San José State University 

appletmagic.com Thayer Watkins Silicon Valley & Tornado Alley USA 


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.
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.
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:
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:
The torque T determines the time rate of change of the angular momentum vector; i.e.,
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 ω_{L} is given by:
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
The induced magnetization is proportional to the applied field H and thus the magnetic susceptibility is constant.
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, CH_{4}, and includes the linear chain molecules of the form C_{n}H_{2n+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 LeastSquares Regression line for the first 11 members of the series gives the following equation for estimating magnetic susceptibility (measured in units of 10^{6} cgsemu:
where n is the number of CH_{2} units in the chain.
The regression equation indicates that each CH_{2} 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 H_{2} of 3.98.
The coefficient of determination, R^{2} = 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.
The diamagnetic susceptibilities of ionic compounds of the form A_{i}B_{j} is presumed to determined as:
The sum of squared errors to be minimized is
The necessary first order conditions for a minimization of S with respect to the a_{i}'s and b_{j}'s are:
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.
Ion  Estimated 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 
Ion  Estimated Diamagnetic Susceptibility 

Mg++  7.375 
Ca++  10.475 
Sr++  18.8 
Ba++  29 
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 socalled 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:
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:
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 (10.947=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:
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. H_{2} would be the analogue of the halic acids. The diamagnetic susceptibility of H_{2} is 3.98, which if H_{2} 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.
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^{+}, Zn^{2+}, Ga^{3+}, and Ge^{4+} 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, ZnCl_{2}, GaCl_{3}, Ge_{4} would be explained by equation:
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:
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.
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.
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, CH_{4}, 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.
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
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/Ion  Diamagnetic Susceptibility 10^{6} cgsemu 
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/Ion  Diamagnetic Susceptibility 10^{6} cgsemu 
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} cgsemu 
C=C  +5.5 
C=C  +0.8 
C=N  +8.2 
C=N  +0.8 
N=N  +0.8 
benzene ring  15.1 
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 CH bond in this system. Instead the carbon atoms are distinguished as to the number of other carbon atoms they are linked to; i.e., C_{1}, C_{2}, C_{3}, C_{4}. There are ABIS constants for C_{1}H, C_{2}H, and C_{3}H bonds.
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.