Jay Martin Anderson1
Bryn Mawr College Bryn Mawr, Pennsylvania
Measurements Based on the Symmetry of NMR Spectra
The purpose of this paper is to examine a numher of ineasurement processes. The specific examples are taken from the field of nuclear magnetic resonance (XAlR) spectroscopy: the measureinents cited have been of significant value in the theory of the N3TR method. Usually, the measurement of a physical quantity requires an analysis of a body of experimental data. The unique feature of the measurements to he discussed is that they all rely on a simple examination of a number of spectral patterns. Furthermore, the theoretical groundwork on which the measurement is based is free of restrictive assunlptions. This theory makes extensive use of the methods of operator algebra, hut requires no lengthy calculation. From a t least a pedagogical point of view, the algebraic theory that yields the measurement processes to he discussed is interesting in its own right. Berause what is detected in each of these measurements is simply the presence or absence of a center of symmetry of a spectral patteru, the physical measurement is equally simple: only the sign of a quantity is measured. This nmy seem a trivial exercise-the measurement of a sigil-except that the sign of a quantity is as much a part of that quantity as its magnitude. In fact, as we shall see later, a prinri calculations yield both sign and magnitude of particular quantities, whereas experiment had heretofore yielded only the magnitude. In these
' D a n f d h k w x i s t e , B r p Mawr College, 1964-66.
situations, a measurement of the sign of the quantity is, in itself, a physical experiment of significant importance for the correlation of a priori quantum-mechanical theory and experiment. In another situation, the NMR experiment acts as a null instrument, much like a galvanometer in an electrical n~easuringdevice. I n this case, detection of the null point (which in a sense is a measurement of sign) allows evaluation of the parameters describing the system. All the results upon which these measurements are based were derived using the methods of operator algebra. Although detailed proofs are given elsewhere (14'),an appendix has been added to this paper which describes the properties of one of the operators used and gives a proof of a typical theorem. Measurement of Resonance Frequency
One of the simplest properties that governs the NMR spectrum of a nuclear species in a molecule is its resonance frequency. The frequency a t which a nucleus will absorb electromagnetic energy from an oscillatory (in NMR, a t radio-frequencies) electromagnetic field depends on the strength of the magnetic field, Ho, in which the molecule is placed, the nature of the particular nucleus, and the nature of its position or environment in the molecule. These latter pieces of information may be thought of as describing the effective magnetic moment of the nucleus in the molecule, fit, where the subscript i refers to a particular nucleus in a
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particular environment. In turn, the magnetic moment r, may be factored into a product of the magnetogyric ratio of the nucleus in its particular environment, y c Planck's constant divided by 2 r , 6,and the component of the spin of the nucleus in the direction of the magnetic field, I, (2) : The energy of a nucleus i in a magnetic field is p a , , or E
=
-r i U f d L i ) ,
(2)
and the resonance frequency, for an absorption of one quantum of radiation to change the spin component by one unit, is One might think that an experiment to measure the resonance frequency would be straightforward: a t a constant field H,,simply vary the frequency of the r.f. field until there is an absorption of energy. Or, alternatively, a t constant frequency, sweep Hountil the resonance is detected. Such an experiment has, in fact, been used to measure v , and thereby y, for a variety of nuclei. However, such an experiment, although simple in theory, is often difficult in practice, for there are other parameters which govern the NMR spectrum than the magnetogyric ratio and the field, and these are not easy to control. Fortunately, there is an indirect method of observing nuclei, namely by means of nuclear magnetic double resonance (NMDR). Generally, the total energy of a collection of nuclei in a molecule may be described by the total energy, or Hamiltonian, operator for the spin system:
Here the first term is merely a sum of the energies of the individual nuclei (compare with eqn. (2)); the second term is the spin-spin coupling (7) term. JU is the spin-spin coupling constant, or simply the coupling constant for the nuclei i and j, which we shall discuss later. As shown in equation (4), the coupling term is expressed by the dot product of the nuclear spin vectors (7). If, as in NMDR, one applies a strong r.f. field to the sample, the energy may be described by a Hamiltoniane (8) X =
AJ,(i)
+
+
J~~l(i).I(j)
BJ,(i),
(b)
i
