Nuclear spin states

For instance, a proton (hydrogen nucleus) has the spin quantum number /= and has two allowed spin slates [2(i) + 1 = 2] for its nucleus and +l For the chlorine nucleus, /=I and there are four allowed 

SPIN QUANTUM NUMBERS OF SOME COMMON NUCLEI 

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In the absence of an applied magnetic field, all the spin states of a given nucleus are of equivalent energy (degenerate), and in a collection of atoms, all of the spin states should be almost equally populated, with the same number of atoms having each of the allowed spins. 

Atomic nuclei consist of neutrons and protons and have a structure that is as rich as the electronic structure of atoms and molecules. In chemistry, though, nuclear structure is often unimportant since it does not change in the course of a chemical reaction. It is frequently appropriate to consider nuclei to be simply positively charged point masses, as we have done to this point. However, there are some important manifestations of nuclear structure that have been exploited in developing powerful types of molecular spectroscopies. 

Protons and neutrons have an intrinsic spin and an intrinsic magnetic moment. Experiments have revealed that there are just two possible orientations of their spin vectors with respect to a z-axis defined by an external field. This means that their intrinsic spin quantum number is 1/2 they are fermions. Whereas the letter S is commonly used for the electron spin quantum number, the letter I is commonly used for nuclear spin quantum numbers. Thus, I = 1/2 for a proton, and the allowed values for the quantum number giving the nuclear spin angular momentum projection on the z-axis, Mj, are + 1/2 and -1/2. 

Another result we can anticipate from electronic structure is the possible values for the total nuclear spin quantum number. For instance, the deuteron (a proton and a neutron) consists of two particles with an intrinsic spin of 1/2. Were these two particles electrons, we would know that the possible values for the total spin quantum number are 1 and 0. It turns out for the deuteron that of the two coupling possibilities, 1=1 and 1 = 0, the 1=1 spin coupling occurs for the ground state. This is a consequence of the interactions that dictate nuclear structure, which is outside the scope of this chapter. For chemical applications, the key information is simply that the deuteron is an / = 1 particle. 

The intrinsic spins of stable nuclei have been determined experimentally, and the values have been explained with modem nuclear structure theory. Tables such as that in Appendix E are available for looking up the spin (I value) of a particular nucleus. The mles of angular momentum coupling are an aid in remembering the intrinsic spins of certain common nuclei. For instance, the helium nucleus, with its even number of protons and neutrons, has an integer spin, I = 0. In terms of the number of protons and neutrons, the carbon-12 nucleus is simply three helium nuclei. It should have an even-integer spin, and it also turns out to be an I = 0 nucleus. The carbon-13 nucleus has one more neutron and thereby has a half-integer spin, I = 1/2. 

Nuclear magnetic moments are small enough to have an almost ignorable effect on atomic and molecular electronic wavefunctions. On the other hand, the electronic structure has a measurable influence on the energies of the nuclear spin states. In this situation, it is an extremely good approximation to separate nuclear spin from the rest of a molecular wavefunction. The electronic, rotational, and vibrational wavefunctions of a molecule can be accurately determined while ignoring nuclear spin, as has been done so far. Then, the effect of the electrons and the effect of vibration and rotation can be incorporated as an external influence on the nuclear spin states. 

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NMR Nuclear magnetic resonance [223, 224] Chemical shift of splitting of nuclear spin states in a magnetic field H [225], C [226, 227], N [228], F [229], 2 Xe [230] Other Techniques Chemical state diffusion of adsorbed species... 

By examining the expression for Q ( equation (B1.16.4)). it should now be clear that the nuclear spin state influences the difference in precessional frequencies and, ultimately, the likelihood of intersystem crossing, tlnough the hyperfme tenn. It is this influence of nuclear spin states on electronic intersystem crossing which will eventually lead to non-equilibrium distributions of nuclear spin states, i.e. spin polarization, in the products of radical reactions, as we shall see below. 

Figure Bl.16.5. An example of the CIDNP net effect for a radical pair with one hyperfme interaction. Initial conditions g > g2, negative and the RP is initially singlet. Polarized nuclear spin states and schematic NMR spectra are shown for the recombination and scavenging products in the boxes.

In this simple case, there are just two nuclear spin states, a and (3. Equation (1.16.5) shows the calculation of the difference in electron precessional frequencies, Q, for nuclear spin states a (equation (B 1.16.5a)) and (3 (equation (B1.16.5h)). 

Since Ag is positive and is negative, Q is larger for the p state than for the a state. Radical pairs in the p nuclear spin state will experience a faster intersystem crossing rate than those in the a state with the result that more RPs in the p nuclear spin state will become triplets. The end result is that the scavenging product, which is fonned primarily from triplet RPs, will have an excess of spins in the p state while the recombination product, which is fonned from singlet RPs, will have an excess of a nuclear spin states. 

FIGURE 13 4 An external magnetic field causes the two nuclear spin states to have different energies The difference in energy AE is proportional to the strength of the applied field... 

Energy difference between nuclear spin states (kJ/mol or kcal/mol)... 

It turns out though that there are several possible variations on this general theme We could for example keep the magnetic field constant and continuously vary the radiofrequency until it matched the energy difference between the nuclear spin states Or we could keep the rf constant and adjust the energy levels by varying the magnetic... 

Section 13 3 In the presence of an external magnetic field the +j and —5 nuclear spin states of a proton have slightly different energies... 

If the radiofrequency spectmm is due to emission of radiation between pairs of states - for example nuclear spin states in NMR spectroscopy - the width of a line is a consequence of the lifetime, t, of the upper, emitting state. The lifetime and the energy spread, AE, of the upper state are related through the uncertainty principle (see Equation 1.16) by... 

No energy difference in nuclear spin states in absence of external magnetic field... 

Nuclear magnetic resonance, NMR (Chapter 13 introduction) A spectroscopic technique that provides information about the carbon-hydrogen framework of a molecule. NMR works by detecting the energy absorptions accompanying the transitions between nuclear spin states that occur when a molecule is placed in a strong magnetic field and irradiated with radiofrequency waves. 

Clearly, if a situation were achieved such that exceeded Np, the excess energy could be absorbed by the rf field and this would appear as an emission signal in the n.m.r. spectrum. On the other hand, if Np could be made to exceed by more than the Boltzmann factor, then enhanced absorption would be observed. N.m.r. spectra showing such effects are referred to as polarized spectra because they arise from polarization of nuclear spins. The effects are transient because, once the perturbing influence which gives rise to the non-Boltzmann distribution (and which can be either physical or chemical) ceases, the thermal equilibrium distribution of nuclear spin states is re-established within a few seconds. 

Methods of disturbing the Boltzmann distribution of nuclear spin states were known long before the phenomenon of CIDNP was recognized. All of these involve multiple resonance techniques (e.g. INDOR, the Nuclear Overhauser Effect) and all depend on spin-lattice relaxation processes for the development of polarization. The effect is referred to as dynamic nuclear polarization (DNP) (for a review, see Hausser and Stehlik, 1968). The observed changes in the intensity of lines in the n.m.r. spectrum are small, however, reflecting the small changes induced in the Boltzmann distribution. 

The origin of postulate (iii) lies in the electron-nuclear hyperfine interaction. If the energy separation between the T and S states of the radical pair is of the same order of magnitude as then the hyperfine interaction can represent a driving force for T-S mixing and this depends on the nuclear spin state. Only a relatively small preference for one spin-state compared with the other is necessary in the T-S mixing process in order to overcome the Boltzmann polarization (1 in 10 ). The effect is to make n.m.r. spectroscopy a much more sensitive technique in systems displaying CIDNP than in systems where only Boltzmann distributions of nuclear spin states obtain. More detailed consideration of postulate (iii) is deferred until Section II,D. 

At time [Case (2)] therefore, the hj rfine energy is approximately equal to the energy difference between the S and T states and can provide the driving force for T-S mixing. Now the h3q>erfine constants and Oj are a function of both nuclear and electronic spin states and thus one particular nuclear spin state for Hj and Hj will induce the T-S mixing more readily than the other. Thus nuclear spin selection occurs during the transition between S and T manifolds. However, this would yield no... 

Furthermore, even though the time dependence of the mixing process has been placed in Jee> ee must approximately equal Uj dming successive encounters in order that the time development of nuclear spin states in... 

E. Quantitative Aspects of Tq-S Mixing 1. The spin Hamiltonian and Tq-S mixing A basic problem in quantum mechanics is to relate the probability of an ensemble of particles being in one particular state at a particular time to the probability of their being in another state at some time later. The ensemble in this case is the population distribution of nuclear spin states. The time-dependent Schrodinger equation (14) allows such a calculation to be carried out. In equation (14) i/ (S,i) denotes the total... 

The spin Hamiltonian is thus generated. In particular it can be used to examine the Tq-S mixing of electron spin states and its relationship to the distributions of populations of nuclear spin states. The total spin Hamiltonian is given in equation (15) which contains both electron and nuclear terms. 

If represents the wavefunction of the radical pair at time t, then can be assumed to be a function of both the S and T wavefunctions and the nuclear spin states at time t as given in equation (25). 

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