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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 consequentally change the minimum energy
configuration.
Abstract:Diamagnetism is
conceptually simple and can be quanitatively 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 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 and the electronic bonds that it forms.
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 universially 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:
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 ωL 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 request 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.
μe = i(πr2)
T = μexH
= μeHsin(θ)
d(mωr2)/dt =
mr2dω/dt = T
-eω(πr
Thus, the precession frequency is independent of the
angle θ.
ωL = |T|/[psin(θ)
= μHsin(θ)/[psin(θ)
= (μ/|p|)H.
iL = -eωL/[2π].
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 10-6cgs-emu:
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.
The diamagnetic susceptibilities of
ionic compounds
of the form AiBj 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 ai's
and bj'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 susceptiblities of Halic acids
are included along with the susceptibilities of alkali
halides.
Least Squares Estimates of Ionic Diamagnetic
Susceptibilities
where ai is the susceptibility of the
A ion and bj is that of the B ion. The
varable u is a random variable, perhaps measurement
error.
xij = ai + bj + u
S = Σ(xij -
ai - bj)2.
njai +
Σjbj
= Σjxi,j
nibj +
Σiai
= Σixi,j
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 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:
There is obviously a close relationship. The regression line for the data is:
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. 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.
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.
Transition Metal Chlorides
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:
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 constantcy. 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,
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.
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.
χ = Σ
χi + λ
Pascal's Constants
Atom/Ion Diamagnetic
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/Ion Diamagnetic
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 |
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
comppounds 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 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 consequentally change the minimum energy
configuration.
C3-H bonds.
Conclusions
References: