Since the bonding of ligands to metals can change the spin state, determination of magnetic properties is important for understanding the properties of metal complexes.
High spin vs low spin
Complexes are high spin if Dq is small : weak field case
Complexes are low spin if Dq is large : strong field case
The spin of the complex is captured in the S quantum number. S is related to the magnetic moment by:
μ = g[S(S+1)]½ μB
g ~ 2 (fundamental constant for free electrons); μB = Bohr magneton
Since each unpaired electron has S = ½
If n = the number of unpaired electrons, then S = n/2
Then, μ = g[n/2(n/2+1)]½ μB
μ = 2[n(n+2)/4]½ μB assuming the g-value for a free electron is maintained in complexes.
μ = [n(n+2)] ½ μB
This allows us to estimate the magnetic moment based of the number of unpaired spins or to determine the spin state if the magnetic moment is measured.
Fundamentally, the magnetic susceptibility is the relationship between the magnetic induction (B), the magnetization (M), and the applied field (Ho).
B = Ho + 4πM
Assuming the vector directions are coincident, then the magnitudes of each quantity can be used. Dividing by Ho gives
B/Ho = 1 + 4πM/Ho = 1 + 4πχv
where χv = M/Ho is a dimensionless quantity known as the magnetic susceptibility per unit volume
B/Ho is known as the magnetic permeability and is the magnetic equivalent of a dielectric constant
Chemists prefer to work with molar quantities, so
χg = gram susceptibility = χv/ρ where ρ is the density
χg typically has units of cm3/g
χ = molar susceptibility = χg×MolWt, where MolWt is the molecular weight of the compound
χ has units of cm3/mol
There are 4 types of magnetic responses that metal complexes may exhibit.
Type
Magnitude (cm3/mol)
External Field Dependence
Diamagnetism
–10–6
Independent
Paramagnetism
0 to +10–4
Independent
Superparamagnetism
0 to +10–3
Dependent, no hysteresis
Ferromagnetism
+10–4 to +10–2
Dependent, with hysteresis
Antiferromagnetism
0 to +10–4
Dependent, with hysteresis
Diamagnetism arises from paired spins and contributes to the susceptibility of all substances.
Paramagnetism arises from unpaired spins and other sources of angular momentum and is normally the parameter of interest.
Superparamagnetism is observed in nanoparticles that are paramagnetic.
Ferromagnetism is a bulk quantity in solids in which all of the microscopic magnetic moments align in the same direction. This is the source of bulk magnetic behavior, such as an iron magnet.
Antiferromagnetism is also a bulk magnetic behavior where all of the microscopic moments align antiparallel.
The diamagnetic contribution to the total magnetic response is nearly constant for any atom or ion. Thus, these values are tabulated and are known as Pascal's Constants. χdiatot = ∑χdiaatom
χdiaatom is the Pascal's Constant for each atom in the molecule and the sum is taken over all atoms.
The paramagnetic susceptibility is found from the total susceptibility: χpara = χ – χdiatot
χ is the measured susceptibility. Since χdia is negative, the consequence is the χpara > χ
In the ideal case, χpara = Nog2β2S(S+1)/3kT where
χpara is the paramagnetic susceptibility
No is Avogadro's constant
g is the g-value
β is the Bohr-Magneton = 9.2741×10–24 J/T (T = tesla)
S is the spin quantum number
k is the Boltzmann constant = 1.3807×10–23 J/K
T is the absolute temperature
Since μ = g[S(S+1)]½ or μ2 = g2S(S+1). Then
χpara = Noμ2β2/3kT
μ = (3kTχpara/Noβ2)½ in units of Bohr-Magnetons
Owing to spin-orbit coupling, the ideal case is rarely attained.
In real systems, the effective magnetic moment is given by
μeff = (3k/Noβ2)½(χparaT)½ = 2.828(χparaT)½
Any deviations from ideal behavior are embodied in the g-value, which becomes an experimental parameter in molecules rather than a fundamental constant as it is for free electrons.
Magnetic moments are, in principal, temperature independent. This means that the susceptibility is inversely related to temperature.
The simplest temperature dependence is the Curie Law, χ = C/T, where C is the Curie constant and T is the absolute temperature. A plot of χ–1 vs T should be linear with a slope of C–1. This can be used to find the effective magnetic moment: μeff = 2.828(C)½.
More frequently, a plot of χ–1 vs T is linear but the intercept is nonzero. This is known as the Curie-Weiss Law: χ = C/(T – θ), where θ is a measure of ferromagnetic or antiferromagnetic interactions at low temperature.
There are many ways to measure the magnetic susceptibility. The classic technique is Faraday's, where the mass of a sample is measured in the absence and presence of a magnetic field. The difference in these mass values is related to magnetic force acting against the gravitational force, which then gives the total susceptibility. The more modern method is to use a SQUID (Superconducting Quantum Interference Device) where the current across a superconductor/insulator/superconductor junction is measured in the presence and absence of the sample. Again, the magnetism of the sample affects the measured current, which can then be used to find the susceptibility.
In solution, the magnetic susceptibility can be measured using Evan's method, which correlates the chemical shift of a reference in the presence and absence of the magnetic material. A paramagnetic compound will create an internal magnetic field that causes the chemical shift to move by many ppm.
The magnetic susceptibility of diamagnetic and paramagnetic compounds are independent of external magnetic field strength. A plot of M vs Ho is linear and the slope of the plot give the susceptibility at that temperature.
Superparamagnetic materials show a nonlinear behavior when M is plotted as a function of Ho. Since χ is the slope of this plot, the susceptibility changes at each field strength. Superparamagnetism is identified by the size of the susceptibility, which at low fields is typically larger than paramagnetism. Further, when the field is reversed the magnetization is reversible. Magnetic moments can reach as high as 10 μB per magnetic site.
Ferromagnetic materials show bulk magnetic behavior and when M is plotted vs. Ho there is a sharp nonlinear rise in M but this asymptotes to some maximum value called the saturation magnetization. When the field is reversed the saturation value is maintained to below Ho = 0 and then eventually drops to the same saturation magnetization but with a negative value. This is known as hysteresis.
Ferromagnetism arises because below some transition temperature, the Curie Temperature (TC), the microscopic magnetic moments all align in the same direction to give a bulk moment: ↑ ↑ ↑ ↑
Antiferromagnetism is similar to ferromagnetism except that the microscopic moments align in an antiparallel fashion: ↑ ↓ ↑ ↓ . The consequence of this is that the field dependence is nonlinear and hysteretic but the magnetic moment is much smaller, close to 0.
Related to this is ferrimagnetism, where there are two microscopic magnetic sites with different spin quantum numbers. The microscopic moments can align parallel or antiparallel: ↑ ↓ ↑ ↓ . The net effect is a bulk ferromagnetic response, although the saturation moment can be much smaller.
Electron Paramagnetic Resonance (EPR) spectroscopy, also known as electron spin resonance spectroscopy (ESR) is a probe magnetic materials that can provide structural, electronic, and magnetic information.
Is analogous to NMR spectroscopy but measures unpaired electron spins rather than unpaired nuclear spins.
Since the magnetic moment of an electron is about 1000 times greater than that of a nucleus, EPR spectroscopy is run in the GHz region rather than the MHz region. Historically, this led to using constant frequency sources and scanning the magnetic field. Further, for resolution reasons, EPR spectra are usually observed as derivative spectra.
In the absence of a magnetic field, a compound with spin S will have 2S+1 degenerate energy levels. With application of an external field this degeneracy is removed and splits into energies determined by ms values.
ΔE = gβHoms
In addition, since the electron density for s electrons (and hybrids using s orbitals) overlap well with the nucleus, there can be coupling between the electron magnetic moment and the nuclear magnetic moment (for nuclei with I ≠ 0), which is called hyperfine coupling (usually denoted by A). This leads to splitting of the main resonance into 2I+1 peaks.
The central peak is defined by the g-factor and is analogous to the chemical shift in NMR spectroscopy. Likewise, the hyperfine coupling is analogous to spin-spin coupling in NMR spectra.
In transition metal complexes, where the symmetry is often lower than Oh, the g-factor and hyperfine constants A are different in different directions, which often complicates spectral interpretation.
Finally, integrating the EPR spectrum gives the magnetic susceptibility, χpara
One of the limitations of EPR spectroscopy on transition metal ions is that typically only systems with half-integer spin (S = 1/2, 3/2, 5/2) are observable at room temperature. Integer spin systems (S = 1, 2, 3) tend to have fast relaxation times, which leads to spectra too broad to be observed except at liquid helium temperatures.
Detailed interpretation of g-values provides support for optical transition assignments. For example, for d3 Oh complexes in the ionic limit it has been shown that g = 2.0023 – 8λ/10Dq, where λ is the spin-orbit coupling constant.