🧲 K₃[Fe(CN)₆] — Magnit tahlili

Fe³⁺ (LS, d⁵, t₂g⁵) • S=1/2 • Paramagnit • μ_eff ≈ 2.3 μ_B • Rombik EPR

K₃[Fe(CN)₆]

Magnit tahlili — to'liq profil

Qizil qon tuzining magnit xususiyatlari va spektroskopiyasi

K₃[Fe(CN)₆] past spinli (LS) Fe³⁺ kompleksidir. Kuchli maydon ligandlari (CN⁻) tufayli t₂g⁵ konfiguratsiyaga ega, ya'ni S = 1/2 (bitta toq elektron). Spin-only nazariyasi bo'yicha μ_so = 1.73 μ_B bo'lishi kerak, ammo eksperimental μ_eff ≈ 2.3 μ_B. Bu farq spin-orbital coupling va Yahn-Teller effekti tufayli yuzaga keladi — orbital momentning ham hissasi bor.

S = 1/2

LS, t₂g⁵

Paramagnit

μ = 2.3 μ_B

298 K da

> spin-only

θ ≈ −5 K

Curie-Weiss

Kuchsiz antiferro

g: 2.76, 2.20, 2.00

EPR

Rombik simmetriya

📚 Magnit xususiyatlarning nazariy asoslari

1. Magnit momentning tarkibi

Umumiy magnit moment ikki qismdan iborat:

μ_eff = g · √[S(S+1)] · μ_B (spin hissa)

+ L·S coupling (orbital hissa)

K₃[Fe(CN)₆] uchun orbital degeneratsiya (t₂g⁵) mavjud — shuning uchun orbital moment to'liq so'nmasdan, μ_eff ni oshiradi.

2. Curie va Curie-Weiss qonunlari

Ideal paramagnet: χ = C/T (Curie)
Real tizimlar: χ = C/(T − θ) (Curie-Weiss)
θ > 0 → ferromagnit o'zaro ta'sir
θ < 0 → antiferromagnit o'zaro ta'sir
K₃[Fe(CN)₆]: θ ≈ −5 K — qo'shni molekulalar orqali kuchsiz AF o'zaro ta'sir.

3. Kotani nazariyasi (t₂g⁵ uchun)

Kotani (1956) t₂gⁿ konfiguratsiyalar uchun μ_eff(T) ni aniq hisobladi. t₂g⁵ uchun (LS Fe³⁺):

μ_eff(T) = f(kT/λ)

T → ∞: μ_eff → 1.73 μ_B

T → 0: μ_eff → 2.1 μ_B (A'₁/₂ ground state)

298 K da kutilayotgan qiymat: ~2.3 μ_B — eksperimental bilan mos keladi.

🧮 Spin-only magnit momenti

Formula:

μ_so = √[n(n+2)] μ_B

Toq elektronlar soni (n):1

1234567

Hisoblash

√(1·3)

Natija

1.732 μ_B

📋 n bo'yicha qiymatlar:

n=1

1.73

n=2

2.83

n=3

3.87

n=4

4.90

n=5

5.92

n=6

6.93

n=7

7.94

⚠️ K₃[Fe(CN)₆] uchun ogohlantirish:

K₃[Fe(CN)₆] da n=1 (t₂g⁵, S=1/2), spin-only μ_so = 1.73 μ_B. Ammo eksperimental μ_eff ≈ 2.3 μ_B — sababi spin-orbital coupling (λ ≠ 0) va past simmetriya. Orbital momentning hissasi hisobga olinishi kerak!

📈 Curie-Weiss qonuni — interaktiv fit

Qonun:

χ = C / (T − θ)

μ_eff = √(8C) = 2.45 μ_B

C (Curie doimiysi):0.75 emu·K/mol

θ (Weiss doimiysi):-5.0 K

Antiferro (−)Ideal (0)Ferro (+)

χ vs T

1/χ vs T (chiziqli)

x-o'qi bilan kesishish → T = θ

💡 K₃[Fe(CN)₆] uchun:

θ ≈ −5 K (kuchsiz antiferromagnit o'zaro ta'sir — spinlar qo'shni molekulalar orqali antiparallel joylashishga moyil). μ_eff ≈ 2.3 μ_B (spin-only qiymatdan yuqori).

🌊 Brillouin funksiyasi — M(B/T)

Magnitlanish (paramagnetizm uchun):

M/M_sat = B_J(x)

x = gμ_BB / k_BT

J (to'liq moment):0.5

B (maydon):1.0 T

T (harorat):300 K

Magnitlanish darajasi0.4%

x = 4.48 × 10⁻³ → B_J(x) = 0.0045

M/M_sat vs B/T

🧪 Evans usuli — NMR orqali μ_eff

Evans formulasi:

μ_eff = 0.061 × √[(Δf × T) / (c × f₀)]

Δf (Hz):215

f₀ (MHz):400

c (mol/L):0.050

T (K):298

Hisoblangan μ_eff:

3.45 μ_B

K₃[Fe(CN)₆] uchun kutilayotgan qiymat: ~2.3 μ_B

💡 Usulning afzalligi:

SQUID magnitometri kerak emas — oddiy NMR spektrometr bilan μ_eff ni aniqlash mumkin. Paramagnit modda eritmadagi TMS signalini siljitadi. Δf = ν_paramagnit − ν_diamagnit.

📡 EPR spektr simulyatori (X-band, 9.5 GHz)

hν = gμ_BB → B = hν/(gμ_B)

g₁

2.76

B = 2459 G

g₂

2.20

B = 3085 G

g₃

2.00

B = 3394 G

Linewidth (ΔB):50 G

💡 K₃[Fe(CN)₆] EPR xulosasi:

g₁ ≠ g₂ ≠ g₃ — rombik simmetriya. Bu t₂g orbitallarining degeneratsiyasi Yahn-Teller effekti tufayli buzilganini ko'rsatadi. g-qiymatlarning 2.00 dan farqi — kuchli spin-orbital o'zaro ta'sir.

📋 Fe³⁺ komplekslarining magnit xususiyatlari

Birikma	Holat	S	n	μ_so	μ_eff	θ (K)	Xulosa
K₃[Fe(CN)₆]	Fe³⁺ (LS, t₂g⁵)	1/2	1	1.73	2.3	−5	Spin-orbital hissa
K₄[Fe(CN)₆]	Fe²⁺ (LS, t₂g⁶)	0	0	0	0	—	Diamagnit
[Fe(H₂O)₆]³⁺	Fe³⁺ (HS, t₂g³eg²)	5/2	5	5.92	5.9	—	Ideal spin-only (⁶A₁g)
[FeF₆]³⁻	Fe³⁺ (HS, t₂g³eg²)	5/2	5	5.92	5.9	kuchsiz −	Past maydon ligandi
[Fe(acac)₃]	Fe³⁺ (HS)	5/2	5	5.92	5.95	—	Klassik HS

* Barcha μ qiymatlari μ_B da. K₃[Fe(CN)₆] diqqatga sazovor — μ_eff > μ_so, ya'ni orbital momentning hissasi bor.

🔬 Eksperimental usullar

⚖️ Gouy usuli

Namunani tortish kuchi maydonidagi o'zgarish orqali o'lchash. Δm = χ · (H² − H₀²) · A / (2g)

✓ Arzon, klassik
✗ Ko'p namuna kerak (~100 mg)
✗ Sezgirligi past

🎯 Faraday usuli

Gradientli maydonda namunaning kuchi o'lchanadi. F = χ · m · H · (dH/dz)

✓ Kamroq namuna (1-10 mg)
✓ Yuqori aniqlik
✗ Murakkab apparat

🌡️ SQUID magnitometr

Superconducting Quantum Interference Device — eng zamonaviy va sezgir. 10⁻⁸ emu gacha sezgirlik, 1.8-400 K haroratda.

✓ Eng sezgir
✓ χ(T), M(H), ZFC/FC
✗ Juda qimmat

🧲 Evans usuli (NMR)

NMR spektrometr orqali. Eritma ichidagi TMS signalining siljishi o'lchanadi. μ_eff = 0.061·√(Δf·T / c·f₀)

✓ Oddiy NMR yetarli
✓ Tez va oson
✗ Faqat eritmalarda

✅ Asosiy xulosalar

K₃[Fe(CN)₆] — paramagnit, past spinli (LS, t₂g⁵, S=1/2)
Spin-only μ_so = 1.73 μ_B, eksperimental μ_eff ≈ 2.3 μ_B
Farq spin-orbital coupling (λ ≠ 0) va Kotani effekti tufayli
Curie-Weiss θ ≈ −5 K — kuchsiz antiferromagnit o'zaro ta'sir
EPR: rombik simmetriya (g₁=2.76, g₂=2.20, g₃=2.00) — Yahn-Teller buzilishi
Evans usuli NMR orqali μ_eff ni tez va arzon aniqlash imkonini beradi
K₄[Fe(CN)₆] (Fe²⁺, t₂g⁶) — diamagnit (μ = 0), magnit tahlil uchun ideal kontrol namunadir

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