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Diamagnetic Susceptibilities

Abstract: Diamagnetism is conceptually simple and can be quantitatively explained to a high degree of accuracy. To a close approximation the diamagnetic susceptibility of a compound is the sum of the susceptibility of its components. But the components of a molecule as far as its diamagnetic susceptibility is concerned are probably its electron bonds rather than its atoms and their electrons. Thus the contribution of an atom to a compound depends upon the other constituents of the compound and the electronic bonds that it forms.

The simple notion that if two entities, atoms or ions, have the same number of electrons then they will have the same electronic structure and hence the same diamagnetism is not always valid. Sometimes the difference of one unit of charge in the nucleus will alter the energy levels and consequently change the minimum energy configuration.


Outline

  • I. The Theory of Diamagnetism
    • A. Classical
    • B. Quantum Theoretic
  • II. The Potential for Precise Prediction of Diamagnetic Susceptibilities
    • A. The Aliphatic Hydrocarbons
    • B. The Alkali Halides
    • C. The Alkaline Halides
  • III. The Limitations on the Predictability of Diamagnetic Susceptibilities
    • A. The Transition Metal Chlorides
    • B. The Transition Metal Dihalides
    • C. The Chloro-alkanes
  • IV. Pascal's Constants
    • A. Accuracy of the Pascal System
    • B. Explaining the Values of Pascal's Constants


Introduction

Diamagnetism is the phenomenon of a magnetic field inducing in a material a magnetic field which opposes it. In other words, a diamagnetic material has a negative magnetic susceptibility. The diamagnetic susceptibilities are very small in magnitude compared to paramagnetic materials, and negligible compared to ferromagnetic materials.

The universally accepted explanation of diamagnetism is the precession of the magnetic moment created by the orbital motion of electrons.

Classical Theory

Consider an electron in a circular orbit of radius r with angular velocity of ω rotating about an axis p which is directed at an angle θ to the magnetic field of strength H. The angular momentum of the electron is mωr2. The direction of the angular momentum is along angular velocity vector ω and in the same direction. The direction of ω is given by the right-hand rule.

The motion of the electron with a charge of -e creates a current equal to -eω/(2π). This effective current creates a magnetic dipole moment, which by Faraday's Law, has a strength equal to:

μe = i(πr2)

The induced magnetic dipole moment of the electron orbit and the external magnetic field H creates a torque T on the angular momentum of electron orbital given by:

T = μexH
= μeHsin(θ)

The torque T determines the time rate of change of the angular momentum vector; i.e.,

d(mωr2)/dt =
mr2dω/dt = T
= -eω(πr2)/(2π)

The angular momentum vector p will precess in the magnetic field and in the process of precession create a current about a vector in the direction of the magnetic field. This induced current then creates a magnetic field opposing H.

The precession frequency ωL is given by:

ωL = |T|/[p·sin(θ)
= μHsin(θ)/[p·sin(θ)
= (μ/|p|)H.

Thus, the precession frequency is independent of the angle θ.

The magnetic moment induced by the precession frequency is the same for all atoms in the substance no matter what is the angle of orientation of the orbit with respect to the magnetic field. The precession of the orbit is equivalent to an electrical current of

iL = -eωL/(2π).

The induced magnetization is proportional to the applied field H and thus the magnetic susceptibility is constant.

Diamagnetism


The Potential for Precise Prediction
of Diamagnetic Susceptibilities

A Least Squares Estimate of
the Diamagnetic Susceptibilities
of Aliphatic Hydrocarbons

The first type of compound that physicists noted as having regular, predictable magnetic susceptibilities is the aliphatic hydrocarbons (the alkanes), the series which starts with methane, CH4, and includes the linear chain molecules of the form CnH2n+2 such as propane, butane and octane. The graph shows that the relationship between magnetic susceptibility and the number of carbon atoms in the chain is very close to linear.

A Least-Squares Regression line for the first 11 members of the series gives the following equation for estimating magnetic susceptibility (measured in units of 10-6 cgs-emu:

χdiam = -4.80 -11.56n
R2 = 0.9958 <

where n is the number of CH2 units in the chain.

The regression equation indicates that each CH2 group contributes -11.56 to the diamagnetic susceptibility. The constant -4.80 represents the contribution of the two hydrogen ions at the ends of the carbon chain. This value is not too far off from the measured susceptibility of gaseous hydrogen H2 of -3.98.

The coefficient of determination, R2 = 0.9958 indicates that 99.58 percent of the variation in the magnetic susceptibility is explained by variation in the length of the carbon chain.

Least Squares Estimates of
Ionic Diamagnetic Susceptibilities

The diamagnetic susceptibilities of ionic compounds of the form AiBj is presumed to determined as:

xij = ai + bj + u

where ai is the susceptibility of the A ion and bj is that of the B ion. The variable u is a random variable, perhaps measurement error.

The sum of squared errors to be minimized is

S = Σ(xij - ai - bj)2.

The necessary first order conditions for a minimization of S with respect to the ai's and bj's are:

njai + Σjbj = Σjxi,j
nibj + Σiai = Σixi,j

There is an equation for each parameter but the equations are not independent and consequently one parameter can be chosen arbitrarily. This degree of arbitrariness does not affect the accuracy of the values of susceptibilities computed from them but it limits the opportunity to explain the empirical parameters from theory. Fortunately there is a theoretical basis for setting the diamagnetic susceptibilty of the H+ ion equal to zero; it has no electrons. Thus the susceptibilities of Halic acids are included along with the susceptibilities of alkali halides.

The graph shows that the diamagnetic susceptibilities of the alkali halides are virtually entirely explained in terms of the diamagnetic susceptibilities of the component ions. The proportion of the variation not explained is only 0.07 of 1 percent.

IonEstimated
Diamagnetic
Susceptibility
Li+-1.825
Na+-8.225
K+-15.925
Rb+-23.675
Cs+-34.8
F--8.6
Cl--22.6
Br--32.9
I--47.7

IonEstimated
Diamagnetic
Susceptibility
Mg++-7.375
Ca++-10.475
Sr++-18.8
Ba++-29


Diamagnetic Susceptibilities of Halide Ions

There are two ways to measure and estimate the magnetic susceptibilities of halide ions. First, the diamagnetic susceptibilities of halic acids should be entirely due to the halide ion because the H+ ion has no orbital electrons and thus no orbital magnetic moment. The second approach makes use of the fact that the halic ions having completed electron outer shells have the same number of electrons as a corresponding noble gas atom. A halide ion would not necessarily have the same electronic configuration as the corresponding noble gas atom because the difference in the charge of the nucleus could sufficiently alter the energy levels of various states to change the configuration which has the minimum energy.

The analysis of this question would be on the assumption, the so-called null hypothesis that the two variables are equal and differ only because of some random variable such as measurement error. Formally the null hypothesis is:

χnoble = χHX + u

The following graph shows the data for magnetic susceptibilities of halic acids and noble gases.

There is obviously a close relationship. The regression line for the data is:

χnoble = +1.71 + 0.947χHX
R2 = 0.999
σ = 0.02086

where σ is the standard error of the regression coefficient.

Although the regression coefficient seems reasonably close to unity the statistical significance of the difference (1-0.947=0.053) has to be judged relative to the standard error of the regression coefficient, 0.02086. The ratio of the difference to the standard error is 0.053/0.02086=2.54 and a ratio of this magnitude would occur due to chance with a probability of less than 5 percent.

If the regression constant is suppressed the regression result is:

χnoble = 0.899χHX
R2 = 0.996
(Standard Error of Coefficient)
σ = 0.0163

This equation indicates even more deviation of the regression coefficient from unity. Thus we are forced to conclude that the magnetic susceptibility of halide ions, while closely related to the susceptibility of the corresponding noble gas atoms, are not identical with the noble gas atom of the same number of electrons.

However with only four observation points it is difficult to use statistical analysis. It would be helpful to have some additional halogens to add to the sample.

Susceptibility measurements are not available for the acid of the other halogen, Astatine. However, hydrogen with its outer shell short one electron could be considered part of the halogen family. Helium is the noble gas corresponding to hydrogen. We need then a susceptibility for a H- ion, which may not exist. H2 would be the analogue of the halic acids. The diamagnetic susceptibility of H2 is -3.98, which if H2 is considered to be (H+)(H-) would mean that the susceptibility of H- is -3.98. The susceptibility of He is -1.88.

Another approach to obtaining an estimate of the susceptibility of H- is to look at the susceptibilities of hydrides. Lithium hydride has a susceptibility of -4.6. If the susceptibility of Li+ as previously estimated is -1.8 then the value for the hydride ion should be -2.8.

Both estimates of the susceptibility of H-, -3.98 and -2.8, are different from the value for He, but are of the same order of magnitude. Over all the evidence is that the diamagnetic susceptibilities and hence electronic configurations of halic ions is the same as that of the corresponding noble gas atoms.


Transition Metal Chlorides

The transition metals and their compounds typically display paramagnetic and ferromagnetic properties. Some metals at the end of the series are diamagnetic. The graph shows the susceptibilities of chlorides of the metals following Nickel in the periodic table.

These ionic compounds have constituent ions which have closed outer electron shells. That is to say, the chloride ions have the same number of electrons as Argon (Ar) atoms. It would seem then that the contribution of the chloride ions to the diamagnetic susceptibility of the compound is the same as the same number of Ar atoms. Cl- ions are roughly equivalent magnetically to Ar atoms.

The metal ions in the series, Cu+, Zn2+, Ga3+, and Ge4+ with their loss of valence electrons would seem to be electronically equivalent to each other and to a Ni atom. But magnetically the ions are diamagnetic whereas Ni is ferromagnetic. Even leaving Ni out of the comparison reveals magnetic differences among the ions.

If diamagnetic susceptibility were the same for isoelectronic atoms and ions the magnetic susceptibility of the series CuCl, ZnCl2, GaCl3, Ge4 would be explained by equation:

χdiam = γ + βn

where n is the number of Chloride ions in the compound and the value of β would be roughly equal to the susceptibility of Argon.

The least squares regression line for the data is:

χdiam = -36.5 -9.4n
R2 = 0.764

The susceptibility of Argon is -19.6, significantly different from the regression coefficient of -9.4. The regression line is not a particularly good fit. The accompanying graph shows the data, along with the susceptibility of the corresponding number of Argon atoms and with the least squares estimates of the susceptibility.

The conclusion to be drawn is that the ions in the series, although they have the same number of electrons, do not have the same electronic configurations and the diamagnetic susceptibilities. There could be shifts in configuration of electrons resulting from the additional positive charge which the ions have compared to Ar. In this test the notion that ions are equivalent to Argon atoms does not show up well.


Transition Metal Dihalides

Since the preceding material indicates that the susceptibility of halide ions is predictable an interesting comparison would involve keeping the metal ion constant and varying the halide ion. The data is for metal dihalides, MX2, where X is a halogen and M stands for Mn, Fe, Co, Ni, Cu or Zn. Thus if the magnetic susceptibilities of MnF2, MnCl2, MnBr2, and MnI2 are adjusted for the susceptibilities of the halic ions then values should be the susceptibility of the Mn2+ ion.

The results, shown in the accompanying 2D and 3D graphs, indicate roughly constant values for the metal ion susceptibilities but with systematic rather than random deviations for constancy. The interpretation of the results is complicated by the paramagnetism of these ions.


Chlorinated Methane

Not all series of compounds have diamagnetic susceptibilities which are simple sums of the component parts. For example, consider the compounds formed by the replacement of the hydrogen atoms in methane, CH4, by chlorine atoms. One might expect that as each successive replacement of H by Cl the diamagnetic susceptibility would change by the amount of the difference in susceptibility of the Cl and H atom. That is to say, one would expect a linear relationship between the diamagnetic susceptibility and the number of Chlorine atoms in the molecule.

The relationship has a definite curvature indicating a quadratic rather than linear relationship.


Pascal's Constants

P. Pascal began in the early part of the twentieth century developing a systematic method for computing diamagnetic susceptibilities amd his work continued up to the 1960's. He was aided and replaced in this endeavor by his student A. Pacault. The Pascal system is based on the proportion that a material composed of constituent atoms A, B, ..., Z will have a diamagnetic susceptibility equal to

χ = Σ χi + λ

where λ stands for corrective factors having to do with the structure of the molecule. The exitence of double bonds in organic compounds is one such structure feature that requires an additional increment to accurately predict susceptibilities.

Pascal deduced his constants from examining the susceptibilities of a large number of compounds. Adjustments were made as more accurate data became available or better methods of deducing the constants. There is a problem of determining what are the definitive versions of Pascal's Constants. The following are recent estimates of Pascal's Constants.

Pascal's Constants
Atom/IonDiamagnetic
Susceptibility
10-6 cgs-emu
H+0
Covalent H-2.93
Alkali Ions
Li+-4.2
Na+-9.2
K+-18.5
Rb+-27.2
Cs+-41.0
Halide Ions
F--11
Cl- 
Br--36
I--52
Nonionic Halides
F-6.3
Cl-20.1
Br-30.6
I-44.6
Alkaline Earth Ions
Mg++-10
Ca++-15.6
Group IIIa Ions
B+++-7
Al+++-13
Pascal's Constants
Atom/IonDiamagnetic
Susceptibility
10-6 cgs-emu
Organic Group
CH2-11.36
Group IVa Ions
C-6
Si-20
Sn(IV)-30
Pb++-46
Group Va Ions
N-2.1
P-26.2
As+++-20.9
Bi-192
Group VIa Ions
O-12
S-15
Se-23
Te-37.3
Transition Metals
Co, Fe, Ni-13
Zn-13.5
Hg++-41.5




Structural
Element
Corrective Term
Susceptibility
10-6 cgs-emu
-C=C-+5.5
-C=C-+0.8
-C=N-+8.2
-C=N-+0.8
-N=N-+0.8
benzene
ring
-15.1


Other Empirical Systems for
Computing Diamagnetic Susceptibilities

Pascal's System has gone through a number of revisions. Other investigators have opted to create whole new systems. Haberditzl and coworkers developed the Atom and Bonding Increment System (ABIS) for organic compounds in which the basic constants are for the various types of bonds rather than for atoms and ions as in Pascal's system. There is not just one C-H bond in this system. Instead the carbon atoms are distinguished as to the number of other carbon atoms they are linked to; i.e., C1, C2, C3, C4. There are ABIS constants for C1-H, C2-H, and C3-H bonds.


Conclusions

As is well known diamagnetism is conceptually simple and can be quanitatively explained to a high degree of accurately. To a close approximation the diamagnetic susceptibility of a compound is the sum of the susceptibility of its components. But the components of a molecule as far as its diamagnetic susceptibility is concerned is probably its electron bonds rather than its atoms and their electrons. Thus the contribution of an atom to a compound depends upon the other constituents of the compound.

The simple notion that if two entities, atoms or ions, have the same number of electrons then they will have the same electronic structure and hence the same diamagnetism is not always valid. Sometimes the difference of one unit of charge in the nucleus will alter the energy levels and consequently change the minimum energy configuration.


References:

  • P. Pascal, Constantes Selectionnees Diamagnetisme et Paramagnetisme
  • L.N. Mulay and E.A. Boudreaux, Theory and Applications of Molecular Diamagnetism, Wiley-Interscience, 1976.