Richard 1. Carlin
Brown University Providence, Rhode Island
Paramagnetic Susceptibilities
A
profusion of recent books is evidence of the current emphasis on the introduction of quantum concepts into undergraduate programs. Particular attention has been paid to topics concerned with chemical bonding as well as to the principles and applications of spectroscopy. A subject where the quantum theory has been particularly successful, that of paramagnetism, has, however, been noticeably absent in recent discussions, and presumably in lectures as well. Since we have had some success in showing the theory of paramagnetic susceptibilities to seniors as an interesting application of quantum mechanics, we have summarized here some of the fundamental ideas in this field. The discussion will be of particular interest to those interested in transition metal complexes. The standard source for the theory of paramagnetism is Van Vleck (1): several of the examnles we illustrate are due to ~ a i l h k s e n(2) and ~ o r a i ; (3).
The derivation requires the use of the Boltzmann factor which allows one to calculate the average behavior of a system in thermal equilibrium which is restricted to discrete energy levels. Thus, the probability of finding the system with energy E, is P,
exp ( -E,/kT) exp (-E./kT)
=
z,
the summation extending over all the energy levels of the system. The application to an assembly of free spins is simple. I n the absence of an external magnetic field, the energy level is twofold degenerate, corresponding to m, = +% or -%; when the field is applied, the degeneracy is removed since the energy in a field is magpH. The following situation obtains:
The Curie Law
Shortly after the topic of electron spin has been introduced, the Curie Law which states that paramagnetic susceptibilities are inversely proportional to temperature, can be developed quite simply. The electron spin generates an angular momentum p, resulting in a magnetic dipole moment N = - g (lel/2me)p, where e is the electronic charge, m the electronic mass, and e is the velocity of light. The spectroscopic splitting factor g has the value of 1 for orbital motion, but the anomalous value of 2.0023 for a free electron spin. The electron spin angular momentum operator is designated by S, yielding finally
Thus the energy separation is AE = gSH,, and for free electrons is of the order of 2.3 ern-' if H. = 25 kgauss. (Incidentally, one may easily introduce electron spin resonance spectroscopy here as well, setting AE = hv = gBH.) Now the magnetic moment of level a is given as r, = -bE,,/bHn; the molar macroscopic magnetic moment M is therefore obtained as the sum over magnetic moments weighted according to the Boltzmann factor.
According to classical electromagnetic theory, the energy of a magnetic dipole P in a magnetic field H is
where 0 is the angle between the dipolar axis and the field, H. The perturbation Hamiltonian for the energy of an electron spin in a magnetic field is
where the principal axis of the system is taken as the The magnetic field is taken as Hz, and the operator 8,has the eigenvalues m,fi, where m, = *% for a one-electron system. The final result is then z axis.
where N is Avogadro's number and the summation in this case extends over only the two states, m, = - M and +%. Then,
and expand the exponentials according to:
E = m,g@H,,
where (3 = (lelfi/2mc) is the Bohr magneton, of magnitude 0.927 X erg/gauss.
which approximation corresponds to moderate fields H, and temperatures T. Volume 43, Number 10, October 1966
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Thus
Retaining only terms linear in H,
Since x and since magnetic susceptibility x = M / H , x = NgZpz/4kT, which is in the form of the Curie Law. Note that x 0 as T a,corresponding to molecular chaos. This is a special case of the more general and more familiar spin-only formula,
= M/H,
the final result is:
- -
where the r-degeneracy of a level must be summed r times. Some Applications
x.
with S = This equation is derived below. A simple problem to assign a t this point is the same derivation for a system with spin 1 and m, = -1, 0, 1; the spin-only result is always obtained. These formulas are completely valid only for isotropic systems. The extension to axial ligand fields will be discussed below. Van Vleck's Equation
Many situations cannot be handled by the simple procedure outlined above. If the energy levels of a system are known, the magnetic susceptibility may always be calculated by application of Van Vleck's equation. The standard derivation ( I ) follows, along with several applications. Let the energy, B,, of a level be developed in a series in the applied field: E , = Eno HE.(') H?F,(2' . . . where, in the standard nomenclature, the term in H is called the first-order Zeeman term, and the term in H 2 the second order Zeeman term. Since rr, = -bE,/bH, the total magnetic moment, M, for the system follows as before:
+
+
The general form of the spin-only susceptibility is obtained as follows. Consider an orbital singlet with 2S 1 spin degeneracy. The energy levels are at magpH, where m, spans the values from +S to -8. Note that the energy levels correspond to
+
since the zero of energy can be taken as the level of lowest energy in the magnetic field, so that
since B
+
C m2 -S
=
'/aS(S
+ 1x2s + 1)
Kotani (4) applied Van Vleck's equation to octahedral complexes of trivalent titanium. The Hamiltonian for this d' ion perturbed by both a magnetic field and spin-orbit coupling is: The resulting energy level diagram follows:
Now,
9
T2
by expansion of the exponential, and
-'e Free Ion
To this approximation we obtain
We limit the derivation to paramagnetic substances, as distinct from ferromagnetic ones so that the absence of permanent polarization in zero magnetic field (i.e. I f 4 at H=O) requires that
Oh Field
H
-4pPHa 3 A
The numbers in parentheses indicate the degeneracy of a particular level. We ignore the contribution to paramagne'tism from the z E states, for they are some 20,000 em-' higher in energy. (The contribution of any energy level n is proportional to exp (-EJlcT), and k T = 205 em-' at room temperature.) Therefore,
which reduces to 522 / Journal of Chemical Education
L.S
with
- -
-
in units of the square of the Bohr magneton. Note that the Curie Law does not hold in this case. I n fact, as T m , pZ 5 while pZ 0 as T -t 0. How can a system with an unpaired electron have zero susceptibility at O0K? The spin and orbital angular momenta cancel each other out. Note also that p z - t 3, the spin-only value, as X 0. It is easy to show that the hyperfine contact interaction, so important to electron spin resonance spectroscopy, does not contribute to the paramagnetic susceptibility. Consider, for example, an electron spin interacting with a nucleus of spin, I, one-half. The total rnague1,icinteraction takes the form where A, the hypedine splitting constant, is a measure of the strength of the interaction. The eigenvalues of this Hamiltonian are
-
For 6/kT> 1, corresponding to a large zerofield splitting, the exponential terms go to zero, yield in^ x = Ng2p2/4kT, which is the spin-only formula . . for iso&pic s = The nickel(ll) ion. When this species is in a cubic field with small tetragonal component it has the following energy level diagram:
x.
\ I Free Ion
where all four combinations of signs are considered. Insertion of these energies into the Van Vleck equation yields a susceptibility independent of A.
(1)
m. = 0; EO= 0
Tetragonal Magnetic Field Field, Hz
Cubic
Field
The parameter D measures the zero-field splitting of the ground state. Our now-familiar procedure yields: XII/N =
Typical Problems
Students have been found capable of solving the following typical problems, to put the foregoing theory into practical application. Cr(III) in an octahedral Jield. The energy level diagram for this species is:
(g.PY ,erp(-D/kT)
+ O.exp (-o/kT) + W kTe x p ( - ~ / k ~ ) I
+ 2 exp (-D/kT)
Assuming D may be large (i.e., the exponentials are not expanded), 2Ngeafla exp ( -D/kT) =
kT 1 + 2 exp(-D/kT)
Generally, however, D T,), the snsceptibility follows a Curie-Weiss law, x=--
4 T
where x
= D/kT.
Spin-spin interaction. A simple model for spin-spin interaction (8, 6) may also be introduced at this point. Let two neighboring ions of spin Sl and Sz interact. The Hamiltonian for this interaction is X' = -2JS1S2, where J , in energy units, is called the exchange coupling constant. The result of the interaction for SI and J negative is to give a spin-singlet = Sa = ground state and a spin-triplet, energy 2J above the 524
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Jovrnal of Chemical Education
+ 0' with B = - J / 2 k
which illustrates the well-known connection between a sizeable Weiss 0 and exchange interaction. Note also that x-Oas
-J-
m
or, as we would expect, as the paramagnetic stat,e gets further away (higher in energy) its effect on the susceptibility must decrease. This calculation also serves as a model for any situation where there is a thermally accessible paramagnetic
state placed not too high above a magnetically inert ground state. The best-documented case concerns some dithiocarbamate complexes of iron(II1). The magnetism arises in this case by virtue of the fact that these molecules are near the cross-over point from spinpaired to spin-free (7). Acknowledgment
The author's ideas on exchange interaction were considerably clarified by correspondence with Professor R. L. Martin.
Literature Cited VAN VLECK,3. H., "The Theory of Eleotrio and Magnetic Susceptibilities," Oxford University Press,London, 1932. BALLHAUBEN, C. J., "Introduction to Ligand Field Theory," McG~aw-HillBook Co., New York, 1962. DORNN,P. B., "Symmetry in Inorganic Chemistry," Addison-Wesley Publishing Co., Reading, Mass., 1965. KOTANI,M., J. Phys SOC.Japan 4, 293 (1949). R. L., J. CHEM.EDUC.40, 135 (1963). CARLIN, FIGGIS.B. N.. AND MARTIN.R. L.. J . Chem. Soc. 1956.3837. See &o: MARTIN, R. L.; to bepuhlished. EWAIJ),A. H., MIRTIN, R. L., ROSS, I. G., AND WEITE, A. H., Proc. Roy. Soc. (London) A280.235 (1964).
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