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Magnetochemistry is concerned with the magnetic properties of chemical compounds. 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 electron 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, ferrimagnetism or antiferrimagnetism, depending on the relative orientations of the individual spins.

Magnetic susceptibility

Main article: 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 $\chi _{v}$ is defined by the relationship

${\overrightarrow {M}}=\chi _{v}{\overrightarrow {H}}$

where, M is the magnetization of the material (the magnetic dipole moment per unit volume), measured in amperes per meter ( SI units), and H 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. (χ_mol) measured in m³•mol⁻¹ (SI) or cm³•mol⁻¹ (CGS) and is defined as

$\chi _{\text{mol}}=M\chi _{v}/\rho$

where ρ is the density in kg•m⁻³ (SI) or g•cm⁻³ (CGS) and M is molar mass in kg•mol⁻¹ (SI) or g•mol⁻¹ (CGS).

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, HgCo(NCS)₄. The Evans balance. is a torsion balance which uses a variable secondary magnet to bring the position of the sample back to its initial position. It, too is calibrated against HgCo(NCS)₄. 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 type of interaction.

1. Diamagnetism. 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. 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.
2. 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 (magnetic moment has both magnitude and direction). This gives rise to ferromagnetism, antiferromagnetism, ferrimagnetism or antiferrimagnetism. 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 a 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" 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. A detailed calculation is shown in Figgis and Lewis, p.417.

Paramagnetism

Temperature dependence

Some substances obey the Curie law with susceptibility being inversely proportional to temperature, in kelvins.

$\chi ={C \over T}$

The proportionality constant, C, is known as the Curie constant. The quantum mechanical conditions required for Curie's law to be obeyed are rather stringent. When these conditions are not met the Curie-Weiss law may apply rather then the Curie law.

$\chi ={\frac {C}{T-T_{c}}}$

T_c is the Curie temperature. The Curie-Weiss law will apply only when the temperature is well above the Curie temperature.

Effective magnetic moment

When the Curie law is obeyed, the product of susceptibility and temperature is a constant. The effective magnetic moment, μ_eff is then calculated as

$\mu _{eff}=constant{\sqrt {\chi \times T}}$

The constant is equal to 2.84 when susceptibility is measured in CGI units. The unit for μ_eff is Bohr magneton. Thus for substances that obey the Curie law, the effective magnetic moment is independent of temperature. For other substances μ_eff will be temperature dependent, but the dependence will be small if 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 interaction between the compound and the magnetic field. 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

Copper(II) acetate dihydrate

Ferrimagnetic ordering

One of the simplest systems to exhibit the result of exchange interactions is crystalline Copper(II) acetate, Cu₂(OAc)₄(H₂O)₂. As the formula indicates, it contains two copper(II) 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 now believed that this is an example of super-exchange, 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, ferrimagnetism or antiferrimagnetism, depending on the nature and relative orientations of the individual spins.

Theoretical calculation for complexes of metal ions

The effective magnetic moment for a compound containing a metal ion with one or more unpaired electrons depends on the total orbital and spin angular momentum of the unpaired electrons, ${\overrightarrow {L}}$ and ${\overrightarrow {S}}$, respectively. "Total" in this context means "vector sum". In the approximation that the electronic states of the metal ions are determined by Russel-Saunders coupling and that spin-orbit coupling is negligible, the magnetic moment is given by

$\mu _{eff}={\sqrt {{\overrightarrow {L}}({\overrightarrow {L}}+1)+4{\overrightarrow {S}}({\overrightarrow {S}}+1)}}$

Spin-only formula

If the paramagnetism can be attributed to electron spin alone, the total orbital angular momentum is zero and the total spin angular momentum is simply half the number of unpaired electrons. The spin-only formula results.

$\mu _{eff}={\sqrt {n(n+2)}}B.M.$

where n is the number of unpaired electrons. The spin-only formula is a good first approximation for high-spin complexes of first-row transition metals.

Ion	Number of unpaired electrons	Spin-only moment /B.M.	observed moment /B.M.
Ti3+	1	1.73	1.73
V4+	1		1.68-1.78
Cu2+	1		1.70-2.20
V3+	2	2.83	2.75-2.85
Ni2+	2		2.8-3.5
V2+	3	3.88	3.80-3.90
Cr3+	3		3.70-3.90
Co2+	3		4.3-5.0
Mn4+	3		3.80-4.0
Cr2+	4	4.90	4.75-4.90
Fe2+	4		5.1-5.7
Mn2+	5	5.92	5.65-6.10
Fe3+	5		5.7-6.0

The small deviations from the spin-only formula may result from the neglect of orbital angular momentum or of spin-orbit coupling. For example, tetrahedral complexes tend to show larger deviations from the spin-only formula than octahedral complexes of the same ion, because "quenching" of the orbital contribution is less effective in the tetrahedral case.

Low-spin complexes

Main article: crystal field theory

Crystal field diagram for octahedral low-spin d⁵

Crystal field diagram for octahedral high-spin d⁵

According to crystal field theory, the d orbitals of a transition metal ion in an octahedal complex are split into two groups in a crystal field. If the splitting is large enough to overcome the energy needed to place electrons in the same orbital, with opposite spin, a low-spin complex will result.

Number of unpaired electrons, octahedral complexes

d-count	high-spin	low-spin	examples
d4	4	2	Cr2+ , Mn3+
d5	5	1	Mn2+, Fe3+
d6	4	0	Fe2+, Co3+
d7	3	1	Co2+

Note that low-spin complexes of Fe²⁺ and Co³⁺ are diamagnetic. Another group of complexes that are diamagnetic are square-planar complexes of d⁸ ions such as Ni²⁺ and Rh⁺ and Au³⁺.

Spin cross-over

When the energy difference between the high-spin and low-spin states is comparable to kT (k is the Boltzmann constant and T the temperature) an equilibrium is established between the spin states, involving what have been called "electronic isomers". Tris-dithiocarbamato iron(III), Fe(S₂CNR₂)₃, is a well-documented example. The effective moment varies from a typical d⁵ low-spin value of 2.25 B.M. at 80K to more than 4 B.M above 300K.

2nd and 3rd row transition metals

Crystal field splitting is larger for complexes of the heavier transition metals than for the transition metals discussed above. A consequence of this is that low-spin complexes are much more common. Spin-orbit coupling constants, ζ, are also larger and cannot be ignored, even in elementary treatments. The magnetic behaviour has been summarized, as below, together with an extensive table of data.

d-count	kT/ζ=0.1 μeff	kT/ζ=0 μeff	Behaviour with large spin-orbit coupling constant, ζnd
d1	0.63	0	μeff varies with T1/2
d2	1.55	1.22	μeff varies with T, approximately
d3	3.88	3.88	Independent of temperature
d4	2.64	0	μeff varies with T1/2
d5	1.95	1.73	μeff varies with T, approximately

Lanthanides and actinides

Russel-Saunders coupling, LS coupling, applies to the lanthanide ions, crystal field effects can be ignored, but spin-orbit coupling is not negligible. Consequently spin and orbital angular momenta have to be combined

${\overrightarrow {L}}=\sum _{i}{\overrightarrow {l}}_{i}$

${\overrightarrow {S}}=\sum _{i}{\overrightarrow {s}}_{i}$

${\overrightarrow {J}}={\overrightarrow {L}}+{\overrightarrow {S}}$

and the calculated magnetic moment is given by

$\mu _{eff}=g{\sqrt {{\overrightarrow {J}}({\overrightarrow {J}}+1)}};g={3 \over 2}+{\frac {{\overrightarrow {S}}({\overrightarrow {S}}+1)-{\overrightarrow {L}}({\overrightarrow {L}}+1)}{2{\overrightarrow {J}}({\overrightarrow {J}}+1)}}$

Magnetic properties of trivalent lanthanide compounds

lanthanide	Ce	Pr	Nd	Pm	Sm	Eu	Gd	Tb	Dy	Ho	Er	Tm	Yb	Lu
Number of unpaired électrons	1	2	3	4	5	6	7	6	5	4	3	2	1	0
calculated moment /B.M.	2.54	3.58	3.62	2.68	0.85	0	7.94	9.72	10.65	10.6	9.58	7.56	4.54	0
observed moment /B.M.	2.3-2.5	3.4-3.6	3.5-3.6		1.4-1.7	3.3-3.5	7.9-8.0	9.5-9.8	10.4-10.6	10.4-10.7	9.4-9.6	7.1-7.5	4.3-4.9	0

In actinides spin-orbit coupling is strong and the coupling approximates to j j coupling.

${\overrightarrow {J}}=\sum _{i}{\overrightarrow {j}}_{i}=\sum _{i}({\overrightarrow {l}}_{i}+{\overrightarrow {s}}_{i})$

This means that it is difficult to calculate the effective moment. To give an example, uranium(IV), f², in the complex [UCl₆]^2- has a measured effective moment of 2.2 B.M., which includes a contribution from temperature-independent paramagnetism.

Main group elements and organic compounds

Very few compounds of main group elements are paramagnetic. Notable examples include: oxygen, O₂; nitric oxide, NO; nitrogen dioxide, NO₂ and chlorine dioxide, ClO₂. In organic chemistry, compounds with an unpaired electron are said to be free radicals. Free radicals, with some exceptions, are short-lived because one free radical will react rapidly with another, so their magnetic properties are difficult to study. However, if the radicals are well separated from each other in a dilute solution in a solid matrix, at low temperature, they can be studied by electron paramagnetic resonance, EPR. Such radicals are generated by irradiation. Extensive EPR studies have revealed much about electron delocalization in free radicals.

Spin labels are long-lived free radicals which can be inserted into organic molecules so that they can be studied by EPR.

Applications

For many years the nature of oxyhemoglobin, Hb-O₂, was highly controversial. It was found experimentally to be diamagnetic. Deoxy-hemoglobin is generally accepted to be a complex of iron in the +2 oxidation state, that is a d⁵ system with a high-spin magnetic moment near to the spin-only value of 5.92 B.M. It was proposed that the iron is oxidized and the oxygen reduced to superoxide.

Fe(II)Hb (high-spin) + O₂ ⇌ [Fe(III)Hb]O₂^-

Pairing up of electrons from Fe³⁺ and O₂^- was then proposed to occur via an exchange mechanism. It has now been shown that in fact the iron(II) changes from high-spin to low-spin when an oxygen molecule donates a pair of electrons to the iron. Whereas in deoxy-hemoglobin the iron atom lies above the plane of the heme, in the low-spin complex the effective ionic radius is reduced and the iron atom lies in the heme plane.

Fe(II)Hb + O₂ ⇌ [Fe(II)Hb]O₂ (low-spin)

This information has an important bearing on research to find artificial oxygen carriers.

The ground state term for the gadolinium ion,Gd³⁺, is ⁸S_7/2. The fact that it is an S state makes it the most suitable for use as a contrast agent for MRI scans even though other lanthanide ion have larger effective moments. The magnetic moments of gadolinium compounds are larger than those of any transition metal ion.

Compounds of gallium(II) were unknown until quite recently, when salts of the dimeric ions such as [Ga₂Cl₆]^2- were synthesized. As the atomic number of gallium is an odd number (31) Ga²⁺ should have an unpaired electron. However, the compounds were found to be diamagnetic. This provides one line of evidence for the existence of a Ga-Ga bond in which all electrons are paired up.

External links

Online available information resources on magnetochemistry

Bibliography

Selwood, P.W. (1943). Magnetochemistry. Interscience Publishers Inc. Available online

Figgis, B.N.; Lewis, J. (1960). "The Magnetochemistry of Complex Compounds". In Lewis. J. and Wilkins. R.G. (ed.). Modern Coordination Chemistry. New York: Wiley.

Earnshaw, Alan (1968). Introduction to Magnetochemistry. Academic Press.

Carlin, R.L. (1986). Magnetochemistry. Springer. ISBN 9783540158165.

Greenwood, Norman N.; Earnshaw, Alan (1997). Chemistry of the Elements (2nd ed.). Butterworth-Heinemann. ISBN 978-0-08-037941-8.

Orchard, A.F. (2003). Magnetochemistry. Oxford Chemistry Primers. Oxford University Press. ISBN 10: 0198792786. {{cite book}}: Check |isbn= value: invalid character (help)

Category:MagnetismCategory:Physical chemistry