Mechanism and temperature dependence

Variation of magnetic susceptibility with temperature

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 ${\displaystyle e^{-\Delta E/kT}}$, 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.^[10]

${\displaystyle \chi ={C \over T}}$

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^[11]

${\displaystyle C={\frac {Ng^{2}S(S+1)\mu _{B}^{2}}{3k}}}$

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.

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

T_c is the Curie temperature. The Curie-Weiss law will apply only when the temperature is well above the Curie temperature. 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^[12] as

${\displaystyle \mu _{\text{eff}}=\mathrm {constant} {\sqrt {T\chi }}}$

Where C has CGS units cm³ mol⁻¹ K, μ_eff is

${\displaystyle \mu _{\text{eff}}={\sqrt {3k \over N\mu _{B}^{2}}}{\sqrt {T\chi }}\approx 2.82787{\sqrt {T\chi }}}$

Where C has SI units m³ mol⁻¹ K, μ_eff is

${\displaystyle \mu _{\text{eff}}={\sqrt {3k \over N\mu _{0}\mu _{B}^{2}}}{\sqrt {T\chi }}\approx 797.727{\sqrt {T\chi }}}$

The quantity μ_eff is effectively dimensionless, but is often stated as in units of Bohr magneton (μ_B).^[12]

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.^[13] It is easier to observe in compounds of the heavier elements, such as uranyl compounds.

Exchange interactions

Copper(II) acetate dihydrate^[14]

Ferrimagnetic ordering in 2 dimensions

Antiferromagnetic ordering in 2 dimensions

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(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.^[14] It is believed that this is an example of superexchange, mediated by the oxygen and carbon atoms of the acetate ligands.^[15] Other dimers and clusters exhibit exchange behaviour.^[16]

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.^[17]

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.^[18]

Spin-only formula

Orbital angular momentum is generated when an electron in an orbital of a degenerate set of orbitals is moved to another orbital in the set by rotation. In complexes of low symmetry certain rotations are not possible. In that case the orbital angular momentum is said to be "quenched" and ${\displaystyle {\vec {L}}}$ is smaller than might be expected (partial quenching), or zero (complete quenching). There is complete quenching in the following cases. Note that an electron in a degenerate pair of d_x²–y² or d_z² orbitals cannot rotate into the other orbital because of symmetry.^[20]

Quenched orbital angular momentum

dn	Octahedral			Tetrahedral
		high-spin	low-spin
d1				e1
d2				e2
d3	t2g3
d4		t2g3eg1
d5		t2g3eg2
d6			t2g6	e3t23
d7			t2g6eg1	e4t23
d8	t2g6eg2
d9	t2g6eg3

legend: t_2g, t₂ = (d_xy, d_xz, d_yz). e_g, e = (d_x²–y², d_z²).

When orbital angular momentum is completely quenched, ${\displaystyle {\vec {L}}=0}$ and the paramagnetism can be attributed to electron spin alone. The total spin angular momentum is simply half the number of unpaired electrons and the spin-only formula results.

${\displaystyle \mu _{\text{eff}}={\sqrt {n(n+2)}}\mu _{B}}$

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.^[21]

Ion	Number of unpaired electrons	Spin-only moment /μB	observed moment /μB
Ti3+	1	1.73	1.73
V4+	1	1.73	1.68–1.78
Cu2+	1	1.73	1.70–2.20
V3+	2	2.83	2.75–2.85
Ni2+	2	2.83	2.8–3.5
V2+	3	3.87	3.80–3.90
Cr3+	3	3.87	3.70–3.90
Co2+	3	3.87	4.3–5.0
Mn4+	3	3.87	3.80–4.0
Cr2+	4	4.90	4.75–4.90
Fe2+	4	4.90	5.1–5.7
Mn2+	5	5.92	5.65–6.10
Fe3+	5	5.92	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 d³, d⁴, d⁸ and d⁹ 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.^[22]

Low-spin complexes

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.

High and low -spin octahedral complexes

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

With one unpaired electron μ_eff values range from 1.8 to 2.5 μ_B and with two unpaired electrons the range is 3.18 to 3.3 μ_B. 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 μ_B at 80 K to more than 4 μ_B above 300 K.^[23]

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.^[24]

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

Russell-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

${\displaystyle {\vec {L}}=\sum _{i}{\vec {l}}_{i}}$

${\displaystyle {\vec {S}}=\sum _{i}{\vec {s}}_{i}}$

${\displaystyle {\vec {J}}={\vec {L}}+{\vec {S}}}$

and the calculated magnetic moment is given by

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

Magnetic properties of trivalent lanthanide compounds^[25]

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 /μB	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 /μB	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.

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

This means that it is difficult to calculate the effective moment. For example, uranium(IV), f², in the complex [UCl₆]²⁻ has a measured effective moment of 2.2 μ_B, which includes a contribution from temperature-independent paramagnetism.^[26]