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❙♣✐♥ ❞②♥❛♠✐❝s s✐♠✉❧❛t✐♦♥ ♦❢ ❡❧❡❝tr♦♥ s♣✐♥ r❡❧❛①❛t✐♦♥ ✐♥ ◆✐2+(aq)

❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ◆▼❘ ❘❡s❡❛r❝❤ ●r♦✉♣ ❉❡♣❛rt♠❡♥t ♦❢ P❤②s✐❝s ❯♥✐✈❡rs✐t② ♦❢ ❖✉❧✉✱ ❋✐♥❧❛♥❞ ❚❤❡ ❊①❛❝t✉s ❞♦❝t♦r❛❧ ♣r♦❣r❛♠ ♦❢ t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❖✉❧✉ ●r❛❞✉❛t❡ ❙❝❤♦♦❧ ❥②r❦✐✳r❛♥t❛❤❛r❥✉❅♦✉❧✉✳❢✐ ❏✳ ❘❛♥t❛❤❛r❥✉✱ ❏✳ ▼❛r❡➨ ❛♥❞ ❏✳ ❱❛❛r❛✱ s✉❜♠✐tt❡❞ ❢♦r ♣✉❜❧✐❝❛t✐♦♥✳

❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✶✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹

■♥tr♦❞✉❝t✐♦♥

• ❚❤❡ ❛❜✐❧✐t② t♦ q✉❛♥t✐t❛t✐✈❡❧② ♣r❡❞✐❝t ❛♥❞ ❛♥❛❧②③❡ t❤❡ r❛t❡ ♦❢ ❡❧❡❝tr♦♥ s♣✐♥ r❡❧❛①❛t✐♦♥ ♦❢

❡❧❡❝tr♦♥✐❝❛❧❧② ♦♣❡♥✲s❤❡❧❧ s②st❡♠s ✐s ✐♠♣♦rt❛♥t ❢♦r ❡❧❡❝tr♦♥ ♣❛r❛♠❛❣♥❡t✐❝ r❡s♦♥❛♥❝❡ ❛♥❞ ♣❛r❛♠❛❣♥❡t✐❝ ♥✉❝❧❡❛r ♠❛❣♥❡t✐❝ r❡s♦♥❛♥❝❡ s♣❡❝tr♦s❝♦♣✐❡s✳

• ❚❤❡ ❇❧♦❝❤✲❘❡❞✜❡❧❞✲❲❛♥❣s♥❡ss t❤❡♦r② ✐s r❛r❡❧② ❛♣♣❧✐❝❛❜❧❡ t♦ ♣❛r❛♠❛❣♥❡t✐❝ s②st❡♠s✳

❙♦❧✈✐♥❣ t❤❡ st♦❝❤❛st✐❝ ▲✐♦✉✈✐❧❧❡ ❡q✉❛t✐♦♥ ❤❛s ❝♦♥st✐t✉t❡❞ ❛ st❛t❡✲♦❢✲t❤❡✲❛rt ♠❡t❤♦❞ ❢♦r ❡❧❡❝tr♦♥ s♣✐♥ r❡❧❛①❛t✐♦♥✳

❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✷✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹

• ■♥ ♣r✐♥❝✐♣❧❡ ✐t ✐s ♣♦ss✐❜❧❡ t♦ s✐♠✉❧❛t❡ ♥✉❝❧❡❛r ❛♥❞ ❡❧❡❝tr♦♥ s♣✐♥ r❡❧❛①❛t✐♦♥ ♦❢ ❛♥② s②st❡♠

❢♦r ✇❤✐❝❤ t❤❡ ✇❡❧❧✲❡st❛❜❧✐s❤❡❞ ▼❉ ❛♥❞ ◗❈ ❡❧❡❝tr♦♥✐❝ str✉❝t✉r❡ ♠❡t❤♦❞s ❛r❡ ❛♣♣❧✐❝❛❜❧❡✳

• ❲❡ ❞❡♠♦♥str❛t❡ t❤✐s ✉s✐♥❣ ❛q✉❡♦✉s s♦❧✉t✐♦♥ ♦❢ ◆✐2+ ✐♦♥s ❛s ❛ ♠♦❞❡❧ s②st❡♠✳
• ❲❡ s❛♠♣❧❡ ❛ ▼❉ tr❛❥❡❝t♦r② ❜② q✉❛♥t✉♠ ❝❤❡♠✐❝❛❧ ✭◗❈✮ ❝❛❧❝✉❧❛t✐♦♥s ❛♥❞✱ ✐♥ t✉r♥✱

♥✉♠❡r✐❝❛❧❧② s♦❧✈❡ t❤❡ ▲✐♦✉✈✐❧❧❡✲✈♦♥ ◆❡✉♠❛♥♥ ❡q✉❛t✐♦♥ ❢♦r t❤❡ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ s♣✐♥ ❞❡♥s✐t② ♠❛tr✐①✳

• ❲❡ ❢♦✉♥❞ ♦♥❧② ♦♥❡ ❛tt❡♠♣t ❬✶❪ ♦❢ s✉❝❤ s✐♠✉❧❛t✐♦♥ ❢r♦♠ ❧✐t❡r❛t✉r❡✳ ❆t ♣r❡s❡♥t ❞❛② ✇❡

❤❛✈❡ ❜❡tt❡r ▼❉ ❛♥❞ ◗❈✳ ❚❤❡ r❡❧❛①❛t✐♦♥ t✐♠❡s ✇❡ ❡①tr❛❝t ❢r♦♠ ♦✉r s✐♠✉❧❛t✐♦♥ ❛r❡ ✐♥ ✈❡r② ❣♦♦❞ ❛❣r❡❡♥♠❡♥t ✇✐t❤ t❤❡ ❛✈❛✐❧❛❜❧❡ ❡①♣❡r✐♠❡♥t❛❧ ❞❛t❛✳ ❬✶❪ ▼✳ ❖❞❡❧✐✉s✱ ❈✳ ❘✐❜❜✐♥❣ ❛♥❞ ❏✳ ❑♦✇❛❧❡✇s❦✐✱ ❙♣✐♥ ❞②♥❛♠✐❝s ✉♥❞❡r t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ✈❛r②✐♥❣ ✇✐t❤ t✐♠❡ ✐♥ ❞✐s❝r❡t❡ st❡♣s✿ ▼♦❧❡❝✉❧❛r ❉②♥❛♠✐❝s ❇❛s❡❞ ❙✐♠✉❧❛t✐♦♥ ♦❢ ❊❧❡❝tr♦♥ ❛♥❞ ◆✉❝❧❡❛r ❙♣✐♥ ❘❡❧❛①❛t✐♦♥ ✐♥ ❆q✉❡♦✉s ◆✐❝❦❡❧✭■■✮✱ ❏♦✉r♥❛❧ ♦❢ ❈❤❡♠✐❝❛❧ P❤②s✐❝s ✶✵✹✱ ✸✶✽✶ ✭✶✾✾✻✮✳

❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✸✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹

❍❛♠✐❧t♦♥✐❛♥

• ❚❤❡ t✇♦ ❡❧❡❝tr♦♥s ♦❢ ◆✐2+ ❝♦♥st✐t✉t❡ ❛♥ ❡✛❡❝t✐✈❡ s♣✐♥✲✶ s②st❡♠✳ ❚♦ t❤❡ s♣✐♥ ❍❛♠✐❧t♦✲

♥✐❛♥ ✇❡ ✐♥❝❧✉❞❡ t❤❡ ❣ ❛♥❞ ❩❋❙ t❡♥s♦rs ˆ H(t) = µBˆ ❙ · ❣(t) · ❇ + ˆ ❙ · ❉(t) · ˆ ❙ = ˆ H0 + ˆ HI(t) ˆ H0 = µBgB ˆ Sz = ¯ hω0 ˆ Sz ˆ HI(t) = µBˆ ❙ · [❣(t) − g✶] · ❇ + ˆ ❙ · ❉(t) · ˆ ❙, ✭✶✮

• ◗❈ ❝❛❧❝✉❧❛t✐♦♥s ♦❢ ▼❉ s♥❛♣s❤♦ts ♣r♦❞✉❝❡ ❛ ♣✐❡❝❡✇✐s❡ ❝♦♥st❛♥t✱ t✐♠❡✲♦r❞❡r❡❞ s❡r✐❡s ♦❢

s♣✐♥ ❍❛♠✐❧t♦♥✐❛♥s ˆ H1, ˆ H2, ˆ H3, . . . , ˆ Hl−1, ˆ Hl, ✇✐t❤ t❤❡ t✐♠❡ st❡♣ τ✳

• ■♥ t❤✐s s✐♠✉❧❛t✐♦♥ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✉s❡❞ ▼❉ tr❛❥❡❝t♦r② ✇❛s 750 ♣s ❛♥❞ ✐t ✇❛s s❛♠♣❧❡❞

t♦ ❛ ♣✐❡❝❡✇✐s❡ ❝♦♥st❛♥t s❡r✐❡s ♦❢ l = 15625 s♣✐♥ ❍❛♠✐❧t♦♥✐❛♥s { ˆ Hi} ✇✐t❤ t❤❡ t✐♠❡ st❡♣ ♦❢ 48 ❢s✳

❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✹✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹

❙✉♣❡r♣r♦♣❛❣❛t♦rs

• ❲❡ tr❛♥s❢♦r♠ t❤❡ { ˆ

Hi} t♦ s✉♣❡r♣r♦♣❛❣❛t♦rs e

ˆ ˆ L1τ, e ˆ ˆ L2τ, e ˆ ˆ L3τ, . . . , e ˆ ˆ Ll−1τ, e ˆ ˆ Llτ, ✇❤❡r❡

t❤❡ ˆ ˆ Lj ❛r❡ ▲✐♦✉✈✐❧❧✐❛♥s ❞❡✜♥❡❞ ❛s ˆ ˆ Lj = i[·, ˆ Hj]

• ❚❤❡ ❞❡♥s✐t② ♦♣❡r❛t♦r ✐s ❞❡✜♥❡❞ ❛s

ˆ ρ(t) = |Ψ(t)Ψ(t)|, ✭✷✮ ✇❤❡r❡ |Ψ(t) ✐s t❤❡ st❛t❡ ♦❢ t❤❡ s♣✐♥ s②st❡♠✳ ˆ ρ ✐s t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ▲✐♦✉✈✐❧❧❡✲✈♦♥ ◆❡✉♠❛♥♥ ❡q✉❛t✐♦♥✱ dˆ ρ(t) dt = ˆ ˆ L(t)ˆ ρ(t), ✭✸✮

❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✺✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹

• ✇❤✐❝❤ ♠❡❛♥s t❤❛t ✇❡ ❝❛♥ ❡①♣r❡ss ˆ

ρ(t)✱ ✐♥ r♦t❛t✐♥❣ ❢r❛♠❡✱ ❛t t❤❡ t✐♠❡ ✐♥st❛♥t t = nτ ❛s ˆ ρr(nτ) = e−ˆ

ˆ L0nτe ˆ ˆ Lnτe ˆ ˆ Ln−1τ · · · e ˆ ˆ L1τ ˆ

ρ(0) ≡ L ˆ ˆ (n)ˆ ρ(0), ✭✹✮ ✇❤❡r❡ ˆ ˆ L0 ✐s t❤❡ ▲✐♦✉✈✐❧❧✐❛♥ ❢♦r♠ ♦❢ ˆ H0✳

❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✻✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹

❊♥s❡♠❜❧❡

• ❈♦♥s✐❞❡r✐♥❣ ❛♥ ❡♥s❡♠❜❧❡ ♦❢ s♣✐♥ s②st❡♠s✱ ✇✐t❤ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t ❛❧❧ ✐ts ♠❡♠❜❡rs ❛r❡ ✐♥✐t✐❛❧❧② ✐♥ t❤❡

s❛♠❡ st❛t❡✱ t❤❡ ❡♥s❡♠❜❧❡ ❛✈❡r❛❣❡ ♦❢ ˆ ρr ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ ❛s ˆ

ρr(nτ) = L ˆ ˆ (n)ˆ ρ(0).

• ❆ss✉♠✐♥❣ ❡r❣♦❞✐❝✐t② ❛♥❞ t❤❛t t❤❡ s❡r✐❡s { ˆ

Hi} ✐s ❧♦♥❣ ❡♥♦✉❣❤✱ t❤❡ ❡♥s❡♠❜❧❡ ❛✈❡r❛❣❡ ♦❢ L

ˆ ˆ (n) ✐s

♦❜t❛✐♥❡❞ ❛s

L ˆ ˆ (n) = 1 m(n)e−ˆ

ˆ L0nτ m(n)

• i=1

e

ˆ ˆ Li+nτe ˆ ˆ Li+n−1τ · · · e ˆ ˆ Liτ,

✭✺✮

✇❤❡r❡ m(n) ✐s t❤❡ ♥✉♠❜❡r ♦❢ t❤❡ s✉❜✲s❡r✐❡s ˆ Hi, . . . , ˆ Hi+n t❤❛t ❝❛♥ ❜❡ ❡①tr❛❝t❡❞ ❢r♦♠ { ˆ Hi}✳ m(n) s❤♦✉❧❞ ❜❡ ❧❛r❣❡ ❡♥♦✉❣❤ t♦ r❡♥❞❡r (1/m(n)) m(n)

i=1

ˆ ˆ Hi(nτ) ❛ ❣♦♦❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ˆ ˆ H0✳

❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✼✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹

▼❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥

• ˆ

ρr ❧✐✈❡s ✐♥ ❛ ♥✐♥❡✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ ❛♥❞ ✐ts ❝♦♠♣♦♥❡♥ts ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ❝♦♥s✐st✐♥❣ ♦❢ t❤❡ s❤✐❢t ♦♣❡r❛t♦rs ❛♥❞ t❤❡ z ❝♦♠♣♦♥❡♥t ♦❢ ˆ ❙✿ { ˆ Bi} = ˆ S− ˆ S− 2 , ˆ S− ˆ Sz + ˆ Sz ˆ S− 2 , ˆ S− 2 , ˆ Sz √ 2, 3 ˆ Sz ˆ Sz − 2ˆ 1 √ 6 , ˆ 1 √ 3, ˆ S+ ˆ Sz + ˆ Sz ˆ S+ 2 , ˆ S+ 2 , ˆ S+ ˆ S+ 2 . ✭✻✮

❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✽✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹

• ˆ

ρ ❝❛♥ ♥♦✇ ❜❡ ❡①♣r❡ss❡❞ ❛s ❛ ❦❡t ✈❡❝t♦r ❛♥❞ L ˆ ˆ (n) ❛s ❛ ♠❛tr✐① ♦♣❡r❛t♦r✱ ✐♥ t❤❡ ♥✐♥❡✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✿ |ˆ ρ(0)) =

9

• j=1

cj| ˆ Bj); ✭✼✮ L ˆ ˆ (n) =

9

• j,k=1

Ljk(n)| ˆ Bj)( ˆ Bk|. ✭✽✮

❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✾✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹

❘❡❧❛①❛t✐♦♥ ❡①♣❡r✐♠❡♥t

• ■♥ t❤❡r♠❛❧ ❡q✉✐❧✐❜r✐✉♠ t❤❡ ♠❛❣♥❡t✐③❛t✐♦♥ ▼ ✐s ❛❧✐❣♥❡❞ ✇✐t❤ t❤❡ ❈❛rt❡s✐❛♥ z ❛①✐s✳ ■♥ ❛

T1 ❡①♣❡r✐♠❡♥t✱ ▼ ✐s ✢✐♣♣❡❞ ❜② 180 ✇✐t❤ ❛ π✲♣✉❧s❡ ❛♥❞ t❤❡ r❡❧❛①❛t✐♦♥ r❛t❡ ♦❢ t❤❡ z ❝♦♠♣♦♥❡♥t✱ Mz✱ ✐s ♠❡❛s✉r❡❞✱ ❛s ✐t ❞❡❝❛②s ❜❛❝❦ t♦ ❡q✉✐❧✐❜r✐✉♠✳ ■♥ ❛ T2 ❡①♣❡r✐♠❡♥t ▼ ✐s r♦t❛t❡❞ ❜② 90 ✇✐t❤ ❛ π/2✲♣✉❧s❡ t♦ t❤❡ x ❛①✐s✱ ❛❢t❡r ✇❤✐❝❤ t❤❡ r❡❧❛①❛t✐♦♥ r❛t❡ ♦❢ Mx ✐s ♠❡❛s✉r❡❞✳

• ■♥ t❤❡ ♣r❡s❡♥t s✐♠✉❧❛t✐♦♥ ♠❡t❤♦❞ t❤❡ ❢❡❡❞❜❛❝❦ ❢r♦♠ t❤❡ ✐♥❞✐✈✐❞✉❛❧ ♠❡♠❜❡rs ♦❢ t❤❡ s♣✐♥

❡♥s❡♠❜❧❡ t♦ t❤❡ s✉rr♦✉♥❞✐♥❣ ❧❛tt✐❝❡ ✐s ♥❡❣❧❡❝t❡❞ ❛♥❞✱ ❛s ❛ r❡s✉❧t✱ ❛❧❧ ❝♦♠♣♦♥❡♥ts ♦❢ ▼ ❞❡❝❛② t♦ ③❡r♦ ✐♥st❡❛❞ ♦❢ t❤❡ ♥♦♥✈❛♥✐s❤✐♥❣ ❡q✉✐❧✐❜r✐✉♠ ✈❛❧✉❡ ♦❢ Mz ✐♥ ❛ r❡❛❧✱ ✐♥t❡r❛❝t✐♥❣ s♣✐♥ s②st❡♠✳ ❆t ❤✐❣❤ t❡♠♣❡r❛t✉r❡s t❤✐s ♠♦❞❡❧ ❣✐✈❡s ❛❝❝✉r❛t❡ r❡s✉❧ts✳

❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✶✵✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹

❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

• ❚❤❡ ❡❧❡♠❡♥ts Ljk(n) = ( ˆ

Bj| ˆ Bk(nτ)) = ( ˆ Bj|L ˆ ˆ (n)| ˆ Bk) ❛r❡ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ♠❡♠❜❡rs ˆ Bi ♦❢ t❤❡ s❤✐❢t ❛♥❞ z ♦♣❡r❛t♦r ❜❛s✐s✳ ❚❤❡ ✐♥♥❡r ♣r♦❞✉❝t ( ˆ Bj| ˆ Bk(nτ)) ✐s ❞❡✜♥❡❞ ❛s Tr[ ˆ B†

j ˆ

Bk(nτ)]✳

• ❚❤❡ ❡①♣r❡ss✐♦♥s ❢♦r t❤❡ ♥♦r♠❛❧✐③❡❞ ❈❛rt❡s✐❛♥ z ❝♦♠♣♦♥❡♥t ♦❢ ▼ ✐♥ ❛ T1 ❡①♣❡r✐♠❡♥t✱

❛s ✇❡❧❧ ❛s t❤❡ x ❝♦♠♣♦♥❡♥t ✐♥ ❛ T2 ❡①♣❡r✐♠❡♥t ✭✇❤❡r❡ ▼ ✐s ✐♥✐t✐❛❧❧② ❛❧✐❣♥❡❞ ❛❧♦♥❣ t❤❡ x ❛①✐s✮ ❛r❡✱ ❛t t✐♠❡ nτ✱ M r

z(nτ)

M r

z(0) = L44(n) ; M r x(nτ)

M r

x(0) = L33(n) + L88(n)

2 , ✭✾✮ r❡s♣❡❝t✐✈❡❧②✳

❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✶✶✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹

• ❚❤❡ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ♥♦r♠❛❧✐③❡❞ ❝♦♠♣♦♥❡♥ts ♦❢ ˆ

S ❛r❡ ❞❡✜♥❡❞ ❛s Cr

µν(nτ) =

( ˆ Sµ| ˆ Sν(nτ))

• ( ˆ

Sµ| ˆ Sµ)( ˆ Sν| ˆ Sν) . ✭✶✵✮ ✳

❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✶✷✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹

❙✐♠✉❧❛t✐♦♥

• ❲❡ ❝❛❧❝✉❧❛t❡ t❤❡ ▲✐♦✉✈✐❧❧✐❛♥ ❢♦r♠ ♦❢ t❤❡ ❍❛♠✐❧t♦✲

♥✐❛♥s { ˆ Hi} ❛s ✇❡❧❧ ❛s t❤❡ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ t❤❡ ▲✐♦✉✈✐❧❧✐❛♥s ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ t❤❡ r♦✉t✐♥❡s ♦❢ ❙♣✐♥❉②♥❛♠✐❝❛ ❬✶❪✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ♣r♦♣❛❣❛t♦rs ❛r❡ ♦❜t❛✐♥❡❞ ✇✐t❤ t❤❡ ▼❛tr✐①❊①♣ r♦✉t✐♥❡ ♦❢ ▼❛t❤❡♠❛t✲ ✐❝❛✳ ❲❡ ✐♠♣❧❡♠❡♥t❡❞ ❛ ▼❛t❤❡♠❛t✐❝❛ r♦✉t✐♥❡ ❢♦r ❝♦♠♣✉t✐♥❣ t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ L

ˆ ˆ (n) ❢r♦♠ t❤❡ ❧✐st

♦❢ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ♣r♦♣❛✲ ❣❛t♦rs✳ ❚❤❡ ♣r♦♣❛❣❛t✐♦♥ ♦❢ L

ˆ ˆ (n) ✐s ❢❛st ❛♥❞ ♠❡♠♦r② ❡✣❝✐❡♥t✳

❬✶❪ ❙♣✐♥❉②♥❛♠✐❝❛ ✐s ❛ ♣❧❛t❢♦r♠ ❢♦r s♣✐♥ ❞②♥❛♠✐❝❛❧ s✐♠✉❧❛t✐♦♥s ✐♥ ▼❛t❤❡♠❛t✐❝❛✱ ♣r♦❣r❛♠♠❡❞ ❜② ▼❛❧❝♦❧♠ ❍✳ ▲❡✈✐tt✱ ✇✐t❤ ❝♦♥tr✐❜✉t✐♦♥s ❜② ❏②r❦✐ ❘❛♥t❛❤❛r❥✉✱ ❆♥❞r❡❛s ❇r✐♥❦♠❛♥♥ ❛♥❞ ❙♦✉♠②❛ ❙✐♥❣❤❛ ❘♦② ✭s❡❡ ✇✇✇✳❙♣✐♥❉②♥❛♠✐❝❛✳s♦t♦♥✳❛❝✳✉❦✮✳ ❊❧❡❝tr♦♥ s♣✐♥ ♠❛❣♥❡t✐s❛t✐♦♥ ❞❡❝❛② Mr z (t)/Mr z (0) ✭❜❧✉❡✮✱ Mr x(t)/Mr x(0) ✭r❡❞✮ ✐♥ T1 ❛♥❞ T2 r❡❧❛①❛t✐♦♥ s✐♠✉❧❛t✐♦♥s✱ r❡s♣❡❝t✐✈❡❧②✱ ❢♦r ◆✐2+(aq) ❛t ✸✵✵ ❑✳ ❆❧s♦ s❤♦✇♥ ❛r❡ t❤❡ r❡s♣❡❝t✐✈❡ s✐♥❣❧❡✲❡①♣♦♥❡♥t✐❛❧ ✜ts✱ ♣❧♦tt❡❞ ✇✐t❤ ❞❛s❤❡❞ ❧✐♥❡s✱ ❛s ✇❡❧❧ ❛s t❤❡ ❝r♦ss✲❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥ Cr zx(t) ✭❜r♦✇♥✮ ✐♥ ❛ 4.5 ❚ ♠❛❣♥❡t✐❝ ✜❡❧❞ ❛❧♦♥❣ t❤❡ z ❛①✐s✳ ❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✶✸✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹

❘❡❧❛①❛t✐♦♥ r❛t❡

• ❙✐♠✉❧❛t❡❞ R1 ✭❜❧✉❡ ♦♣❡♥ ❝✐r❝❧❡s✮✱ R2 ✭r❡❞

♦♣❡♥ ❝✐r❝❧❡s✮ r❡❧❛①❛t✐♦♥ r❛t❡s ♦❢ ❡❧❡❝tr♦♥ s♣✐♥ ✐♥ ◆✐2+(aq) ❛t ✸✵✵ ❑ ✐♥ t❤❡ r❛♥❣❡ ❢r♦♠ 0.001 ❚ t♦ 100 ❚✳ ❆❧s♦ s❤♦✇♥ ❛r❡ t❤❡✐r ✜ts t♦ ❊q✳ ✭✾✮ ❛s ❜❧✉❡ ❛♥❞ r❡❞ s♦❧✐❞ ❧✐♥❡s✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ❞❛t❛ ❛r❡ ♣r❡s❡♥t❡❞ ❛s ❢✉♥❝t✐♦♥s ♦❢ t❤❡ str❡♥❣t❤ ♦❢ t❤❡ ♠❛❣♥❡t✐❝ ✜❡❧❞✳

• ❚❤❡ r❛t❡s ❢♦❧❧♦✇ t❤❡ ❡q✉❛t✐♦♥

R1,2(B) = Y1,2 + A1,2 U1,2 + B2, ✭✶✶✮ ✇❤❡r❡ Y1,2, A1,2✱ ❛♥❞ U1,2 ❛r❡ ✜tt❡❞ ❝♦♥✲ st❛♥ts✳

❙✐♠✉❧❛t❡❞ R1 ✭❜❧✉❡ ♦♣❡♥ ❝✐r❝❧❡s✮✱ R2 ✭r❡❞ ♦♣❡♥ ❝✐r❝❧❡s✮ r❡❧❛①❛t✐♦♥ r❛t❡s ♦❢ ❡❧❡❝tr♦♥ s♣✐♥ ✐♥ ◆✐2+(aq) ❛t ✸✵✵ ❑ ✐♥ t❤❡ r❛♥❣❡ ❢r♦♠ 0.001 ❚ t♦ 100 ❚✳ ❆❧s♦ s❤♦✇♥ ❛r❡ t❤❡✐r ✜ts t♦ ❊q✳ ✭✾✮ ❛s ❜❧✉❡ ❛♥❞ r❡❞ s♦❧✐❞ ❧✐♥❡s✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ❞❛t❛ ❛r❡ ♣r❡s❡♥t❡❞ ❛s ❢✉♥❝t✐♦♥s ♦❢ t❤❡ str❡♥❣t❤ ♦❢ t❤❡ ♠❛❣♥❡t✐❝ ✜❡❧❞✳ ❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✶✹✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹

❘❡❧❛①❛t✐♦♥ t✐♠❡

❙✐♠✉❧❛t❡❞ T1 ✭❜❧✉❡✮ ❛♥❞ T2 ✭r❡❞✮ ❡❧❡❝tr♦♥ s♣✐♥ r❡❧❛①❛t✐♦♥ t✐♠❡s ✐♥ Ni2+(aq) ❛t ✸✵✵ ❑ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ♠❛❣♥❡t✐❝ ✜❡❧❞✳ ❚❤❡ ♦♣❡♥ ❝✐r❝❧❡s ❛r❡ t❤❡ s✐♠✉❧❛t✐♦♥ r❡s✉❧ts ❛♥❞ t❤❡ s♦❧✐❞ ❧✐♥❡s ❛r❡ ✜tt❡❞✳ ❚❤❡ ✜❧❧❡❞ ❝✐r❝❧❡s ❝♦rr❡s♣♦♥❞ t♦ s✐♠✉❧❛t✐♦♥s ❞♦♥❡ ✇✐t❤♦✉t t❤❡ ✢✉❝t✉❛t✐♥❣ ♣❛rt ♦❢ t❤❡ g t❡♥s♦r✳ ❚❤❡ st❛rs r❡♣r❡s❡♥t t❤❡ ❛✈❛✐❧❛❜❧❡ ❡①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts✿ T1 = T2 = 2.9 ♣s ✭❜❧✉❡✮ ❛t B = 0 ❚ ❬✶❪✱ T1 = 5.7 ♣s ✭❜❧✉❡✮ ❛t 1.39 ❚ ❛t t❤❡ t❡♠♣❡r❛t✉r❡ ♦❢ 223 ❑ ❬✷❪✱ ❛♥❞ T1 = 3.4 ♣s ✭❜❧✉❡✮ ✭✐❢ T1 ≫ T2✮ ❛♥❞ T1 = 1.9 ♣s ✭r❡❞✮ ✭✐❢ T1 = T2✮ ❛t 2.11 ❚ ❛♥❞ 243 ❑ ❬✸❪✳

❬✶❪ ❍✳ ▲✳ ❋r✐❡❞♠❛♥♥✱ ▼✳ ❍♦❧③✱ ❍✳ ●✳ ❍❡rt③✱ ❊P❘ ❘❡❧❛①❛t✐♦♥s ♦❢ ❆q✉❡♦✉s ◆✐2+ ■♦♥✱ ❚❤❡ ❏♦✉r♥❛❧ ♦❢ ❈❤❡♠✐❝❛❧ P❤②s✐❝s ✼✵✱ ✸✸✻✾ ✭✶✾✼✾✮✳ ❬✷❪ ❉✳ ❋✐❛t✱ ❆✳ ▼✳ ❈❤♠❡❧♥✐❝❦✱ ❖①②❣❡♥✲✶✼ ▼❛❣♥❡t✐❝ ❘❡s♦♥❛♥❝❡ ❙t✉❞✐❡s ♦❢ t❤❡ ❍②❞r❛t✐♦♥ ♦❢ t❤❡ ❋❡rr♦✉s ❛♥❞ ◆✐❝❦❡❧♦✉s ■♦♥s✱ ❏♦✉r♥❛❧ ♦❢ ❆♠❡r✐❝❛♥ ❈❤❡♠✐❝❛❧ ❙♦❝✐❡t② ✾✸✱ ✷✽✼✺ ✭✶✾✼✶✮✳ ❬✸❪ ❏✳ ●r❛♥♦t✱ ❆✳ ▼✳ ❆❝❤❧❛♠❛✱ ❉✳ ❋✐❛t✱ Pr♦t♦♥ ❛♥❞ ❉❡✉t❡r✐✉♠ ▼❛❣♥❡t✐❝ ❘❡s♦♥❛♥❝❡ ❙t✉❞② ♦❢ t❤❡ ❆q✉❡♦✉s ◆✐❝❦❡❧♦✉s ❈♦♠♣❧❡①✱ ❚❤❡ ❏♦✉r♥❛❧ ♦❢ ❈❤❡♠✐❝❛❧ P❤②s✐❝s ✻✶✱ ✸✵✹✸ ✭✶✾✼✹✮✳ ❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✶✺✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹

❙❝r❡❡♥ ❝❛♣t✉r❡ ❢r♦♠ ♣✉❜❧✐❝❛t✐♦♥ ❍✳ ▲✳ ❋r✐❡❞♠❛♥♥✱ ▼✳ ❍♦❧③✱ ❍✳ ●✳ ❍❡rt③✱ ❊P❘ ❘❡❧❛①❛t✐♦♥s ♦❢ ❆q✉❡♦✉s ◆✐2+ ■♦♥✱ ❚❤❡ ❏♦✉r♥❛❧ ♦❢ ❈❤❡♠✐❝❛❧ P❤②s✐❝s ✼✵✱ ✸✸✻✾ ✭✶✾✼✾✮✳✳ ❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✶✻✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹

❆❝❦♥♦✇❧❡❞❣♠❡♥ts

❲❡ ❛r❡ ❣r❛t❡❢✉❧ t♦

• ▼❛❧❝♦❧♠ ❍✳ ▲❡✈✐tt ✭❙♦✉t❤❛♠♣t♦♥✮✱
• Pär ❍å❦❛♥ss♦♥ ✭❙♦✉t❤❛♠♣t♦♥✮✱
• ▼✐❝❤❛❡❧ ❈✳ ❉✳ ❚❛②❧❡r ✭❘❛❞❜♦✉❞✮✱
• ❏♦③❡❢ ❑♦✇❛❧❡✇s❦✐ ✭❙t♦❝❦❤♦❧♠✮✱

❢♦r ✉s❡❢✉❧ ❞✐s❝✉ss✐♦♥s✳ ❋✐♥❛♥❝✐❛❧ s✉♣♣♦rt ❤❛s ❜❡❡♥ ♦❜t❛✐♥❡❞ ❢r♦♠

• t❤❡ ▼❛❣♥✉s ❊❤r♥r♦♦t❤ ❋♦✉♥❞❛t✐♦♥ ✭❏❘✮✱
• t❤❡ ❊①❛❝t✉s ❞♦❝t♦r❛❧ ♣r♦❣r❛♠ ♦❢ t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❖✉❧✉ ❣r❛❞✉❛t❡

s❝❤♦♦❧ ✭❏❘✮✱

• t❤❡ ❆❝❛❞❡♠② ♦❢ ❋✐♥❧❛♥❞✱ t❤❡ ❊✉r♦♣❡❛♥ ❯♥✐♦♥ ❙❡✈❡♥t❤ ❋r❛♠❡✲

✇♦r❦ Pr♦❣r❛♠♠❡ ✭❋P✼✴✷✵✵✼✲✷✵✶✸✮ ✉♥❞❡r ❣r❛♥t ❛❣r❡❡♠❡♥ts ♥♦✳ ✷✺✹✺✺✷ ✭❛❧❧ ❛✉t❤♦rts✮ ❛♥❞ ✸✶✼✶✷✼ ✭❏▼✮✱

• ❯♥✐✈❡rs✐t② ♦❢ ❖✉❧✉ ✭❏❱✮✱
• ❛♥❞ t❤❡ ❚❛✉♥♦ ❚ö♥♥✐♥❣ ❋♦✉♥❞❛t✐♦♥ ✭❏❱✮✳

❈♦♠♣✉t❛t✐♦♥❛❧ r❡s♦✉r❝❡s ❞✉❡ t♦ ❈❙❈ ✭❊s♣♦♦✱ ❋✐♥❧❛♥❞✮ ❛♥❞ t❤❡ ❋✐♥♥✐s❤ ●r✐❞ ■♥✐t✐❛t✐✈❡ ♣r♦❥❡❝t ✇❡r❡ ✉s❡❞✳ ❏✳ ❘❛♥t❛❤❛r❥✉✱ ❏✳ ▼❛r❡➨ ❛♥❞ ❏✳ ❱❛❛r❛✱ ❙♣✐♥ ❞②♥❛♠✐❝s s✐♠✉❧❛t✐♦♥ ♦❢ ❡❧❡❝tr♦♥ s♣✐♥ r❡❧❛①❛t✐♦♥ ✐♥ ◆✐2+(aq)✱ s✉❜♠✐tt❡❞ ❢♦r ♣✉❜❧✐❝❛t✐♦♥✳ ❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✶✼✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹