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Electronic Structure
The electronic structure of coordination complexes can lead to several different properties that involve different responses to magnetic fields. These properties can vary between related compounds because of differences in electron counts, geometry, or donor strength. As a result, magnetic measurement of these materials can be used as a tool to provide insight into the structure of a coordination complex.
Diamagnetic and paramagnetic compounds are two common categories defined by interaction with a magnetic field. Diamagnetic compounds display a very slight repulsion from magnetic fields. The interaction is quite subtle and requires the proper instrumentation if it is to be observed; it’s not something you would normally notice by waving a magnet in front of a vial of coordination complex. In contrast, paramagnetic compounds display a very slight attraction to magnetic fields. In terms of electronic structure, diamagnetic compounds have all of their electrons in pairs whereas paramagnetic compounds have one or more unpaired electrons. These definitions are not limited to coordination complexes. Atmospheric oxygen, O2, is a common example of a paramagnetic compound. Atmospheric nitrogen, N2, is a common example of a diamagnetic compound.
These differences arise because of a quantum mechanical property of electrons and other subatomic particles: spin. Spin does not have a direct macroscopic analogy; physical chemists often urge caution about equating it to a phenomenon on the level at which we observe the universe. However, we do know that spin is related to magnetic phenomena. We can think of an electron as having a magnetic moment. There are two allowed values of this magnetic moment (\(+\frac{1}{2}\) and \(-\frac{1}{2}\)) that differ in orientation; spin is a vector quantity. Normally, there is no preference for one orientation over the other. If a coordination complex has one unpaired electron, and we have a million of those coordination complexes in a very tiny sample, about 500,000 of the unpaired electrons would have spin = \(+\frac{1}{2}\) and about 500,000 of them would have spin = \(-\frac{1}{2}\). The spin of each electron is randomly oriented. As a result, the material would have no net magnetic moment. In the presence of a magnetic field, however, there is a small energy difference between those two spin states. As a result, a majority of the spins adopt the energetically more favorable orientation. Although the material by itself has no net magnetic field, one has temporarily been induced by the influence of an external magnetic field. This is what we mean by magnetic susceptibility.
Measuring Magnetic Susceptibility
All diamagnetic compounds have a roughly similar response to a magnetic field; either their electrons are all paired or they are not. Paramagnetic compounds, however, can respond quite differently to a magnetic field, depending on the number of unpaired electrons. The more unpaired electrons there are, the stronger the magnetic susceptibility will be, and so the stronger the attraction between the compound and the magnetic field. For this reason, measuring the strength of attraction of a compound for a magnetic field can reveal the number of unpaired electrons in the compound. This relationship is most straightforward for complexes of first row transition metals (\(3d\) metals), becoming a little more complicated for the \(4d\) and \(5d\) series.
A Guoy balance is probably the simplest example of a method for determining the presence of unpaired electrons in a coordination complex.1 It uses a balance, a sample holder and a magnet. Essentially, the magnet is weighed in the presence and absence of the sample. The discrepancy between the two measurements arises because of the interaction of the sample with the magnetic field. Any diamagnetic sample causes slight downward repulsion, registering as a heavier weight. A paramagnetic sample suspended between the poles of the magnet causes a slight upward pull on the magnet, which registers as a lighter weight. The more unpaired electrons there are, the greater the induced magnetic moment, and so the lighter the weight becomes.
From that measurement, we can extract a parameter that we normally refer to as the effective magnetic moment of the material, \(\mu_{eff}\). The magnetic moment is expressed in units called Bohr magnetons. A Bohr magneton (BM or \(\mu_{B}\)) is defined as:
\[1 \text{BM} =\frac{e h}{4 \pi m c} \nonumber \]
in which e is the charge on an electron; \(h\) is Planck’s constant; \(m\) is the mass of the electron; \(c\) is the speed of light.
This experimental quantity is fitted to a simple relationship that depends on the number of unpaired electrons. This mathematical model gives a prediction of the “spin-only” magnetic moment. If the magnetic moment arises purely from the number of unpaired electrons without additional complicating factors, the fit to the experimental data is pretty good.
\[\mu_{ eff } \approx \mu_{ so }=g \sqrt{S(S+1)} \nonumber \]
Here, g is the gyromagnetic ratio, which is a proportionality constant between the angular momentum of the electron and the magnetic moment. It has a value of 2.00023, or approximately 2.0. The term \(\sqrt{S(S+1)}\) is the value of the angular momentum, which depends on the number of unpaired electrons; S is the absolute value of the sum of the individual spins of the valence electrons. Of course, the spins would cancel out in electrons that were paired, because one would have \(m_s = \frac{1}{2}\) and the other would have \(m_s = -\frac{1}{2}\). For unpaired electrons, Hund’s rule states that they would have parallel spins; for example, two unpaired electrons gives \(S=1\).
\[\begin{array}{|c|c|c|} \hline
\text{Number of} & \text{Maximum total} & \text{Spin-only magnetic moment, } \\
\text{unpaired electrons, } n & \text{spin, S} & \mu_{so} (BM) \\ \hline
1 & \frac{1}{2} & 1.73 \\
2 & 1 & 2.83 \\
3 & \frac{3}{2} & 3.87 \\
4 & 2 & 4.90 \\
5 & \frac{5}{2} & 5.92\\ \hline
\end{array} \nonumber \]
The table above shows how the magnetic moment changes with the number of unpaired electrons. The maximum number of unpaired electrons given is five, because for a transition metal a sixth electron would have to pair up in a previously occupied orbital. In f-block elements, there could be seven unpaired electrons before pairing occurs because there are seven f orbitals rather than just five d orbitals.
Note that the expression for spin-only magnetic moment is sometimes written in an alternative way, based directly on the number of unpaired electrons, \(n\), rather than \(S\).
\[\mu_{ eff } \approx \mu_{ so }=\sqrt{n(n+2)} \nonumber \]
There is an additional approximation in these cases, based on how the values of the spin-only magnetic moment correlate with the number of unpaired electrons. If we always round the value of \(\mu_{so}\), then it is one greater than the number of unpaired electrons. Thus, \(\mu_{so} \approx n + 1\), provided of course that there are any unpaired electrons at all; the relationship doesn’t hold if n = 0 because then \(\mu_{so}=0\), not 1.
In reality, observed magnetic moments are slightly different than spin-only magnetic moments. In some cases, the observed magnetic moment is smaller than expected, but those cases are more complicated and we won’t consider them here. Very often, the observed magnetic moments are larger than predicted because orbital angular momentum also plays a role in determining the magnitude of the overall magnetic moment. We may use a modified expression that takes this part into account.
\[\mu_{ eff } \approx \mu_{ s + L }=g \sqrt{S(S+1)+\frac{1}{4} L(L+1)} \nonumber \]
Just as S is the absolute value of the sum of the spin quantum numbers in the ion, \(m_s\), L is the absolute value of the sum of the orbital quantum numbers, \(m_l\). There are five d orbitals, with \(m_l= 2, 1, 0, -1, -2\), and the value of L has to be maximized according to Hund’s rule. That means the value of L can be 3, 2, or 0.
As an example, suppose we have a \(\ce{Co^2+}\) ion. \(\ce{Co^2+}\) ion has a \(d^7\) configuration. It has five \(d\) orbitals, so four of these electrons are paired, leaving only three unpaired electrons with parallel spins, so \(S = 3/2\). In order to maximize the orbital quantum number, \(L\), two electrons will be in an orbital with \(m_l= 2\), two electrons will be in an orbital with \(m_l=1\), and one electron will be in each orbital with \(m_l\) = 0, -1, and -2. When we take the sum, we find \( L = 2 + 2 + 1 + 1 + 0 -1 -2 = 3\). That gives us:
\[\mu_{ s + L }=g \sqrt{3 / 2(3 / 2+1)+\frac{1}{4} 3(3+1)}=5.20 \nonumber \]
In comparison, \(m_{so} = 3.87\) in this case. Observed values of \(\mu_{eff}\) vary between different Co2+ complexes but are generally in the range 4.1 – 5.2 BM.2 The orbital contribution is often smaller than expected, so magnetic susceptibilities frequently fall somewhere between \(\mu_{so}\) and \(\mu_{S+L}\).
The use of a simple Guoy balance is an historically important method of determining magnetic susceptibility, and it is sometimes used in undergraduate laboratory experiments. Other methods are frequently used to measure magnetic susceptibility in the research laboratory. A magnetic susceptibility balance operates on a similar principle to the one behind the Guoy balance. A sample is placed within the poles of an electromagnet, causing the electromagnet to move very slightly. The current in the electromagnet is adjusted, changing its magnetic field, until the magnet comes back to its initial position. The magnitude of the current adjustment is proportional to the magnetic susceptibility of the sample.
A superconducting quantum interference device (SQUID) uses a superconducting loop in an external magnetic field to measure magnetic susceptibility of a sample.3 The sample is mechanically moved though the superconducting loop, inducing a change in current and magnetic field that are proportional to the magnetic susceptibility of the sample. Commercially produced SQUID magnetometers often have variable temperature controls, allowing magnetic susceptibility to be measured across a range of temperatures.
The Evans NMR method is quite common because of the widespread availability of high-field NMR spectrometers in research labs. A capillary containing the paramagnetic sample and a reference analyte is placed in an NMR tube containing the same reference analyte but without the paramagnetic sample. The analyte in the presence of the paramagnetic material will experience a local, induced magnetic field owing to the effect of the superconducting magnet of the NMR instrument on the paramagnetic sample. Its NMR signals will shift as a result. The NMR signal of the analyte outside the capillary will undergo no such shift, and the difference in the two signals is an indicator of the magnetic susceptibility of the sample.
There are other types of magnetic behavior in addition to diamagnetism and paramagnetism. Ferromagnetic materials have long-range order with spins oriented parallel to each other, even in the absence of an external magnetic field. Common, permanent magnets are made from ferromagnetic materials. Antiferromagnetic materials also have long-range order, but the magnetic moments are arranged in opposing pairs. These three different magnetic behaviors are diagnosed by the temperature dependence of the magnetic susceptibility. Paramagnetic materials display magnetic susceptibility (\(\chi_M\), related to \(\mu_{eff}\)_) that increases with the inverse of temperature. Ferromagnetic materials have a critical temperature below which magnetic susceptibility rapidly rises. Antiferromagnetic materials have a critical temperature below which magnetic susceptibility rapidly falls. However, these behaviors are largely beyond the scope of the current discussion.
Problems
Exercise \(\PageIndex{1}\)
Show that, given g = 2.0, then \(g \sqrt{S(S+1)}=\sqrt{n(n+2)}\).
- Answer
-
\[\begin{aligned}
\mu_{ so } &=g \sqrt{S(S+1)} \\
&=2 \sqrt{S(S+1)} \\
&=\sqrt{4 S(S+1)} \\
&=\sqrt{2 S(2 S+2)} \end{aligned} \nonumber \]but \(S=n(1 / 2)\) or \(n=2 S\) then
\[\mu_{ so }=\sqrt{n(n+2)} \nonumber \]
Exercise \(\PageIndex{2}\)
Calculate the value of \(\mu_{so}\) in the following cases:
a) V4+ b) Cr2+ c) Ni2+ d) Co3+ e) Mn2+ f) Fe2+
- Answer
-
a) \(V ^{4+} \text{ is } d ^{1} ; n=1 ; \sqrt{n(n+2)}=1.73\)
b) \(Cr ^{2+} \text{ is } d ^{4} ; n=4 ; \sqrt{n(n+2)}=4.90\)
c) \(Ni ^{2+} \text{ is } d ^{8} ; n=2 ; \sqrt{n(n+2)}=2.83\)
d) \(Co ^{3+} \text{ is } d ^{6} ; n=0 ; \sqrt{n(n+2)}=0\)
e) \(Mn ^{2+} \text{ is } d ^{5} ; n=5 ; \sqrt{n(n+2)}=5.92\)
f) \(Fe ^{2+} \text{ is } d ^{6} ; n=4 ; \sqrt{n(n+2)}=4.90\)
g) \(Cr ^{3+} \text{ is } d ^{3} ; n=3 ; \sqrt{n(n+2)}=3.87\)
h) \(V ^{3+} \text{ is } d ^{2} ; n=2 ; \sqrt{n(n+2)}=2.83\)
Exercise \(\PageIndex{3}\)
Calculate the value of \(\mu_{eff}\) in the following cases:
a) V4+ b) Cr2+ c) Ni2+ d) Co3+ e) Mn2+ f) Fe2+
g) Cr3+ h) V3+
- Answer
-
a) \(V ^{4+}\text{ is } d ^{1} ; S=1 / 2 ; L=2 ; \mu_{ s + L }=g \sqrt{1 / 2(1 / 2+1)+\frac{1}{4} 2(2+1)}=3.00\)
b) \(Cr ^{2+}\text{ is } d ^{4} ; S=2 ; L=2+1+0-1=2 ; \mu_{ s + L }=g \sqrt{2(2+1)+\frac{1}{4} 2(2+1)}=5.48\)
c) \(Ni ^{2+}\text{ is } d ^{8} ; S=1 ; L=2+2+1+1+0+0-1-2=3 ; \mu_{ s + L }=g \sqrt{1(1+1)+\frac{1}{4} 3(3+1)}=2.24\)
d) \(Co ^{3+}\text{ is } d ^{6} ; S=2 ; L=2+2+1+0-1-2=2 ; \mu_{ s + L }=g \sqrt{(2+1)+\frac{1}{4} 2(2+1)}=5.48\)
e) \(Mn ^{2+}\text{ is } d ^{5} ; S=5 / 2 ; L=2+1+0-1-2=0 ; \mu_{ s + L }=g \sqrt{5 / 2(5 / 2+1)+\frac{1}{4} 0(0+1)}=5.92\)
f) \(Fe ^{2+}\text{ is } d ^{6} ; S=2 ; L=2+2+1+0-1-2=2 ; \mu_{ s + L }=g \sqrt{2(2+1)+\frac{1}{4} 2(2+1)}=5.48\)
g) \(Cr ^{3+}\text{ is } d ^{3} ; S=3 / 2 ; L=2+1+0=3 ; \mu_{ s + L }=g \sqrt{3 / 2(3 / 2+1)+\frac{1}{4} 3(3+1)}=5.20\)
h) \(V ^{3+}\text{ is } d ^{2} ; S=1 ; L=2+1=3 ; \mu_{ s + L }=g \sqrt{1(1+1)+\frac{1}{4} 3(3+1)}=2.24\)
References
- Lancashire, R. J. Magnetic Susceptibility https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Map%3A_Inorganic_Chemistry_(Housecroft)/04%3A_Experimental_techniques/4.14%3A_Magnetism/Magnetic_Susceptibility_Measurements (accessed Jun 27, 2021).
- Cotton, F.A.; Wilkinson, G. Advanced Inorganic Chemistry, 4th Ed. John Wiley & Sons: New York, 1980, p 628.
- Raja, P. M. V.; Barron, A. R. Magnetism https://chem.libretexts.org/@go/page/55872 (accessed Jun 27, 2021).
FAQs
What is magnetic susceptibility greater than 1? ›
A positive relative permeability greater than 1 implies that the material magnetizes in response to the applied magnetic field. The quantity χm is called magnetic susceptibility, and it is just the permeability minus 1. The magnetic susceptibility is then zero if the material does not respond with any magnetization.
What is magnetic susceptibility χ? ›The magnetic susceptibility of a material, commonly symbolized by χm, is equal to the ratio of the magnetization M within the material to the applied magnetic field strength H, or χm = M/H.
What if magnetic susceptibility is 1? ›Superconductors have magnetic susceptibility -1. Examples of superconductors: niobium-titanium, germanium-niobium, and niobium nitride, mercury, lead. Any perfect conductor would exhibit perfect diamagnetism is called superconductors.
When magnetic susceptibility is less than 1? ›Since magnetic susceptibility of given material is less than one. Hence, the given magnetic material is a diamagnetic material.
What is the susceptibility of a magnetic material is 1.9 * 10? ›The susceptibility of a magnetic material is 1.9×10−5. Name the type of magnetic materials it represents. As the susceptibility of given magnetic material (x=1.9×10−5) is extremely small but positive, the material is a paramagnetic material.
What has the highest magnetic susceptibility? ›Ferromagnetic materials like iron and cobalt, has very large magnetic susceptibility that is positive susceptibility, unlike diamagnetic material which has negative susceptibility.
What is magnetic susceptibility low and positive? ›Magnetic susceptibility is defined as the ratio of magnetization and the magnetic intensity i,e X=MH The magnetic susceptibility is low and positive for paramagnetic materials like : Sodium chloride, silicon, bismuth etc.
What is the range of magnetic susceptibility? ›EXPLANATION: The range of magnetic susceptibility of diamagnetic materials is -1 ≤ χ ≤ 0.
How is magnetic susceptibility measured? ›Magnetic susceptibility is defined as ratio between magnetization M of the material in the magnetic field and the field intensity H: M = χm H.
What is magnetic susceptibility negative? ›Diamagnetic material are the magnetic materials that align against the magnetic field. The magnetic susceptibility of diamagnetic material is less than zero. This means that a diamagnetic material always has a negative value of magnetic susceptibility.
What is positive magnetic susceptibility? ›
If χ is positive, a material can be paramagnetic. In this case, the magnetic field in the material is strengthened by the induced magnetization. Alternatively, if χ is negative, the material is diamagnetic. In this case, the magnetic field in the material is weakened by the induced magnetization.
What is the susceptibility of magnetism at 200k? ›The susceptibility of magnesium at 200 K is 18 × 10^-5 .
Can magnetic susceptibility be zero? ›If magnetic susceptibility is less than zero then it is anti-aligned then the material is diamagnetic. If the magnetic susceptibility is more than zero then it is aligned and the material is paramagnetic. Note: If χ is positive, a material can be paramagnetic.
What has a magnetic susceptibility of zero? ›Magnetic susceptibility of vacuum is 0 since vacuum cannot be magnetized.
What has very weak and negative susceptibility to magnetic field? ›Diamagnetic materials have a weak, negative susceptibility to magnetic fields. Diamagnetic materials are slightly repelled by a magnetic field and do not retain the magnetic properties when the external field is removed.
What is the susceptibility of a magnetic material 2.6 10? ›As susceptibility of given magnetic material is −2.6×10−5 the material is a diamagnetic material. Two important properties of these materials are: (i) These are feebly repelled by a magnetic field and tend to move from stronger to weaker region of field.
What is the magnetic material of susceptibility 3 * 10 4? ›The magnetic susceptibility of a paramagnetic substance is 3 × 10^-4 Am^-1 . The intensity of magnetisation will be : ( H = 4 × 10^3 )
What is the susceptibility of magnetism at 300 K is 1.2 10? ›The susceptibility of Mg at 300K is 1.2×10−5. The temperature at which susceptibility of Mg will be 1.8×10−5 is: Q. The susceptibility of magnesium at 200K is 18×10−5.
What is the magnetic susceptibility of steel? ›All austenitic stainless steels are paramagnetic (nonmagnetic) in the fully austenitic condition as occurs in well-annealed alloys. The DC magnetic permeabilities range from 1.003 to 1.005 when measured at magnetizing forces of 200 oersteds (16k A/m).
For which material magnetic susceptibility is low? ›For paramagnetic material the magnetic susceptibility is low an positive.
What are the three types of magnetic susceptibility? ›
Magnetic susceptibility will determine whether a material will be attracted to or repelled from a given magnetic field. Magnetic materials may be classified as one of three types; diamagnetic, paramagnetic or ferromagnetic, depending of their susceptibilities.
Why is magnetic susceptibility important? ›Magnetic susceptibility is used to study the properties of magnetic materials and their behavior in magnetic fields. This information is used to design new materials with specific magnetic properties for applications such as data storage, energy conversion, and sensors.
What affects magnetic susceptibility? ›Magnetic susceptibility is the magnetic response of a substance to a magnetic field and can result in local magnetic field inhomogeneities and signal loss. These effects are proportional to field strength and the differences in susceptibility of two regions. Magnetic susceptibility increases with field strength.
What is magnetic susceptibility for which material is it high and negative? ›Step1: The magnetic susceptibility denotes whether a material is repelled out or attracted of a magnetic field. It is negative only for diamagnetic material and it is positive for paramagnetic material and ferromagnetic material has large positive magnetic susceptibility.
What is the value of susceptibility? ›The value of magnetic susceptibility for diamagnetic materials is χ<0. The relation signifies that for diamagnetic material, the value of magnetic susceptibility is always a negative one. The magnets do not attract these materials but prefer to repel.
What does negative susceptibility signify? ›Negative susceptibility of a substance signifies that the substance will be repelled by a strong magnet or opposite feeble magnetism induced in the substance. Such a substance is called diamagnetic.
For which material magnetic susceptibility is positive? ›It means paramagnetic materials are weakly effected by the field. Therefore the susceptibility of paramagnetic materials is positive but small.
What are the units for susceptibility? ›Sometimes the mass susceptibility (χm) is quoted and this has the units of m3kg-1 and can be calculated by dividing the susceptibility of the material by the density.
How is magnetic susceptibility reduced? ›They can be minimized by using shorter TE values (less time for dephasing) and by using fast spin-echo instead of gradient-echo sequences. Susceptibility artifacts can also be reduced by increasing gradient strength for a given field-of-view and avoiding narrow bandwidth techniques.
What is magnetic susceptibility in MRI? ›Magnetic susceptibility artifacts (or just susceptibility artifacts) refer to a variety of MRI artifacts that share distortions or local signal change due to local magnetic field inhomogeneities from a variety of compounds.
What is the susceptibility of a magnetic material at 300k? ›
The susceptibility of a paramagnetic material at 300 K is 1.4 x 109.
Which metal has negative susceptibility? ›Copper is an element that has a negative magnetic susceptibility. Iron is an element that has a large positive magnetic susceptibility. From the above relation, we can obtain the magnetization of the material placed in the external magnetic field as $M = \chi H$ .
Is magnetic susceptibility a number? ›Magnetic susceptibility is χ>0 which means it is always a small positive value for paramagnetic materials.
What is an example of negative susceptibility? ›Which means that a diamagnetic material always has a negative value of magnetic susceptibility. They easily get repelled by magnets and move from a stronger to weaker field. These are temperature independent but do have a small amount of intensity. Some examples are gold, tin and water.
What is the magnetic susceptibility of water? ›Water is reported as a diamagnetic substance with a susceptibility of − 9 × 10 − 6 .
What can weaken a magnetic field? ›- They Get Old. While the passage of time does weaken the strength of a magnet, the changes are very slow. ...
- They Get Very Cold (Or Hot) ...
- Reluctance Changes. ...
- External Charges.
The magnetic field is weakest at the center and strongest between the two poles just outside the bar magnet. The magnetic field lines are densest at the center and least dense between the two poles just outside the bar magnet.
Can you have negative magnetic field strength? ›Answer and Explanation: No. The magnetic field is a vector and the magnitude of a vector is positive by definition, since it denominates the length of the vector.
What is positive and negative magnetic susceptibility? ›Step1: The magnetic susceptibility denotes whether a material is repelled out or attracted of a magnetic field. It is negative only for diamagnetic material and it is positive for paramagnetic material and ferromagnetic material has large positive magnetic susceptibility.
Why is magnetic susceptibility 1 for superconductors? ›-1, Because superconductor are perfect diamagnetic substances.
What is magnetic susceptibility for which material is too low and positive? ›
For paramagnetic materials, the magnetic susceptibility is low and positive. In paramagnetic materials, electrons align with the applied field and are attracted to regions of greater magnetic field.
Which category of magnetic susceptibility is considered very high positive? ›Ferromagnetic materials have a large, positive susceptibility to an external magnetic field.
What does susceptibility mean on an MRI? ›Susceptibility weighted imaging (SWI) is an magnetic resonance imaging (MRI) technique that exploits the magnetic susceptibility differences of various compounds, such as blood, iron, and diamagnetic calcium, thus enabling new sources of MR contrast[1-3].
What determines magnetic susceptibility? ›The determination of a magnetic susceptibility depends on the measurement of B/H. Experimentally the Gouy method involves measuring the force on the sample by a magnetic field and is dependent on the tendency of a sample to concentrate a magnetic field within itself.