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Dr M.D. SPICER
4. MAGNETISM
4.1 Origins of Magnetic Behaviour
The magnetic properties of materials can be very useful in understanding their chemistry. It should
be noted that although the origins of magnetism are based in single magnetic centres, unlike the
other techniques we have studied, the interaction of these centres gives rise to bulk properties which
can be measured.
Let us consider what happens when a substance is placed in an external magnetic field of strength H
– a magnetic field of strength B is induced in the material given by the following expression:
B = H + 4πI
where I is the intensity of magnetisation. Traditionally, magnetism is discussed in terms of
susceptibilities which may be defined as follows:
Volume susceptibility

= I/H
Gram susceptibility,  =

/
M
=
 . M.Wt.
Molar susceptibility,

(where  = density)
M is also defined by the Lande equation, such that
M
N A 2

3kT
Where NA = Avagadro’s number, μ = magnetic moment, k = Boltzmann constant and T =
temperature. Rearranging this equation gives:
 3kT 
. M
 NA 
  
μ is usually referred to as the effective magnetic moment, and by enumerating the constants the
equation becomes
 eff  2.83  M .T
There are two major types of behaviour when a substance is placed in a magnetic field:
If B < H the substance is said to be diamagnetic. It is repelled by the field and  is negative.
Diamagnetism occurs as a result of electrons in closed shells circulating under the influence of the
external magnetic field to generate a local field which is in opposition to the external field.
If B > H the substance is said to be paramagnetic. It is attracted by the field and  is positive.
Paramagnetism arises from the interaction of the spin and angular momentum of unpaired electrons
with the external field. (If the spins are paired the spin and angular momentum cancel each other
out). Note that paramagnetism is roughly 3 orders of magnitude greater than diamagnetism – hence,
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Dr M.D. SPICER
although paramagnetic species will have a diamagnetic contribution to the magnetic moment due to
the core electrons in closed shells, the magnitude of this effect is relatively small (ca 1 – 5 %).
4.2 Types of Magnetism
4.2.1 Paramagnetism
Paramagnetic species can display a number of subclasses of magnetic behaviour. In simple
paramagnetic materials the magnetic centres are separated by diamagnetic species and are said to be
magnetically dilute. This means that the individual magnetic centres do not interact with one
another. In such species the magnetic moment is given by the expression
  4S S  1  LL  1
Where S is the total spin (ie the sum of values of ms) and L is the total orbital angular momentum
(ie the sum of the values of ml).
For many compounds the sum of the orbital angular momentum is zero, so we can use the spin-only
formula as a good approximation to the magnetic moment as given by:
 spinonly  4S S  1
This can also be expressed in terms of the number of unpaired electrons, n (since n = 2S)
 spinonly  nn  2
4.2.2 Ferromagnetism and Antiferromagnetism
There are, however, many compounds in which the neighbouring magnetic centres can interact (or
couple) with each other, leading to magnetic ordering of the bulk material. This can take two main
forms, ferromagnetism and antiferromagnetism (you will also discuss a third form, ferrimagnetism,
in Dr Kennedy’s Solid State Chemistry course).
If the magnetic moments all line up in parallel (as shown below), then the species is said to be
ferromagnetic.

In ferromagnetic compounds μeff is generally much greater than μspin-only due to the cooperative
effect of the spins coupling in parallel which reinforces the bulk magnetic moment. The most
common ferromagnetic materials are metals such as iron and cobalt and their alloys. CrO2 is a rare
example of a binary compound which is ferromagnetic at room temperature.
If the magnetic moments line up antiparallel with respect to one another (as shown below) then the
substance in said to be antiferromagnetic.

In antiferromagnetic compounds μeff tends to be somewhat less than μspin-only since the coupling of
the spins into an anti-parallel arrangement results in the individual magnetic moments cancelling
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one another out.
materials.
Dr M.D. SPICER
Antiferromagnetic materials are much more common than ferromagnetic
Many metal halides and oxides are antiferromagnetic. Examples include:
α-TiCl3
μeff = 1.31 B.M. (μspin-only = 1.73 B.M.)
VCl3
μeff = 2.42 B.M. (μspin-only = 2.83 B.M.)
The coupling of the magnetic moments arises via a mechanism known as super-exchange, in which
the ligands help to order the spins on the metal centres.
Antiferromagetic
Ferromagnetic
Ferromagnetic and antiferromagnetic coupling is not restricted to ionic lattice type compounds, but
can also be observed in polymetallic molecular compounds. So simple dimers and larger clusters
can have the magnetic moments from neighbouring metal ions coupled via bridging ligands giving
rise to discrete magnetic molecular entities which exhibit a degree of magnetic ordering.
The field of molecular magnetism is rapidly growing and it beyond the scope of this course.
However, there is a a lot of information available on the web which gives a background to the
subject. If you want too find out more you can try the following website:
http://www.molmag.de/
Research group interested in molecular magnets tend to form large cluster molecules with multiple
paramagnetic centres which can couple together to give extremely strong ferromagnetic coupling.
These species are touted as having potential in molecular quantum computing and in refrigeration
applications.
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4.2.3 Temperature Dependence of Magnetic Susceptibilities.
The magnetic moments (and susceptibilities) of paramagnetic materials on the whole show marked
temperature dependence as shown in the idealised plot below.
At high temperatures both ferromagnetic and antiferromagnetic materials behave as simple
paramagnets, any magnetic coupling is broken down by thermal effects (sometimes referred to as
thermal randomisation). As the temperature is lowered, the effect of coupling can overcome thermal
randomisation and magnetic ordering is observed. This means that there is a significant rise
(ferromagnetic) or fall (anti-ferromagnetic) of the susceptibility and magnetic moment as the
coupling takes effect. The points at which this occurs are known as the Curie point (TC, for
ferromagnetic compounds) and the Néel point (TN, for antiferromagnetic compounds).
4.3 Deviations from Spin-Only Behaviour
The spin-only description of magnetic moments often holds good for transition metal compounds,
but when the electrons in the metal valence shell have non-zero orbital angular momentum then we
see deviations of the observed magnetic moments from the spin only values.
4.3.1 Orbital Angular Momentum
How does this orbital angular momentum arise? In order for an electron to have orbital angular
momentum it must be possible to transform it from the orbital it occupies into an exactly equivalent
and degenerate orbital by means of a rotation and it must retain the same spin quantum number.
Such a case can occur in the p-orbitals, where px, py and pz are identical in form and can be
transformed into one-another by simple 90º rotation as seen in the diagram below.
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If there are one, two, four or five electrons in the p-orbitals they will have orbital angular
momentum – in each case an electron can be transferred from one orbital to another by a simple
rotation. Consider the p1 case. The electron can reside in px, py or pz orbital. Each is equivalent in
shape and energy and so there are three possible identical arrangements of the electrons. In the case
of three electrons they will all have parallel spin and thus to move an electron from one orbital to
another would require a change of spin (and thus energy). Similarly, with six electrons the porbitals are full and therefore transfer from one orbital to another is not possible.
In the d-orbitals, dxy, dxz, dyz and dx2-y2 are all equivalent in form (shape). In the free ion, all are
degenerate and therefore there is potential for orbital angular momentum by transferring an electron
between any of these orbitals, but in metal complexes the dx2-y2 orbital is never degenerate with the
other three orbitals. Remember the orbital splitting in tetrahedral and octahedral complexes:
The set of orbitals with t2g symmetry consist of the dxy, dxz and dyz orbitals, while the eg set consist
of dx2-y2 and dz2 orbitals. It can thus be seen that the dx2-y2 orbital is degenerate with the dz2 orbital,
which is of different shape and therefore electrons in the eg orbitals cannot have orbital angular
momentum. So, in transition metal complexes electrons in the eg (octahedral) or e (tetrahedral) do
not give rise to orbital angular momentum whereas electrons in t2g (octahedral) or t2 (tetrahedral)
sets of orbitals can give rise to an orbital momentum contribution, but only if the electronic ground
state has a T-term (ie it is triply degenerate). Consider the following diagram showing electron
arrangements in an octahedral complex:
In the first case (d3) there is only one possible arrangement of the electrons (A term, singly
degenerate) while in the second case (d1) there are three possible arrangements of the electrons (Tterm – triply degenerate). We can treat the tetrahedral complexes in similar fashion.
Consider the table below, which lists the number of d-orbitals, the electron configuration, the
ground term and the presence or absence of an orbital contribution for both tetrahedral and
octahedral complexes.
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Table 4.1 Summary of Ground Terms and Orbital Contributions in First Row Transition Metal Ions.
Number
Octahedral Complexes
Tetrahedral Complexes
of d
Electron
Ground
Orbital
Electron
Ground
Orbital
electrons
Configuration
Term
Contribution Configuration
Term
Contribution
1
t2g1
2
T2g
Yes
e1
2
t2g2
3
T1g
Yes
e2
3
3
t2g3
4
A2g
No
e2 t21
4
T1
Yes
4
t2g3 eg1
t2g4
5
Eg
T1g
No
Yes
e2 t22
5
T2
Yes
A1g
T2g
No
Yes
e2 t23
6
A1
No
Yes
No
e3 t23
5
E
No
T1g
Eg
Yes
No
e4 t23
4
A2
No
A2g
No
e4 t24
3
T1
Yes
2
No
e4 t25
2
T2
Yes
5
6
3
t2g3 eg2
t2g5
6
t2g4 eg2
t2g6
5
2
1
T2g
A1g
t2g5 eg2
t2g6 eg1
4
8
T2g6 eg2
3
9
T2g6 eg3
7
2
Eg
2
E
No
A2
No
It can be seen that complexes with a T ground term do have an orbital angular momentum
contribution, while those with A or E ground terms do not. You should work through the electron
configurations and satisfy yourself that the assignments are correct
Let us now consider these two groups of magnetic behaviour.
4.3.2 Magnetic Properties of Complexes with A and E ground terms.
Such complexes have no orbital contribution to the ground state, so we would expect the magnetic
moment to follow the spin-only formula. Now, it happens that for a number of the transition metals
in commonly occurring oxidation states the d-electron configurations give rise to A or E ground
terms (e.g. d3 Cr3+; d5 (Mn2+, Fe3+), d8 (Ni2+), d9 (Cu2+)) and thus should give magnetic moments in
good agreement with the spin-only formula. This is indeed the case in many (but not all) instances
(See Table 4.2). Furthermore, since the spin-only formula does not have a temperature dependent
term, complexes with A and E ground terms are expected to show no temperature dependence.
However, we can see from table 4.2 (below) that the fit to spin-only magnetic moments is not
always good. This is because, although the ground state has no orbital contribution, excited
electronic states may have T terms, and if they have the same multiplicity these can mix with the
ground state.
If the ground term is A1 this is derived from a free ion S term and does not give rise to higher T
terms. Hence, ions with A1 ground terms fit the spin-only model very well (e.g. d5, Mn2+). Electron
configurations with A2 ground terms (from a free ion F term) and E ground terms (from a free ion D
term) will have higher T terms which can mix with the ground states and thus will show deviations
from the spin only formula. In such cases the magnetic moment is given by:


 eff   spinonly 1 
 

E 
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Where α = 2 for an E term and α = 4 for an A2 term. The spin-orbit coupling constant, λ, has a
positive sign for less than half-filled d-shells and and a negative sign for greater than half-filled dshells. ΔE is the energy of the lowest energy d-d transition in cm-1. We therefore get values for the
observed magnetic moments which are slightly larger (later transition metals) or smaller (early
transition metals) than the spin-only values. Typically the values will deviate by a few tenths of a
Bohr Magneton.
Table 4.2. Magnetic Moments of Selected Compounds with A or E Ground Terms
Number
of d
electrons
µeff (B.M.)
Compound
Geometry
1
VCl4
3
80 K
300 K
Spin only
Tetrahedral
1.6
1.6
1.73
KCr(SO4)2.12 H2O
Octahedral
3.8
3.8
3.87
4
CrSO4.6 H2O
Octahedral
4.8
4.8
4.90
5
K2Mn(SO4)2.6 H2O
Octahedral
5.9
5.9
5.92
7
Cs2CoCl4
Tetrahedral
4.5
4.6
3.87
8
(NH4)2Ni(SO4)2.6H2O
Octahedral
3.3
3.3
2.83
9
(NH4)2Cu(SO4)2.6H2O Octahedral
1.9
1.9
1.73
Example:
Calculate the effective magnetic moment of the [Ni(en)3]2+ ion (en = 1,2-diaminoethane). The
lowest energy band in the electronic spectrum is at 11500 cm-1 and the spin-orbit coupling constant
is -315 cm-1.
The complex is octahedral and Ni2+ has a 3d8 electron configuration. Thus this species has an 3A2g
ground state. Now,


 eff   spinonly 1 
 

E 
and
 spinonly  n(n  2)
therefore since there are two unpaired electrons in an octahedral 3d8 complex
= 2.83 Bohr Magnetons (BM)
 spinonly  2(2  2)
Since this species has an A term, α = 4. λ is negative, since there are > 5 d electrons and therefore:
 4  315 

11500 

= 3.14 BM
 eff  2.831 
The resulting magnetic moment is a little larger than the spin-only value.
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4.3.3 Magnetic Properties of Complexes with T Ground Terms.
The magnetic behaviour of complexes with T ground terms is rather more complex. The T terms are
split by spin-orbit coupling to produce levels whose energy differences are often of the order kT (kT
 200 cm-1 at room temperature). Consequently, temperature will affect the populations of these
levels in a magnetic field, and so the magnetic moments of such materials will vary with
temperature. Some examples of compounds with T ground terms are given in Table 4.3.
Table 4.3. Magnetic Moments of Selected Compounds with T Ground Terms
Number
of d
electrons
µeff (B.M.)
Compound
Geometry
1
Cs2VCl6
2
80 K
300 K
Spin only
Octahedral
1.4
1.8
1.73
(NH4)V(SO4)2.12 H2O
Octahedral
2.7
2.7
2.83
4
K3[Mn(CN)6]
Octahedral
3.1
3.2
2.83
5
K3[Fe(CN)6]
Octahedral
2.2
2.4
1.73
6
(NH4)2Fe(SO4)2.6H2O
Octahedral
5.4
5.5
4.90
7
(NH4)2Co(SO4)2.6H2O
Octahedral
4.6
5.1
3.87
8
(Et4N)2NiCl4
Tetrahedral
3.2
3.8
2.83
It can be seen that, with one exception (V3+) the room temperature magnetic moment is slightly
larger than the spin-only value, with the deviation being of the order of a few tenths of a Bohr
Magneton. In addition there is a small difference in the magnetic moments at 80 K and 300 K
(again, a few tenths of a Bohr magneton), confirming the temperature dependence in these species.
So, we have a way of determining the geometry and spin state of almost every first row transition
metal, since the magnetic behaviour of octahedral and tetrahedral compounds of a metal in a given
oxidation state will vary – either in its deviation (or not) from the spin-only value or in its
temperature dependence. Usually the oxidation state can be determined as well from the value of
the magnetic moment.
4.4 High Spin- Low Spin Equilibria
Up to now we have assumed that the compounds are in a static electronic arrangement. However,
this is not always the case. Complications occur when a compound has a fixed stereochemistry, but
is in high spin – low spin equilibrium, or when a compound is in structural equilibrium and thus has
different electronic states. We will consider each of these in turn.
4.4.1 Thermal Equilibria between Spin States (Spin Crossover).
Ions with d4 to d7 electron configurations can exist as either high spin or low spin complexes,
depending on the magnitude of the ligand field. Whether the ion adopts a high spin or low spin
configuration depends on the balance between the orbital splitting energy, E, and the spin pairing
energy (SPE) which is the energy required to overcome the repulsion between two negatively
charged electrons occupying one orbital. Consider the following:
E >> SPE
SPE >> E
Low Spin
High Spin
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If the energy required to promote an electron from the lower to higher set of orbitals is greater than
the spin-pairing energy then the electron will prefer to pair and be in the lower set of orbitals,
resulting in a low spin complex. If however the spin pairing energy is greater than the energy gap
between the orbital sets then a high spin complex will result.
However, if the ligand field splitting E is approximately equal to the spin pairing energy, known
as the cross-over point, then small changes in conditions (e.g. temperature or pressure) can result in
the complex switching from high spin to low spin, or vice versa.
This situation is well known for Iron(II) complexes (3d6 electron
configuration), which are ideal, as the low spin state is diamagnetic
and the high spin state has four unpaired electrons as shown on the
right
Consequently, the change in magnetic moment is large. In addition,
the switching between spin states should occur abruptly as the
conditions change.
Consider the following pair of octahedral iron(II) phenanthroline
complexes and their magnetic behaviour.
It can be seen that at room temperature the magnetic moment is around 5 B.M. Both the NCS and
NCSe complexes exist in a 5T2 ground state and therefore have magnetic moments which are
greater than the spin only value for four unpaired electrons. At 174 K the magnetic moment of the
SCN complex falls dramatically to below 1 B.M. indicative of a low spin, diamagnetic compound,
consistent with the expected 1A1g ground state. Notice that the selenium analogue undergoes
transition at a much higher temperature of 232 K.
A range of other species also exhibit such behaviour, including iron(II) complexes of
tris(pyrazolyl)borate ligands. Iron(III) complexes in a sulphur coordination sphere, such as the tris
dithiocarbamate complexes [Fe(S2C=NR2)3] also show this type of spin equilibrium, only now the
two states have five and one unpaired electrons respectively.
4.4.2 Structural Equilibrium between Spin States
When the energy difference between two stereochemical arrangements of a set of ligands around a
metal centre is small then small changes in conditions can cause a structural change and this will
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Dr M.D. SPICER
result in a change in the magnetic properties. Nickel(II) compounds are particularly prone to this
type of behaviour, and we will take our examples from among such compounds.
4.4.2.a Octahedral-Square planar equilibrium
Many nickel(II) compounds dissolved in organic solvents have magnetic moments that vary
significantly with concentration, temperature and the nature of the solvent. One example of this is
the complexes Ni(en)2X2 (en = 1,2-diaminoethane, X = halide). These can be obtained as both
yellow, diamagnetic compounds and as deep blue paramagnetic compounds. The former are square
planar complexes, [Ni(en)2] X2, while the latter are pseudo-octahedral complexes, [Ni(en)2X2]. In
solution mixtures of both can occur, and the study of their magnetic moments gives information
about the position of the equilibrium under differing conditions.
4.4.2.b Monomer-polymer Equilibrium
A number of square planar nickel(II) complexes which are diamagnetic in the solid state form
solutions in non-coordinating organic solvents which are paramagnetic. This arises by the
aggregation of monomers into dimers and polymers in solution. Examples include the bis(Nalkylsalicylaldimine)Ni(II) complexes and the nickel(II) β-ketoenolate complexes which form
solutions of different colours and magnetic properties, dependent on the concentration, temperature
and nature of the solvent.
4.4.2.c Tetrahedral_Square Planar Equilibrium
The nickel(II) phosphine halide complexes, [Ni(PR3)2X2], can form both square planar and
tetrahedral complexes, dependent on the identity of R and X and the crystallisation conditions.
For instance, [Ni(PPh3)2Cl2] when synthesised at elevated temperatures from ethanol gives dark
blue crystals of the tetrahedral isomer. This is paramagnetic with a magnetic moment of
approximately 3.2 B.M., arising from a 3d8 T2 ground term. However, crystallisation from
dichloromethane at low temperatures gives a red isomer, which is square planar and diamagnetic. In
solution at room temperature, the square planar isomer rapidly converts to the tetrahedral isomer.
The kinetics of this isomerisation can be followed by monitoring the magnetic moment.
Similarly, solid square planar [Ni(PiPr)2Br2] is stable indefinitely at 0 °C while at 25 °C it
isomerises to the tetrahedral form in about a day.
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Finally, [Ni(PBzPh2)2Br2] crystallises in a red diamagnetic form, which is square planar, and a
green paramagnetic form, which has a magnetic moment of 2.7 B.M. It has been shown by X-ray
crystallography to contain nickel in both square planar and tetrahedral forms in a 1:2 ratio. Analysis
of the magnetic properties, taking into account a third of the molecules being diamagnetic gives a
good agreement for the magnetic moment (3.3 B.M. for the tetrahedral component).