Magnetochemistry

Magnetochemistry is concerned with the magnetic properties of chemical compounds and elements. Magnetic properties arise from the spin and orbital angular momentum of the electrons contained in a compound. Compounds are diamagnetic when they contain no unpaired electrons. Molecular compounds that contain one or more unpaired electrons are paramagnetic. The magnitude of the paramagnetism is expressed as an effective magnetic moment, μ_eff. For first-row transition metals the magnitude of μ_eff is, to a first approximation, a simple function of the number of unpaired electrons, the spin-only formula. In general, spin–orbit coupling causes μ_eff to deviate from the spin-only formula. For the heavier transition metals, lanthanides and actinides, spin–orbit coupling cannot be ignored. Exchange interaction can occur in clusters and infinite lattices, resulting in ferromagnetism, antiferromagnetism or ferrimagnetism depending on the relative orientations of the individual spins.

Magnetic susceptibility

The primary measurement in magnetochemistry is magnetic susceptibility. This measures the strength of interaction on placing the substance in a magnetic field. The volume magnetic susceptibility, represented by the symbol is defined by the relationship
where, is the magnetization of the material, measured in amperes per meter, and is the magnetic field strength, also measured in amperes per meter. Susceptibility is a dimensionless quantity. For chemical applications the molar magnetic susceptibility is the preferred quantity. It is measured in m³·mol⁻¹ or cm³·mol⁻¹ and is defined as
where ρ is the density in kg·m⁻³ or g·cm⁻³ and M is molar mass in kg·mol⁻¹ or g·mol⁻¹.
A variety of methods are available for the measurement of magnetic susceptibility.

With the Gouy balance the weight change of the sample is measured with an analytical balance when the sample is placed in a homogeneous magnetic field. The measurements are calibrated against a known standard, such as mercury cobalt thiocyanate, HgCo₄. Calibration removes the need to know the density of the sample. Variable temperature measurements can be made by placing the sample in a cryostat between the pole pieces of the magnet.
The Evans balance. is a torsion balance which uses a sample in a fixed position and a variable secondary magnet to bring the magnets back to their initial position. It, too, is calibrated against HgCo₄.
With a Faraday balance the sample is placed in a magnetic field of constant gradient, and weighed on a torsion balance. This method can yield information on magnetic anisotropy.
SQUID is a very sensitive magnetometer.
For substances in solution NMR may be used to measure susceptibility.
Types of magnetic behaviour

When an isolated atom is placed in a magnetic field there is an interaction because each electron in the atom behaves like a magnet, that is, the electron has a magnetic moment. There are two types of interaction.

Diamagnetism. When placed in a magnetic field the atom becomes magnetically polarized, that is, it develops an induced magnetic moment. The force of the interaction tends to push the atom out of the magnetic field. By convention diamagnetic susceptibility is given a negative sign. Very frequently diamagnetic atoms have no unpaired electrons ie each electron is paired with another electron in the same atomic orbital. The moments of the two electrons cancel each other out, so the atom has no net magnetic moment. However, for the ion Eu³⁺ which has six unpaired electrons, the orbital angular momentum cancels out the electron angular momentum, and this ion is diamagnetic at zero Kelvin.
Paramagnetism. At least one electron is not paired with another. The atom has a permanent magnetic moment. When placed into a magnetic field, the atom is attracted into the field. By convention paramagnetic susceptibility is given a positive sign.

When the atom is present in a chemical compound its magnetic behaviour is modified by its chemical environment. Measurement of the magnetic moment can give useful chemical information.
In certain crystalline materials individual magnetic moments may be aligned with each other. This gives rise to ferromagnetism, antiferromagnetism or ferrimagnetism. These are properties of the crystal as a whole, of little bearing on chemical properties.

Diamagnetism

Diamagnetism is a universal property of chemical compounds, because all chemical compounds contain electron pairs. A compound in which there are no unpaired electrons is said to be diamagnetic. The effect is weak because it depends on the magnitude of the induced magnetic moment. It depends on the number of electron pairs and the chemical nature of the atoms to which they belong. This means that the effects are additive, and a table of "diamagnetic contributions", or Pascal's constants, can be put together. With paramagnetic compounds the observed susceptibility can be adjusted by adding to it the so-called diamagnetic correction, which is the diamagnetic susceptibility calculated with the values from the table.

Paramagnetism

Mechanism and temperature dependence

A metal ion with a single unpaired electron, such as Cu²⁺, in a coordination complex provides the simplest illustration of the mechanism of paramagnetism. The individual metal ions are kept far apart by the ligands, so that there is no magnetic interaction between them. The system is said to be magnetically dilute. The magnetic dipoles of the atoms point in random directions. When a magnetic field is applied, first-order Zeeman splitting occurs. Atoms with spins aligned to the field slightly outnumber the atoms with non-aligned spins. In the first-order Zeeman effect the energy difference between the two states is proportional to the applied field strength. Denoting the energy difference as ΔE, the Boltzmann distribution gives the ratio of the two populations as, where k is the Boltzmann constant and T is the temperature in kelvins. In most cases ΔE is much smaller than kT and the exponential can be expanded as 1 – ΔE/kT. It follows from the presence of 1/T in this expression that the susceptibility is inversely proportional to temperature.
This is known as the Curie law and the proportionality constant, C, is known as the Curie constant, whose value, for molar susceptibility, is calculated as
where N is the Avogadro constant, g is the Landé g-factor, and μ_B is the Bohr magneton. In this treatment it has been assumed that the electronic ground state is not degenerate, that the magnetic susceptibility is due only to electron spin and that only the ground state is thermally populated.
While some substances obey the Curie law, others obey the Curie-Weiss law.
T_c is the Curie temperature. The Curie-Weiss law will apply only when the temperature is well above the Curie temperature. At temperatures below the Curie temperature the substance may become ferromagnetic. More complicated behaviour is observed with the heavier transition elements.

Effective magnetic moment

When the Curie law is obeyed, the product of molar susceptibility and temperature is a constant. The effective magnetic moment, μ_eff is then defined as
Where C has CGS units cm³ mol⁻¹ K, μ_eff is
Where C has SI units m³ mol⁻¹ K, μ_eff is
The quantity μ_eff is effectively dimensionless, but is often stated as in units of Bohr magneton.
For substances that obey the Curie law, the effective magnetic moment is independent of temperature. For other substances μ_eff is temperature dependent, but the dependence is small if the Curie-Weiss law holds and the Curie temperature is low.

Temperature independent paramagnetism

Compounds which are expected to be diamagnetic may exhibit this kind of weak paramagnetism. It arises from a second-order Zeeman effect in which additional splitting, proportional to the square of the field strength, occurs. It is difficult to observe as the compound inevitably also interacts with the magnetic field in the diamagnetic sense. Nevertheless, data are available for the permanganate ion. It is easier to observe in compounds of the heavier elements, such as uranyl compounds.

Exchange interactions

Exchange interactions occur when the substance is not magnetically dilute and there are interactions between individual magnetic centres. One of the simplest systems to exhibit the result of exchange interactions is crystalline copper acetate, Cu₂₄₂. As the formula indicates, it contains two copper ions. The Cu²⁺ ions are held together by four acetate ligands, each of which binds to both copper ions. Each Cu²⁺ ion has a d⁹ electronic configuration, and so should have one unpaired electron. If there were a covalent bond between the copper ions, the electrons would pair up and the compound would be diamagnetic. Instead, there is an exchange interaction in which the spins of the unpaired electrons become partially aligned to each other. In fact two states are created, one with spins parallel and the other with spins opposed. The energy difference between the two states is so small their populations vary significantly with temperature. In consequence the magnetic moment varies with temperature in a sigmoidal pattern. The state with spins opposed has lower energy, so the interaction can be classed as antiferromagnetic in this case. It is believed that this is an example of superexchange, mediated by the oxygen and carbon atoms of the acetate ligands. Other dimers and clusters exhibit exchange behaviour.
Exchange interactions can act over infinite chains in one dimension, planes in two dimensions or over a whole crystal in three dimensions. These are examples of long-range magnetic ordering. They give rise to ferromagnetism, antiferromagnetism or ferrimagnetism, depending on the nature and relative orientations of the individual spins.
Compounds at temperatures below the Curie temperature exhibit long-range magnetic order in the form of ferromagnetism. Another critical temperature is the Néel temperature, below which antiferromagnetism occurs. The hexahydrate of nickel chloride, NiCl₂·6H₂O, has a Néel temperature of 8.3 K. The susceptibility is a maximum at this temperature. Below the Néel temperature the susceptibility decreases and the substance becomes antiferromagnetic.