Spin-1 Systems

It is often helpful to categorize spin systems in terms of the chemical and magnetic equivalence of coupled protons. Protons are chemically equivalent if they have the same chemical environment and dius the same chemical shift. Chemical equivalence can result from eidier identical environments or rapid rotations which yield an average environment for a group of protons. Considering toluene, it 

Protons are magnetically equivalent if they have the same chemical shift and are coupled equally to otiier equivalent nuclei in die molecule. This is similar to chemical equivalence but is a more rigorous definition of equivalence. For example, the methyl protons of isobutane are chemically and magnetically equivalent since they absorb at die same frequency and are all coupled equally to the methine proton (which should be split into a 10-line multiplet ). Likewise die two mediyl groups of /i-xylene are chemically and magnetically equivalent because they are coupled equally (J = 0) to the aromatic protons both ortho and meta to diem. 

We would expect that the spectrum of the latter compound would consist of two signals a two-proton triplet in the vinyl region and a four-proton doublet in the allylic region. This is because the coupling constant, / 3 is zero. It if were not zero, then a more complicated spectrum would result. Thus magnetic nonequivalence can lead to much more complicated spectra. 

Protons that are chemically equivalent but magnetically nonequivalent are indicated by, for example, A A. The examples of such systems given below illustrate the medtod. This system for designating spin systems is merely a labeling device. The appearance of actual spectra will depend on die magnitude of die various J values. Nevertheless this is a convenient and common way of categorizing coupled proton systems. 

Another structural factor which can lead to nonequivalence of aliphatic protons the symmetry properties of protons  

When the chemical shifts of two protons coupled to each other are very different, or more precisely when their difference in chemical shifts (in Hz) divided by the coupling constant in Hz (A5/J) is greater than 7-10, then essentially first-order AX spectra are observed the resonance of each proton is split into a doublet and the intensity of the lines is about the same. The 

As the center point of the AB pattern is at 66 Hz, and as 5 and 8 are 8 Hz apart and equal distances from the center, 5 can be calculated to be 66 Hz - 4 Hz = 62 Hz, while will be 66 Hz + 4 Hz = 70 Hz from TMS. Now if we record the same spectrum at 400 MHz, since the chemical shift (but not the coupling constants) is proportional to the operating frequency, proton B will appear at (70 x 400)/60 = 466.6 Hz while proton A will appear at (62 X 400)/60 = 413.3 Hz from TMS. The precise positions of the four lines can be derived from the formula 

It may be noted that a double doublet obtained in an AB system should not be confused with a quartet obtained as a part of an AX3 system. As mentioned earlier, the distances between the lines in a quartet will all be identical, and the relative intensities of the peaks will be in a ratio of 1 3 3 1, though distortions in the intensities may occur as A5/J becomes smaller. In an AB system, on the other hand, the distance between the two central peaks is different from the distance between the outer and inner peaks. 

In an i42 system both nuclei are equivalent and therefore do not show any spin-spin coupling. 

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In 1972 Wegner [25] derived a power-series expansion for the free energy of a spin system represented by a Flamiltonian roughly equivalent to the scaled equation (A2.5.28). and from this he obtained power-series expansions of various themiodynamic quantities around the critical point. For example the compressibility... 

Tjs which is dependent upon the strength of the dipole couplmg between the / and. S spin systems, is the... 

Whilst MAS is effective (at least for spin/ = d) in narrowing inliomogeneous lines, for abundant spin systems... 

Introductory text, fairly mathematical, concentrates on spin/ = systems, good references. 

Figure Bl.13.1. Energy levels and transition probabilities for anIS spin system. (Reproduced by pennission of Academic Press from Kowalewski J 1990 Annu. Rep. NMR Spectrosc. 22 308-414.)...

Figure Bl.13.7. Simulated NOESY peak intensities in a homoniielear two-spin system as a fiinetion of the mixing time for two different motional regimes. (Reprodiieed by pennission of Wiley from Neiihaiis D 1996 Encyclopedia of Nuclear Magnetic Resonance ed D M Grant and R K Harris (Chiehester Wiley) pp 3290-301.)...

Di Bari L, Kowalewski J and Bodenhausen G 1990 Magnetization transfer modes in scalar-coupled spin systems investigated by selective 2-dimensional nuclear magnetic resonance exchange experiments J. Chem. Rhys. 93 7698-705... 

Application of an oscillating magnetic field at the resonance frequency induces transitions in both directions between the two levels of the spin system. The rate of the induced transitions depends on the MW power which is proportional to the square of oi = (the amplitude of the oscillating magnetic field) (see equation (bl.15.7)) and also depends on the number of spins in each level. Since the probabilities of upward ( P) a)) and downward ( a) p)) transitions are equal, resonance absorption can only be detected when there is a population difference between the two spin levels. This is the case at thennal equilibrium where there is a slight excess of spins in the energetically lower p)-state. The relative population of the two-level system in thennal equilibrium is given by the Boltzmaim distribution... 

A second type of relaxation mechanism, the spin-spm relaxation, will cause a decay of the phase coherence of the spin motion introduced by the coherent excitation of tire spins by the MW radiation. The mechanism involves slight perturbations of the Lannor frequency by stochastically fluctuating magnetic dipoles, for example those arising from nearby magnetic nuclei. Due to the randomization of spin directions and the concomitant loss of phase coherence, the spin system approaches a state of maximum entropy. The spin-spin relaxation disturbing the phase coherence is characterized by T. ... 

The practical goal of EPR is to measure a stationary or time-dependent EPR signal of the species under scrutiny and subsequently to detemiine magnetic interactions that govern the shape and dynamics of the EPR response of the spin system. The infomiation obtained from a thorough analysis of the EPR signal, however, may comprise not only the parameters enlisted in the previous chapter but also a wide range of other physical parameters, for example reaction rates or orientation order parameters. 

ELDOR is tlie acronym for electron-electron double resonance. In an ELDOR experiment [28] one observes a rednction in the EPR signal intensity of one hyperfme transition that results from the saturation of another EPR transition within the spin system. ELDOR measurements are still relatively rare bnt the experiment is fimily established in the EPR repertoire. 

The practical goal for pulsed EPR is to devise and apply pulse sequences in order to isolate pieces of infomiation about a spin system and to measure that infomiation as precisely as possible. To achieve tliis goal it is necessary to understand how the basic instnunentation works and what happens to the spins during the measurement. 

Figure Bl.15.13. Pulsed ENDOR spectroscopy. (A) Top energy level diagram of an. S-/=i spin system (see also figure Bl,15,8(A)). The size of the filled circles represents the relative population of the four levels at different times during the (3+1) Davies ENDOR sequence (bottom). (B) The Mims ENDOR sequence.

Figure B2.4.1 shows the lineshape for intennediate chemical exchange between two equally populated sites without scalar coupling. For more complicated spin systems, the lineshapes are more complicated as well, since a spin may retain its coupling infonnation even though its chemical shift changes in the exchange.

Figure B2.4.3 shows an example of this in the aldehyde proton spectnim of N-labelled fonnamide. Some lines in the spectnim remain sharp, while others broaden and coalesce. There is no frmdamental difference between the lineshapes in figures B2.4.1 and figures B2.4.3—only a difference in the size of the matrices involved. First, the uncoupled case will be discussed, then the extension to coupled spin systems.

In this equation, 01 is the ifeqiieney of the RF irradiation, oiq is the Lannor ifeqiieney of the spin, is the spm-spm relaxation time andM is the z magnetization of the spin system. The notation ean be simplified somewhat by defining a eomplex magnetization, AY, as in equation (B2.4.3). 

Binsch [6] provided the standard way of calculating these lineshapes in the frequency domain, and implemented it in the program DNMR3 [7], Fonnally, it is the same as the matrix description given in section (B2.4.2.3). The calculation of the matrices L, R and K is more complex for a coupled spin system, but that should not interfere witii the understanding of how the method works. This work will be discussed later, but first the time-domain approach will be developed. 

For a coupled spin system, the matrix of the Liouvillian must be calculated in the basis set for the spin system. Usually this is a simple product basis, often called product operators, since the vectors in Liouville space are spm operators. The matrix elements can be calculated in various ways. The Liouvillian is the conmuitator with the Hamiltonian, so matrix elements can be calculated from the commutation rules of spin operators. Alternatively, the angular momentum properties of Liouville space can be used. In either case, the chemical shift temis are easily calculated, but the coupling temis (since they are products of operators) are more complex. In section B2.4.2.7. the Liouville matrix for the single-quantum transitions for an AB spin system is presented. 

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