How to Calculate Spin-Only Magnetic Moment | Chemistry

This section explains why experimental magnetic moments often differ from spin-only values due to orbital contribution. When experimental μ > spin-only μ, it indicates positive orbital contribution; when experimental μ < spin-only μ, it indicates negative orbital contribution. Orbital contribution occurs when: (1) Ground state has orbital degeneracy; (2) Ligand field does not completely quench orbital angular momentum; (3) Energy gap between ground and excited states is small enough for thermal population. The spin-orbit coupling constant (λ) determines contribution magnitude and sign. For 3d metals, λ is small (100-1000 cm⁻¹), making orbital contribution often negligible. For 4d and 5d metals, λ is larger, making orbital contribution more significant. For [Fe(H₂O)₆]²⁺, λ ≈ 400 cm⁻¹, giving positive orbital contribution (experimental μ = 5.4 BM > spin-only μ = 4.90 BM). For [Fe(CN)₆]³⁻, λ ≈ -400 cm⁻¹, giving negative orbital contribution (experimental μ = 2.3 BM < spin-only μ = 1.73 BM).

Spin-orbit coupling in coordination chemistry occurs when excited states with orbital angular momentum mix with the ground state term, causing deviations from the spin-only formula; this mixing happens when the energy gap between the ground state and excited states is small, allowing the ground state to acquire an orbital contribution to its magnetic moment, which is calculated using the formula μ_eff = μ_s × [1 - α(Λ/Δ₀)] where μ_s is the spin-only value, α is the spin-orbit coefficient, Λ is the spin-orbit coupling constant, and Δ₀ is the crystal field splitting energy.

For 4d, 5d transition metals and copper complexes, spin-only predictions deviate due to significant spin-orbit coupling. The correction formula is μ_eff = μ_spin × [1 - (αλ/Δ)], where α is the spin-orbit coupling constant, λ is the spin-orbit coupling parameter, and Δ is crystal field splitting energy (calculated from absorption wavelength: Δ = hc/λ). This explains why Cu²⁺ complexes show μ > 1.73 BM instead of exactly √3. Recognizing when to apply this formula versus others is critical for exam success.

The spin-only magnetic moment formula is μs = √[4S(S+1)]. However, when the orbital angular momentum (L) is non-zero, there is deviation from the spin-only value. For d8 (Ni2+) and d7 (Co2+) configurations, L values are 2 and 3 respectively. Higher L values cause greater deviation because the orbital and spin angular momenta couple, requiring consideration of total angular momentum (J) for accurate magnetic moment calculation.

The effective magnetic moment combines orbital and spin contributions: μ_effective = √(μ_L² + μ_S²). For free ions, the Landé g-factor accounts for LS coupling: μ_effective = √[J(J+1)] × g_J × Bohr magneton. For coordination complexes, the spin-only formula μ = √[n(n+2)] × Bohr magneton applies when orbital quenching occurs. Orbital quenching suppresses orbital angular momentum in crystal fields, leaving only spin contributions. Deviations from spin-only behavior occur when quenching relaxes or LS coupling appears, allowing orbital contributions to emerge. Orbital contribution happens when t₂g or e_g orbitals are asymmetrically filled; symmetric filling prevents it. The d⁵ high-spin configuration uniquely shows no orbital contribution because symmetric filling cancels orbital effects.

For coordination number 4, geometry depends on ligand field strength: (1) Square planar (dsp2 hybridization) occurs with strong field ligands (e.g., [Ni(CN)4]2- is diamagnetic); (2) Tetrahedral (sp3 hybridization) occurs with weak field ligands (e.g., [NiCl4]2- is paramagnetic). For Ni2+ (3d8), the ligand field strength determines whether the complex is square planar or tetrahedral.

Magnetic susceptibility is a material property that indicates how strongly a material responds to a magnetic field. In coordination compounds, magnetic susceptibility measurements are used to determine the number of unpaired electrons experimentally. The magnetic moment (μ) is related to the number of unpaired electrons (n) by the formula μ = √[n(n+2)] BM. This experimental method is used to determine the electronic configuration of the central metal ion.

Hybridization determines the geometry of coordination compounds. sp³ hybridization results in tetrahedral geometry, sp³d in trigonal bipyramidal, and sp³d² in octahedral geometry. Magnetic properties depend on the number of unpaired electrons: paramagnetic compounds have unpaired electrons and are attracted to magnetic fields, while diamagnetic compounds have all electrons paired and are not attracted. For example, [NiCl₄]²⁻ is paramagnetic (unpaired electrons) with tetrahedral geometry, while [Fe(CN)₆]³⁻ is diamagnetic (all electrons paired) with octahedral geometry.

The magnetic behavior of coordination compounds depends on the number of unpaired electrons: (1) Octahedral geometry requires coordination number 6, (2) Tetrahedral geometry requires coordination number 4, (3) Square planar geometry requires coordination number 4 with dsp^2 hybridization. Strong field ligands cause pairing of electrons (low spin), while weak field ligands result in unpaired electrons (high spin).

For coordination compounds with coordination number 4, the geometry can be determined by the electronic configuration of the central metal ion: if the configuration is d0-d7 or d10, the complex is tetrahedral (sp3 hybridized), while if it is d8 or d9, the complex is square planar (dsp2 hybridized). This rule applies to 3d series metals with +2 charge, and for 4d/5d series metals with +2, +3, or +4 charge, the complexes are always square planar regardless of ligand field strength.

Cooperative magnetism describes collective behavior of magnetic moments in materials. In ferromagnetism, neighboring spins align parallel, producing strong net magnetization. The Curie temperature (Tc) marks the transition from ferromagnetic to paramagnetic behavior. In antiferromagnetism, neighboring spins align antiparallel, causing net magnetization to cancel. The Néel temperature (TN) marks this transition. Cooperative effects significantly increase magnetic susceptibility compared to non-cooperative paramagnetic behavior. Temperature reduction promotes cooperative alignment, while thermal energy disrupts it.

Ferromagnetic materials (Fe, Co, Ni, Gd) have domains of aligned magnetic moments that strongly attract to magnets. Antiferromagnetic materials (MnO) have oppositely oriented domains that cancel each other's magnetic moments. Ferrimagnetic materials have unequal parallel and antiparallel domain alignments, showing weaker attraction than ferromagnetic materials (examples: magnetite, ferrites like MgFe₂O₄ and ZnFe₂O₄).

Magnetic materials respond to external fields through attraction (paramagnetic) or repulsion (diamagnetic). Paramagnetism arises from unpaired electron spins and orbital motions, while diamagnetism results from paired electrons. Magnetic susceptibility measures response, with positive values indicating paramagnetism and negative values indicating diamagnetism. Curie's law describes the inverse temperature dependence of paramagnetic susceptibility (χ ∝ 1/T). Cooperative magnetism emerges when magnetically active atoms interact in solids, manifesting as ferromagnetism (parallel alignment), antiferromagnetism (opposite alignment), or ferrimagnetism (unequal alignment). Critical temperatures (Curie and Néel) mark transitions between ordered and disordered magnetic states. The effective magnetic moment combines orbital and spin contributions: μ_effective = √(μ_L² + μ_S²). For free ions, the Landé g-factor accounts for LS coupling: μ_effective = √[J(J+1)] × g_J × Bohr magneton.

Solids exhibit different magnetic properties: (1) Paramagnetic - attracted by magnetic field due to unpaired electrons (O2); (2) Diamagnetic - weakly repelled by magnetic field, all electrons paired (H2O, NaCl, C6H6, TiO2, P2O3); (3) Ferromagnetic - strongly attracted, retain magnetism, domains form (Fe, Co, Ni, CrO2); (4) Antiferromagnetic - opposite moments cancel (MnO, FeO, Fe2O3); (5) Ferrimagnetic - unequal opposite moments. Domains are regions with aligned magnetic moments. Curie temperature (Tc) is when ferromagnetic materials become paramagnetic.

Ferromagnetic substances are strongly attracted by magnetic fields due to aligned magnetic moments in domains. They retain magnetization after external field removal, making them suitable for permanent magnets. Examples include iron, cobalt, and nickel. Antiferromagnetic substances have unpaired electrons but zero net magnetic moment because adjacent magnetic moments align in opposite directions with equal magnitude, causing complete cancellation. Examples include MnO, FeO, and NiO. Ferrimagnetic substances are weakly attracted due to unequal numbers of parallel and antiparallel magnetic moments. Examples include Fe3O4 and ferrites. When parallel and antiparallel moments become equal, net magnetization becomes zero.