱ ❛s ♠♦st ♦❢ t❤❡ ❝r②st❛❧❧✐♥❡ r❡❣✐♦♥ ❝♦♥s✐sts ♦❢ ❛❧❛♥✐♥❡ ❛♥❞ ❣❧②❝✐♥❡✳ ❆ ❜❛❝❦❜♦♥❡ ❝❛r❜♦♥ ✐♥ ❛ ♥♦♥ ❝r②st❛❧❧✐♥❡ str✉❝t✉r❡ ❤❛s ♠✉❝❤ ♠♦r❡ ❢r❡❡❞♦♠ ♦❢ ♠♦t✐♦♥✱ ✇❤✐❝❤ s❤♦✉❧❞ r❡s✉❧t ✐♥ s❧♦✇❡r ❛♥❞ ❧❛r❣❡r ♠♦t✐♦♥s✳ ❍♦✇❡✈❡r✱ ❢♦r r❡❛s♦♥s t❤❛t ✇✐❧❧ ❜❡ ❞✐s❝✉ss❡❞ ✐♥ ❙❡❝t✐♦♥ ✼✳✷✳✸✱ t❤❡ ♦❜s❡r✈❛t✐♦♥s ♦❢ ❙✐♠♠♦♥s ♠❛② ❤❛✈❡ ♠✐ss❡❞ ❛ ❝♦♠♣♦♥❡♥t ♦❢ r❡❧❛①❛t✐♦♥ ✐♥ t❤✐s ❝❤❡♠✐❝❛❧ s❤✐❢t r❛♥❣❡ t❤❛t ❤❛♣♣❡♥❡❞ t♦ ❤❛✈❡ t❤❡ s❛♠❡ T1 ❛s t❤❡ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥ ✐♥ ❛ ✷✵✵ ▼❍③ ✜❡❧❞✳ ❍♦✇❡✈❡r✱ ❛s ✇✐❧❧ ❛❧s♦ ❜❡ s❤♦✇♥✱ t❤✐s ❡①tr❛ r❡❧❛①❛t✐♦♥ t❤❛t ✇❛s ♠✐ss❡❞ ✐s ❛❧s♦ ✐♥ t❤❡ s❧♦✇ ❧✐♠✐t✱ s♦ t❤❡ r❡s✉❧t ♦❢ ❣❧✉t❛♠✐♥❡ ✐♥ t❤❡ s❧♦✇ ♠♦t✐♦♥ r❡❣✐♠❡ st❛♥❞s✱ ✐❢ ♥♦t ❜② s❤❡❡r ❧✉❝❦✳ ❚❤❡ ❣❧✉t❛♠✐♥❡ β ❛♥❞ γ ❝❛r❜♦♥ ♣❡❛❦ ✐♥ s♣✐❞❡r s✐❧❦ ✇❛s ❢♦✉♥❞ t♦ r❛♥❣❡ ❢r♦♠ ✷✽✲✸ ✵ ♣♣♠✳ ❉✉❡ t♦ t❤❡ ❢❛❝t t❤❛t ❛❧❧ ♦❢ t❤❡ ❣❧✉t❛♠✐♥❡ γ ❝♦♥❢♦r♠❛t✐♦♥s ❧✐❡ ✐♥ t❤✐s r❛♥❣❡✱ ✐t ✇❛s ✐♠♣♦ss✐❜❧❡ t♦ ❞❡❞✉❝❡ ❛♥②t❤✐♥❣ ❛❜♦✉t t❤❡ str✉❝t✉r❡ ❢r♦♠ t❤✐s✳ ❚❤❡r❡ ✇❛s ❛❧s♦ ❛ ✈❡r② s❧✐❣❤t ❧❡❞❣❡ ❛t ✷✺ ♣♣♠ t❤❛t ❡①t❡♥❞❡❞ ❞♦✇♥✜❡❧❞ ❢r♦♠ t❤❡ ❛❧❛♥✐♥❡ β ❝❛r❜♦♥ t❤❛t ✇❛s ✇❡❧❧ ❜❡②♦♥❞ t❤❡ r❛♥❣❡ ♦❢ ❝❤❡♠✐❝❛❧ s❤✐❢ts ❛❧❧♦✇❡❞✳ ❚❤✐s ♠♦st ❧✐❦❡❧② ❝♦rr❡s♣♦♥❞s t♦ ❛ ❣❧✉t❛♠✐♥❡ r❡s✐❞✉❡ ✐♥ t❤❡ α✲❤❡❧✐① ✺✽  Intensity  Spider Silk Glutamine C-beta/gamma Relaxation Data Single exponential fit Stretched exponential fit Double exponential fit  0  5  10  15  20  25  30  35  40  D2 (s)  ❋✐❣✉r❡ ✻✳✼✿ ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ s♣✐❞❡r s✐❧❦ ❣❧✉t❛♠✐♥❡ β ✴γ ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳ ❝♦♥❢♦r♠❛t✐♦♥ ✇❤✐❝❤ ❤❛s ❛ ❝❤❡♠✐❝❛❧ s❤✐❢t ♦❢ ✷✺✳✻ ♣♣♠✱ ❛❧t❤♦✉❣❤ t❤❡ 31 ✲❤❡❧✐① ❝❛♥♥♦t ❜❡ r✉❧❡❞ ♦✉t ❛s ✐t ✇♦✉❧❞ ❛♣♣❡❛r ♥❡❛r ✷✻✳✺ ♣♣♠✳ ❚❤❡ ♣❡❛❦ r❛♥❣❡ ♦❢ ✷✽✲✸✸ ♣♣♠ ✇❛s ✉s❡❞ ❢♦r t❤❡ ✐♥t❡❣r❛t✐♥❣ ♦✈❡r ❢♦r t❤❡ ❣❧✉t❛♠✐♥❡ β/γ ♣❡❛❦✳ ❚❤❡ ❜❡st ✜t ✇❛s ♦❜s❡r✈❡❞ t♦ ❜❡ ❛ ❞♦✉❜❧❡ ❡①♣♦♥❡♥t✐❛❧✱ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✻✳✼ ❝♦♥s✐st✐♥❣ ♦❢ ❛ ❢❛st r❡❧❛①❛t✐♦♥ t✐♠❡ ♦❢ T1f ast = 1.7 ± 0.2 s ✇❤✐❝❤ ♠❛❞❡ ✉♣ 50 ± 4% ❛♥❞ ❛ s❧♦✇ ❝♦♠♣♦♥❡♥t ♦❢  T1slow = 26 ± 3 s✳ ❚♦ ✉s❡ ❛s ❛ ♣♦✐♥t ♦❢ ❝♦♠♣❛r✐s♦♥ ✇✐t❤ t❤❡ s✐♥❣❧❡ ❝♦♠♣♦♥❡♥t r❡❧❛①❛t✐♦♥ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✱ t❤❡ ♦♥❡ ❝♦♠♣♦♥❡♥t ✜t ②✐❡❧❞❡❞ T1 = 9.8 ± 1.5 s✳ ■♥ t❤❡ ✷✵✵ ▼❍③ s♣❡❝tr♦♠❡t❡r t❤❛t ✇❛s ✉s❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✱ t❤❡ s✐♥❣❧❡ ❝♦♠♣♦♥❡♥t T1 = 4 s ❲❤✐❧❡ ✇❡ ❤❛✈❡ str♦♥❣ r❡❛s♦♥ t♦ s✉s♣❡❝t t❤❛t t❤❡r❡ ✐s ♠♦r❡ t❤❛♥ ♦♥❡ str✉❝t✉r❡ ❢r♦♠ t✇♦ ✈❡r② ❞✐✛❡r❡♥t ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ r❡❧❛①❛t✐♦♥ t✐♠❡s✱ t❤❡ ✐♥❝r❡❛s❡ ✐♥ T1 ✉♥❞❡r ❛ ❧❛r❣❡r ✜❡❧❞ ❢♦r t❤❡ ♦♥❡ ❝♦♠♣♦♥❡♥t s❤♦✇s t❤❛t t❤❡r❡ ❛t ❧❡❛st t❤❡ s❧♦✇ ❝♦♠♣♦♥❡♥t ✐s ✇❡❧❧ ♣❛st t❤❡ s❧♦✇ s✐❞❡ ♦❢ t❤❡ ❢❛st✴s❧♦✇ tr❛♥s✐t✐♦♥ ♦♥ t❤❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ✈❡rs✉s ✐♥✈❡rs❡ r❡❧❛①❛t✐♦♥ t✐♠❡ ❝✉r✈❡✳ ❲❡ ❦♥♦✇ ❛t ❧❡❛st t❤❡ s❧♦✇ ❝♦♠♣♦♥❡♥t ❤❛s ❤❛❞ t♦ ❜❡ ✐♥ t❤❡ s❧♦✇ r❡❣✐♠❡ ❜❡❝❛✉s❡ ✐❢ ✐t ✇❡r❡ ❥✉st t❤❡ ❢❛st ❝♦♠♣♦♥❡♥t✱ t❤❡ r❡❧❛①❛t✐♦♥ t✐♠❡ ✇♦✉❧❞ ♥♦t ✐♥❝r❡❛s❡ ❛s ♠✉❝❤ ✭t❤❡ ♦♥❡ ❝♦♠♣♦♥❡♥t ✐s ❧❛r❣❡r ✐♥ ❜♦t❤ ✜❡❧❞s t❤❛♥ t❤❡ ❢❛st ❝♦♠♣♦♥❡♥t✮✳ ❚❤✐s str♦♥❣ ✜❡❧❞ ❞❡♣❡♥❞❡♥❝❡ ❝♦✉❧❞ ❡①♣❧❛✐♥ ✇❤② t❤❡ ♣❡❛❦ ♥❡❛r ✷✺ ♣♣♠ ❤❛s ♥♦t ❜❡❡♥ ✐❞❡♥t✐✜❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✳ ❚❤❡ ♦♥❧② ♦t❤❡r T1 r❡❧❛①❛t✐♦♥ ❡①♣❡r✐♠❡♥t ✇❛s ❝♦♥❞✉❝t❡❞ ✐♥ ❛ ✷✵✵ ▼❍③ ✜❡❧❞✱ ✇❤✐❝❤ ✇♦✉❧❞ ❤❛✈❡ ❛ r❡❧❛①❛t✐♦♥ t✐♠❡ ❛t ❧❡❛st ✷✳✺ t✐♠❡s ❢❛st❡r✱ ❞❡♣❡♥❞✐♥❣ ♦♥ ✐❢ t❤❡ ❢❛st ❝♦♠♣♦♥❡♥t ✇❛s ✐♥ t❤❡ ❢❛st r❡❣✐♠❡ ♦r ♥♦t✳ ❆t t❤✐s s♣❡❡❞ ♦❢ r❡❧❛①❛t✐♦♥✱ ✐t ✐s q✉✐t❡ ♣♦ss✐❜❧❡ t♦ ♠✐st❛❦❡ t❤✐s ❝♦♠♣♦♥❡♥t ❢♦r r❡❧❛①✐♥❣ ❛t t❤❡ s♣❡❡❞ ♦❢ t❤❡ ❛❧❛♥✐♥❡ β ❝❛r❜♦♥✳ ❚❤❡ s♣❡❝tr❛ ❢r♦♠ t❤❡ r❡❧❛①❛t✐♦♥ ❡①♣❡r✐♠❡♥t ❛t ✹✵ o ❈ ❣❛✈❡ r❡❧❛①❛t✐♦♥ t✐♠❡s ♦❢ T1f ast = 0.89 ±  0.30 s ✇❤✐❝❤ ♠❛❞❡ ✉♣ 44 ± 7% ❛♥❞ ❛ s❧♦✇ ❝♦♠♣♦♥❡♥t ♦❢ T1slow = 20.9 ± 3.7 s✳ ❲❤✐❧❡ t❤❡  ✺✾  ❡rr♦r ❞♦❡s s❧✐❣❤t❧② ♦✈❡r❧❛♣ ✇✐t❤ t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡ ✈❛❧✉❡s✱ t❤✐s r❡❞✉❝t✐♦♥ ✐♥ t❤❡ r❡❧❛①❛t✐♦♥ t✐♠❡s ♦❢ ❜♦t❤ ❝♦♠♣♦♥❡♥ts ✇❤❡♥ ❤❡❛t❡❞ ❛♥❞ t❤❡ ❢❛❝t t❤❛t t❤❡ s✐♥❣❧❡ ❝♦♠♣♦♥❡♥t T1 ✇❛s ✜❡❧❞ ❞❡♣❡♥❞❡♥t ✐s ❝♦♥s✐st❡♥t ✇✐t❤ ❜♦t❤ ❝♦♠♣♦♥❡♥ts ❤❛✈✐♥❣ ❝♦rr❡❧❛t✐♦♥ t✐♠❡s ✐♥ t❤❡ s❧♦✇ ♠♦t✐♦♥ r❡❣✐♠❡✳  ✻✵  ❈❤❛♣t❡r ✼  ❘❡❝♦♠❜✐♥❛♥t ▼❛♠♠❛❧✐❛♥ ❙✐❧❦  ❚❤❡ ♣r✐♠❛r② ❣♦❛❧ ♦❢ t❤✐s ✇♦r❦ ✇❛s t❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ▼❛❙♣✶ ❛♥❞ ▼❛❙♣✷ ♣r♦t❡✐♥s ♣r♦❞✉❝❡❞ ❜② ◆❡①✐❛✳ ❚❤❡ ✜rst s❡❝t✐♦♥ ❞❡t❛✐❧s t❤❡ ❛♥❛❧②s✐s ♦❢ ▼❛❙♣✶ ♣♦✇❞❡r✱ t❤❡ s❡❝♦♥❞ s❡❝t✐♦♥ ♣r♦✈✐❞❡s t❤❡ ❛♥❛❧②s✐s ♦❢ ▼❛❙♣✷ ♣♦✇❞❡r✱ ❛♥❞ t❤❡ t❤✐r❞ s❡❝t✐♦♥ ♣r♦✈✐❞❡s t❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶✳  ✼✳✶  ▼❛❙♣✶  ❲❡ ♥♦✇ t✉r♥ ❢r♦♠ ♥❛t✐✈❡ s♣✐❞❡r s✐❧❦ t♦ r❡❝♦♠❜✐♥❛♥t ▼❛❙♣✶ ✐♥ ♣♦✇❞❡r ❢♦r♠✱ t❤❡ ✐♥t❡r❡st ❤❡r❡ ✐s t❤❛t t❤✐s ✐s t❤❡ ♣r✐♠❛r② ♣r♦t❡✐♥ ✐♥ s♣✐❞❡r s✐❧❦✳  ✼✳✶✳✶  ❆❧❛♥✐♥❡  ❚❤❡ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥ ✇❛s ❢♦✉♥❞ t♦ ❤❛✈❡ ❛ ❝❤❡♠✐❝❛❧ s❤✐❢t ♦❢ ✹✾✳✻ ♣♣♠ ✇✐t❤ ❛ ♣❡❛❦ ✇✐❞t❤ ♦❢ ✸ ♣♣♠✳ ❚❤❡ ♣❡❛❦ ✇✐❞t❤ ✇❛s ❢♦✉♥❞ ❜② ✜tt✐♥❣ t❤❡ s♣❡❝tr✉♠ t♦ ❛ ●❛✉ss✐❛♥ t❤❡ s❛♠❡ r❛♥❣❡ ❛s t❤❡ ♦♥❡ ✉s❡❞ ❢♦r s♣✐❞❡r s✐❧❦✳ ❚❤✐s ❝❤❡♠✐❝❛❧ s❤✐❢t ✐s ✇✐t❤✐♥ t❤❡ ❡①♣❡❝t❡❞ r❛♥❣❡ ❢♦r ❞r❛❣❧✐♥❡ s✐❧❦✳ ❚❤❡ r❡❧❛①❛t✐♦♥ s♣❡❝tr❛ ✇❡r❡ ✐♥t❡❣r❛t❡❞ ❢r♦♠ ✹✼ t♦ ✺✵✳✺ ♣♣♠✳ ❚❤❡ st❛t✐st✐❝❛❧❧② ❜❡st ✜t ✇❛s ❢♦✉♥❞ t♦ ❜❡ ❛ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧✱ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✼✳✶ ✇✐t❤ ❛ r❡❧❛①❛t✐♦♥ t✐♠❡ ♦❢  T1∗ = 16.3 ± 0.8 s ❛♥❞ β = 0.70 ± 0.02 ②✐❡❧❞✐♥❣ T1 = 21 ± 1.0 s✳ ■t ✐s ❛❧s♦ ✐♥t❡r❡st✐♥❣ ✻✶  Intensity  SpI Alanine C-alpha Relaxation Data Single exponential fit Stretched exponential fit Double exponential fit  0  5  10  15  20  25  30  35  40  45  50  D2 (s)  ❋✐❣✉r❡ ✼✳✶✿ ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ▼❛❙♣✶ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳ t❤❛t t❤❡ ♠❡❛s✉r❡❞ r❡❧❛①❛t✐♦♥ t✐♠❡ ✐s ❝❧♦s❡ t♦ t❤❛t ❢♦✉♥❞ ✐♥ t❤❡ s✐❧❦✳ ❚❤✐s s✉❣❣❡sts ❜♦t❤ ❛ s✐♠✐❧❛r ♠❡❛♥ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ❛♥❞ ♦r❞❡r ♣❛r❛♠❡t❡r✳ ❚❤❡ s♠❛❧❧❡r β ♠❡❛♥s t❤❛t t❤❡r❡ ✐s ❛ ❧❛r❣❡r ❞✐str✐❜✉t✐♦♥ ♦❢ ✈✐❜r❛t✐♦♥❛❧ ♠♦❞❡s ♦r t❤❡② ❛r❡ s❤✐❢t❡❞ ❛✇❛② ❢r♦♠ t❤❡ ❢❛st✴s❧♦✇ ♠♦t✐♦♥ tr❛♥s✐t✐♦♥ ♣♦✐♥t✳ ■t ✐s ✐♥t❡r❡st✐♥❣ t♦ ♥♦t❡ t❤❛t t❤❡ ❝❤❡♠✐❝❛❧ s❤✐❢t ❢r♦♠ t❤❡ ❝r♦ss ♣♦❧❛r✐③❛t✐♦♥ s♣❡❝tr✉♠✱ ❛❧♦♥❣ ✇✐t❤ t❤❡ str✐❦✐♥❣❧② s✐♠✐❧❛r s❝❛❧❡ ♦❢ r❡❧❛①❛t✐♦♥ t✐♠❡ t♦ t❤❛t ♦❢ ♥❛t✉r❛❧ s✐❧❦ s✉❣❣❡st ❛ ❧❛r❣❡ ❛♠♦✉♥t ♦❢ β s❤❡❡ts✳ ❚❤❡ ❛❧❛♥✐♥❡ β ❝❛r❜♦♥ ✐♥ ▼❛❙♣✶ ✇❛s ❢♦✉♥❞ t♦ ❤❛✈❡ ❛ ❝❤❡♠✐❝❛❧ s❤✐❢t ♦❢ ✷✶✳✶ ♣♣♠✳ ❚❤✐s ♣❡❛❦ αβ ❝♦rr❡s♣♦♥❞s t♦ β s❤❡❡ts✳ ❚❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ♣❡❛❦ s❡♣❛r❛t✐♦♥ ✇❛s ♠❡❛s✉r❡❞ t♦ ❜❡ ∆CSala = 28.5  ♣♣♠✱ ✇❤✐❝❤ ✐s ✐♥ t❤❡ r❛♥❣❡ ♦❢ β s❤❡❡ts ❛♥❞ ❢❛r ❢r♦♠ ❛♥② ♦t❤❡r str✉❝t✉r❡✳ ❚❤❡ β ❝❛r❜♦♥ ❛❧s♦ ❛♣♣❡❛rs t♦ ❤❛✈❡ ❛ ❜❛r❡❧② ✈✐s✐❜❧❡ ❧❡❞❣❡ ❜❡t✇❡❡♥ ✶✺✲✶✻ ♣♣♠✳ ❚❤✐s ❧❡❞❣❡ ❝♦rr❡s♣♦♥❞s t♦ α ❤❡❧✐❝❡s ❜✉t ❛s ❛ s✐♠✐❧❛r ❧❡❞❣❡ ❝❛♥ ❜❡ s❡❡♥ ✐♥ t❤❡ s♣✐❞❡r s✐❧❦ ❞❛t❛ ❛❜♦✈❡ s♦ ✐s ♥♦t ❛ ❞✐st✐♥❣✉✐s❤✐♥❣ ❝❤❛r❛❝t❡r✐st✐❝✳ ❚❤❡ ♣❡❛❦ ✇❛s ✐♥t❡❣r❛t❡❞ ♦✈❡r ❛ r❛♥❣❡ ♦❢ ✶✺✲✷✶ ♣♣♠✳ ❚❤❡ ❜❡st ✜t ✇❛s ❢♦✉♥❞ t♦ ❜❡ ❛ ❞♦✉❜❧❡ ❡①♣♦♥❡♥t✐❛❧ ❛s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✼✳✷✱ ✇✐t❤ T1f ast = 0.40 ± 0.09 s ❛♥❞ ❛ s❧♦✇ r❡❧❛①❛t✐♦♥ t✐♠❡ ♦❢ T1slow = 1.8 ± 0.7 s ✇✐t❤ 70 ± 16% ♦❢ t❤❡ s✐❣♥❛❧ ✐♥ t❤❡ ❢❛st r❡❣✐♠❡✳ ❚❤❡s❡ r❡s✉❧ts ❛r❡ ❝♦♥s✐st❡♥t ❢♦r t❤♦s❡ ❢♦✉♥❞ ✐♥ s♣✐❞❡r s✐❧❦✬s ❛❧❛♥✐♥❡ r❡s✐❞✉❡ ❛♥❞ str♦♥❣❧② s✉❣❣❡sts t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ s✐❣♥✐✜❝❛♥t β s❤❡❡t ❝♦♠♣♦♥❡♥t ✐♥ ▼❛❙♣✶✳  ✻✷  Intensity  Sp1 Alanine C-beta Relaxation Data Single exponential fit Stretched exponential fit Double exponential fit  0  2  4  6  8  10  12  14  16  18  20  D2 (s)  ❋✐❣✉r❡ ✼✳✷✿ ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ▼❛❙♣✶ ❛❧❛♥✐♥❡ β ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳ ✼✳✶✳✷  ●❧②❝✐♥❡  α  ❈❛r❜♦♥  ❚❤❡ ❣❧②❝✐♥❡ α ❝❛r❜♦♥ ✐♥ ▼❛❙♣✶ ✇❛s ❢♦✉♥❞ t♦ ❤❛✈❡ ❛ ❝❤❡♠✐❝❛❧ s❤✐❢t ♦❢ ✹✸✳✹ ♣♣♠✱ ✇✐t❤ ❛ ✇✐❞t❤ ♦❢ ✹ ♣♣♠✳ ❚❤❡ ✇✐❞t❤ ✇❛s ♠❡❛s✉r❡❞ ❜② ✜tt✐♥❣ ❛ ●❛✉ss✐❛♥ t♦ t❤❡ ♣❡❛❦✳ ❚❤❡ s❤✐❢t ✐s ❝♦♥s✐st❡♥t ✇✐t❤ t❤❡ ✈❛❧✉❡s r❡♣♦rt❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✱ ❜✉t ❞♦❡s ♥♦t ✐♥ ✐ts❡❧❢ ✐♠♣❧② ❛ str♦♥❣ ♣r❡s❡♥❝❡ ♦❢  β s❤❡❡ts ❛s t❤✐s ✐s ❛❧s♦ ✇✐t❤ t❤❡ r❛♥❣❡ ♦❢ t❤❡ r❛♥❞♦♠ ❝♦✐❧ ❝❤❡♠✐❝❛❧ s❤✐❢t✱ ❛♥❞ t❤❡ ♣❡❛❦ ✇✐❞t❤ ❞♦❡s ❝♦✈❡r ❛ s✐❣♥✐✜❝❛♥t r❡❣✐♦♥ ♦❢ t❤❡ 31 ❤❡❧✐① ❝❤❡♠✐❝❛❧ s❤✐❢t✳ ❚❤❡ ♣❡❛❦ r❛♥❣❡ ❢♦r ✐♥t❡❣r❛t✐♦♥ ♦❢ t❤❡ r❡❧❛①❛t✐♦♥ s♣❡❝tr❛ ✇❛s ✹✶✳✺✲✹✺ ♣♣♠✳ ❚❤❡ ❜❡st ✜t t♦ t❤❡ r❡❧❛①❛t✐♦♥ ✇❛s ❢♦✉♥❞ t♦ ❜❡ ❛ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧✱ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✼✳✸ ✇✐t❤ ♣❛r❛♠❡t❡rs  T1 = 15.3 ± 2.2 s ❛♥❞ β = 0.64 ± 0.10✳ ❚❤✐s ②✐❡❧❞❡❞ ❛ ♠❡❛♥ r❡❧❛①❛t✐♦♥ t✐♠❡ ♦❢ < T1 >= 21.0 ± 5.0 s✳ ❚❤✐s s❤♦rt❡♥✐♥❣ ♦❢ t❤❡ r❡❧❛①❛t✐♦♥ t✐♠❡ ✇♦✉❧❞ ❜❡ ❜❡st ❡①♣❧❛✐♥❡❞ ❜② ❛ s♠❛❧❧❡r ♦r❞❡r ♣❛r❛♠❡t❡r t❤❛♥ t❤❡ ♦♥❡ ❢♦✉♥❞ ✐♥ s♣✐❞❡r s✐❧❦✱ s✉❣❣❡st✐♥❣ ❛ ❧♦♦s❡r str✉❝t✉r❡ t❤❛♥ t❤❛t ♦❢ r✐❣✐❞ β s❤❡❡ts✳  ✼✳✶✳✸  ●❧✉t❛♠✐♥❡  ▲✐❦❡ t❤❡ s♣✐❞❡r s✐❧❦✱ t❤❡ ❣❧✉t❛♠✐♥❡ α ❝❛r❜♦♥ ♣❡❛❦ ❞♦❡s ♥♦t ❛♣♣❡❛r ✐♥ t❤❡ ❝r♦ss ♣♦❧❛r✐③❛t✐♦♥ s♣❡❝tr❛✳ ❍♦✇❡✈❡r ✐♥ t❤❡ s♣❡❝tr✉♠ ✇✐t❤ ❉✷❂✺ s✱ ❛ ♣❡❛❦ ❜❡❣✐♥s t♦ ❡♠❡r❣❡ ✇✐t❤ ❛ ❝❤❡♠✐❝❛❧ s❤✐❢t ♦❢ ✺✹ ♣♣♠ ❛s ❝❛♥ ❜❡ s❡❡♥ ✐♥ ❋✐❣✉r❡ ✼✳✹✱ ❛♥❞ ❜② ❉✷❂✹✵ s ✐t ❛♣♣❡❛rs t♦ ❜❡ ♦♥ t❤❡ s❛♠❡ s❝❛❧❡ ❛s t❤❛t ♦❢ t❤❡ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥✳ ❚❤✐s s✉❣❣❡sts t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ ♠✉❝❤ s❧♦✇❡r r❡❧❛①✐♥❣ ❣❧✉t❛♠✐♥❡ α ❝❛r❜♦♥ ♣❡❛❦✳ ✻✸  Intensity  SpI Glycine C-alpha Relaxation Data Single exponential fit Stretched exponential fit Double exponential fit  0  5  10  15  20  25  30  35  40  45  50  D2 (s)  ❋✐❣✉r❡ ✼✳✸✿ ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ▼❛❙♣✶ ❣❧②❝✐♥❡ α✲❈ r❡❧❛①❛t✐♦♥ ❞❛t❛  Intensity  D2=.1s D2=.2s D2=.5s D2=1s D2=2s D2=5s D2=10s D2=20s D2=40s  70  60  50  40  30  20  10  0  Chemical Shift (ppm)  ❋✐❣✉r❡ ✼✳✹✿ ▼✉❧t✐♣❧❡ s♣❡❝tr❛ ❢r♦♠ t❤❡ ▼❛❙♣✶ T1 r❡❧❛①❛t✐♦♥ ❡①♣❡r✐♠❡♥t✳ ❲❤✐❧❡ ♠♦st ✐♥✲ ❢♦r♠❛t✐♦♥ ❢r♦♠ t❤✐s ❞❛t❛ r❡q✉✐r❡❞ ✜tt✐♥❣ ❛♥❞ ❛♥❛❧②s✐s✱ ♦♥❡ r❡s✉❧t t❤❛t ✇❛s ♦❜✈✐♦✉s ❢r♦♠ ♦❜s❡r✈❛t✐♦♥ ✇❛s t❤❡ ❡♠❡r❣❡♥❝❡ ♦❢ t❤❡ ❧❡❞❣❡ ❛t ✷✺✳✸ ♣♣♠ ✐♥t♦ ❛ ❢✉❧❧ ♣❡❛❦✳  ✻✹  Intensity  Sp1 Glutamine C-beta/gamma Relaxation Data Single exponential fit Stretched exponential fit Double exponential fit  0  5  10  15  20  25  30  35  40  D2 (s)  ❋✐❣✉r❡ ✼✳✺✿ ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ▼❛❙♣✶ ❣❧②❝✐♥❡ β ✴γ ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳ ❚❤❡ ❣❧✉t❛♠✐♥❡ β ❛♥❞ γ ❝❛r❜♦♥ ♣❡❛❦ ✐♥ ▼❛❙♣✶ ❝❤❡♠✐❝❛❧ s❤✐❢t ✇❛s ❢♦✉♥❞ t♦ r❛♥❣❡ ❢r♦♠ ✷✽✲✸✷ ♣♣♠✳ ❉✉❡ t♦ t❤❡ ❢❛❝t t❤❛t ❛❧❧ ♦❢ t❤❡ ❣❧✉t❛♠✐♥❡ γ ❝♦♥❢♦r♠❛t✐♦♥s ❧✐❡ ✐♥ t❤✐s r❛♥❣❡✱ ✐t ✇❛s ✐♠♣♦ss✐❜❧❡ t♦ ❞❡❞✉❝❡ ❛♥②t❤✐♥❣ ❛❜♦✉t t❤❡ str✉❝t✉r❡ ❢r♦♠ t❤✐s✳ ❆❧s♦✱ ❛s ✐♥ t❤❡ s✐❧❦✱ t❤❡r❡ ✇❛s ❛ ❧❡❞❣❡ ❛t ✷✺ ♣♣♠ t❤❛t ❡①t❡♥❞❡❞ ❞♦✇♥✜❡❧❞ ❢r♦♠ t❤❡ ❛❧❛♥✐♥❡ β ❝❛r❜♦♥ t❤❛t ✇❛s ✇❡❧❧ ❜❡②♦♥❞ t❤❡ r❛♥❣❡ ♦❢ ❝❤❡♠✐❝❛❧ s❤✐❢ts ❡①♣❡❝t❡❞ ❢♦r ❛❧❛♥✐♥❡ β ❝❛r❜♦♥✳ ❚❤❡ ♠♦st ❧✐❦❡❧② str✉❝t✉r❡ ❢♦r t❤✐s ✐s t❤❡ ❣❧✉t❛♠✐♥❡ α ❤❡❧✐① ✇❤✐❝❤ ❤❛s ❛ ❝❤❡♠✐❝❛❧ s❤✐❢t ♦❢ ✷✺✳✻ ♣♣♠✱ ❛❧t❤♦✉❣❤ t❤❡ 31 ✲❤❡❧✐① ❝❛♥♥♦t ❜❡ r✉❧❡❞ ♦✉t ❛s ✐t ✇♦✉❧❞ ❜❡ ♥❡❛r ❛t ✷✻✳✾ ♣♣♠ ❚❤❡ ♣❡❛❦ ✇❛s ✐♥t❡❣r❛t❡❞ ♦✈❡r ❛ r❛♥❣❡ ♦❢ ✷✼✲✸✵ ♣♣♠✳ ❚❤❡ ❜❡st ✜t ✇❛s ♦❜s❡r✈❡❞ t♦ ❜❡ ❛ ❞♦✉❜❧❡ ❡①♣♦♥❡♥t✐❛❧✱ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✼✳✺ ❝♦♥s✐st✐♥❣ ♦❢ ❛ ❢❛st r❡❧❛①❛t✐♦♥ t✐♠❡ ♦❢ T1f ast = 0.79 ± 0.38 s ✇❤✐❝❤ ♠❛❞❡ ✉♣ 18±4% ❛♥❞ ❛ s❧♦✇ ❝♦♠♣♦♥❡♥t ♦❢ T1slow = 18.2±1.3 s✳ ❲❤✐❧❡ t❤❡s❡ r❡❧❛①❛t✐♦♥ t✐♠❡s ❛r❡ q✉✐t❡ ❞✐✛❡r❡♥t ❢r♦♠ t❤❛t ✐♥ s♣✐❞❡r s✐❧❦ t❤❡② ❛r❡ str✐❦✐♥❣❧② s✐♠✐❧❛r t♦ t❤♦s❡ ❢♦✉♥❞ ✐♥ t❤❡ ✹✵◦ ❈ s❛♠♣❧❡✳ ❚❤❡ ❜✐❣ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ▼❛❙♣✶ ❛♥❞ s♣✐❞❡r s✐❧❦✱ ❜♦t❤ ❤❡❛t❡❞ ❛♥❞ ❛t r♦♦♠ t❡♠♣❡r❛t✉r❡✱ ✐s t❤❡ r❛t✐♦ ♦❢ ❢❛st t♦ s❧♦✇ r❡❧❛①❛t✐♦♥ ❢♦r t❤❡ ❣❧✉t❛♠✐♥❡ β ✱γ ❝❛r❜♦♥s✳ ❲❤❡r❡ s♣✐❞❡r s✐❧❦ ❤❛s ❛ r❛t✐♦ ♦❢ ❛r♦✉♥❞ ✶✿✶ ❢♦r ❢❛st t♦ s❧♦✇✱ t❤❡ ❣❧✉t❛♠✐♥❡ ❤❛s ❛ r❛t✐♦ ♦❢ ✶✿✹✱ s✉❣❣❡st✐♥❣ t❤❛t t❤❡ str✉❝t✉r❡ ❝❛✉s✐♥❣ t❤❡ ❢❛st r❡❧❛①❛t✐♦♥ ✐s ❢❛r ♠♦r❡ ♣r♦♠✐♥❡♥t ✐♥ s♣✐❞❡r s✐❧❦✳  ✼✳✶✳✹  ❉✐s❝✉ss✐♦♥ ♦❢ ▼❛❙♣✶  ❚❤❡ ❛❧❛♥✐♥❡ α ❛♥❞ β ❝❛r❜♦♥s ❡①❤✐❜✐t❡❞ ❜♦t❤ ❝❤❡♠✐❝❛❧ s❤✐❢t ❛♥❞ r❡❧❛①❛t✐♦♥ ❜❡❤❛✈✐♦r t❤❛t str♦♥❣❧② s✉❣❣❡sts ❛ str♦♥❣ ♣r❡s❡♥❝❡ ♦❢ β s❤❡❡ts✳ ❇♦t❤ ❝❛r❜♦♥s ♣❡❛❦❡❞ ✐♥ t❤❡ r❛♥❣❡ ❛❝❝❡♣t❡❞ αβ ❢♦r t❤❡ ♣r❡s❡♥❝❡ ♦❢ β s❤❡❡ts ❛♥❞ ♥♦ str✉❝t✉r❡ ♦t❤❡r t❤❛♥ β s❤❡❡ts ❤❛❞ ❛ ∆CSala ❡✈❡♥ ❝❧♦s❡ t♦  ✻✺  A B  C  D E 60  40 20 Chemical Shift (ppm)  0  ❋✐❣✉r❡ ✼✳✻✿ ❙♣❡❝tr❛ ♦❢ ❧②♦♣❤✐❧✐③❡❞ ❣❧❛♥❞ s✐❧❦ ✭❇✮ ❛♥❞ ❧②♦♣❤✐❧✐③❡❞ ❞❡♥❛t✉r❡❞ s✐❧❦ ✭❉✮✱ ❝♦♠♣❛r❡❞ t♦ ▼❛❙♣✶ ✭❆✮ ❛♥❞ ▼❛❙♣✷ ✭❈✮ ❛♥❞ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶✳ ▲②♦♣❤✐❧✐③❡❞ s♣❡❝tr❛ ❢r♦♠ ❍✐❥✐r✐❞❛ ❡t ❛❧✳ ❬✷✵❪✳ t❤❡ ✈❛❧✉❡ ♦❢ ✷✽✳✺ ♣♣♠ ❢♦✉♥❞ ❢♦r ▼❛❙♣✶✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡ r❡❧❛①❛t✐♦♥ ♣❛r❛♠❡t❡rs ❢♦r ❜♦t❤ t❤❡  α ❛♥❞ β ❝❛r❜♦♥s ❛♣♣❡❛r❡❞ t♦ t❛❦❡ ❛❧♠♦st ✐❞❡♥t✐❝❛❧ ✈❛❧✉❡s t♦ t❤♦s❡ ♠❡❛s✉r❡❞ ✐♥ s♣✐❞❡r s✐❧❦✳ ❍♦✇❡✈❡r✱ t❤❡ ❣❧②❝✐♥❡ r❡s✉❧ts t❡❧❧ ❛ ❞✐✛❡r❡♥t st♦r②✳ ❚❤❡ ❞r♦♣ ✐♥ r❡❧❛①❛t✐♦♥ t✐♠❡ s✉❣❣❡sts ❛ r❡❞✉❝t✐♦♥ ✐♥ t❤❡ ♦r❞❡r ♣❛r❛♠❡t❡r✱ ♦r ❧♦♦s❡♥✐♥❣ ♦❢ t❤❡ r✐❣✐❞ str✉❝t✉r❡ t❤❛t ✇❛s ♠❡❛s✉r❡❞ ✐♥ s♣✐❞❡r s✐❧❦✳ ❚❤✐s s✉❣❣❡sts t❤❛t t❤❡ ♠❡❝❤❛♥✐s♠s ♦r r❡q✉✐r❡♠❡♥ts ❢♦r t❤❡ ❢♦r♠❛t✐♦♥ ♦❢ β s❤❡❡ts ♠❛② ❜❡ ❞✐✛❡r❡♥t ❢♦r ❛❧❛♥✐♥❡ t❤❛♥ ❣❧②❝✐♥❡✳ ❆♥♦t❤❡r ✐♥t❡r❡st✐♥❣ ❝♦♠♣❛r✐s♦♥ ✐s t♦ 13 ❈ ◆▼❘ ❞❛t❛ ♦❢ ❧②♦♣❤✐❧✐③❡❞ ❞❡♥❛t✉r❡❞ ❛♥❞ ❣❧❛♥❞ s♣✐❞❡r s✐❧❦ ❢r♦♠ ❍✐❥✐r✐❞❛ ❡t ❛❧✳ ❬✷✵❪✱ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✼✳✻✳ ■♥ t❤❡✐r ✇♦r❦✱ t❤❡ s✐❧❦ ✇❛s ❞❡♥❛t✉r❡❞ ❜② ❢♦r♠✐❝ ❛❝✐❞ ❜❡❢♦r❡ ❧②♦♣❤✐❧✐③❛t✐♦♥✱ ✇❤✐❧❡ t❤❡ s✐❧❦ ❣❧❛♥❞s ✇❡r❡ ❡①tr❛❝t❡❞ ❢r♦♠ s♣✐❞❡rs✳ ❚❤❡② ❝♦♥❝❧✉❞❡❞ t❤❛t ✐♥ ❜♦t❤ st❛t❡s✱ t❤❡r❡ ✇❡r❡ ♥♦ β s❤❡❡ts ♣r❡s❡♥t✳ ❚❤✐s ✐s ♣❛rt✐❝✉❧❛r❧② ✐♥t❡r❡st✐♥❣ ❜❡❝❛✉s❡ t❤✐s s✉❣❣❡sts t❤❛t ♣❡r❤❛♣s ❛t ♦♥❡ ♣♦✐♥t ❞✉r✐♥❣ ♣r♦❞✉❝t✐♦♥ ♦❢ ▼❛❙♣✶ ❛ st❡♣ ✇❛s t❛❦❡♥ t❤❛t ❛❧❧♦✇❡❞ ❛❧❛♥✐♥❡ β s❤❡❡ts t♦ ❢♦r♠✱ ❛s ♦♣♣♦s❡❞ t♦ t❤❡✐r ❢♦r♠❛t✐♦♥ s✐♠♣❧② ❜❡✐♥❣ ❛ ♥❛t✉r❛❧❧② ♣r❡❢❡rr❡❞ st❛t❡✳  ✻✻  Intensity  Sp2 Alanine C-alpha Relaxation Data Single exponential fit Stretched exponential fit Double exponential fit  0  10  20  30  40  50  60  70  80  90  100  D2 (s)  ❋✐❣✉r❡ ✼✳✼✿ ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ▼❛❙♣✷ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳ ✼✳✷  ▼❛❙♣✷  ❆s ▼❛❙♣✷ ❡①✐sts ✐♥ ♦♥❧② s♠❛❧❧ ❛❜✉♥❞❛♥❝❡s ✐♥ s♣✐❞❡r s✐❧❦✱ ❜❡✐♥❣ ❛❜❧❡ t♦ ❛♥❛❧②③❡ ❛♥ ✐s♦❧❛t❡❞ s❛♠♣❧❡ ❝♦✉❧❞ ❤❡❧♣ ♣r♦✈✐❞❡ ✐♥❢♦r♠❛t✐♦♥ ✐♥t♦ ✇❤❛t r♦❧❡ ✐t ♣❧❛②s ✐♥ t❤❡ ❢♦r♠❛t✐♦♥ ♦❢ s♣✐❞❡r s✐❧❦✬s s❡❝♦♥❞❛r② str✉❝t✉r❡✳  ✼✳✷✳✶  ❆❧❛♥✐♥❡  ❚❤❡ ▼❛❙♣✷ ❛❧❛♥✐♥❡ α✲❝❛r❜♦♥ ✐s ♦❜s❡r✈❡❞ ❛t ✹✾✳✷ ♣♣♠ ✇✐t❤ ❛ ♣❡❛❦ ✇✐❞t❤ ♦❢ ✸ ♣♣♠✱ ✇❤✐❝❤ ❢❛❧❧s ✇✐t❤✐♥ t❤❡ r❛♥❣❡ ♦❢ ❜♦t❤ β ✲s❤❡❡ts ❛♥❞ t❤❡ 31 ❤❡❧✐①✳ ❚❤✐s ❝❤❡♠✐❝❛❧ s❤✐❢t ✐s ✇✐t❤✐♥ t❤❡ ♣r❡❞✐❝t❡❞ r❛♥❣❡ ❢♦r ❞r❛❣❧✐♥❡ s✐❧❦✳ ❚❤❡ r❡❧❛①❛t✐♦♥ s♣❡❝tr❛ ✇❡r❡ ✐♥t❡❣r❛t❡❞ ❢r♦♠ ✹✽ t♦ ✺✵✳✺ ♣♣♠✳ ❚❤❡ st❛t✐st✐❝❛❧❧② ❜❡st ✜t ✇❛s ❢♦✉♥❞ t♦ ❜❡ ❛ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧✱ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✼✳✼✱ ✇✐t❤ ❛ r❡❧❛①❛t✐♦♥ t✐♠❡ ♦❢  T1∗ = 15.3 ± 0.6 s ❛♥❞ β = 0.71 ± 0.02 ②✐❡❧❞✐♥❣ T1 = 19.2 ± 0.8 s✳ ■t ✐s ❛❧s♦ ✐♥t❡r❡st✐♥❣ t❤❛t t❤❡ ♠❡❛s✉r❡❞ r❡❧❛①❛t✐♦♥ t✐♠❡ ✐s ❝❧♦s❡ t♦ t❤❛t ❢♦✉♥❞ ✐♥ t❤❡ s✐❧❦✳ ❚❤✐s s✉❣❣❡sts ❜♦t❤ ❛ s✐♠✐❧❛r ♠❡❛♥ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ❛♥❞ ♦r❞❡r ♣❛r❛♠❡t❡r✳ ❚❤❡ s♠❛❧❧❡r β ♠❡❛♥s t❤❛t t❤❡r❡ ✐s ❛ ❧❛r❣❡r ❞✐str✐❜✉t✐♦♥ ♦❢ ✈✐❜r❛t✐♦♥❛❧ ♠♦❞❡s ♦r t❤❡② ❛r❡ s❤✐❢t❡❞ ❛✇❛② ❢r♦♠ t❤❡ ❢❛st✴s❧♦✇ ♠♦t✐♦♥ tr❛♥s✐t✐♦♥ ♣♦✐♥t✳ ❚❤❡ ❛❧❛♥✐♥❡ β ❝❛r❜♦♥ ✐♥ ▼❛❙♣✷ ✇❛s ❢♦✉♥❞ t♦ ❤❛✈❡ ❛ ❝❤❡♠✐❝❛❧ s❤✐❢t ♦❢ ✷✵✳✽ ♣♣♠ ❛♥❞ ❛ ❧❡❞❣❡ ❛t ❢r♦♠ ✶✻✳✺✲✶✺✳✺ ♣♣♠✳ ❚❤❡ ♣❡❛❦ ❝♦rr❡s♣♦♥❞s t♦ β s❤❡❡ts✱ ✇❤✐❧❡ t❤❡ ❧❡❞❣❡ ❝♦✈❡rs ❛ r❛♥❣❡ ♦❢ ✻✼  Intensity  Sp2 Alanine C-beta Relaxation Data Single exponential fit Stretched exponential fit Double exponential fit  0  1  2  3  4  5  6  7  8  9  10  D2 (s)  ❋✐❣✉r❡ ✼✳✽✿ ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ▼❛❙♣✷ ❛❧❛♥✐♥❡ β ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳ ❝♦♥❢♦r♠❛t✐♦♥s t❤❛t ✐♥❝❧✉❞❡ r❛♥❞♦♠ ❝♦✐❧✱ 31 ❤❡❧✐❝❡s✱ ❛♥❞ α ❤❡❧✐❝❡s✳ ❚❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ✭♠❛❥♦r✮ αβ ♣❡❛❦ s❡♣❛r❛t✐♦♥ ✇❛s ♠❡❛s✉r❡❞ t♦ ❜❡ ∆CSala = 28.4 ♣♣♠✱ ✇❤✐❝❤ ✐s ✐♥ t❤❡ r❛♥❣❡ ♦❢ β s❤❡❡ts  ❛♥❞ ❢❛r ❢r♦♠ ❛♥② ♦t❤❡r str✉❝t✉r❡s✳ ❚❤❡ β ❝❛r❜♦♥ ♣❡❛❦ ✇❛s ✐♥t❡❣r❛t❡❞ ♦✈❡r ❛ r❛♥❣❡ ♦❢ ✶✺✲✷✶ ♣♣♠ ❚❤❡ ❜❡st ✜t ✇❛s ❢♦✉♥❞ t♦ ❜❡ ❛ ❞♦✉❜❧❡ ❡①♣♦♥❡♥t✐❛❧✱ ❛s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✼✳✽✱ ❝♦♥s✐st✐♥❣ ♦❢ ❛ ❢❛st r❡❧❛①❛t✐♦♥ t✐♠❡ ♦❢  T1f ast = 0.26 ± 0.07 s t❤❛t ♠❛❞❡ ✉♣ 53 ± 12% ♦❢ t❤❡ s✐❣♥❛❧✱ ❛♥❞ ❛ s❧♦✇ r❡❧❛①❛t✐♦♥ t✐♠❡ ♦❢ T1slow = 1.7 ± .04 s✳ ❚❤❡s❡ ♣❛r❛♠❡t❡rs ❛r❡ ✇✐t❤✐♥ t❤❡ ❡①♣❡❝t❡❞ ✈❛❧✉❡s ♦❢ t❤❡ ❛❝❝❡♣t❡❞ ♠♦❞❡❧ ❢♦r t❤❡ t✇♦ ❝♦♠♣♦♥❡♥t ♥❛t✉r❡ ♦❢ s♣✐❞❡r s✐❧❦✬s ❝r②st❛❧❧✐♥❡ r❡❣✐♦♥✳  ✼✳✷✳✷  ●❧②❝✐♥❡  α  ❈❛r❜♦♥  ❚❤❡ ❣❧②❝✐♥❡ α ❝❛r❜♦♥ ✐♥ ▼❛❙♣✷ ✇❛s ❢♦✉♥❞ t♦ ❤❛✈❡ ❛ ❝❤❡♠✐❝❛❧ s❤✐❢t ♦❢ ✹✷✳✾ ♣♣♠ ✇✐t❤ ❛ ✇✐❞t❤ ♦❢ ✻ ♣♣♠✳ ❚❤❡ ❝❤❡♠✐❝❛❧ s❤✐❢t ✐s s❧✐❣❤t❧② ❜❡❧♦✇ t❤❡ ✈❛❧✉❡s ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ❢♦r s♣✐❞❡r s✐❧❦✳ ❚❤❡ r❡❧❛①❛t✐♦♥ s♣❡❝tr❛ ✇❡r❡ ✐♥t❡❣r❛t❡❞ ♦✈❡r t❤❡ ❣❧②❝✐♥❡ ♣❡❛❦ ✇✐t❤ ❛ r❛♥❣❡ ❢r♦♠ ✹✶✲✹✹ ♣♣♠✳ ❚❤❡ ❜❡st ✜t t♦ t❤❡ r❡❧❛①❛t✐♦♥ ❞❛t❛ ✇❛s ❢♦✉♥❞ t♦ ❜❡ ❛ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧✱ ❛s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✼✳✾✱ ✇✐t❤ ♣❛r❛♠❡t❡rs T1∗ = 18 ± 1 s ❛♥❞ β = 0.70 ± 0.05✳ ❚❤✐s ②✐❡❧❞❡❞ ❛ ♠❡❛♥ r❡❧❛①❛t✐♦♥ t✐♠❡ ♦❢ T1 = 23 ± 2 s✳ ❇♦t❤ t❤❡ β ❞✐str✐❜✉t✐♦♥ ❛♥❞ t❤❡ ♠❡❛♥ r❡❧❛①❛t✐♦♥ t✐♠❡ ❛r❡ ✇✐t❤✐♥ t❤❡ r❛♥❣❡ ♦❢ ✉♥❝❡rt❛✐♥t② ❢♦r t❤❛t ♦❢ ▼❛❙♣✶✳  ✻✽  Intensity  Sp2 Glycine C-alpha Relaxation Data Single exponential fit Stretched exponential fit Double exponential fit  0  5  10  15  20  25  30  35  40  45  50  D2 (s)  ❋✐❣✉r❡ ✼✳✾✿ ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ▼❛❙♣✷ ❣❧②❝✐♥❡ α ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳ ✼✳✷✳✸  ●❧✉t❛♠✐♥❡ ❛♥❞ ❙❡r✐♥❡  α  ❈❛r❜♦♥s  ■♥ ▼❛❙♣✷ ❛ ♣❡❛❦ ✐s ♦❜s❡r✈❡❞ ❛t ✺✹ ♣♣♠ t❤❛t ✐s ❜r♦❛❞ ❜✉t ✇❡❧❧ ❞❡✜♥❡❞✳ ■❞❡♥t✐❢②✐♥❣ t❤❡ r❡s✐❞✉❡s ✐♥ t❤✐s ♣❡❛❦ ✐s ♣r♦❜❧❡♠❛t✐❝ ❛s t❤❡r❡ ✐s s✐❣♥✐✜❝❛♥t ♦✈❡r❧❛♣ ❜❡t✇❡❡♥ t❤❡ s❡r✐♥❡ α ❝❛r❜♦♥ ♣❡❛❦✱ ✇❤✐❝❤ ❤❛s ✐ts β s❤❡❡t ❝❤❡♠✐❝❛❧ s❤✐❢t ❛t ✺✹✳✹✲✺✺✳✵ ♣♣♠ ❛♥❞ ❣❧✉t❛♠✐♥❡ α ❝❛r❜♦♥ ✇❤❡r❡ t❤❡ 31 ❤❡❧✐① str✉❝t✉r❡ ❛t ✺✹ ♣♣♠✳ ❆s s❡r✐♥❡ ❤❛s ❛ ♠✉❝❤ ❧❛r❣❡r ❛❜✉♥❞❛♥❝❡ ✐♥ ▼❛❙♣✷✱ ✐t ❝❛♥♥♦t ❜❡ ❞✐s❝♦✉♥t❡❞✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ✺✹ ♣♣♠ ♣❡❛❦ ❝♦rr❡s♣♦♥❞s t♦ ❜♦t❤ ❣❧✉t❛♠✐♥❡  α ❝❛r❜♦♥s ✐♥ 31 ❤❡❧✐❝❡s ❛♥❞ t❤❡ s❡r✐♥❡ α ✐♥ β s❤❡❡ts✳ ❚❤❡ r❡❧❛①❛t✐♦♥ ❞❛t❛ ❢r♦♠ t❤✐s ♣❡❛❦ r❡s✉❧t❡❞ ✐♥ ❛ t✇♦ ❝♦♠♣♦♥❡♥t ✜t ❛♥❞ ✐s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ❄❄✳  ❚❤❡ t✇♦ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ✜t ✇❡r❡ T1f ast = 12 ± 2 s ❛♥❞ T1slow = 65 ± 16 s ✇✐t❤ t❤❡  ❢❛st ❝♦♠♣♦♥❡♥t ♠❛❦✐♥❣ ✉♣ 65 ± 5% ♦❢ t❤❡ s✐❣♥❛❧✳ ❲❤✐❧❡ t❤❡ s❧♦✇ T1 ♠❛② s❡❡♠ s✉r♣r✐s✐♥❣❧② ❧♦♥❣✱ ♣❛rt✐❝✉❧❛r❧② ✇✐t❤ ✐t ♥♦t ❜❡✐♥❣ ♦❜s❡r✈❡❞ ✐♥ t❤❡ ✷✵✵ ▼❍③ ♠❛❣♥❡t ❬✸✸❪✱ ✐t ✐s ♣♦ss✐❜❧❡ t❤❛t ✐t✬s r❡❧❛①❛t✐♦♥ s♣❡❡❞ ✇❛s s♦ ❝❧♦s❡ t♦ t❤❛t ♦❢ t❤❡ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥ ❢r♦♠ t❤❡ ❧✐t❡r❛t✉r❡ t❤❛t ✐t ✇❛s ♥♦t ❛❝❝♦✉♥t❡❞ ❢♦r✳ ■t ✐s ♠♦st ❧✐❦❡❧② t❤❛t t❤❡ ❣❧✉t❛♠✐♥❡ α r❡♣r❡s❡♥ts ❛t ❧❡❛st t❤❡ s❧♦✇ r❡❧❛①❛t✐♦♥✱ ❛s t❤❡ s❧♦✇ r❡❧❛①❛t✐♦♥ r❡❣✐♠❡ ✐s s❡❡♥ ✐♥ ▼❛❙♣✶ ❛s ✐t ❝♦♥t❛✐♥s ✈❡r② ❧✐tt❧❡ s❡r✐♥❡✱ ❛s s❤♦✇♥ ✐♥ ❚❛❜❧❡ ✹✳✸✳ ■t ✐s ❛❧s♦ ❧✐❦❡❧② t❤❛t t❤❡ T1f ast ✐s ❛❧s♦ ✐♥ ♦r ♥❡❛r t❤❡ s❧♦✇ ♠♦t✐♦♥ ❧✐♠✐t✳ ❚❤❡ r❡❛s♦♥ ❢♦r t❤✐s ✐s t❤❛t ❢♦r ✐t t♦ ❜❡ ✐❣♥♦r❡❞✱ ✐t ✇♦✉❧❞ ❤❛✈❡ t♦ ❤❛✈❡ t♦ r❡❧❛① ❛t ❧❡❛st t❤r❡❡ ♦r ❢♦✉r t✐♠❡s q✉✐❝❦❡r t❤❛♥ t❤❡ s♣✐❞❡r s✐❧❦ ❢♦r ✐t t♦ ❜❡ r❡❛s♦♥❛❜❧② ♥❡❣❧❡❝t❡❞ ❢r♦♠ t❤❡ ✜t✳ ❇❡❝❛✉s❡ t❤❡ ✷✵✵ ▼❍③ T1 ✇❛s ✶✷ s❡❝♦♥❞s ❢♦r t❤❡ ❛❧❛♥✐♥❡✱ ❡①♣❡❝t✐♥❣ ✐t t♦ ❤❛✈❡ ❜❡❡♥ ✷ ♦r ✸✳✺ s❡❝♦♥❞s ✇♦✉❧❞ ❜❡ ❛ r❡❛s♦♥❛❜❧❡ ❣✉❡ss✳ ❚❤❡ ♠❛①✐♠✉♠ ✜❡❧❞ ❞❡♣❡♥❞❡♥❝❡ t❤❛t t❤❡ r❡❧❛①❛t✐♦♥ t✐♠❡ ❝♦✉❧❞ ❤❛✈❡ ✇♦✉❧❞ ❜❡ ❛ ❢❛❝t♦r ♦❢ ❢♦✉r ✐♥ t❤❡ s❧♦✇ ♠♦t✐♦♥ ❧✐♠✐t✳  ✻✾  Intensity  Sp2 Glutamine C-alpha Relaxation Data Single exponential fit Stretched exponential fit Double exponential fit  0  5  10  15  20  25  30  35  40  45  50  D2 (s)  ❋✐❣✉r❡ ✼✳✶✵✿ ❚❤❡ r❡❧❛①❛t✐♦♥ ❝✉r✈❡ ♦❢ t❤❡ ❣❧✉t❛♠✐♥❡ α ❝❛r❜♦♥s ✭❛♥❞ ♣♦ss✐❜❧② s❡r✐♥❡ α ❝❛r❜♦♥s✮ ✐♥ ▼❛❙♣✷✳ ✼✳✷✳✹  Pr♦❧✐♥❡  α  ❛♥❞ ❙❡r✐♥❡  β  ❈❛r❜♦♥s  ❚❤✐s ♣❡❛❦ ❛t ✻✵✳✽ ♣♣♠ s✐ts ♦✈❡r t❤❡ ❝❤❡♠✐❝❛❧ s❤✐❢t r❛♥❣❡ ❢♦r t❤❡ ♣r♦❧✐♥❡ α ❛♥❞ t❤❡ s❡r✐♥❡ β ❝❛r❜♦♥s✳ ❚❤❡r❡ ✐s ♥♦ ♣❡❛❦ ✐♥ ▼❛❙♣✶ ♦r t❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ✜❜❡rs ❛t t❤✐s ♣♦✐♥t ❛s ♥❡✐t❤❡r r❡s✐❞✉❡ ❤❛s s✉❜st❛♥t✐❛❧ ❛❜✉♥❞❛♥❝❡s ✐♥ t❤❡s❡ s❛♠♣❧❡s✳ ❚❤❡ ♣r♦❧✐♥❡ ✇♦✉❧❞ ❤❛✈❡ t♦ ❜❡ ✐♥ ❛♥ ✉♣✜❡❧❞ ❝♦♥✜❣✉r❛t✐♦♥ ❛s t❤❡ ✐♥t❡♥s✐t② ❞r♦♣s ♦✛ s✉❜st❛♥t✐❛❧❧② ✐♥ t❤❡ ❤✐❣❤❡r ❝❤❡♠✐❝❛❧ s❤✐❢ts t❤❛t t❤❡ ♣r♦❧✐♥❡ α ❝❛r❜♦♥ ❝❛♥ ❤❛✈❡ ❛❞♦♣t✳ ❚❤❡ ❝❧♦s❡st ❝❤❡♠✐❝❛❧ s❤✐❢t t♦ t❤✐s ✐s t❤❡ 31 ❤❡❧✐① ❛t ✻✷✳✸ ♣♣♠✱ ✇❤✐❝❤ ✐s q✉✐t❡ ❢❛r ❢r♦♠ t❤❡ ♣❡❛❦ ❝❡♥t❡r✳ ❚❤❡ s❡r✐♥❡ β ❝❛r❜♦♥ ♣❡❛❦ ❝❤❡♠✐❝❛❧ s❤✐❢t s❡❡♠s t♦ ❝♦♥tr❛❞✐❝t t❤❡ r❡s✉❧ts ❢r♦♠ t❤❡ s❡r✐♥❡ α ♣❡❛❦✳ ❚❤❡ t✇♦ ♠♦st ❧✐❦❡❧② ❝❛♥❞✐❞❛t❡ ♣❡❛❦s ❢♦r t❤❡ str✉❝t✉r❛❧ ❛ss✐❣♥♠❡♥t ❛r❡ t❤❡ α ❤❡❧✐❝❡s ♦r β t✉r♥s✳ ❍♦✇❡✈❡r✱ t❤❡s❡ str✉❝t✉r❡s ❢♦r t❤❡ s❡r✐♥❡ α ❝❛r❜♦♥ ❛r❡ ✐♥ ❧♦❝❛❧ ♠✐♥✐♠❛✳ ❇❡❝❛✉s❡ t❤❡ t✇♦ s✐t❡s✱ ♣r♦❧✐♥❡ α ❛♥❞ s❡r✐♥❡ β ❝❛r❜♦♥s✱ ❛♣♣❡❛r ✇✐t❤ s✐♠✐❧❛r ❛❜✉♥❞❛♥❝❡s✱ ❛♥❞ t❤❡ ♣❡❛❦ ✐s r❡❧❛t✐✈❡❧② ❢❡❛t✉r❡❧❡ss✱ ✐t ✐s ❞✐✣❝✉❧t t♦ ❝♦♥❝❧✉❞❡ ✇❤❛t ❝♦♥❢♦r♠❛t✐♦♥ t❤❡ t✇♦ ❛♠✐♥♦ ❛❝✐❞s ♠✐❣❤t ❛❞♦♣t✳ ❚❤♦✉❣❤ ✇❡ ❝❛♥ s❛② ✇✐t❤ s♦♠❡ ❝❡rt❛✐♥t② t❤❛t t❤❡ ♣r♦❧✐♥❡ ✇✐❧❧ ♥♦t ❛❞♦♣t t❤❡ α ❤❡❧✐①✱ ✇❤✐❝❤ ❤❛s ❛ ❝❤❡♠✐❝❛❧ s❤✐❢t ♦❢ ✻✻✳✸ ♣♣♠✳ ❚❤❡ ♣❡❛❦ ✇❛s ✐♥t❡❣r❛t❡❞ ♦✈❡r t❤❡ r❛♥❣❡ ♦❢ ✺✾✲✻✸ ♣♣♠✳ ❚❤❡ ❜❡st ✜t t♦ t❤❡ r❡❧❛①❛t✐♦♥ ❝✉r✈❡ ✇❛s ❛ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧✱ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✼✳✶✶✱ ✇✐t❤ r❡❧❛①❛t✐♦♥ ❝♦♠♣♦♥❡♥ts T1∗ = 16.93 ± 0.6 s ❛♥❞ β = 0.60 ± 0.02 ②✐❡❧❞✐♥❣ ❛ r❡❧❛①❛t✐♦♥ t✐♠❡ ♦❢ T1 = 25.3 ± 1.0 s✳ ❚❤✐s r❡s✉❧t s✉❣❣❡sts t❤❛t ❡✐t❤❡r t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ♣❡❛❦ ✐♥ t❤✐s r❡❣✐♦♥ ❛♥❞ t❤❛t ♦♥❡ ♦❢ t❤❡ ❝❤♦✐❝❡s ♠❛❞❡ ✐♥ t❤❡ ❝❤❡♠✐❝❛❧ s❤✐❢t ❛♥❛❧②s✐s ❛❜♦✈❡ ✐s ❡✐t❤❡r ♥♦t ♣r❡s❡♥t ♦r ❞✐❞ ♥♦t ❝r♦ss ♣♦❧❛r✐③❡ ✇❡❧❧✱ ♦r t❤❛t t❤❡  T1 s ♦❢ t❤❡ t✇♦ ❞✐✛❡r❡♥t ❝❛r❜♦♥s ❛r❡ s✐♠✐❧❛r ❡♥♦✉❣❤ t❤❛t t❤❡② ❝❛♥ ♥♦t ❜❡ ❞✐st✐♥❣✉✐s❤❡❞✳  ✼✵  Intensity  Sp2 Pro C-alpha Relaxation Data Single exponential fit Stretched exponential fit Double exponential fit  0  10  20  30  40  50  60  70  80  90  100  D2 (s)  ❋✐❣✉r❡ ✼✳✶✶✿ ❚❤❡ r❡❧❛①❛t✐♦♥ ❝✉r✈❡ ♦❢ t❤❡ ♣r♦❧✐♥❡ α ❝❛r❜♦♥s ✐♥ ▼❛❙♣✷✳ ✼✳✷✳✺  ●❧✉t❛♠✐♥❡  β ✴γ  ❚❤❡ ❣❧✉t❛♠✐♥❡ β ❛♥❞ γ ❝❛r❜♦♥ ♣❡❛❦ ✐♥ ▼❛❙♣✶ ✇❛s ❢♦✉♥❞ t♦ r❛♥❣❡ ❢r♦♠ ✷✾✲✸✷ ♣♣♠✳ ❉✉❡ t♦ t❤❡ ❢❛❝t t❤❛t ❛❧❧ ♦❢ t❤❡ ❣❧✉t❛♠✐♥❡ γ ❝♦♥❢♦r♠❛t✐♦♥s ❧✐❡ ✐♥ t❤✐s r❛♥❣❡✱ ✐t ✇❛s ✐♠♣♦ss✐❜❧❡ t♦ ❞❡❞✉❝❡ ❛♥②t❤✐♥❣ ❛❜♦✉t t❤❡ str✉❝t✉r❡ ❢r♦♠ t❤✐s✳ ❚❤❡ ❧❡❞❣❡ ♦❜s❡r✈❡❞ ❛t ✷✺ ♣♣♠ ✐♥ t❤❡ ♦t❤❡r s♣❡❝tr❛ ✐s ❛ ✇❡❧❧ ❞❡✜♥❡❞ ♣❡❛❦ ✐♥ ▼❛❙♣✷✳ ❚❤✐s ✐s ✉♥❞❡rst❛♥❞❛❜❧❡ ❛s ▼❛❙♣✷ ✇❛s t❤❡ ♦♥❧② s❛♠♣❧❡ t♦ ❤❛✈❡ ❛ ✇❡❧❧ ❞❡✜♥❡❞ ❣❧✉t❛♠✐♥❡ α ♣❡❛❦✳ ❚❤❡ ♠♦st ❧✐❦❡❧② str✉❝t✉r❡ ❢♦r t❤✐s ✐s ❛♥ α ❤❡❧✐① ✇❤✐❝❤ ❤❛s ❛ ❝❤❡♠✐❝❛❧ s❤✐❢t ♦❢ ✷✺✳✻ ♣♣♠✱ ❛❧t❤♦✉❣❤ t❤❡ 31 ✲❤❡❧✐① s❤♦✉❧❞ ♥♦t ❜❡ ❜❡ r✉❧❡❞ ♦✉t ❛s ✐t ✐t ✇♦✉❧❞ ❜❡ ♥❡❛r ✷✻✳✽ ♣♣♠✳ ❚❤❡ ♣❡❛❦ ✇❛s ✐♥t❡❣r❛t❡❞ ♦✈❡r ❛ r❛♥❣❡ ❢r♦♠ ✷✽✲✸✷ ♣♣♠✳ ❚❤❡ ❜❡st ✜t ✇❛s ❢♦✉♥❞ t♦ ❜❡ ❛ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧✱ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✼✳✶✷✱ ✇✐t❤ ♣❛r❛♠❡t❡rs ♦❢ T1∗ = 1.7 ± 0.2 s✱ ✇❤✐❝❤ ♠❛❞❡ ✉♣ 44 ± 4% ♦❢ t❤❡ s✐❣♥❛❧✱ ❛♥❞ ❛ s❧♦✇ ❝♦♠♣♦♥❡♥t ♦❢ T1slow = 23 ± 2 s✳ ❚❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢ t❤❡s❡ ✈❛❧✉❡s ✐s t❤❛t t❤❡② ❛r❡ ❝❧♦s❡r t♦ s♣✐❞❡r s✐❧❦ t❤❛♥ ▼❛❙♣✶✱ ✇❤✐❝❤ ✐s ✉♥❡①♣❡❝t❡❞ ❛s ▼❛❙♣✶ ✐s t❤❡ ❞♦♠✐♥❛♥t ♣r♦t❡✐♥ ✐♥ s✐❧❦✳ ❆s t❤❡ ❣❧✉t❛♠✐♥❡ ✐s ♣r✐♠❛r✐❧② ✐♥ t❤❡ ❛♠♦r♣❤♦✉s r❡❣✐♦♥s ♦❢ s♣✐❞❡r s✐❧❦✱ t❤✐s ❝♦✉❧❞ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ▼❛❙♣✷ ♣❧❛②✐♥❣ ❛ ✈✐t❛❧ r♦❧❡ ✐♥ t❤❛t r❡❣✐♦♥ ❛s ✇❡❧❧ ❛s t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ t❤❡ ❝r②st❛❧❧✐♥❡ r❡❣✐♦♥ ❛s t❤❡ ❛❝❝❡♣t❡❞ ♠♦❞❡❧ ♣♦st✉❧❛t❡s✳ ❚❤❛t s❛✐❞✱ t❤❡ ❝❤❡♠✐❝❛❧ s❤✐❢t ❢♦r t❤✐s ♣❡❛❦ ✐♥ s♣✐❞❡r s✐❧❦ ✐s ❝❧♦s❡r ✐♥ ❧✐♥❡ ✇✐t❤ ▼❛❙♣✶ t❤❛♥ ▼❛❙♣✷✳  ✼✳✷✳✻  ▼❛❙♣✷ ❘❡s✉❧ts  ▼❛❙♣✷ ❤❛s t✇♦ ❞❡✜♥✐♥❣ ❝❤❛r❛❝t❡r✐st✐❝s✳ ❚❤❡ ✜rst ✐s t❤❛t ✐ts ❛❧❛♥✐♥❡ ❛♥❞ ❣❧②❝✐♥❡ ❜❡❤❛✈❡ ❛❧♠♦st ❡①❛❝t❧② ❧✐❦❡ ▼❛❙♣✶✳ ❚❤❡ r❡❧❛①❛t✐♦♥ ♣❛r❛♠❡t❡rs ❜❡t✇❡❡♥ t❤❡ t✇♦ ♣r♦t❡✐♥s ❛r❡ ✐♥❞✐s✲ ✼✶  Intensity  Sp2 Glutamine C-beta/gamma Relaxation Data Single exponential fit Stretched exponential fit Double exponential fit  0  5  10  15  20  25  30  35  40  45  50  D2 (s)  ❋✐❣✉r❡ ✼✳✶✷✿ ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ▼❛❙♣✷ ❣❧✉t❛♠✐♥❡ β ✴γ ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳ t✐♥❣✉✐s❤❛❜❧❡ ❛♥❞ t❤❡ ❝❤❡♠✐❝❛❧ s❤✐❢t r❡s✉❧ts s❤♦✇ ❛ str♦♥❣ ♣r❡s❡♥❝❡ ♦❢ ❛❧❛♥✐♥❡ ♦❢ β s❤❡❡ts✳ ▲✐❦❡ t❤❡ ▼❛❙♣✶✱ t❤❡ ❞r♦♣ ✐♥ r❡❧❛①❛t✐♦♥ t✐♠❡ ❢♦r ❣❧②❝✐♥❡ s✉❣❣❡sts ❛ r❡❞✉❝t✐♦♥ ✐♥ t❤❡ ♦r❞❡r ♣❛r❛♠❡t❡r✱ ♦r ❧♦♦s❡♥✐♥❣ ♦❢ t❤❡ r✐❣✐❞ str✉❝t✉r❡ t❤❛t ✇❛s ♠❡❛s✉r❡❞ ✐♥ s♣✐❞❡r s✐❧❦✳ ❚❤❡ s✐♠✐❧❛r✲ ✐t✐❡s ✐♥ t❤❡ ♣r✐♠❛r② str✉❝t✉r❡ ♦❢ t❤❡s❡ t✇♦ ♣r♦t❡✐♥s s✉❣❣❡st t❤❛t t❤❡ ❡①❛❝t ♣r✐♠❛r② str✉❝t✉r❡ ♦❢ t❤❡ ♣r♦t❡✐♥ ♣❧❛②s ❛ ❧❡ss s✐❣♥✐✜❝❛♥t r♦❧❡ ✐♥ t❤❡ ❢♦r♠❛t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞❛r② str✉❝t✉r❡ ❛♥❞ r❡s✉❧t✐♥❣ ♠❡❝❤❛♥✐❝❛❧ ♣r♦♣❡rt✐❡s t❤❛♥ t❤❡ s♣✐♥♥✐♥❣ ♣r♦❝❡❞✉r❡ ✐ts❡❧❢✳ ■t ✐s ♣♦ss✐❜❧❡ t❤❛t t❤❡ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ t❤❡ t✇♦ ❧✐❡ ✐♥ t❤❡ ❛♠♦r♣❤♦✉s r❡❣✐♦♥✱ ✇❤✐❝❤ ❛r❡ ♠♦r❡ ❞✐✣❝✉❧t t♦ ♣r♦❜❡ ✉s✐♥❣ t❤❡s❡ t✇♦ ♦♥❡ ❞✐♠❡♥s✐♦♥❛❧ ◆▼❘ t❡❝❤♥✐q✉❡s✳ ❚❤❡ s❡❝♦♥❞ ❝❤❛r❛❝t❡r✐st✐❝ ✐s t❤❡ t✇♦ ♣❡❛❦s ✈✐s✐❜❧❡ ✐♥ t❤❡ ❣❧✉t❛♠✐♥❡ r❛♥❣❡✳ ❖♥❡ ✇♦✉❧❞ ❜❡ t❡♠♣t❡❞ t♦ ❛ss✐❣♥ t❤❡ ❞✐✛❡r❡♥❝❡ ✐♥ ❧✐♥❡✲s❤❛♣❡ ❝♦♠♣❛r❡❞ t♦ ▼❛❙♣✶ t♦ t❤❡ ❞✐✛❡r❡♥t ❝♦♠♣♦✲ s✐t✐♦♥s ♦❢ t❤❡ t✇♦ ♣r♦t❡✐♥s✳ ❍♦✇❡✈❡r ✇❤❡♥ ▼❛❙♣✷ ✐s ❝♦♠♣❛r❡❞ t♦ ❧②♦♣❤✐❧✐③❡❞ ❣❧❛♥❞ s✐❧❦✱ ✇❤✐❝❤ ❛❧t❤♦✉❣❤ ♣r✐♠❛r✐❧② ❙♣✶✱ ❛❧s♦ ❤❛s ❛ s✐♠✐❧❛r ❧✐♥❡✲s❤❛♣❡ ❢♦r t❤❡ r❡❣✐♦♥ ❢r♦♠ ✷✺✲✹✵ ♣♣♠✱ ❛s s❡❡♥ ✐♥ ❋✐❣✉r❡ ✼✳✻✳ ❚❤✐s s✉❣❣❡sts t❤❛t s♦♠❡ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡s ✐♥ t❤❡ ❧✐♥❡✲s❤❛♣❡ ❜❡t✇❡❡♥ t❤❡ ▼❛❙♣✶ ❛♥❞ ▼❛❙♣✷ ❛r❡ ♥♦t ❛ r❡s✉❧t ♦❢ t❤❡✐r ❞✐✛❡r❡♥t ❝♦♠♣♦s✐t✐♦♥s ❜✉t ♣♦ss✐❜❧② ❛ r❡s✉❧t ♦❢ ❞✐✛❡r❡♥t ♣r♦❝❡ss✐♥❣ ❝♦♥❞✐t✐♦♥s✳  ✼✳✸  ❊❧❡❝tr♦s♣✉♥ ▼❛❙♣✶  ❆♥❛❧②③✐♥❣ t❤❡ str✉❝t✉r❡ ♦❢ t❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ✐s ♣❛rt✐❝✉❧❛r❧② ✐♠♣♦rt❛♥t✱ ❛s ✐t ❝♦✉❧❞ ❤❡❧♣ t♦ ❡①♣❧❛✐♥ ✐ts ♣♦♦r ♠❡❝❤❛♥✐❝❛❧ ♣r♦♣❡rt✐❡s ❛♥❞ r❡s♦❧✈❡ t❤❡ ✉♥❝❡rt❛✐♥t② s✉rr♦✉♥❞✐♥❣ t❤❡ ❋❚■❘ r❡s✉❧ts✳ ✼✷  Intensity  Electrospun Sp1 Alanine C-alpha Relaxation Data Single exponential fit Stretched exponential fit Double exponential fit  0  5  10  15  20  25  30  35  40  45  50  D2 (s)  ❋✐❣✉r❡ ✼✳✶✸✿ ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳ ✼✳✸✳✶  ❆❧❛♥✐♥❡  ❚❤❡ ❝❤❡♠✐❝❛❧ s❤✐❢t ❢♦r t❤❡ ❛❧❛♥✐♥❡ α ❝❛r❜♦♥ ♦❢ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ✇❛s ❢♦✉♥❞ t♦ ❜❡ ✺✷✳✸ ♣♣♠✱ ❢❛r ♦✉ts✐❞❡ ♦❢ t❤❡ ❡①♣❡❝t❡❞ ✈❛❧✉❡s ❢♦r β s❤❡❡ts✳ ❚❤❡ ♠♦st ❧✐❦❡❧② ❝❛♥❞✐❞❛t❡ ❢♦r t❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶✬s ❛❧❛♥✐♥❡ α ❝❛r❜♦♥ ✐s t❤❡ α ❤❡❧✐①✱ ✇❤✐❝❤ r❛♥❣❡s ❢r♦♠ ✺✷✳✸✲✺✷✳✽ ♣♣♠✳ ◆♦ ♦t❤❡r ♠❛❥♦r str✉❝t✉r❡ ❢♦r ❛❧❛♥✐♥❡ α ❝❛r❜♦♥s ❡①t❡♥❞s ✐♥t♦ t❤✐s r❡❣✐♦♥✳ ❚❤✐s ✐s ❞✐✈❡r❣❡♥t ❢r♦♠ t❤❡ r❡s✉❧ts ♦❢ ●❛♥❞❤✐✱ ✇❤♦ ♦❜s❡r✈❡❞ ❛ str♦♥❣ β s❤❡❡t ❝♦♠♣♦♥❡♥t ✐♥ t❤❡ ❡❧❡❝tr♦s♣✉♥ s✐❧❦✳ ❚❤❡ r❡❧❛①❛t✐♦♥ s♣❡❝tr❛ ✇❡r❡ ✐♥t❡❣r❛t❡❞ ❛❝r♦ss t❤❡ ♣❡❛❦✳ ❉✉❡ t♦ t❤❡ ❞♦✇♥✜❡❧❞ s❤✐❢t ♦❢ t❤❡ ♣❡❛❦✱ t❤❡r❡ ✇❛s ♥♦ ✇❛② t♦ ❛✈♦✐❞ s♦♠❡ ❝♦♥t❛♠✐♥❛t✐♦♥ ❢r♦♠ t❤❡ ❣❧✉t❛♠✐♥❡ α ♣❡❛❦✳ ❚❤❡ ❜❡st ✜t t♦ t❤❡ ❞❛t❛ ✇❛s t❤❡ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧✱ ❛s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✼✳✸✳✶✱ ✇✐t❤ ♣❛r❛♠❡t❡rs  T1∗ = 21 ± 4 s ❛♥❞ β = 0.7 ± 0.1 ②✐❡❧❞✐♥❣ ❛ ♠❡❛♥ r❡❧❛①❛t✐♦♥ t✐♠❡ ♦❢ T1 = 27 ± 7 s✳ ❲❤✐❧❡ t❤❡ ❧❛r❣❡ ✉♥❝❡rt❛✐♥t② ♠❛❦❡s ❛♥② ❞❡t❛✐❧❡❞ ❛♥❛❧②s✐s ♦❢ t❤✐s ✈❛❧✉❡ ❞✐✣❝✉❧t✱ t❤✐s ♣❡❛❦ ❞♦❡s ❛♣♣❡❛r t♦ ❞❡❝❛② s❧♦✇❡r t❤❛♥ t❤♦s❡ ♦❢ s♣✐❞❡r s✐❧❦ ❛♥❞ t❤❡ ▼❛❙♣✶ ❛♥❞ ▼❛❙♣✷ ♣♦✇❞❡r s♣❡❝tr❛ s❤♦✇♥ ✐♥ ❋✐❣✉r❡s ✻✳✶✱ ✼✳✶✱ ❛♥❞ ✼✳✼✳ ❲❤✐❧❡ t❤✐s ❝♦✉❧❞ ❜❡ ❝❛✉s❡❞ ❜② ❛ ❧❛r❣❡r✱ ♠♦r❡ r❡str✐❝t✐✈❡✱ ♦r❞❡r ♣❛r❛♠❡t❡r✱ t❤✐s ✐s ♠♦r❡ ❧✐❦❡❧② t❤❡ r❡s✉❧t ♦❢ ❛ ♠✉❝❤ s❧♦✇❡r ❝♦rr❡❧❛t✐♦♥ t✐♠❡ t❤❛t ✇♦✉❧❞ r❡s✉❧t ❢r♦♠ t❤❡ s❧♦✇❡r ♠♦t✐♦♥ t❤❛t ✇♦✉❧❞ r❡s✉❧t ❢r♦♠ ❛ ❧❡ss ♦r✐❡♥t❡❞✱ ♠♦r❡ ❛♠♦r♣❤♦✉s str✉❝t✉r❡✳ ❚❤❡ ❛❧❛♥✐♥❡ β ❝❛r❜♦♥ ✐♥ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ✇❛s ❢♦✉♥❞ t♦ ❤❛✈❡ ❛ ❝❤❡♠✐❝❛❧ s❤✐❢t ♦❢ ✶✻✳✶ ♣♣♠✳ ❚❤✐s ♣❡❛❦ ✇❛s ❡①tr❡♠❡❧② ❜r♦❛❞✱ ♦♥❧② ❞r♦♣♣✐♥❣ s❧✐❣❤t❧② ❜② ✷✹ ♣♣♠✳ ❚❤✐s ✐s ♦❢ ♣❛rt✐❝✉❧❛r ✐♥t❡r❡st ❜❡❝❛✉s❡ t❤❛t α ❤❡❧✐❝❡s ❛♣♣❡❛r t♦ ❜❡ ❛ ❞♦♠✐♥❛♥t str✉❝t✉r❡✱ ❛s t❤❡② ❛r❡ t❤❡ ♦♥❧② ❝♦♥❢♦r♠❛t✐♦♥ t❤❛t ❤❛s ❛ ❝❤❡♠✐❝❛❧ s❤✐❢t ❜❡❧♦✇ ✶✻✳✺ ♣♣♠✳ ❚❤❡ ♣❡❛❦ s❡♣❛r❛t✐♦♥ ✇❛s ❢♦✉♥❞ t♦ αβ ❜❡ ∆CSala = 35.5 − 36.5 ♣♣♠✱ ✇❤✐❝❤ ✐s t❤❡ r❛♥❣❡ ♦❢ ❜❡✐♥❣ ❡✐t❤❡r α ❤❡❧✐❝❡s ♦❢ β t✉r♥s✱ ❜✉t  ✼✸  Intensity  Electrospun Sp1 Alanine C-beta Relaxation Data Single exponential fit Stretched exponential fit Double exponential fit  0  2  4  6  8  10  12  14  16  18  20  D2 (s)  ❋✐❣✉r❡ ✼✳✶✹✿ ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ❛❧❛♥✐♥❡ β ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳ ❢❛r ❢r♦♠ t❤❡ ❡st❛❜❧✐s❤❡❞ ✈❛❧✉❡ ❢♦r β s❤❡❡ts✳ ❚❤❡r❡ ✐s ❛❧s♦ ❧✐❦❡❧② ❛ ❧❛r❣❡ ❝♦♥tr✐❜✉t✐♦♥ ❢r♦♠  β t✉r♥s ❛♥❞ r❛♥❞♦♠ ❝♦✐❧s ❛s t❤❡ ♣❡❛❦ ✭❜❡❢♦r❡ t❤❡ ♣❧❛t❡❛✉✮ ❞♦❡s ❡①t❡♥❞ ✐♥t♦ t❤❡✐r ❝❤❡♠✐❝❛❧ s❤✐❢t r❛♥❣❡s ❛s ✐♥ ✜❣✉r❡ ✺✳✷✳ ❚❤❡ r❡♠❛r❦❛❜❧② s♠❛❧❧❡r ❝❤❡♠✐❝❛❧ s❤✐❢t ❛❞❞s ♠♦r❡ ✇❡✐❣❤t t♦ t❤❡ ❝❧❛✐♠ t❤❛t t❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ❛❧❛♥✐♥❡s ❤❛✈❡ ❛ ❞♦♠✐♥❛♥t α ❤❡❧✐① ❝♦♠♣♦♥❡♥t✱ ❛s ✇❛s ❛❧s♦ ♦❜s❡r✈❡❞ ✐♥ t❤❡ α ❝❛r❜♦♥✳ ❚❤✐s ✐s ❛ ❞❡♣❛rt✉r❡ ❢r♦♠ t❤❡ str✉❝t✉r❡s ♦❢ ❜♦t❤ s✐❧❦ ❛♥❞ t❤❡ ♣♦✇❞❡r❡❞ ▼❛❙♣✶✳ ■t ✐s ❛❧s♦ ✐♥❝♦♥s✐st❡♥t ✇✐t❤ t❤❡ r❡s✉❧ts ♦❢ t❤❡ ❋❚✲■❘ ❡①♣❡r✐♠❡♥ts ✇❤❡r❡ ✐t ✇❛s s❤♦✇♥ t❤❛t t❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ❤❛❞ ❛ str♦♥❣ β s❤❡❡t ❝♦♠♣♦♥❡♥t✱ ♠❛❦✐♥❣ ✉♣ ✹✷% ♦❢ t❤❡ str✉❝t✉r❡✳ ■t ✐s ❤♦✇❡✈❡r ❝♦♥s✐st❡♥t t♦ t❤❡ r❡s✉❧ts ❢r♦♠ ❜♦t❤ ❧②♦♣❤✐❧✐③❡❞ ❞❡♥❛t✉r❡❞ ❛♥❞ ❧②♦♣❤✐❧✐③❡❞ ❣❧❛♥❞ s♣✐❞❡r s✐❧❦ t❤❛t ❛r❡ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✼✳✻ ❬✷✵❪✳ ❚❤❡ ♣❡❛❦ ✇❛s ✐♥t❡❣r❛t❡❞ ♦✈❡r ❛ r❛♥❣❡ ♦❢ ✶✺✲✷✶ ♣♣♠✳ ❚❤❡ ❜❡st ✜t ❢♦r t❤✐s ✇❛s ❛ s✐♥❣❧❡ ❡①♣♦♥❡♥t✐❛❧✱ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✼✳✶✹✱ ❛♥❞ ✐s ✐♥❞✐st✐♥❣✉✐s❤❛❜❧❡ ❢r♦♠ t❤❡ ✜ts ❣❡♥❡r❛t❡❞ ❜② t❤❡ str❡t❝❤❡❞ ❛♥❞ ❞♦✉❜❧❡ ❡①♣♦♥❡♥t✐❛❧s✳ ❚❤❡ r❡❧❛①❛t✐♦♥ t✐♠❡ ✇❛s ♠❡❛s✉r❡❞ t♦ ❜❡ T1 = 0.77 s✳ ■t ✐s ✐♥t❡r❡st✐♥❣ t❤❛t t❤✐s ✐s ❜❡t✇❡❡♥ t❤❡ t✇♦ ✈❛❧✉❡s ❢r♦♠ s♣✐❞❡r s✐❧❦✱ ▼❛❙♣✶✱ ❛♥❞ ▼❛❙♣✷✳ ❚❤✐s ✐s ❛♥♦t❤❡r ❞❡♣❛rt✉r❡ ❢r♦♠ ❜♦t❤ t❤❡ ♣♦✇❞❡r❡❞ ♣r♦t❡✐♥s ❛♥❞ s♣✐❞❡r s✐❧❦✱ ❛❧❧ ♦❢ ✇❤✐❝❤ ✜t t♦ ❞♦✉❜❧❡ ❡①♣♦♥❡♥t✐❛❧s✳ ❚❤✐s s✉❣❣❡sts t❤❛t t❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ❞♦❡s ♥♦t ❝♦♥t❛✐♥ t✇♦ ❞✐st✐♥❣✉✐s❤❛❜❧❡ st❛t❡s ♦❜s❡r✈❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✳ ❖❢ ❝♦✉rs❡ t❤✐s ✐s ♥♦t s♦ s✉r♣r✐s✐♥❣✱ ❛s t❤❡s❡ t✇♦ st❛t❡s s✉♣♣♦s❡❞❧② ❡①✐st ❛s t✇♦ ❢♦r♠s ♦❢ β str❛♥❞s✱ ✇❤✐❝❤ ❞♦ ♥♦t ❛♣♣❡❛r t♦ ❜❡ ❛ ❞♦♠✐♥❛♥t ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶✳  ✼✹  Intensity  Electrospun Sp1 Glycine C-alpha Relaxation Data Single exponential fit Stretched exponential fit Double exponential fit  0  5  10  15  20  25  30  35  40  45  50  D2 (s)  ❋✐❣✉r❡ ✼✳✶✺✿ ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ❣❧②❝✐♥❡ α ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳ ✼✳✸✳✷  ●❧②❝✐♥❡  ❚❤❡ ❣❧②❝✐♥❡ α ❝❛r❜♦♥ ✐♥ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ✇❛s ❢♦✉♥❞ t♦ ❤❛✈❡ ❛ ❝❤❡♠✐❝❛❧ s❤✐❢t ♦❢ ✹✷✳✹ ♣♣♠ ✇✐t❤ ❛ ♣❡❛❦ ✇✐❞t❤ ♦❢ ✾✳✼ ♣♣♠✳ ❚❤❡ ❝❤❡♠✐❝❛❧ s❤✐❢t ✐s s✉❜st❛♥t✐❛❧❧② ❜❡❧♦✇ t❤❡ ✈❛❧✉❡s r❡♣♦rt❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ❛s ✇❡❧❧ ❛s ❝♦♠♣❛r❡❞ t♦ t❤❡ ♦t❤❡r s❛♠♣❧❡s ❞✐s❝✉ss❡❞ ❛❜♦✈❡ ❛♥❞ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ 31 ❤❡❧✐①✳ ❍♦✇❡✈❡r✱ ❛s t❤❡ ♣❡❛❦ ✐s ✐♥❝r❡❞✐❜❧② ❜r♦❛❞✱ t❤✐s tr❛♥s❧❛t❡s ✐♥t♦ ❧✐tt❧❡ s♣❡❝✐✜❝ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ t②♣❡s ♦❢ str✉❝t✉r❡✱ ♦t❤❡r t❤❛♥ t❤❛t t❤❡r❡ ❛r❡ ❧♦ts ♦❢ t❤❡♠✳ ❚❤❡ ♣❡❛❦ ✇❛s ✐♥t❡❣r❛t❡❞ ♦✈❡r ❛ r❛♥❣❡ ♦❢ ✹✵✲✹✹ ♣♣♠✳ ❚❤❡ r❡s✉❧t ✇❛s ❛ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧✱ ❛s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✼✳✶✺✱ ✇✐t❤ ♣❛r❛♠❡t❡rs ♦❢ T1∗ = 15.6 ± 2.3 s ❛♥❞ β = 0.63 ± 0.09✱ ②✐❡❧❞✐♥❣ ❛ ♠❡❛♥ r❡❧❛①❛t✐♦♥ t✐♠❡ ♦❢ T1 = 21.9 ± 5 s✳ ▲✐❦❡ ❜♦t❤ t❤❡ ▼❛❙♣✶ ❛♥❞ ▼❛❙♣✷ ♣♦✇❞❡rs✱ t❤✐s ❢❛❧❧s ❜❡❧♦✇ ❜♦t❤ t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡ ❛♥❞ ❤♦t r❡❧❛①❛t✐♦♥ t✐♠❡ ❢♦r s♣✐❞❡r s✐❧❦✳  ✼✳✸✳✸  ●❧✉t❛♠✐♥❡  β/γ  ❈❛r❜♦♥s  ❚❤❡ ❣❧✉t❛♠✐♥❡ β ❛♥❞ γ ❝❛r❜♦♥ ♣❡❛❦ ✐♥ ❊❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ❝❤❡♠✐❝❛❧ s❤✐❢t ✇❛s ❢♦✉♥❞ t♦ r❛♥❣❡ ❢r♦♠ ✷✼✲✷✾ ♣♣♠✳ ❉✉❡ t♦ t❤❡ ❢❛❝t t❤❛t ❛❧❧ ♦❢ t❤❡ ❣❧✉t❛♠✐♥❡ γ ❝♦♥❢♦r♠❛t✐♦♥s ❧✐❡ ✐♥ t❤✐s r❛♥❣❡✱ ✐t ✇❛s ✐♠♣♦ss✐❜❧❡ t♦ ❞❡❞✉❝❡ ❛♥②t❤✐♥❣ ❛❜♦✉t t❤❡ str✉❝t✉r❡ ❢r♦♠ t❤✐s✳ ❚❤❡r❡ ✐s ❛ ♣❡❛❦ ♦❜s❡r✈❡❞ ❛t ✷✹ ♣♣♠✱ ✇❤✐❝❤ ✐s ♦✉ts✐❞❡ ♦❢ t❤❡ r❛♥❣❡ ❢♦r ❜♦t❤ t❤❡ ❣❧✉t❛♠✐♥❡ ❛♥❞ ❛❧❛♥✐♥❡ β ❝❛r❜♦♥s✳ ❲❤❛t ❝❛r❜♦♥ t❤✐s ♣❡❛❦ r❡♣r❡s❡♥ts ✐s ❛ ♠②st❡r②✳ ❚❤❡ ❣❧②❝✐♥❡ β/γ ♣❡❛❦ ✇❛s ✐♥t❡❣r❛t❡❞ ♦✈❡r t❤❡ r❛♥❣❡ ❢r♦♠ ✷✼✲✸✷ ♣♣♠✳ ❚❤❡ ❜❡st ✜t ✇❛s ✼✺  Intensity  Electrospun Sp1 Glutamine C-beta/gamma Relaxation Data Single exponential fit Stretched exponential fit Double exponential fit  0  5  10  15  20  25  30  35  40  45  50  D2 (s)  ❋✐❣✉r❡ ✼✳✶✻✿ ❙✐♥❣❧❡✱ ❞♦✉❜❧❡✱ ❛♥❞ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ✜ts t♦ t❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ❣❧✉✲ t❛♠✐♥❡ β ✴γ ❝❛r❜♦♥ r❡❧❛①❛t✐♦♥ ❞❛t❛✳ ♦❜s❡r✈❡❞ t♦ ❜❡ ❛ ❞♦✉❜❧❡ ❡①♣♦♥❡♥t✐❛❧✱ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✼✳✶✻ ❝♦♥s✐st✐♥❣ ♦❢ ❛ ❢❛st r❡❧❛①❛t✐♦♥ t✐♠❡ ♦❢ T1f ast = 3.4 ± 1.1 s ✇❤✐❝❤ ♠❛❞❡ ✉♣ 58 ± 12% ♦❢ t❤❡ s✐❣♥❛❧ ❛♥❞ ❛ s❧♦✇ ❝♦♠♣♦♥❡♥t ♦❢  T1slow = 36 ± 12 s✳ ❉✉❡ t♦ t❤❡ ❧❛r❣❡ ✉♥❝❡rt❛✐♥t② ✐♥ t❤✐s ✜t ✐t ✐s ❞✐✣❝✉❧t t♦ ♠❛❦❡ ❛♥② str✉❝t✉r❛❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s ❢r♦♠ t❤❡ r❡s✉❧ts✳  ✼✳✸✳✹  ❊❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ❈♦♥❝❧✉s✐♦♥s  ❚❤❡ ♠♦st ❞❡✜♥✐♥❣ ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ t❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ✇❛s t❤❡ ❜r♦❛❞♥❡ss ♦❢ ✐ts ♣❡❛❦s✳ ❚❤✐s s✉❣❣❡sts ❛ ❞❡♥❛t✉r❡❞ str✉❝t✉r❡ t❤❛t ♦❝❝✉♣✐❡s ❛ ✇✐❞❡ ✈❛r✐❡t② ♦❢ t♦rs✐♦♥ ❛♥❣❧❡s✳ ❲❤✐❧❡ ♠✉❝❤ ♦❢ t❤❡ r❡❧❛①❛t✐♦♥ ❞❛t❛ ✇❛s ✉♥✐❧❧✉♠✐♥❛t✐♥❣ ❛s t❤❡ ✭r❛t❤❡r ❧❛r❣❡✮ ✉♥❝❡rt❛✐♥t② ✇❛s ❣r❡❛t❡r t❤❛♥ t❤❡ ❞✐s❝r❡♣❛♥❝✐❡s ✇✐t❤ ❜♦t❤ t❤❡ ❞r❛❣❧✐♥❡ s✐❧❦ ❛♥❞ t❤❡ ♣r♦t❡✐♥s ♣♦✇❞❡rs✱ t❤❡ s✐♥❣❧❡ ❝♦♠♣♦♥❡♥t ✜t ❢♦r t❤❡ ❛❧❛♥✐♥❡ β ❝❛r❜♦♥ ❞✐❞ r❡✈❡❛❧ t❤❛t t❤❡ t✇♦ ❝♦♠♣♦♥❡♥t ♥❛t✉r❡ ♦❢ t❤❡  β s❤❡❡ts ♦❜s❡r✈❡❞ ✐♥ ❜♦t❤ ❞r❛❣❧✐♥❡ s✐❧❦ ✇❛s ♥♦t ♣r❡s❡♥t ✐♥ t❤❡ ❡❧❡❝tr♦s♣✉♥ ✜❜❡rs✳ ❚❤✐s ✐♥ αβ ❝♦♠❜✐♥❛t✐♦♥ ✇✐t❤ t❤❡ ❧❛r❣❡ ✐♥❝r❡❛s❡ ✐♥ t❤❡ ✈❛❧✉❡ ♦❢ ∆CSala str♦♥❣❧② s✉❣❣❡sts t❤❛t t❤❡r❡ ❛r❡ ❡✐t❤❡r ♥♦ ♦r ✈❡r② ❢❡✇ β s❤❡❡ts ♣r❡s❡♥t ✇✐t❤✐♥ t❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶✳ ❲❤✐❧❡ t❤✐s ❞♦❡s ♥♦t ❛♣♣❡❛r t♦ ❜❡ ❝♦♥s✐st❡♥t ✇✐t❤ ●❛♥❞❤✐✬s ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ❡❛r❧✐❡r ❋❚■❘ ♠❡❛s✉r❡♠❡♥ts✱ ✐t ✐s ❝♦♥s✐st❡♥t ✇✐t❤ t❤❡ r❡✲❛♥❛❧②③❡❞ ❞❛t❛ ❢r♦♠ ●❛♥❞❤✐✬s ❋❚■❘ ❡①♣❡r✐♠❡♥ts ♦♥ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ❢r♦♠ ❚❛❜❧❡ ✹✳✻ ❛♥❞ ❋✐❣✉r❡ ✹✳✻ ✇❤✐❝❤ s❤♦✇❡❞ ❜♦t❤ ❛ ❧♦❝❛❧ ♠✐♥✐♠❛ ❢♦r t❤❡ ♣r❡s❡♥❝❡ ♦❢ ♣❛r❛❧❧❡❧ β s❤❡❡ts ❛♥❞ ❛ ❧❛r❣❡ ♣❡❛❦ ❢♦r t❤❡ ❛❣❣r❡❣❛t❡❞ str❛♥❞s t❤❛t ❛r❡ ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ ❞❡♥❛t✉r❡❞ ♣r♦t❡✐♥s✳ ❚❤✐s ❞❡♥❛t✉r❡❞ str✉❝t✉r❡ ❞♦❡s ❡①♣❧❛✐♥ t❤❡ ♣♦♦r ♠❡✲ ❝❤❛♥✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❡❧❡❝tr♦s♣✉♥ s✐❧❦ ❞♦♥❡ ❜② ●❛♥❞❤✐✳ ❆❧♠♦st ❡✈❡r② ♠♦❞❡❧ ♦❢ s♣✐❞❡r ✼✻  s✐❧❦✬s ♠❡❝❤❛♥✐❝❛❧ ♣r♦♣❡rt✐❡s ❞❡♣❡♥❞s ❛t ❧❡❛st ♣❛rt✐❛❧❧② ♦♥ t❤❡ ❝r②st❛❧❧✐♥❡ r❡❣✐♦♥s t❤❛t ❢♦r♠ ❬✷✶✱ ✶✺✱ ✺❪✳ ❲✐t❤♦✉t t❤✐s r✐❣✐❞ ❜❛❝❦❜♦♥❡✱ ✐t ✐s ♥♦t s✉r♣r✐s✐♥❣ t❤❛t ✐ts str❡♥❣t❤ ❛♥❞ ♠♦❞✉❧✉s ❞r♦♣ ❞r❛♠❛t✐❝❛❧❧②✳ ❚❤✐s ✐s ❛❧s♦ ❝♦♥✜r♠❡❞ ❜② t❤❡ ❢❛❝t t❤❛t ✇❤✐❧❡ t❤❡ str❡♥❣t❤ ❛♥❞ ♠♦❞✉❧✉s ♦❢ t❤❡ ❡❧❡❝tr♦s♣✉♥ ✜❜❡rs ❞r♦♣♣❡❞ ❜② s❡✈❡r❛❧ ♦r❞❡rs ♦❢ ♠❛❣♥✐t✉❞❡ ❢r♦♠ t❤❛t ♦❢ ♥❛t✉r❛❧ s✐❧❦✱ ❛s ❣✐✈❡♥ ✐♥ ❚❛❜❧❡ ✹✳✹✱ t❤❡ ❡①t❡♥❞❛❜✐❧✐t② ✇❛s ♦♥❧② r❡❞✉❝❡❞ ♠❛r❣✐♥❛❧❧②✳ ❚❤❡s❡ r❡s✉❧ts ❞♦ ♥♦t ❤❡❧♣ r❡s♦❧✈❡ ✇❤❡t❤❡r ♦r ♥♦t ❡❧❡❝tr♦s♣✐♥♥✐♥❣ ✐s ❛ ♣r❛❝t✐❝❛❧ ♠❡t❤♦❞ ❢♦r ♠❛❦✐♥❣ ❛rt✐✜❝✐❛❧ s✐❧❦ ✜❜❡rs✳ ❲❤✐❧❡ t❤❡ ♣r♦❝❡ss ❞✐❞ ②✐❡❧❞ ✉s❡❢✉❧ ✜❜❡rs✱ t❤❡ ❧✐♥❡✲s❤❛♣❡ ♦❢ t❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ✐s ✈❡r② s✐♠✐❧❛r t♦ t❤❡ ❞❡♥❛t✉r❡❞ ❧②♦♣❤✐❧✐③❡❞ s♣✐❞❡r s✐❧❦ ❢♦✉♥❞ ❜② ❍✐❥✐r✐❞❛ ❡t ❛❧✳ ❬✷✵❪✱ ❛s ❝❛♥ ❜❡ s❡❡♥ ✐♥ ❋✐❣✉r❡ ✼✳✻✳ ❚❤✐s ✐s r❛t❤❡r s✐❣♥✐✜❝❛♥t ❛s ✐t s✉❣❣❡sts t❤❛t t❤❡ str✉❝t✉r❛❧ ✐♥❛❞❡q✉❛❝✐❡s ♦❢ t❤❡ ✜❜❡rs ❛♥❞ t❤❡ r❡s✉❧t✐♥❣ ♣♦♦r ♠❡❝❤❛♥✐❝❛❧ ♣r♦♣❡rt✐❡s ♠❛② ❜❡ ❛ r❡s✉❧t ♦❢ t❤❡ ❞❡♥❛t✉r✐♥❣ ✈✐❛ ❢♦r♠✐❝ ❛❝✐❞ ❛s ♦♣♣♦s❡❞ t♦ t❤❡ ❡❧❡❝tr♦s♣✐♥♥✐♥❣ ♣r♦❝❡ss ✐ts❡❧❢✳ ■t ✐s ♣♦ss✐❜❧❡ t❤❛t ♣♦st s♣✐♥♥✐♥❣ tr❡❛t♠❡♥ts ❝♦✉❧❞ ❜❡ ✉s❡❞ t♦ ❛❧❧♦✇ ❢♦r ❝r②st❛❧❧✐③❛t✐♦♥✳ ❖♥❡ s✉❝❤ t❡❝❤♥✐q✉❡ ♦❢ ♣❧❛st✐❝✐③❛t✐♦♥ ✇❛s ❞♦♥❡ ❜② ❍✐❥✐r✐❞❛ ❡t ❛❧✳ ❬✷✵❪ ♦♥ t❤❡✐r ❞❡♥❛t✉r❡❞ ❧②♦♣❤✐❧✐③❡❞ s✐❧❦ ♣r♦t❡✐♥ ❛♥❞ s❤♦✇❡❞ ❛ s✐❣♥✐✜❝❛♥t ✐♥❝r❡❛s❡ ✐♥ β s❤❡❡t ❝r②st❛❧s✳ ❚❤✐s ✐s ♣❛rt✐❝✉❧❛r❧② ♣r♦♠✐s✐♥❣ ❞✉❡ t♦ t❤❡ s✐♠✐❧❛r✐t✐❡s ✐♥ ❈P✲▼❆❙ ❧✐♥❡✲s❤❛♣❡ ❜❡t✇❡❡♥ t❤❡ ❡❧❡❝tr♦s♣✉♥ s✐❧❦ ❛♥❞ t❤❡✐r s❛♠♣❧❡✳ ❚❤✐s ❞♦❡s ♥♦t ❤♦✇❡✈❡r ♥❡❝❡ss❛r✐❧② ✈✐♥❞✐❝❛t❡ ❡❧❡❝tr♦s♣✐♥♥✐♥❣ ❛s ❛ ❝❤♦✐❝❡ t❡❝❤♥✐q✉❡ ❢♦r t❤✐s ♣r♦❝❡ss✳ ❚❤❡ ❡①tr❡♠❡ ♠♦t✐♦♥s ❞✉r✐♥❣ t❤❡ ♣r♦❝❡ss ❛r❡ ✈❡r② ❞✐✛❡r❡♥t t♦ t❤❛t ♦❢ t❤❡ r❡❧❛t✐✈❡❧② s❧♦✇ s❡❝r❡t✐♦♥ t❤r♦✉❣❤ s♣✐❞❡r✬s s✐❧❦ ❣❧❛♥❞s ❬✷✶❪✳ ❍♦✇ t❤✐s ❛❜✉s❡ ✇♦✉❧❞ ❛✛❡❝t t❤❡ ♣r♦t❡✐♥s ✇❤✐❧❡ ✐♥ ❛ ❞❡♥❛t✉r❡❞ st❛t❡ r❡♠❛✐♥s ✉♥r❡s♦❧✈❡❞✱ ❛♥❞ ✐t ✐s ♣♦ss✐❜❧❡ t❤❛t ♣r♦t❡✐♥s ❝❤❛✐♥s ❝♦✉❧❞ ❜❡ t♦r♥ ❞✉r✐♥❣ t❤✐s ♣r♦❝❡ss✱ ✇❡❛❦❡♥✐♥❣ t❤❡ ✜♥❛❧ ♣r♦❞✉❝t r❡❣❛r❞❧❡ss ♦❢ ❝r②st❛❧ ❢♦r♠❛t✐♦♥✳  ✼✼  ❈❤❛♣t❡r ✽  ❈♦♥❝❧✉❞✐♥❣ ❘❡♠❛r❦s  13  ❈ ❈P✲▼❆❙ s♣❡❝tr❛ ❛♥❞ r❡❧❛①❛t✐♦♥ t✐♠❡s ✇❡r❡ ♠❡❛s✉r❡❞ ❢♦r s♣✐❞❡r s✐❧❦ ♣r♦t❡✐♥s ✐♥ s✐❧❦✱  ♣♦✇❞❡r✱ ❛♥❞ ❡❧❡❝tr♦s♣✉♥ ✜❜❡r ❢♦r♠s✳ ❋♦r s♣✐❞❡r s✐❧❦✱ t❤✐s s❤♦✇❡❞ t❤❛t t❤❡ ♦♣t✐♠❛❧ r❡❧❛①❛t✐♦♥ ✜t ❢♦r t❤❡ ❜❛❝❦❜♦♥❡ ❝❛r❜♦♥s ✇❛s t♦ ❛ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧✱ ✐♠♣❧②✐♥❣ ❛ ❞✐str✐❜✉t✐♦♥ ♦❢ r❡❧❛①❛t✐♦♥ t✐♠❡s✱ ❛♥❞ ❜② ✐♠♣❧✐❝❛t✐♦♥ ❝♦rr❡❧❛t✐♦♥ t✐♠❡s✳ ❚❤❡s❡ r❡s✉❧ts ✇❡r❡ ✉s❡❞ t♦ ❞❡t❡r♠✐♥❡ t❤❡ ♦r❞❡r ♦❢ t❤❡ ❝♦rr❡❧❛t✐♦♥ t✐♠❡ ♦❢ t❤❡ t❤❡r♠❛❧ ♠♦t✐♦♥ ♦❢ t❤❡ ♣r♦t❡✐♥ ❜❛❝❦❜♦♥❡✱ s❤♦✇✐♥❣ ✐t t♦ ❜❡ ❝❧♦s❡ t♦ t❤❡ ❢❛st✴s❧♦✇ tr❛♥s✐t✐♦♥ ♣♦✐♥t✳ ■♥t❡r❡st✐♥❣❧②✱ t❤❡ ♣♦✇❞❡r❡❞ s❛♠♣❧❡s ♦❢ ▼❛❙♣✶ ❛♥❞ ▼❛❙♣✷ ✇❡r❡ s❤♦✇♥ t♦ ❜❡ ♠♦r❡ s✐♠✐❧❛r t♦ ❡❛❝❤ ♦t❤❡r t❤❛♥ ♥♦t✱ ❛♥❞ ❤❛❞ ♠♦t✐♦♥ ♦❢ t❤❡ ❛❧❛♥✐♥❡ ❜❛❝❦❜♦♥❡s ❝♦♠♣❛r❛❜❧❡ t♦ t❤❛t ♦❢ s♣✐❞❡r s✐❧❦✳ ❚❤❡ r❡❧❛①❛t✐♦♥ t✐♠❡ ♠❡❛s✉r❡❞ ❢♦r t❤❡✐r ❣❧②❝✐♥❡ ❜❛❝❦❜♦♥❡s ✇❡r❡ s✐❣♥✐✜❝❛♥t❧② ❢❛st❡r t❤❛♥ t❤❛t ♦❢ t❤❡ s♣✐❞❡r s✐❧❦✱ s✉❣❣❡st✐♥❣ t❤❡ ❣❧②❝✐♥❡s t♦ ❜❡ ✐♥ ❛ ❧❡ss r✐❣✐❞ str✉❝t✉r❡ t❤❛♥ t❤❛t ♦❢ t❤❡ s✐❧❦✳ ❚❤❡ s✐♠✐❧❛r✐t✐❡s t♦ t❤❡ ❛❧❛♥✐♥❡ ♦❢ s✐❧❦✱ ❛♥❞ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ t❤❡ ❣❧②❝✐♥❡ s✉❣❣❡st t❤❛t t❤❡ s♣✐❞❡r✬s s♣✐♥♥✐♥❣ ♣r♦❝❡ss t♦ s✐❧❦ ✐s ♠♦r❡ ❝r✉❝✐❛❧ ❢♦r t❤❡ ♦r✐❡♥t❛t✐♦♥ ❛♥❞ ♦r❞❡r✐♥❣ ♦❢ t❤❡ ♣♦❧② ❣❧②❝✐♥❡ t❤❛♥ ✐t ✐s t♦ t❤❡ ❝r②st❛❧❧✐③❛t✐♦♥ ♦❢ t❤❡ ♣♦❧②❛❧❛♥✐♥❡ ♠♦t✐❢s✳ ❚❤❡ ❡❧❡❝tr♦s♣✉♥ ▼❛❙♣✶ ✇❛s ❢♦✉♥❞ t♦ ❤❛✈❡ ✈❡r② ❧✐tt❧❡ ❝r②st❛❧❧✐③❛t✐♦♥ ❛♥❞ ❛♣♣❡❛r❡❞ t♦ ❤❛✈❡ ❛ str✉❝t✉r❡ ❝♦♥s✐st❡♥t ✇✐t❤ t❤❛t ♦❢ ❞❡♥❛t✉r❡❞ ♣r♦t❡✐♥s✳ ❚❤✐s ❧❛❝❦ ♦❢ ❝r②st❛❧❧✐♥✐t② ❞♦❡s ❡①♣❧❛✐♥ t❤❡ ♣♦♦r ♠❡❝❤❛♥✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❡❧❡❝tr♦s♣✉♥ s♣✐❞❡r s✐❧❦✳ ❚❤✐s r❡s✉❧t st❛♥❞s ✐♥ ❝♦♥tr❛❞✐❝t✐♦♥ t♦ ♣r❡✈✐♦✉s ❋❚■❘ ♠❡❛s✉r❡♠❡♥ts ♦❢ s✐♠✐❧❛r❧② ♣r❡♣❛r❡❞ ✜❜❡rs✱ ❛❧t❤♦✉❣❤ r❡✈✐s✐t✐♥❣ t❤❡ ❋❚■❘ ❞❛t❛ r❛✐s❡s ♠❛♥② q✉❡st✐♦♥s ❛❜♦✉t t❤❛t ❞❛t❛✬s ✐♥t❡r♣r❡t❛t✐♦♥✳ ❚❤❡r❡ ❛r❡ ♠✉❧t✐♣❧❡ ❞✐r❡❝t✐♦♥s t❤❛t ❢✉t✉r❡ r❡s❡❛r❝❤ ❢♦r t❤✐s ❝❛♥ ❣♦ ✐♥✳ ❚❤❡ ✜rst ✐s ❛ ❝♦♥t✐♥✉❛t✐♦♥ ♦❢ t❤❡ st✉❞② ♦❢ s♣✐❞❡r s✐❧❦✬s ❝♦rr❡❧❛t✐♦♥ t✐♠❡✳ ❆s t❤❡ ❞❛t❛ ✉s❡❞ ❢♦r t❤❡ ❧♦✇ ✜❡❧❞ r❡❧❛①❛t✐♦♥ t✐♠❡ ✇❛s ♥♦t ✜tt❡❞ t♦ ❛ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧✱ r❡❧❛①❛t✐♦♥ ♠❡❛s✉r❡♠❡♥ts ✐♥ ❛ s♣❡❝tr♦♠❡t❡r ✼✽  ✇✐t❤ ❛ ❞✐✛❡r❡♥t ✜❡❧❞ ✇♦✉❧❞ ♠❛❦❡ t❤❡ r❡s✉❧ts ♠♦r❡ ❛❝❝✉r❛t❡✳ ❚❤✐s ❝♦✉❧❞ ❛❧s♦ ❜❡ ❞♦♥❡ ❜② ♠❡❛s✉r❡♠❡♥ts ❛t ❞✐✛❡r❡♥t t❡♠♣❡r❛t✉r❡s✱ ❛❧t❤♦✉❣❤ t❤❡ ❡①❛❝t ✜t ♠❛② ❜❡ ❞✐✣❝✉❧t ✇✐t❤♦✉t ✐❞❡♥t✐❢②✐♥❣ t❤❡ ♦r❞❡r ♣❛r❛♠❡t❡r ✈✐❛ ❞✐✛❡r❡♥t ♠✉❧t✐ ✜❡❧❞ str❡♥❣t❤ ❡①♣❡r✐♠❡♥ts✳ ❲❤✐❧❡ t❤✐s ✇♦✉❧❞ ❜❡ ❛♥ ✐♥t❡❧❧❡❝t✉❛❧❧② ✐♥t❡r❡st✐♥❣ ❡①❡r❝✐s❡✱ ✐ts ✉s❡❢✉❧♥❡ss ✐s ❞❡❜❛t❛❜❧❡ ❛s ✐t ♠✐❣❤t ♥♦t ♣r♦✈✐❞❡ ♠✉❝❤ ✉s❛❜❧❡ ✐♥❢♦r♠❛t✐♦♥ t♦ t❤❡ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ s♣✐❞❡r s✐❧❦✳ ❆ s❡❝♦♥❞ ❛✈❡♥✉❡ t♦ ❡①♣❧♦r❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ ▼❛❙♣✶ ❛♥❞ ▼❛❙♣✷✳ ◆▼❘ ❡①♣❡r✐♠❡♥ts ♦❢ s♣✐❞❡r s✐❧❦ ❜② ❍♦❧❧❛♥❞ ❡t ❛❧✳ ❬✸✵❪ ✉s❡❞ ❛ ✷✲❞✐♠❡♥s✐♦♥❛❧ ❈❛r❜♦♥✲❈❛r❜♦♥ ❝♦rr❡❧❛t✐♦♥ s♣❡❝tr✉♠ t♦ ❜❡tt❡r ♠❛♣ t❤❡ s❡❝♦♥❞❛r② str✉❝t✉r❡s ✐♥ s♣✐❞❡r s✐❧❦✳ ❚❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ t❤✐s ❡①♣❡r✐♠❡♥t ✐s t❤❛t ✐t s❤♦✇s ❝♦rr❡❧❛t✐♦♥s ❜❡t✇❡❡♥ ❝❛r❜♦♥s ✇✐t❤✐♥ ❛ r❡s✐❞✉❡ ❛❧❧♦✇✐♥❣ str✉❝t✉r❡s t♦ ❜❡ ♠♦r❡ ❡❛s✐❧② ✐❞❡♥t✐✜❡❞ ❞❡s♣✐t❡ t❤❡ ❜r♦❛❞ ♣❡❛❦s ♣r♦❞✉❝❡❞ ❜② s♣✐❞❡r s✐❧❦✳ ■t ❛❧s♦ ❛❧❧♦✇s ♦✈❡r❧❛♣♣✐♥❣ r❡❣✐♦♥s t♦ ❜❡ ✐s♦❧❛t❡❞ ❛♥❞ s❡♣❛r❛t❡❞✳ ❚❤✐s t❡❝❤♥✐q✉❡ ✇♦✉❧❞ s❤♦✇ ❛ ❜❡tt❡r ♣✐❝t✉r❡ ♦❢ ❛♥② str✉❝t✉r❛❧ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ t❤❡ ▼❛❙♣✶ ❛♥❞ ▼❛❙♣✷ ♣♦✇❞❡rs✳ ■t ✇♦✉❧❞ ❜❡ ♣❛rt✐❝✉❧❛r❧② ✐♥t❡r❡st✐♥❣ t♦ ✉s❡ t❤✐s t❡❝❤♥✐q✉❡ ❢♦r ▼❛❙♣✷ ❛s ✐t ✇♦✉❧❞ ❤❡❧♣ t♦ s❡♣❛r❛t❡ t❤❡ ♦✈❡r❧❛♣♣✐♥❣ r❡❣✐♦♥s ♦❢ t❤❡ ❣❧✉t❛♠✐♥❡✱ ♣r♦❧✐♥❡ α ❝❛r❜♦♥s ❛♥❞ t❤❡ s❡r✐♥❡ α ❛♥❞ β ❝❛r❜♦♥s✳ ❚❤✐s ❝♦✉❧❞ ♣r♦✈❡ ♣❛rt✐❝✉❧❛r❧② ✉s❡❢✉❧ ✐♥ ✉♥❞❡rst❛♥❞✐♥❣ t❤❡ r♦❧❡ ♣❧❛②❡❞ ❜② ▼❛❙♣✷ ✐♥ s♣✐❞❡r s✐❧❦ ❛♥❞ ♣♦ss✐❜❧② ❤❡❧♣ r❡s♦❧✈❡ t❤❡ ✉♥r❡s♦❧✈❡❞ q✉❡st✐♦♥s r❡❣❛r❞✐♥❣ t❤❡ ♣❡❛❦s ❛t ✺✹ ❛♥❞ ✻✵ ♣♣♠✳ ❆ ✜♥❛❧ ❛✈❡♥✉❡ t♦ ❡①♣❧♦r❡ ✐s ❝♦♥t✐♥✉✐♥❣ ❡①♣❡r✐♠❡♥ts ♦♥ ❡❧❡❝tr♦s♣✉♥ s♣✐❞❡r s✐❧❦ ♣r♦t❡✐♥s✳ ❲❤✐❧❡ t❤❡ ✐♥✐t✐❛❧ r❡s✉❧ts ✇❡r❡ ♥♦t ✈❡r② ♣r♦♠✐s✐♥❣✱ ❡❧❡❝tr♦s♣✐♥♥✐♥❣ ✐s t❤❡ s✐♠♣❧❡st ♠❡t❤♦❞ ❢♦r ♣r♦❞✉❝✲ ✐♥❣ ♥❛♥♦✜❜❡rs✳ ❚❤✐s ✐s ✐♠♣♦rt❛♥t ❛s t❤❡ ♠❡❝❤❛♥✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✐❧❦s ❛♣♣❡❛r t♦ ❜❡ ❞✐❛♠❡t❡r ❞❡♣❡♥❞❡♥t✱ ❛♥❞ t❡❝❤♥✐q✉❡s t❤❛t ❛♣♣❡❛r t♦ r❡♣❧✐❝❛t❡ s❡❝♦♥❞❛r② str✉❝t✉r❡s ❤❛✈❡ ❞✐✣❝✉❧t② ♣r♦✲ ❞✉❝✐♥❣ ✜❜❡rs ♦❢ ❛ s♠❛❧❧ ❡♥♦✉❣❤ ❞✐❛♠❡t❡r ❬✸✺❪✳ ■t ✐s ♣♦ss✐❜❧❡ t❤❛t ✇✐t❤ t❤❡ ❝♦rr❡❝t ♣♦st tr❡❛t♠❡♥t t❡❝❤♥✐q✉❡✱ s✉❝❤ ❛s t❤❛t ✉s❡❞ ♦♥ t❤❡ ❞❡♥❛t✉r❡❞ ❧②♦♣❤✐❧✐③❡❞ s✐❧❦ ❜② ❍✐❥✐r✐❞❛ ❡t ❛❧✳ ❬✷✵❪✱ t❤❛t t❤❡ ❡❧❡❝tr♦s♣✉♥ s✐❧❦ ❝♦✉❧❞ t❛❦❡ ♦♥ t❤❡ ❝♦rr❡❝t s❡❝♦♥❞❛r② str✉❝t✉r❡✳ ❖t❤❡r ♣♦ss✐❜❧❡ ♣♦st tr❡❛t♠❡♥t t❡❝❤♥✐q✉❡s ❝♦✉❧❞ ❜❡ tr✐❡❞✱ s✉❝❤ ❛s ♠❡t❤❛♥♦❧ ✇❤✐❝❤ s❤♦✇❡❞ ♣r♦♠✐s✐♥❣ r❡s✉❧ts ❝r②st❛❧❧✐③✐♥❣ ❡❧❡❝tr♦s♣✉♥ ❇♦♠❜②① ♠♦r✐✱ ❛s ❞♦♥❡ ❜② ●❛♥❞❤✐ ❬✶✵❪✳ ❆ ♠♦r❡ ❛♠❜✐t✐♦✉s ❣♦❛❧ ❝♦✉❧❞ ❜❡ t♦ ✜♥❞ ❛ ♠❡t❤♦❞ t♦ ❡❧❡❝tr♦s♣✐♥ ❢r♦♠ ❛♥ ❛q✉❡♦✉s ♣r♦t❡✐♥ s♦❧✉t✐♦♥ ✐♥st❡❛❞ ♦❢ ✉s✐♥❣ ❢♦r♠✐❝ ❛❝✐❞✳ ❚❤❡ r❡s✉❧ts ♦❢ t❤❡ ◆▼❘ ❡①♣❡r✐♠❡♥ts s❤♦✇❡❞ t❤❛t t❤❡ ❡❧❡❝tr♦s♣✉♥ ✜❜❡r ❛r❡ ❝♦♥s✐st❡♥t ✇✐t❤ t❤❛t ♦❢ t❤❡ ❞❡♥❛t✉r✐♥❣ ❝❛✉s❡❞ ❜② ❢♦r♠✐❝ ❛❝✐❞✳ ❚❤✐s s✉❣❣❡sts t❤❛t ❡❧❡❝tr♦s♣✐♥♥✐♥❣ ❢r♦♠ ✇❛t❡r ♠❛② ♥♦t ❤❛✈❡ t❤❡ s❛♠❡ ❞r❛st✐❝ ❡✛❡❝ts ♦♥ t❤❡ ♣r♦t❡✐♥ s❡❝♦♥❞❛r② str✉❝t✉r❡✳  ✼✾  ❇✐❜❧✐♦❣r❛♣❤②  ❬✶❪ ▼✳❍✳ ▲❡✈✐tt✳  ❙♣✐♥ ❉②♥❛♠✐❝s✿ ❇❛s✐❝s ♦❢ ◆✉❝❧❡❛r ❘❡s♦♥❛♥❝❡  ❬✷❪ ❆✳ ❆❜r❛❣❛♠✳  ❚❤❡ Pr✐♥❝✐♣❧❡s ♦❢ ◆✉❝❧❡❛r ▼❛❣♥❡t✐❝ ❘❡s♦♥❛♥❝❡  ❬✸❪ ❈✳P✳ ❙❧✐❝❤t❡r✳  Pr✐♥❝✐♣❧❡s ♦❢ ▼❛❣♥❡t✐❝ ❘❡s♦♥❛♥❝❡ ✷♥❞ ❊❞✐t✐♦♥  ❬✹❪ ▼✳ ❉✉❡r✳  ✳ ❲✐❧❡②✱ ✷✵✵✶✳ ✳ ❖①❢♦r❞✱ ✶✾✻✶✳  ✳ ❙♣r✐♥❣❡r✱ ✶✾✼✽✳  ✳ ❇❧❛❝❦✇❡❧❧ ❙❝✐❡♥❝❡✱ ✷✵✵✹✳  ■♥tr♦❞✉❝t✐♦♥ t♦ ❙♦❧✐❞✲❙t❛t❡ ◆▼❘ ❙♣❡❝tr♦s❝♦♣②  ❬✺❪ ❏✳❉✳ ✈❛♥ ❇❡❡❦✱ ❙✳ ❍❡ss✱ ❋✳ ❱♦❧❧r❛t❤✱ ❛♥❞ ❇✳❍✳ ▼❡✐❡r✳ ❚❤❡ ♠♦❧❡❝✉❧❛r str✉❝t✉r❡ ♦❢ s♣✐✲ ❞❡r ❞r❛❣❧✐♥❡ s✐❧❦✿ ❋♦❧❞✐♥❣ ❛♥❞ ♦r✐❡♥t❛t✐♦♥ ♦❢ t❤❡ ♣r♦t❡✐♥ ❜❛❝❦❜♦♥❡✳ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ◆❛t✐♦♥❛❧ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s ♦❢ t❤❡ ❯♥✐t❡❞ ❙t❛t❡s ♦❢ ❆♠❡r✐❝❛✱ ✾✾✭✶✻✮✿✶✵✷✻✻✕✶✵✷✼✶✱ ✷✵✵✷✳ ❬✻❪ P✳❆✳ ❇❡❝❦♠❛♥✳ ❙♣❡❝tr❛❧ ❞❡♥s✐t✐❡s ❛♥❞ ♥✉❝❧❡❛r s♣✐♥ r❡❧❛①❛t✐♦♥ ✐♥ s♦❧✐❞s✳ ✶✼✶✿✸✿✽✺✕✶✷✽✱ ✶✾✽✽✳  ✱  P❤②s✐❝❛❧ ❘❡♣♦rts  ❬✼❪ ●✳ ▲✐♣❛r✐ ❛♥❞ ❆✳ ❙③❛❜♦✳ ▼♦❞❡❧✲❢r❡❡ ❛♣♣r♦❛❝❤ t♦ t❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ♥✉❝❧❡❛r ♠❛❣♥❡t✐❝ r❡s♦♥❛♥❝❡ r❡❧❛①❛t✐♦♥ ✐♥ ♠❛❝r♦♠♦❧❡❝✉❧❡s✳ ✶✳ t❤❡♦r② ❛♥❞ r❛♥❣❡ ✈❛❧✐❞✐t②✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ❈❤❡♠✐❝❛❧ ❙♦❝✐❡t②✱ ✶✵✹✿✹✺✹✻✕✹✺✺✾✱ ✶✾✽✷✳ ❬✽❪ ❚✳❆✳ ❉❡♣❡✇✳ ❙♦❧✐❞ st❛t❡ ♥♠r ✐♥✈❡st✐❣❛t✐♦♥s ♦❢ ♣r♦t❡✐♥ ❜❛s❡❞ ❜✐♦♠❛t❡r✐❛❧s✳ ▼❛st❡r✬s t❤❡s✐s✱ ❯♥✐✈❡rs✐t② ♦❢ ❇r✐t✐s❤ ❈♦❧✉♠❜✐❛✱ ✷✵✵✽✳ ❬✾❪ ❑✳ ❙❝❤♠✐❞t✲❘♦❤r ❛♥❞ ❍✳❙✳ ❙♣✐❡ss✳ ❆❝❛❞❡♠✐❝ Pr❡ss✱ ✶✾✾✹✳ ❬✶✵❪ ▼✳❘✳ ●❛♥❞❤✐✳  ▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❙♦❧✐❞✲❙t❛t❡ ◆▼❘ ❛♥❞ P♦❧②♠❡rs  ✳  ❙✐❧❦ Pr♦t❡✐♥ ❛s ❛ ❇✐♦♠❛t❡r✐❛❧ ❢♦r ❚✐ss✉❡ ❊♥❣✐♥❡❡r✐♥❣ ❆♣♣❧✐❝❛t✐♦♥✿ ❚❤❡✲  ✳ P❤❉ t❤❡s✐s✱ ❉r❡①❡❧ ❯♥✐✈❡rs✐t②✱ ✷✵✵✻✳  ♦r❡t✐❝❛❧ ❛♥❞ ❊❝♣❡r✐♠❡♥t❛❧ ❙t✉❞②  ❬✶✶❪ ❈✳ ▼✐❝❤❛❧✳  ❙♦❧✐❞✲❙t❛t❡ ❉❡✉t❡r✐✉♠ ❛♥❞ ❘❡❞♦r ◆▼❘ ❙tr✉❝t✉r❛❧ ❙t✉❞✐❡s ♦❢ ❙♣✐❞❡r ❉r❛❣❧✐♥❡  ❙✐❧❦ ❛♥❞ ❛ ◆❡✇ ❊①♣❡r✐♠❡♥t ❢♦r ▼❡❛s✉r✐♥❣ ▼✉❧t✐♣❧❡ ❍❡t❡r♦♥✉❝❧❡❛r ❉✐♣♦❧❛r ❈♦✉♣❧✐♥❣s ✐♥ ❙♦❧✐❞s  ✳ P❤❉ t❤❡s✐s✱ ❈♦r♥❡❧❧ ❯♥✐✈❡rs✐t②✱ ✶✾✾✼✳  ❬✶✷❪ ❚✳❊✳ ❈r❡✐❣❤t♦♥✳ ❈♦♠♣❛♥②✱ ✶✾✹✵✳  Pr♦t❡✐♥s✿  ❙tr✉❝t✉r❡s ❛♥❞ ▼♦❧❡❝✉❧❛r Pr♦♣❡rt✐❡s  ✽✵  ✳ ❲✳❍✳ ❋r❡❡♠❛♥ ❛♥❞  ❬✶✸❪ ❚✳ ❆❝❦❜❛r♦✇✱ ❳✳ ❈❤❡♥✱ ❙✳ ❑❡t❡♥✱ ❛♥❞ ▼✳❏✳ ❇✉❡❤❧❡r✳ ❍✐❡r❛r❝❤✐❡s✱ ♠✉❧t✐♣❧❡ ❡♥❡r❣② ❜❛rr✐✲ ❡rs✱ ❛♥❞ r♦❜✉st♥❡ss ❣♦✈❡r♥ t❤❡ ❢r❛❝t✉r❡ ♠❡❝❤❛♥✐❝s ♦❢ ❛❧♣❤❛✲❤❡❧✐❝❛❧ ❛♥❞ ❜❡t❛✲s❤❡❡t ♣r♦✲ t❡✐♥ ❞♦♠❛✐♥s✳ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ◆❛t✐♦♥❛❧ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s ♦❢ t❤❡ ❯♥✐t❡❞ ❙t❛t❡s ♦❢ ❆♠❡r✐❝❛✱ ✶✵✹✭✹✷✮✿✶✻✹✶✵✕✶✻✹✶✺✱ ✷✵✵✼✳ ❬✶✹❪ ●✳ ❋✐♥❞❧❛②✳  ✳ ❲❡✐❣❧✱ ✷✵✵✺✳  ■♥❞✐❣❡♥♦✉s P❡♦♣❧❡s ▼♦♥❣♦❧s  ❬✶✺❪ ❙✳❇✳ ❲❛r♥❡r✱ ▼✳ P♦❧❦✱ ❛♥❞ ❑✳ ❏❛❝♦❜✳ ❙♣✐❞❡r ❞r❛❣❧✐♥❡ s✐❧❦✳ ✶✾✾✾✳ ❬✶✻❪ ❆✳ ●❧✐s♦❝✐❝ ❛♥❞ ❋✳ ❱♦❧r❛t❤✳  P♦❧②♠❡r ❘❡✈✐❡✇s  ✱ ✸✾✿✻✹✸✕✻✺✸✱  ■♥❞✉str✐❛❧ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ◆❛t✉r❛❧ ❋✐❜r❡s✿ ❙tr✉❝t✉r❡✱ Pr♦♣✲  ✳ ❲✐❧❡②✱ ✷✵✶✵✳  ❡rt✐❡s✳ ❛♥❞ ❚❡❝❤♥✐❝❛❧ ❆♣♣❧✐❝❛t✐♦♥s  ❬✶✼❪ ▼✳ ❳✉ ❛♥❞ ❘✳❱✳ ▲❡✇✐s✳ ❙tr✉❝t✉r❡ ♦❢ ❛ ♣r♦t❡✐♥ s✉♣❡r✜❜❡r✿ s♣✐❞❡r ❞r❛❣❧✐♥❡ s✐❧❦✳ Pr♦❝❡❡❞✲ ✐♥❣s ♦❢ t❤❡ ◆❛t✐♦♥❛❧ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s ♦❢ t❤❡ ❯♥✐t❡❞ ❙t❛t❡s ♦❢ ❆♠❡r✐❝❛✱ ✽✼✿✼✶✷✵✕✼✶✷✹✱ ✶✾✾✵✳ ❬✶✽❪ ▼✳❇✳ ❍✐♥♠❛♥ ❛♥❞ ❘✳❱✳ ▲❡✇✐s✳ ■s♦❧❛t✐♦♥ ♦❢ ❛ ❝❧♦♥❡ ❡♥❝♦❞✐♥❣ ❛ s❡❝♦♥❞ ❞r❛❣❧✐♥❡ s✐❧❦ ✜❜r♦✐♥✳ ♥❡♣❤✐❧❛ ❝❧❛✈✐♣❡s ❞r❛❣❧✐♥❡ s✐❧❦ ✐s ❛ t✇♦✲♣r♦t❡✐♥ ✜❜❡r✳ ❏♦✉r♥❛❧ ♦❢ ❇✐♦❧♦❣✐❝❛❧ ❈❤❡♠✐str②✱ ✷✻✼✿✶✾✸✷✵✕✶✾✸✷✹✱ ✶✾✾✷✳ ❬✶✾❪ ▼✳❧ ❊❧✐❝❡s✳ ❙tr✉❝t✉r❛❧ ❇✐♦❧♦❣✐❝❛❧ s❤✐♣s✳ P❡r❣❛♠♦♥✱ ✷✵✵✵✳  ▼❛t❡r✐❛❧s✿  ❉❡s✐❣♥ ❛♥❞ ❙tr✉❝t✉r❡✲Pr♦♣❡rt② ❘❡❧❛t✐♦♥✲  ❬✷✵❪ ❉✳❍✳ ❍✐❥✐r✐❞❛✱ ❑✳●✳ ❉♦✱ ❈✳ ▼✐❝❤❛❧✱ ❙✳ ❲♦♥❣✱ ❉✳ ❩❛①✱ ❛♥❞ ▲✳❲✳ ❏❡❧✐♥s❦②✳ ✶✸❝ ♥♠r ♦❢ ♥❡♣❤✐❧❛ ❝❧❛✈✐♣❡s ♠❛❥♦r ❛♠♣✉❧❧❛t❡ s✐❧❦ ❣❧❛♥❞✳ ❇✐♦♣❤②s✐❝❛❧ ❏♦✉r♥❛❧✱ ✼✶✿✸✹✹✷✕✸✹✹✼✱ ✶✾✾✻✳ ❬✷✶❪ ❋✳ ❱♦❧❧r❛t❤ ❛♥❞ ❉✳P✳ ❑♥✐❣❤t✳ ▲✐q✉✐❞ ❝r②st❛❧❧✐♥❡ s♣✐♥♥✐♥❣ ♦❢ s♣✐❞❡r s✐❧❦✳ ✺✹✶✕✺✹✾✱ ✷✵✵✷✳ ❬✷✷❪ ❘✳❱✳ ▲❡✇✐s✳ ❙♣✐❞❡r s✐❧❦✿ t❤❡ ✉♥r❛✈❡❧✐♥❣ ♦❢ ❛ ♠②st❡r②✳ ✷✺✭✾✮✿✸✾✷✕✸✾✽✱ ✶✾✾✷✳  ✱ ✹✶✵✿  ◆❛t✉r❡  ❆❝❝♦✉♥ts ♦❢ ❈❤❡♠✐❝❛❧ ❘❡s❡❛r❝❤  ✱  ❬✷✸❪ ●✳P✳ ❍♦❧❧❛♥❞✱ ❏✳❊✳ ❏❡♥❦✐♥s✱ ▼✳❙✳ ❈r❡❛❣❡r✱ ❘✳❱✳ ▲❡✇✐s✱ ❛♥❞ ❘✳❏✳ ❨❛r❣❡r✳ ◗✉❛♥t✐❢②✐♥❣ t❤❡ ❢r❛❝t✐♦♥ ♦❢ ❣❧②❝✐♥❡ ❛♥❞ ❛❧❛♥✐♥❡ ✐♥ ❜❡t❛✲s❤❡❡t ❛♥❞ ❤❡❧✐❝✐❝❛❧ ❝♦♥❢♦r♠❛t✐♦♥s ✐♥ s♣✐❞❡r ❞r❛❣❧✐♥❡ s✐❧❦ ✉s✐♥❣ s♦❧✐❞✲st❛t❡ ♥♠r✳ ❈❤❡♠✳ ❈♦♠♠✉♥✱ ✷✵✵✽✳ ❬✷✹❪ ❆✳ ▲❛③❛r✐s✱ ❙✳ ❆r❝✐❞✐❛❝♦♥♦✱ ❨✳ ❍✉❛♥❣✱ ❏✳❋✳ ❩❤♦✉✱ ❋✳ ❉✉❣✉❛②✱ ◆✳ ❈❤r❡t✐❡♥✱ ❊✳❆✳ ❲❡❧s❤✱ ❏✳❲✳ ❙♦❛r❡s✱ ❛♥❞ ❈✳◆✳ ❑❛r❛t③❛s✳ ❙♣✐❞❡r s✐❧❦ ✜❜❡rs s♣✉♥ ❢r♦♠ s♦❧✉❜❧❡ r❡❝♦♠❜✐♥❛♥t s✐❧❦ ♣r♦❞✉❝❡❞ ✐♥ ♠❛♠♠❛❧✐❛♥ ❝❡❧❧s✳ ❙❝✐❡♥❝❡✱ ✺✺✺✹✿✹✼✷✕✹✼✻✱ ✷✵✵✷✳ ❬✷✺❪ ▼✳ ❏❛❝❦s♦♥ ❛♥❞ ❍✳❍✳ ▼❛♥ts❝❤✳ ❚❤❡ ✉s❡ ❛♥❞ ♠✐s✉s❡ ♦❢ ❢t✐r s♣❡❝tr♦s❝♦♣② ✐♥ t❤❡ ❞❡t❡r✲ ♠✐♥❛t✐♦♥ ♦❢ ♣r♦t❡✐♥ str✉❝t✉r❡✳ ❈r✐t✐❝❛❧ ❘❡✈✐❡✇s ✐♥ ❇✐♦❝❤❡♠✐str② ❛♥❞ ▼♦❧❡❝✉❧❛r ❇✐♦❧♦❣②✱ ✸✵✿✾✺✕✶✷✵✱ ✶✾✾✺✳ ❬✷✻❪ ❲✳❑✳ ❙✉r❡✇✐❝s✱ ❍✳❍✳ ▼❛♥st❝❤✱ ❛♥❞ ❉✳ ❈❤❛♣♠❛♥✳ ❉❡t❡r♠✐♥❛t✐♦♥ ♦❢ ♣r♦t❡✐♥ s❡❝♦♥❞❛r② str✉❝t✉r❡ ❜② ❢♦✉r✐❡r tr❛♥s❢♦r♠ ✐♥❢❛r❡❞ s♣❡❝tr♦s❝♦♣②✿ ❆ ❝r✐t✐❝❛❧ ❛ss❡ss♠❡♥t✳ P❡rs♣❡❝t✐✈❡s ✐♥ ❇✐♦❝❤❡♠✐str②✱ ✸✷✿✸✽✾✕✸✾✹✱ ✶✾✾✸✳  ✽✶  ❬✷✼❪ ❉✳❆✳ ❚♦r❝❤✐❛✳ ❚❤❡ ♠❡❛s✉r❡♠❡♥t ♦❢ ♣r♦t♦♥✲❡♥❤❛♥❝❡❞ ❝❛r❜♦♥✲✶✸ t✶ ✈❛❧✉❡s ❜② ❛ ♠❡t❤♦❞ ✇❤✐❝❤ s✉♣♣r❡ss❡s ❛rt✐❢❛❝ts✳ ❏♦✉r♥❛❧ ♦❢ ▼❛❣♥❡t✐❝ ❘❡s♦♥❛♥❝❡✱ ✸✵✿✻✶✸✕✻✶✻✱ ✶✾✼✽✳ ❬✷✽❪ ❈✳P✳ ▲✐♥❞s❡② ❛♥❞ ●✳❉✳ P❛tt❡rs♦♥✳ ❉❡t❛✐❧❡❞ ❝♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ✇✐❧❧✐❛♠✲✇❛tts ❛♥❞ ❝♦❧❡✲ ❞❛✈✐❞s♦♥ ❢✉♥❝t✐♦♥s✳ ❏♦✉r♥❛❧ ♦❢ ❈❤❡♠✳ P❤②s✱ ✼✸✭✼✮✿✸✸✹✽✕✸✸✺✻✱ ✶✾✽✵✳ ❬✷✾❪ P✳P✳ ❇❡✈✐♥❣t♦♥ ❛♥❞ ❉✳❑✳ ❘♦❜✐♥s♦♥✳ ❙❝✐❡♥❝❡s✳ ▼❝●r❛✇ ❍✐❧❧✱ ✶✾✾✷✳  ❉❛t❛ ❘❡❞✉❝t✐♦♥ ❛♥❞ ❊rr♦r ❆♥❛❧②s✐s ❢♦r t❤❡ P❤②s✐❝❛❧  ❬✸✵❪ ●✳P✳ ❍♦❧❧❛♥❞✱ ▼✳❙✳ ❈r❡❛❣❡r✱ ❏✳❊✳ ❏❡♥❦✐♥s✱ ❘✳❱✳ ▲❡✇✐s✱ ❛♥❞ ❏✳▲✳ ❨❛r❣❡r✳ ❉❡t❡r♠✐♥✐♥❣ s❡❝♦♥❞❛r② str✉❝t✉r❡ ✐♥ s♣✐❞❡r ❞r❛❣❧✐♥❡ s✐❧❦ ❜② ❝❛r❜♦♥✲❝❛r❜♦♥ ❝♦rr❡❧❛t✐♦♥ s♦❧✐❞✲st❛t❡ ♥♠r s♣❡❝tr♦s❝♦♣②✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ❈❤❡♠✐❝❛❧ ❙♦❝✐❡t②✱ ✶✸✵✭✸✵✮✿✾✽✼✶✕✾✽✼✼✱ ✷✵✵✽✳ ❬✸✶❪ ❉✳❙✳ ❲✐s❤❛rt ❛♥❞ ❆✳▼✳ ◆✐♣✳ Pr♦t❡✐♥ ❝❤❡♠✐❝❛❧ s❤✐❢t ❛♥❛❧②s✐s✿ ❛ ♣r❛❝t✐❝❛❧ ❣✉✐❞❡✳ ❈❡❧❧✳ ❇✐♦✱ ✼✻✿✶✺✸✕✶✻✸✱ ✶✾✾✽✳  ❇✐♦❝❤❡♠✳  ❬✸✷❪ ▼✳ ■✇❛❞❛t❡✱ ❚✳ ❆s❛❦✉r❛✱ ❛♥❞ ▼✳ ❲✐❧❧✐❛♠s♦♥✳ ❈ ❛❧♣❤❛ ❛♥❞ ❝ ❜❡t❛ ❝❛r❜♦♥✲✶✸ ❝❤❡♠✐❝❛❧ s❤✐❢ts ✐♥ ♣r♦t❡✐♥ ❢r♦♠ ❛♥ ❡♠♣✐✐❝❛❧ ❞❛t❛❜❛s❡✳ ❏♦✉r♥❛❧ ♦❢ ❇✐♦♠♦❧❡❝✉❧❛r ◆▼❘✱ ✶✸✿✶✾✾✕✷✶✶✱ ✶✾✾✾✳ ❬✸✸❪ ❆✳ ❙✐♠♠♦♥s✱ ❊✳ ❘❛②✱ ❛♥❞ ▲✳❲✳ ❏❡❧✐♥s❦✐✳ ❙♦❧✐❞✲st❛t❡ ✶✸❝ ♥♠r ♦❢ ♥❡♣❤✐❧❛ ❝❧❛✈✐♣❡s ❞r❛❣❧✐♥❡ s✐❧❦ ❡st❛❜❧✐s❤❡s str✉❝t✉r❡ ❛♥❞ ✐❞❡♥t✐t② ♦❢ ❝r②st❛❧❧✐♥❡ r❡❣✐♦♥s✳ ▼❛❝r♦♠♦❧❡❝✉❧❡s✱ ✷✼✭✶✽✮✿ ✺✷✸✺✕✺✷✸✼✱ ✶✾✾✹✳ ❬✸✹❪ ❉✳❆✳ ❚♦r❝❤✐❛ ❛♥❞ ❆✳ ❙③❛❜♦✳ ❙♣✐♥✲❧❛tt✉❝❡ r❡❧❛①❛t✐♦♥ ✐♥ s♦❧✐❞s✳ ❘❡s♦♥❛♥❝❡✱ ✹✾✿✶✵✼✕✶✷✶✱ ✶✾✽✷✳  ❏♦✉r♥❛❧ ♦❢ ▼❛❣♥❡t✐❝  ❬✸✺❪ ▲✳❲✳ ❏❡❧✐♥s❦✐✱ ❆✳ ❇❧②❡✱ ❖✳ ▲✐✐✈❛❦✱ ❈✳ ▼✐❝❤❛❧✱ ●✳ ▲❛❱❡r❞❡✱ ❆✳ ❙❡✐❞❡❧✱ ◆✳ ❙❤❛❤✱ ❛♥❞ ❩✳ ❨❛♥❣✳ ❖r✐❡♥t❛t✐♦♥✱ str✉❝t✉r❡✱ ✇❡t✲s♣✐♥♥✐♥❣✱ ❛♥❞ ♠♦❧❡❝✉❧❛r ❜❛s✐s ❢♦r s✉♣❡r❝♦♥tr❛❝t✐♦♥ ♦❢ s♣✐❞❡r ❞r❛❣❧✐♥❡ s✐❧❦✳ ❇✐♦❧♦❣✐❝❛❧ ▼❛❝r♦♠♦❧❡❝✉❧❡s✱ ✷✹✿✶✾✼✕✷✵✶✱ ✶✾✾✾✳  ✽✷  

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